UNIVERSITY OF CALIFORNIA
Santa Barbara
Multi-Gigabit Reception with Time-interleaved
Analog-to-Digital Converters
A Dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
in
Electrical and Computer Engineering
by
Sandeep Ponnuru
Committee in Charge:
Professor Upamanyu Madhow, Chair
Professor James Buckwalter
Professor Shiv Chandrasekharan
Professor Mark Rodwell
Professor Kenneth Rose
December 2011
The Dissertation of
Sandeep Ponnuru is approved:
Professor James Buckwalter
Professor Shiv Chandrasekharan
Professor Mark Rodwell
Professor Kenneth Rose
Professor Upamanyu Madhow, Committee Chairperson
November 2011
Multi-Gigabit Reception with Time-interleaved Analog-to-Digital Converters
c 2011
Copyright by
Sandeep Ponnuru
iii
To my grandfather
Ponnuru Kedareswara Rao
iv
Acknowledgements
Thanks to many a individual who went out of their way to make this dissertation
a reality. First I acknowledge my advisor Prof. Upamanyu Madhow for his vision in
deciding the important and timely problems, his advice on how to write well and finally, his motivation when things seemed very hard for me. I also thank Prof. James
Buckwalter, Prof. Shiv Chandrasekharan, Prof. Mark Rodwell and Prof. Kenneth
Rose for being on my thesis committee providing with critical feedback on the research
directions. I acknowledge Dr. Munkyo Seo for helping me to perform hardware experiments to complement the simulation studies. I sincerely thank my colleagues Jaspreet
Singh, Sumit Singh, Eric Torkildson, Sriram Venkateswaran and Hong Zhang for many
valuable discussions on research and on life. Finally, I thank NSF for supporting this
research through their grants CNS-0832154 and CCF-0729222.
v
Curriculum Vitæ
Sandeep Ponnuru
Education
PhD, Electrical Engineering, University of California, Santa Barbara,
Sep. 2006 - Dec 2011.
Master of Science (MS), Electrical Engineering, University of California, Santa Barbara, Sep. 2006 - June 2011.
Bachelor of Technology (B. Tech), Electrical Engineering, Indian Institute of Technology (IIT) Kanpur, India, Aug. 2002 - May. 2006
Publications
Joint mismatch and channel compensation for high-speed OFDM
receivers with time-interleaved ADCs by S. Ponnuru, U. Madhow,
M. Seo, M. Rodwell; IEEE Transactions on Communications, Aug.
2010.
On the scaling of joint mismatch and channel estimation for timeinterleaved ADCs by S. Ponnuru and U. Madhow; presented at Allerton Conference on Communication, Control, and Computing, Sept
2010, Allerton Retreat Center, Illinois.
vi
Scalable mismatch compensation for time-interleaved A/D converters in OFDM reception by S. Ponnuru and U. Madhow; presented
at Wireless Communications and Networking Conference (WCNC)
2010, Sydney, Australia.
Multi-Gigabit Communication: the ADC Bottleneck by J. Singh,
S. Ponnuru and U. Madhow; invited paper, Proc. 2009 IEEE International Conference on Ultra-Wideband (ICUWB), Vancouver, Canada.
Joint Channel and Mismatch Correction for OFDM Reception with
Time-interleaved ADCs: Towards Mostly Digital MultiGigabit Transceiver
Architectures by S. Ponnuru, U. Madhow, M. Seo, M. Rodwell; Proc.
2008 IEEE Global Telecommunications Conference (GLOBECOM),
New Orleans, USA
On the convergence of joint channel and mismatch estimation for
time-interleaved data converters by S. Ponnuru and U. Madhow;
Proc. 2011 IEEE Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA.
Awards and Honors
Dissertation fellowship awarded by the Electrical and Computer Engineering department, UCSB.
Qualstar award for summer internship at Qualcomm, San Diego, 2010.
vii
Abstract
Multi-Gigabit Reception with Time-interleaved
Analog-to-Digital Converters
Sandeep Ponnuru
Moore’s law drives the economies of scale in modern communication systems, with
most receiver functionalities being implemented in digital signal processing (DSP) after analog to digital conversion. Extending the computational advantage provided by
Moore’s law to multi-Gigabit communication systems requires analog-to-digital converters (ADCs) with high sampling rate and output resolution.
A promising approach to realize such ADCs at reasonable power consumption is
to employ a time- interleaved (TI) architecture with slower (but power-efficient) subADCs in parallel. However, mismatch among the sub-ADCs, if left uncompensated,
can cause error floors in receiver performance. Traditionally, mismatch is compensated
either by employing larger transistors, by analog adjustments, or by dedicated digital
mismatch compensation whose complexity increases with the desired resolution at the
output of the TI-ADC. In this thesis, we investigate a novel approach, in which mismatch and channel dispersion are compensated jointly, with the performance metric
being overall link reliability rather than ADC performance.
We first characterize the structure of mismatch-induced interference for an OFDM
system, and demonstrate the efficacy of a frequency-domain interference suppression
viii
scheme whose complexity is independent of constellation size (which determines the
desired resolution). Numerical results from computer simulation and from experiments
on a hardware prototype show that the performance with the proposed joint mismatch
and channel compensation technique is close to that without mismatch.
Next, we explore time-domain mismatch compensation approaches that scale with
the number of sub-ADCs and the desired resolution. We show that low-complexity
linear mismatch compensation is possible if we employ oversampling. We establish a
strong analogy between the role of oversampling for mismatch compensation and for
channel equalization, even though the structure of the interference due to mismatch is
different from that due to a dispersive channel.
While the proposed compensation architectures work with offline estimates of mismatch parameters, we provide an iterative, online method for joint estimation of mismatch and channel parameters which leverages the training overhead already available
in communication signals. We provide a closed form solution for each iteration, for
both channel and mismatch estimates, based on a linear approximation for the nonlinear effect of timing mismatch. We investigate the scalability and convergence of
this joint estimation algorithm, and derive rules of thumb relating the required length
of pseudorandom training to the number of sub-ADCs. Further, we design periodic
training sequences with significantly enhanced convergence rates.
ix
Contents
Acknowledgements
v
Curriculum Vitæ
vi
Abstract
viii
List of Figures
xiii
1
2
Multi-Gigabit Communication and the ADC Bottleneck
1.1 Multi-Gigabit communication . . . . . . . . . . . . . . . .
1.2 The ADC bottleneck . . . . . . . . . . . . . . . . . . . . .
1.3 Solution: Time-interleaved ADC . . . . . . . . . . . . . . .
1.4 Mismatches in a TI-ADC . . . . . . . . . . . . . . . . . .
1.5 Thesis contributions . . . . . . . . . . . . . . . . . . . . .
1.5.1 Mismatch compensation for OFDM systems . . . .
1.5.2 Scalable mismatch compensation using oversampling
1.5.3 Scalability of joint channel and mismatch estimation
1.6 Literature Survey . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Mismatch estimation . . . . . . . . . . . . . . . . .
1.6.2 Mismatch compensation . . . . . . . . . . . . . . .
1.6.3 Oversampling for mismatch compensation . . . . .
1.6.4 Drastic Quantization . . . . . . . . . . . . . . . . .
Frequency domain Mismatch Compensation
2.1 Organization and Notation . . . . . . . .
2.2 TI-ADC in an OFDM receiver . . . . . .
2.2.1 TI-ADC Mismatch Model . . . .
2.2.2 OFDM Model . . . . . . . . . .
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2.3
2.4
2.5
2.6
3
4
The Structure of Mismatch-Induced Interference . . . . . . . . . .
Frequency domain Joint Mismatch Compensation and Demodulation
2.4.1 Interference structure when L divides M . . . . . . . . . .
2.4.2 Noise Enhancement . . . . . . . . . . . . . . . . . . . . .
Joint Mismatch and Channel Estimation . . . . . . . . . . . . . . .
2.5.1 Channel estimation given mismatch estimates . . . . . . . .
2.5.2 Mismatch estimation given channel estimates . . . . . . . .
Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Mismatch-induced Error Floors . . . . . . . . . . . . . . .
2.6.2 Structure of Mismatch-induced Interference . . . . . . . .
2.6.3 Suppression of Mismatch-induced interference . . . . . . .
2.6.4 Experimental results using hardware TI-ADC prototype . .
Scalable Mismatch Compensation by Oversampling
3.1 Organization . . . . . . . . . . . . . . . . . . . . . .
3.2 System Model . . . . . . . . . . . . . . . . . . . . .
3.2.1 TI-ADC model . . . . . . . . . . . . . . . .
3.2.2 Zero-forcing mismatch compensation . . . . .
3.2.3 Running Example . . . . . . . . . . . . . . .
3.3 Oversampling for Scalable Mismatch Compensation .
3.3.1 Oversampling factor = 2 . . . . . . . . . . . .
3.3.2 Running Example . . . . . . . . . . . . . . .
3.4 Application to an OFDM receiver employing a TI-ADC
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Scalability of Joint Channel and Mismatch Estimation
4.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Dispersive channel and Gain mismatch . . . . . . . . . . . .
4.3 Joint channel and mismatch estimation: convergence behaviour
4.3.1 Progress of iterations . . . . . . . . . . . . . . . . .
4.3.2 Rate of Convergence . . . . . . . . . . . . . . . . .
4.3.3 Geometry of estimate progression . . . . . . . . . . .
4.4 Joint estimation with gain and timing mismatches . . . . . . .
4.4.1 Iterative algorithm for joint estimation . . . . . . . . .
4.4.2 Convergence behaviour . . . . . . . . . . . . . . . .
4.5 Progression of Channel estimates: Analysis . . . . . . . . . .
4.5.1 Progression along the dominant direction . . . . . . .
4.5.2 Convergence rate formulae . . . . . . . . . . . . . .
4.5.3 Simulation Results . . . . . . . . . . . . . . . . . .
4.6 Effects of noise on convergence behaviour . . . . . . . . . . .
4.6.1 Comparison with CRLB . . . . . . . . . . . . . . .
xi
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4.7
4.8
5
Training sequence design for fast convergence . . . . . . . . . . . . . 96
Rules of Thumb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Conclusions
102
5.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A Invertibility of matrix A
B
110
Structure of the matrix A when L divides M
C Convergence rate formulae: derivations
C.1 Asymptotics of the convergence rate . . . . . . . . .
C.1.1 Accounting for gain scaling . . . . . . . . .
C.2 First order approximation . . . . . . . . . . . . . .
C.3 Long pseudo-random training sequence and channel
Bibliography
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xii
List of Figures
1.1 [a] DSP-centric transmitter [b] DSP-centric receiver . . . . . . . . .
1.2 Ideal time-interleaved ADC with 4 sub-ADCs. The sampling instants
(d = integer, T0 = sampling period) of the sub-ADCs are staggered such that
each sub-ADC operates at one-fourth of the TI-ADC’s net sampling rate.
Red and green indicate the analog and digital components, respectively. . .
1.3 Effect of TI-ADC mismatch on the sampling of the input analog signal:
the four consecutive samples (from left to right) experience gain, timing,
voltage offset and no mismatches, respectively. The samples obtained by
the TI-ADC with mismatches, indicated by •, are compared with the “◦”
samples; the latter being obtained when there are no mismatches. . . . . .
2.1 Orthogonal Frequency Division Multiplexing (OFDM): A popular communication scheme where the information-bearing symbols are encoded in
the frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 [a] OFDM Transmitter (DAC = Digital-to-Analog Converter) [b] OFDM
Receiver [c] Baseband signal model (time-domain). Reprinted from our subc
mission [1] with permission, [2010]
IEEE. . . . . . . . . . . . . . . . .
2.3 The concept of a zero-forcing equalizer, where the received signal vector is projected onto a space orthogonal to the interference. . . . . . . . . .
2.4 Effect of mismatches in a TI-ADC (with 8 sub-ADCs) for OFDM
transmission employing 16-QAM constellation. Perfect channel knowledge
is assumed. Reprinted from our journal submission [1] with permission,
c
[2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Structure of mismatch-induced interference: the relative interference
for the real part of the constellation symbol of first subcarrier (signal) from
all the subcarriers is shown. The signal level is normalized to 0dB, and the
values of “L” in the legend indicate the number of sub-ADCs interleaved.
c
Reprinted from our submission [1] with permission, [2010]
IEEE. . . . .
xiii
2
5
6
15
18
24
36
37
2.6 BER after estimation and correction of 10% mismatches in a TI-ADC
(with 8 sub-ADCs) used for OFDM signal reception employing 16 QAM
constellation. The number of iterations of the estimation algorithm is shown
c
in the legend. Reprinted from our submission [1] with permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 BER after estimation and correction of 10% mismatches in a TI-ADC
(with L=6 sub-ADCs) used for OFDM signal reception employing 16 QAM
constellation. The complexity (in terms of number of real-valued multiplications per sample) is indicated in the legend and the number of iterations
of the estimation algorithm is fixed at 3. Reprinted from our submission [1]
c
with permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . .
2.8 TI-ADC experimental prototype with four interleaved ADCs. D/A and
LPF refer to digital-to-analog converter and low pass filter, respectively. We
have To = 2.5ns and that d takes integral values. Reprinted from our subc
mission [1] with permission, [2010]
IEEE. . . . . . . . . . . . . . . . .
2.9 Experimental results from the prototype confirm the efficacy of the proposed mismatch estimation and equalization scheme. Reprinted from our
c
submission [1] with permission, [2010]
IEEE. . . . . . . . . . . . . . .
3.1 Linear model for mismatch in a TI-ADC and zero-forcing based mismatch compensation (k=integer, To =sampling period). All symbols indicate
the z-transforms of discrete streams at the symbol rate To−1 . Reprinted from
c
our conference submission [2] with permission, [2010]
IEEE. . . . . . .
3.2 Discrete responses for two sub-ADCs with timing mismatch as 10%
(solid arrows) and −10% (dashed arrows), respectively. Popular bandlimited
functions like Sinc and Raised cosine resemble the shape of h(t) (red curve)
to the first sidelobe. Reprinted from our conference submission [2] with
c
permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Zeros of the determinants of the matrices B(z) and C(z) (obtained by
the decomposition of the system matrix A(z)), plotted as a function of the
relative timing mismatch. Since the imaginary parts are insignificant, only
the real parts of the zeros are shown. There are some zeros of det B(z), but
not of det C(z), beyond the depicted range. Reprinted from our conference
c
submission [2] with permission, [2010]
IEEE. . . . . . . . . . . . . . .
3.4 BER in a 64-QAM, 128-subcarrier OFDM system employing a sloppy
TI-ADC with 10% timing mismatch. For the left subfigure (a), Nyquist rate
sampling is performed and the TI-ADC interleaving factor L is 8. On the
other hand, we assume sampling at twice the Nyquist rate and L = 32 in
the right subfigure (b). Reprinted from our conference submission [2] with
c
permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
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64
4.1 Base-band model for transmission over dispersive channel using a timeinterleaved ADC with 4 sub-ADCs at the receiver (d =integer, T0 = sampling
period). Reprinted from our conference submission [3] with permission,
c
[2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2 Progress of iterations: Multiple iterations of joint channel and mismatch algorithm decrease the MSE linearly in the dB scale (or exponentially
in the absolute scale). Here M = 256, N = 20 and L = 64. . . . . . . . .
74
4.3 Mean convergence rates versus number of sub-ADCs: Increasing the
training length for a given number of sub-ADCs increases the convergence
rate of joint channel and mismatch estimation algorithm. Here N = 20
channel taps and the convergence rate is defined with respect to the channel
estimate error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.4 Approximately linear trajectories of the channel estimate with iterations of joint estimation algorithm: Trajectories for several randomly generated channels (length N = 20 taps) are plotted in the two-dimensional plane
containing the true channel, h (indicated by black diamond at origin), the
initial channel estimate, h0 , and the channel estimate after the first iteration,
h1 . The “coefficients” represent the projections onto the orthogonal singular vectors for the matrix, that is obtained for each realization, with columns
h − h0 and h − h1 . M denotes the training sequence length and L = 8
ADCs are interleaved. Reprinted from our conference submission [4] with
c
permission, [2011]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.5 Progress of the joint estimation algorithm: The estimate errors for
channel and mismatches decrease exponentially. Here, M = 256, N = 20
and L = 32. Reprinted from our conference submission [3] with permission,
c
[2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.6 Variation of mean convergence rate with L for different values of M
(N = 20). Reprinted from our conference submission [3] with permission,
c
[2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.7 Progress with iterations for the joint estimation algorithm with (M, L, N ) =
(32, 16, 4). Reprinted from our conference submission [3] with permission,
c
[2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.8 A geometric insight behind the convergence rate formula in (4.28). The
convergence rate is significantly determined by θi (for each i ∈ {0, 1, · · · , L−
1}), and is given by (4.32). . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.9 Analytical results for the mean convergence rate of joint channel and
mismatch estimation: Training sequence length M increases as we move
away from origin as 32, 64, 128 and 256. Reprinted from our conference
c
submission [4] with permission, [2011]
IEEE. . . . . . . . . . . . . . . 91
xv
4.10 Simple formula for convergence rate: Mean convergence rate is well
approximated as 20 log10 (M/L) dB/iteration (Equation 4.33) when L ≥ 32
for a pseudorandom training sequence of length M = 256. Reprinted from
c
our conference submission [4] with permission, [2011]
IEEE. . . . . . .
4.11 Performance of the iterative algorithm in the presence of thermal noise
with (M, L, N ) = (256, 64, 20). Reprinted from our conference submission
c
[3] with permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . .
4.12 Comparison of the algorithm’s MSE with Cramer-Rao lower bounds
for (M, L, N ) = (256, 64, 20). We considered 10 iterations for the alternating minimization algorithm. Reprinted from our conference submission [3]
c
with permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . .
4.13 Comparison of the progress of the iterative algorithm with the pseudorandom and the proposed training sequences. (M, L, N ) = (1216, 64, 19).
c
Reprinted from our conference submission [3] with permission, [2010]
IEEE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
93
95
99
5.1 Proposed system architecture for multi-Gigabit OFDM transceiver (4
Gbps uncoded data rate) employing a time-interleaved ADC: [a] transmitter
[b] receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Proposed system architecture for a more demanding multi-Gigabit OFDM
transceiver (12 Gbps uncoded data rate) employing a time-interleaved ADC:
[a] transmitter [b] receiver . . . . . . . . . . . . . . . . . . . . . . . . . 105
xvi
Chapter 1
Multi-Gigabit Communication and the
ADC Bottleneck
“Mostly-digital” transceiver architectures have been key to the economies of scale
that propel the growth of mass market communication systems. Examples of these systems include the ubiquitous cellular networks, wireless local area networks, broadband
DSL and cable modems.
A typical realization of such a mostly digital transceiver, where a Digital Signal
Processor (DSP) forms the core, is given in Fig. 1.1 (b). Here, analog processing is
restricted to the radio frequency section, while all baseband operations (e.g., transmit
precoding, receive synchronization, equalization, decoding) are performed in the DSP
core. Hence, a mostly digital transceiver can exploit “Moore’s law” for digital electronics (that is the exponential downscaling of the per transistor size) to realize a fast,
low-cost hardware implementation [5].
1
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
[a]
DIGITAL
ANALOG
Mixer
Data bits
Low
Pass Filter
D/A
DSP
Interpolated
signal
Power Amp
Analog
signal
RF
signal
Local
Oscillator
[b]
Low Noise
Amp
Mixer
Low
Pass Filter
Local
Oscillator
A/D
Baseband
Signal
DSP
Digital
Signal
Data bit
estimates
Figure 1.1: [a] DSP-centric transmitter [b] DSP-centric receiver
1.1
Multi-Gigabit communication
Extending the computational advantage provided by Moore’s law to current and
emerging multi-Gigabit systems is of tremendous importance in making them economically attractive.
Current multi-Gigabit systems include optical communication over fiber-optic cables where the data transmission rates can be as high as 40 Gbps per fiber. Traditional
optical communication did not employ many electronic components owing to the ex-
2
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
tremely fast data rates in these systems (50 GSa/sec in 2010). However, increased
attention is being given to DSP-centric optical communication due to the shift to using
coherent optical systems. In these systems, both amplitude and phase information are
available, and hence offer increased receiver sensitivity [6]. In these receivers, DSP
could be used to track phase and frequency offsets that are needed for coherent detection [7, 8]. Moreover, coherent receivers enable an accurate compensation of fiber
dispersion because both amplitude and phase information is available [6, 9].
Examples of emerging wireless multi-Gigabit systems include ultra-wideband systems in the 3.1 -10.6 GHz band [10] and millimeter-wave systems in the 60 GHz band
(e.g. unlicensed spectrum from 57-64 GHz is available in the US) [11]. These systems
are similar to other traditional communication systems and could benefit from having
powerful DSPs at the transmitter and the receiver.
A prerequisite for making the above mentioned multi-Gigabit transceivers mostly
digital is an Analog-to-Digital Converter (ADC) of desired specifications. This is because the ADC forms the front end of the DSP core (See Fig. 1.1), converting the
received analog signal into digital format. The two important specifications of an ADC
are the sampling rate, the rate at which the input analog signal is sampled, and the
resolution, the number of bits used to represent the digital output. For the abovementioned multi-Gigabit systems, operating over wide bandwidths, sampling rates ranging
between 1-5 GHz for mm-wave communication, and between 10-100 GHz for opti-
3
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
cal communication are needed. In addition, these systems could employ large constellations (such as 16-QAM) for increased data rates and could operate over highly
dispersive channels, both of which increase the dynamic range of the received signal.
The ADC must therefore provide relatively high resolution (e.g., 8-10 bits) in such
scenarios. In essence, emerging multi-Gigabit receivers require ADCs with both high
sampling rate and high resolution.
1.2
The ADC bottleneck
A recent survey across popular ADC architectures (Table 1.1) suggests that it is
difficult to achieve both high rate and resolution from the ADC. For example, Flash
ADC is conventionally the architecture of choice for multi-GHz sampling rates, but
its power consumption scales exponentially with resolution, and it is difficult to obtain
more than 5-6 bits of resolution for power levels under 1 W.
Table 1.1: Trends on ADC rate, resolution and power consumption [12]
SigmaSuccessive
Pipelined
Flash
Delta
Approximation
Sampling rate 100 kHz
10 MHz
100 MHz
1 GHz
Resolution
24 bit
18 bit
15 bit
8 bit
Power
1-10 mW
10-100 mW
100 mW-1 W 1-10 W
4
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
1.3
Solution: Time-interleaved ADC
t = 4d T0
ADC
t = (4d+1) T0
ADC
Analog
Input
4-to-1
MUX
t = (4d+2) T0
Digital
Output
ADC
t = (4d+3) T0
ADC
Figure 1.2: Ideal time-interleaved ADC with 4 sub-ADCs. The sampling instants (d = integer,
T0 = sampling period) of the sub-ADCs are staggered such that each sub-ADC operates at
one-fourth of the TI-ADC’s net sampling rate. Red and green indicate the analog and digital
components, respectively.
A possible solution for achieving both high speed and high resolution at reasonable
power consumption is the time-interleaved (TI) ADC (Fig. 1.2). The idea is to first
Sunday, May 1, 2011
build a high resolution ADC at relatively low speed using a power-efficient architecture
(e.g., 50 MHz successive approximation ADC in [13] or 50 MHz pipelined ADC in
[14]). These power- efficient ADCs are then interleaved in parallel to synthesize a
high-rate ADC (See Fig. 1.2). We refer to each constituent low-rate ADC as a subADC.
An example application of the preceding architecture is a TI-ADC designed by
Fujitsu for an optical communication receiver, which has a net sampling rate of 56 GHz
5
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
[15], and offers 8 bits of resolution at a power consumption of 2 W. This state-of-theart ADC, built in 65 nm CMOS technology, interleaves 320 successive approximation
ADCs, each sampling at 175 MHz. Thus, time-interleaving offers significant promise
over the conventional Flash architecture, where the sampling rate is only a few GHz for
similar power consumption.
gain
timing
vol.-offset
none
y + o
3
g y
3
y
1 1
3
y’
y
y
2
4
1
y
2
T
0
3T
2T
0
0
4T
0
Time
Figure 1.3: Effect of TI-ADC mismatch on the sampling of the input analog signal: the four
consecutive samples (from left to right) experience gain, timing, voltage offset and no mismatches, respectively. The samples obtained by the TI-ADC with mismatches, indicated by •,
are compared with the “◦” samples; the latter being obtained when there are no mismatches.
1.4
Mismatches in a TI-ADC
A major bottleneck in the data conversion accuracy provided by a TI-ADC is mismatch among the sub-ADCs. For example, Fig. 1.3 depicts mismatches in gain, timing
and voltage offset. Causes of mismatch include imperfect clock distribution, variation
6
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
of transistor size, signal path differences, and parasitic effects. When used in a communication receiver, we demonstrate in Chapter 2 that uncompensated mismatch leads
to error floors. Such error floors can however be removed by estimating and compensating for the mismatch parameters. The goal of this thesis is to investigate effective
techniques for mismatch compensation, especially in the context of time-interleaved
ADCs employed in communication receivers.
1.5
Thesis contributions
The thesis takes a novel alternative to mismatch compensation for a time-interleaved
analog-to-digital converter: the “sloppy” TI-ADC is viewed as part of the overall communication link, and the goal then is to jointly compensate for mismatch and channel
dispersion. Thus, we focus on the overall link reliability, rather than on the performance of the ADC as an isolated component, which simplifies the task of mismatch
compensation.
1.5.1
Mismatch compensation for OFDM systems
We first illustrate our ideas for an Orthogonal Frequency Division Multiplexing
(OFDM) receiver, where two separate TI-ADCs are employed for sampling the in-
7
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
phase and quadrature phase components. Each TI-ADC may have gain, timing and
voltage offset mismatches among its interleaved sub-ADCs.
We model the effect of mismatch in the frequency domain, after the discrete Fourier
transform performed by the OFDM receiver. The interference due to voltage offset mismatch is signal-independent, but the interference due to gain and timing mismatch is
signal-dependent, and is shown to cause inter-subcarrier interference between nominally orthogonal OFDM subcarriers. If left uncompensated, this leads to an error floor
in the Bit Error Rate (BER).
For the special case when the number of sub-ADCs, L, divides the number of subcarriers, the subcarriers form interference groups of size at most 2L, such that only
subcarriers within a group interfere with each other. We show that zero-forcing interference suppression within an interference group, which requires 4L real-valued coefficients, is effective for joint mismatch compensation and demodulation, regardless of
the desired resolution. In contrast, the complexity (in terms of the number of multiplications) of conventional time-domain mismatch compensation for TI-ADCs increases
with the desired resolution.
We also report on results from an experimental prototype built by interleaving four
commercially available ADCs, each sampling at 100 MSa/s with 14 bits of nominal
resolution. The experimental results demonstrate the efficacy of the proposed mismatch
8
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
and channel compensation technique, and show excellent agreement with theoretical
predictions.
1.5.2
Scalable mismatch compensation using oversampling
The zero-forcing frequency domain approach has complexity that scaled with the
number of sub-ADCs. As the number of sub-ADCs increases, therefore, it becomes
attractive to consider alternative approaches. In particular, we consider oversampling
along with zero-forcing mismatch compensation in the time-domain in such settings.
We prove that, analogous to dispersive channels, a Bezout-like identity holds for mismatch compensation, so that perfect zero-forcing compensation can be guaranteed using a finite number of taps when sampled at twice the Nyquist rate. Furthermore, we
demonstrate via simulations that even smaller oversampling ratios (such as 25%) are
effective in reducing the number of taps.
1.5.3
Scalability of joint channel and mismatch estimation
The proposed mismatch compensation schemes require estimates of the mismatch
parameters, which we calculate jointly with the channel parameters using the training
sequences provided for channel estimation. This involves estimating mismatch parameters and channel parameters alternately using an iterative algorithm. The final contri-
9
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
bution of this thesis is an investigation of the behaviour of this algorithm as a function
of system parameters.
For pseudorandom training sequences with length M , where M is a sufficiently
large multiple of L (e.g., M = 4L), we observe that the estimation error decreases
exponentially with iterations. However, when the training sequence length M = 2L
or smaller, we provide some examples showing that the iterative algorithm can get
stuck away from the true parameter values. Moreover, we demonstrate that by using
well-designed periodic training sequences, the channel and mismatch parameters can
be estimated very well within the first iteration.
Further, we use an observation that the channel estimates in successive iterations are
well approximated as proceeding along a straight line (in N -dimensional space, where
N is the number of channel taps) to derive analytical expressions for the convergence
rate. Specifically, for (pseudo)random training sequences of length M , we obtain that
the convergence rate is well- approximated by 20 log10 (M/L) for typical ranges of
interest for M , the training length and L, the number of sub-ADCs. Moreover, the
iterative algorithm is demonstrated to be robust to noise: the estimation errors first
decrease exponentially (with a rate similar to the noiseless setting), and then settle to
a residual value that depends on the noise level. We evaluate the Cramer-Rao lower
bounds for the estimation error, and observe that the residual mean-squared error in our
estimates is close to these bounds.
10
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
1.6
Literature Survey
The time-interleaving architecture has attracted much attention due to its potential
for realizing high-speed, high-resolution converters with reduced power dissipation. A
number of mismatch models have been proposed, ranging from simple models including gain, timing and voltage offset mismatches, to more general models of mismatched
frequency responses [16, 17, 18, 19].
Some recently proposed designs use efficient clocking schemes and redundant elements as a means of overcoming mismatch [20, 21, 22]. The scalability of such approaches to higher speeds and resolution requires further investigation. Other than these
references, prior approaches typically consist of first estimating the mismatch parameters, and then compensating the output of the ADC. We now list some popular schemes
for mismatch estimation and compensation.
1.6.1
Mismatch estimation
Mismatch estimation can be performed blindly by comparing the statistics of the
digital output between the mismatched and the ideal TI-ADCs [23, 18, 24, 25, 26, 27].
However, reliable blind estimation needs a large sample size. Training-based schemes
are faster, but for general-purpose ADCs for which the input can be arbitrary, they
11
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
require the additional expense of separate hardware for test sequence generation [19,
28]. Furthermore, all of these studies restrict the number of sub-ADCs to less than 16.
1.6.2
Mismatch compensation
Once the mismatch parameters are estimated, the compensation scheme depends on
the type of mismatch. Voltage-offset and gain mismatches can be compensated using
simple subtraction and scalar gain, respectively. On the other hand, timing mismatch,
or more generally, frequency response mismatches, need a set of parallel finite-impulse
response (FIR) filters for compensation [29, 30, 31, 23, 32, 33, 18, 24, 19]. In addition
to such digital compensation schemes, a number of analog techniques have also been
proposed [25, 28].
In [34], communication training sequences are utilized for mismatch compensation,
but these are restricted to estimating and subtracting the signal-independent interference
due to voltage offset mismatch. Prior work on characterizing the effect of gain, timing
and voltage offset mismatches on a single-carrier communication system includes [35].
1.6.3
Oversampling for mismatch compensation
Oversampling has been used previously to deal with mismatch-induced impairments [24, 36, 31]. The main motivation to use oversampling is to limit the complexity
of mismatch compensation. In the absence of oversampling, the exact zero-forcing
12
Chapter 1. Multi-Gigabit Communication and the ADC Bottleneck
equalizers for mismatch compensation are of infinite length, with slowly decaying taps.
Truncated/least-squares solutions have been employed in [24, 30, 19]. However, the
number of taps required can still be prohibitive if the required resolution and/or the
mismatch range is large.
In [36, 31], modest oversampling is used to achieve higher resolution, but these
schemes require the calculation of many FFTs (equal to the number of sub-ADCs),
which is expensive for a typical communication receiver setting. On the other hand, in
[24], the mismatch estimation is facilitated by the use of oversampling, but the compensation still required 41-tap FIR filters.
1.6.4
Drastic Quantization
A radically different approach to overcome the ADC bottleneck at high sampling
rates is to use an extremely low resolution ADC (1-4 bits). However, this introduces a
significant non-linearity at the start of the receiver chain and hence, demands a significant redesign of the transceiver [37, 38]. While such an approach might be appropriate
for some applications (e.g., a small constellation used over a line-of-sight channel), the
TI-ADC approach is applicable more generally, especially to settings that require larger
dynamic range.
13
Chapter 2
Frequency domain Mismatch
Compensation
The advent of low cost digital signal processing has made Orthogonal Frequency
Division Multiplexing (OFDM) a popular transmission scheme for dispersive channels
[39]. In this scheme, as shown in Fig. 2.1, the information-bearing symbols are encoded
onto the frequency domain by modulating orthogonal sinusoids referred to as “subcarriers”. Since sinusoids (or precisely, complex exponentials) are eigenfunctions of any
linear, time-invariant system, the channel can change only the gain or phase of the input
sinusoid. Thus, a dispersive channel is converted to a set of parallel scalar channels, reducing the problem of channel compensation to undoing a complex gain on each scalar
channel. This transmission scheme requires an inverse-Fast Fourier transform (IFFT)
at the transmitter to encode data onto the frequency domain, and an FFT at the receiver
to correlate with the desired sinusoid. These operations can be implemented in DSP, so
c
Parts of this chapter are reprinted from our journal submission [1] with permission, [2010]
IEEE
14
Chapter 2. Frequency domain Mismatch Compensation
Data symbols
Subcarriers
Channel
gain
Channel
phase
0.5
0
B2 = 1
0.75
90
B8 = 1
1
180
B1 = 1
Received
(relative to carrier)
Reference phase
Figure 2.1: Orthogonal Frequency Division Multiplexing (OFDM): A popular communication
scheme where the information-bearing symbols are encoded in the frequency domain.
Sunday, May 29, 2011
that OFDM is highly effective in leveraging the Moore’s law. Further details regarding
OFDM transmission are included in Section 2.2.2.
In this chapter, we employ a time-interleaved ADC in an OFDM receiver, and
present low-complexity mismatch compensation in the frequency domain.
2.1
Organization and Notation
We describe the system model, including the TI-ADC with mismatch, and the
OFDM transceiver in Section 2.2. Mismatch-induced interference is characterized in
15
Chapter 2. Frequency domain Mismatch Compensation
Section 2.3, and we propose a frequency-domain joint mismatch compensation and demodulation scheme in Section 2.4. An iterative scheme for joint mismatch and channel
estimation is presented in Section 2.5. Numerical results from simulation and hardware
experiments are presented in Section 2.6.
Bold-faced symbols are used for vectors and matrices. For example, an M × 1
column vector X equals (X[0], · · · , X[M − 1])t , where the superscript t denotes trans∗
pose. We use the notation X to represent a vector/matrix obtained by taking the
complex conjugates of all the individual elements in X. We use <[·] and =[·] to represent the real and imaginary parts of a vector/matrix. The concatenation of the real and
imaginary parts of a vector X, that is the vector (<[X]t =[X]t )t , is denoted by X̃.
2.2
TI-ADC in an OFDM receiver
In this section, we first give a model for the TI-ADC with gain, timing and voltage
offset mismatches. Next, we give details about the received signal in OFDM transmission, which would serve as the analog input for the TI-ADC.
2.2.1
TI-ADC Mismatch Model
In Fig. 1.3, we illustrate the effect of mismatch on the sampling of the input analog
signal r(t). Gain and voltage offset mismatches result in memoryless effects on the
16
Chapter 2. Frequency domain Mismatch Compensation
ideal samples (simple multiplicative and additive effects, respectively). On the other
hand, the effect of timing mismatch depends on the actual signal (or specifically, on
the values it takes between the ideal sampling instants). Letting
1
To
denote the nominal
sampling rate, the output of the TI-ADC can be written as [16]
r[m] = (1 + gm mod L )r((m + δm mod L )To ) + µm mod L
(2.1)
where r[m] denotes the mth sample, sampled by the sub-ADC with index m mod L,
where L denotes the number of sub-ADCs and mod denotes the modulo operation.
The gain, timing and voltage offset mismatches of the sub-ADC with index m mod L
are denoted by gm mod L , δm mod L , and µm mod L , respectively, where the timing mismatches δm mod L have been normalized with respect to To . In the settings of interest
to us [19], mismatch-induced interference and receiver thermal noise dominate quantization noise (which is small for a moderately high sub-ADC resolution), hence we
neglect the latter from our analysis. Since drift in mismatch parameters occurs over durations of hours, we can neglect it in our framework. We therefore model the mismatch
parameters as constant between successive training phases. In what follows, we omit
explicitly writing “mod L” in the subscripts in order to simplify notation.
17
Chapter 2. Frequency domain Mismatch Compensation
B[y]
b[m]
IFFT
Append
cyclic prefix
Transmit
Filter
DAC
Upconverter
[a]
DownConverter
Receive
Filter
r(t)
r[m]
ADC
R[k]
FFT
Receiver
noise
[b]
Tranmsitted
time-domain
samples
Transmit
Filter
Channel
response
Receive
Filter
Received
time-domain
samples
Cascade impulse response = h(t)
[c]
Figure 2.2: [a] OFDM Transmitter (DAC = Digital-to-Analog Converter) [b] OFDM Receiver
[c] Baseband signal model (time-domain). Reprinted from our submission [1] with permission,
c
[2010]
IEEE.
2.2.2
OFDM Model
We briefly review the standard OFDM transceiver operating over a dispersive channel, as shown in Fig. 2.2 [39]. Information-bearing symbols (drawn from a constellation such as QPSK) are encoded onto M subcarriers by using the inverse-Fast Fourier
Transform (IFFT) operation (of size M ) at the transmitter. The IFFT output (M timedomain symbols) constitutes one OFDM frame, and a cyclic prefix is conventionally
added before each frame to convert the linear convolution between the communication
18
Chapter 2. Frequency domain Mismatch Compensation
channel and the time-domain symbols into a circular one. This converts the vector
channel equalization to a scalar problem. After downconversion, the receiver performs
FFT on the Nyquist-sampled received baseband signal, and then performs scalar equalization and demodulation of the constellation symbols separately for each subcarrier.
We denote the Nyquist sample period by T , so that the baseband signal (assuming
1 1
, 2T . To simplify notation, we use the
no excess bandwidth) occupies the band − 2T
band 0, T1 for analysis. We refer to this as the signal band. The transmit and receive
filters in Fig. 2.2 are assumed to be ideal lowpass filters in the signal band. The receiver
input noise, w(t), is assumed to be proper, complex, zero-mean, white, Gaussian process with a power spectral density No . The noise at the output of the receive-filter, n(t),
follows the same statistics but with a non-zero power spectral density (equal to No ) only
in the signal band. The impulse response of the cascade of the transmitter, channel and
receive filters is denoted by h(t). We assume that h(t) is zero outside [0, N T ], where
N M is a measure of the delay-spread of the communication channel. The length
of the cyclic prefix (in Nyquist rate samples) should be at least N , in order to maintain
the orthogonality of the subcarriers after the FFT.
19
Chapter 2. Frequency domain Mismatch Compensation
2.3
The Structure of Mismatch-Induced Interference
Assuming a long enough cyclic prefix, we ignore the effect of samples from adjacent frames on the current OFDM frame, and consider an isolated frame for analysis.
The input to the TI-ADC is then given in terms of time-domain symbols {b[m]} as
r(t) =
M
−1
X
b[m mod M ]h(t − mT ) + n(t)
(2.2)
m=−N
Since the information-bearing symbols are encoded onto frequency domain subcarriers,
we rewrite the time-domain symbols {b[m]} in terms of the frequency domain symbols
{B[y]} (which are related by the energy-preserving FFT operation):
M
−1
M −1
X
1 X
h(t − mT )ej2πym/M + n(t)
B[y]
r(t) = √
M y=0
m=−N
(2.3)
Since h(t) ≈ 0 outside [0, N T ], all the significant samples of h (at rate T −1 ) lie in
[0, N T ], and it can be readily observed from (2.3) that the summation over m accounts
for all these significant samples, as long as t is restricted to lie in [0, (M − 1)T ]. Hence,
we can convert the finite summation over m in (2.3) to an infinite summation,
r(t) =
M
−1
X
y=0
B[y]φy (t)
∞
X
j2πy
h(t − mT )e− M T (t−mT ) + n(t)
m=−∞
20
(2.4)
Chapter 2. Frequency domain Mismatch Compensation
where φy (t) =
j2πyt
√1 e M T .
M
Let H(f ) denote the Fourier transform of h(t). Since the
transmit filter is bandlimited to the signal band, H(f ) is also bandlimited, hence we can
apply the Poisson’s summation formula for the summation over m in (2.4) to obtain,
r(t) =
M
−1
X
H[y]B[y]φy (t) + n(t)
(2.5)
y=0
where H[y] =
1
H
MT
y
MT
or alternatively, {H[y]} are related to the sequence of the
symbol-rate channel taps in the time-domain, {h[m] = h(mT )}, by the discrete Fourier
transform of size M . We can rewrite the signal part of (2.5) in terms of the In-Phase
(I) and Quadrature-Phase (Q) components as follows:
r(t) =
M
−1
X
<[S[y]φy (t)] + j
M
−1
X
=[S[y]φy (t)]
(2.6)
y=0
y=0
where S[y] = H[y]B[y]. Now, we perform Nyquist rate sampling of I and Q waveforms using two different TI-ADCs with mismatch parameter sets {(gI,m , δI,m , µI,m )}
and {(gQ,m , δQ,m , µQ,m )} , respectively. Using (2.1) in (2.6) with To = T , the ordered
pair of the samples of I and Q components is given as,
1
(rI [m], rQ [m]) = √
M
(1 + gI,m )
(1 + gQ,m )
M
−1
X
j2πy(m+δI,m )
M
< S[y]e
+ µI,m
y=0
M
−1
X
!
j2πy(m+δQ,m )
M
= S[y]e
+ µQ,m
y=0
21
,
(2.7)
Chapter 2. Frequency domain Mismatch Compensation
We now obtain the complex-valued samples r = {rI [m] + jrQ [m]} as
1 ∗
∗
∗
r= √
(∆I + ∆Q )S + (∆I − ∆Q )S + µ,
2 M
(2.8)
where µ = {µI,m +jµQ,m }, and we have S = {S[y]} following the bold-faced notation
in Section 2.1. The (m, y)th element of the matrix ∆I (and ∆Q ) is given as
∆I (m, y) = (1 + gI,m )e
j2πy
(m+δI,m )
M
, ∆Q (m, y) = (1 + gQ,m )e
j2πy
(m+δQ,m )
M
(2.9)
In (2.9), the indices m and y take integer values between 0 and M − 1. We now take
the (energy-preserving) FFT of the complex samples r in (2.8) to obtain
R=
1 ∗
∗
∗
F (∆I + ∆Q )S + F (∆I − ∆Q )S + Υ
2M
(2.10)
where F denotes the standard FFT matrix of size M , and the vectors R and Υ represent
the (energy-preserving) FFT of the vectors r and µ , respectively.
It is worth checking that the model in (2.10) reduces to an OFDM system with no
∗
mismatch by setting ∆I = ∆Q = F and Υ = 0. This gives R = S, since the product
∗
F F equals M times the identity matrix. Estimates for the information-bearing symbols B can be obtained by correlating with the corresponding channel gains, followed
by the demodulation of constellation symbols.
22
Chapter 2. Frequency domain Mismatch Compensation
With mismatch, however, the model for R is more complicated, and we observe
from (2.10) the effect of different types of mismatch. Voltage offset mismatch adds
signal-independent interference to the subcarriers, and can be compensated by simple
subtraction. Gain and timing mismatches cause inter-subcarrier interference, and require an equalizer for compensation. A simpler approach to compensate gain mismatch
is to apply time-varying gains to the time domain samples in (2.7), but timing mismatch
cannot be compensated for in such a simple fashion in either the time or frequency domain, and requires an equalizer. We combine the tasks of gain and timing mismatch
compensation, since the complexity is not reduced by addressing gain mismatch separately.
2.4
Frequency domain Joint Mismatch Compensation
and Demodulation
In this section, we design zero-forcing equalizers to compensate for the signaldependent interference induced by the gain and timing mismatches following the model
given in (2.10).
Zero-forcing is a simple linear technique that is often employed to suppress the
inter-symbol interference induced by channel dispersion [39]. In essence, the interference is eliminated by projecting the received signal vector onto a space perpendicular
23
Chapter 2. Frequency domain Mismatch Compensation
Orthogonal
to interference space
Signal vector
Interference space
Figure 2.3: The concept of a zero-forcing equalizer, where the received signal vector is projected
onto a space orthogonal to the interference.
to the interference space (See Fig. 2.3). Since noise power is likely to be same in all
Saturday, May 21, 2011
dimensions, projecting the received signal in the direction of the desired signal maximizes the signal-to-noise power ratio. When we project along any other direction (as
in Fig. 2.3), it can only degrade the signal-to-noise ratio, or equivalently enhance the
noise for a normalized signal power. However, we later provide a heuristic argument
in the setting of mismatch compensation indicating that the noise enhancement from a
zero-forcing scheme is minimal. Since the equalizer operates on the FFT outputs R,
we refer to it as a frequency-domain zero-forcing equalizer.
For the model in (2.10), we first note that S and S ∗ are dependent. In order to formulate an unconstrained problem, we rewrite (2.10) in terms of the real and imaginary
components of S as
R̃ = AS̃ + Υ̃
24
(2.11)
Chapter 2. Frequency domain Mismatch Compensation
The matrix A in (2.11) is defined as


A


 <[∆I ] −=[∆I ] 
 <[F ] −=[F ] 

 and ∆
˜ =
˜ where F̃ = 
:= F̃ ∆,




=[∆Q ] <[∆Q ]
=[F ] <[F ]
(2.12)
If the matrix A is invertible, we obtain a zero-forcing equalizer for S̃ as follows:
S̃ = A−1 (R̃ − Υ̃)
(2.13)
When A is not invertible, we can still obtain an equalizer using the Moore-Penrose
pseudo-inverse of A. In this case, there would be some residual mismatch-induced
interference at the equalizer output which would lead to an error floor. In all of our
numerical experiments, we have found that A is invertible, in which case perfect zeroforcing equalization is possible. While we have not been able to find general analytical
conditions for invertibility, we provide insight into what such conditions might look
like by considering the special case in which the I and Q channel TI-ADCs have the
same mismatch parameters; that is, ∆I = ∆Q . In this case, we are able to show (see
Appendix A) that the matrix A is invertible as long as the gains are non-zero and the
(normalized) timing mismatches have absolute values less than 1. Deriving conditions
for the more general scenario of ∆I 6= ∆Q is left as an open problem.
25
Chapter 2. Frequency domain Mismatch Compensation
The complexity of the zero-forcing solution in (2.13) depends on the number of nonzero entries in A−1 which, in general, can be as large as 4M 2 . This results in 4M 2 real
multiplications, which can be excessive for large M . Fortunately, the complexity (in
terms of the number of real multiplications) is significantly smaller when the number of
interleaved ADCs, L, divides the number of OFDM subcarriers M , as discussed next.
2.4.1
Interference structure when L divides M
When L divides M , we prove in Appendix B that the set {0, · · · , 2M -1}, which
can be used to index the rows (columns) of A, can be partitioned into disjoint groups
such that A(k, y) is non-zero only when the row and column indices, k and y, belong
to the same group. Hence, we can divide the system in (2.11) into parallel systems
of interfering constellation symbols. Also, we observe from (B.4) of Appendix B that
whenever the parallel system contains the real part of the constellation symbol encoded
on a particular subcarrier, it also contains the corresponding imaginary part. We refer
to these groups of subcarriers as “interference groups.” Equalization is performed separately for each interference group. We show that the total complexity, summed over
all interference groups is bounded by 8M L or equivalently, the average complexity per
sample is less than 4L real-valued multiplications. When L M , this is a significant
reduction in complexity, since in general, the average complexity can be as high as 2M
real-valued multiplications per sample. Thus, it makes sense to restrict attention to the
26
Chapter 2. Frequency domain Mismatch Compensation
special case when L divides M in system design. We note that the complexity of our
interference suppression scheme does not scale up with the desired resolution, and is
fixed at 4L real-valued multiplications per sample. We return to this observation in
Section 2.6, where we provide explicit comparisons with other available options.
2.4.2
Noise Enhancement
In order to obtain an understanding of the noise enhancement caused by our linear
interference suppression scheme, we first derive the approximate structure for the noise
in (2.5) after it is sampled at Nyquist rate using the I and Q TI-ADCs. The bandlimited
noise process in (2.5) can be represented as a “sinc” function interpolation (in a meansquared sense) of its Nyquist rate samples [40]. We then consider the timelimited noise
waveform within the OFDM frame interval of [0, (M − 1)T ], and approximate it as an
interpolation of samples within that window (neglecting edge effects, assuming large
frame sizes). This yields
n(t) ≈
M
−1
X
n[m]sinc(t − mT ),
m=0
27
t ∈ [0, (M − 1)T ]
(2.14)
Chapter 2. Frequency domain Mismatch Compensation
Now, we replace the time-domain noise samples, {n[m]}, in (2.14) by their (energypreserving) FFT coefficients, {N [y]}, to obtain
M −1
M
−1
X
j2πym
1 X
N [y]
sinc(t − mT )e M
n(t) ≈ √
M y=0
m=0
(2.15)
Since the noise samples {n[m] = n(mT )} are zero-mean, i.i.d complex Gaussian random variables, {N [y]} also have the same statistics. Assuming small side-lobes for
the “sinc” function, we can approximate the summation over m given in (2.15) by an
infinite summation, which in turn can be simplified using the Poisson’s summation formula,
n(t) ≈
M
−1
X
N [y]φy (t)
(2.16)
y=0
where φy (t) =
j2πyt
√1 e M T .
M
Substituting (2.16) into the received signal model (2.5), we
obtain
r(t) =
M
−1
X
(H[y]B[y] + N [y]) φy (t)
(2.17)
y=0
Noting the similarity in the structure of (2.17) and (2.5), it can be shown that the
frequency-domain model (2.10) holds even when thermal noise is added at the ADC
input, but with S[y] = H[y]B[y] + N [y]. When we assume perfect knowledge of
the mismatch parameters, this implies that the presence of noise does not degrade the
estimates of S, or equivalently there is no noise enhancement for the proposed zero-
28
Chapter 2. Frequency domain Mismatch Compensation
forcing equalizer. Although this claim is obtained as an approximation, the simulation
results presented in Section 2.6, which do not use the approximations in (2.14) and
(2.16), are consistent with the claim, since they show only a small degradation from the
ideal performance with no mismatch.
2.5
Joint Mismatch and Channel Estimation
In this section, we estimate the mismatch parameters jointly with the communication channel parameters by using the channel estimation training sequences. We use
the model in (2.10) to obtain the estimates, and include the effect of thermal noise by
taking S[y] = H[y]B[y] + N [y]. Since an approximate model is used for the noise
samples, the obtained estimates are suboptimal. The channel taps in the frequency domain are correlated (the number of subcarriers, M , is much larger than the number of
symbol-rate taps in the time domain channel N + 1), and neglecting these correlations
in frequency domain channel estimation would lead to performance degradation. We
therefore adopt the simpler strategy of estimating the channel taps in the time domain.
As shown in [41], frequency domain correlations can be easily extracted from such
time-domain estimates.
29
Chapter 2. Frequency domain Mismatch Compensation
We first rewrite (2.11) in terms of the time-domain channel taps, denoted by h̃, as
follows:
R̃ = A(D h̃ + Ñ ) + Υ̃
(2.18)
where the matrix D, that depends on the training sequence, is given as

1
D= 
2


∗
diag(B )   FN +1 jFN +1 


 ∗

∗
∗
−j diag(B) j diag(B )
FN +1 −jFN +1
diag(B)
(2.19)
where FN +1 denotes the matrix formed by the first N + 1 columns of the FFT matrix of
size M and “diag(B)” denotes the diagonal matrix formed by placing all the frequencydomain symbols of the training sequence B along the diagonal.
We observe from (2.18) that the unknown parameters A (a matrix with 4M 2 entries) and Υ̃ (a vector with 2M entries) actually depend on the mismatch parameters,
a collection of 6L independent unknowns. In order to obtain an unconditioned problem formulation, we estimate the mismatch parameters instead of the entries of A and
Υ̃. Direct joint estimation of the channel taps and the mismatch parameters is complicated, hence we resort to an iterative optimization strategy in which we alternately
optimize over the channel taps and the mismatch parameters, while keeping the other
fixed. Next, we discuss the alternating steps in the iterative optimization.
30
Chapter 2. Frequency domain Mismatch Compensation
2.5.1
Channel estimation given mismatch estimates
Given the estimates of A and Υ̃ after the (k − 1)th iteration, denoted by A(k−1) and
Υ̃(k−1) , we can use (2.18) to obtain the least-squares (LS) estimates of the channel taps
in the k th iteration,
h̃(k) = (D H D)−1 D H A(k−1)
−1
(R̃ − Υ̃(k−1) )
(2.20)
where D H denotes the complex conjugate transpose of D. We note that the LS estimates obtained in (2.20) are also the ML estimates because the entries of the noise
vector Ñ in (2.18) are modelled as independent, Gaussian random variables.
2.5.2
Mismatch estimation given channel estimates
We consider the time-domain samples of the lth sub-ADC of the I-channel, denoted by rI,l , to calculate the corresponding mismatch parameters (gI,l , δI,l , µI,l ). Using (2.7)-(2.9), given the channel estimates at the k th iteration as h̃(k) , we obtain the
ML estimates of the gain, timing and voltage offset mismatches as,
(k)
(k)
(k)
(gI,l , δI,l , µI,l ) = argmin ||rI,l − <[∆I,l (gI,l , δI,l )S (k) ] − µI,l ||2
31
(2.21)
Chapter 2. Frequency domain Mismatch Compensation
where ∆I,l , a function of gI,l and δI,l , represents the matrix formed by the rows of
∆I with indices of the form 1 + l + dL (integer d). In (2.21), the vector S (k) has
entries H (k) [m]B[m]. We note that H (k) , the estimate (in k th iteration) of the vector of
frequency domain channel taps, can be obtained in terms of h̃(k) .
The norm in (2.21) should ideally take into account the correlation among the noise
samples. By considering the standard Euclidean norm, we are implicitly assuming that
the noise samples are uncorrelated. This holds when there is no mismatch, but need
not be true with mismatch. However, our numerical results in Section 2.6 show that
ignoring the noise correlations leads to little degradation in estimation performance for
the mismatch levels considered.
With the Euclidean norm, if we fix the timing mismatch, the objective function
in (2.21) is quadratic in the gain and voltage offset mismatches, hence we can obtain
closed-form expressions for the optimal estimates of these mismatch parameters and
plug them in. It remains to estimate the timing mismatch parameters, which we do
with a one-dimensional numerical search for each sub-ADC. In the latter, we assume
that the normalized timing mismatch is bounded to within an interval (e.g., [−0.1, 0.1]
corresponds to 10% mismatch).
We perform the optimization in (2.21) for l in {0, · · · , L − 1} to estimate the mismatches for all the sub-ADCs in the I-channel TI-ADC. Exactly the same approach
32
Chapter 2. Frequency domain Mismatch Compensation
holds for the Q channel as well; the corresponding expressions are obtained by replacing I in (2.21) with Q.
An important design issue is the number of iterations required. In this chapter, we
fix the number of iterations. An alternative, data-dependent criterion might be to stop
when the change from the prior iteration is “small enough”. We do not consider this
here, since as few as three iterations are found to work well in the settings we have
considered. For a more detailed investigation of convergence, we refer to Chapter 4.
Table 2.1: System parameters for simulations and experimental prototype. Reprinted from our
c
journal submission [1] with permission, [2010]
IEEE.
Variable
OFDM subcarriers
Modulation
Channel taps
Real
Symbol
M
Value
128
16-QAM
hI
0.5, -1.8 , 1.7, -0.6,
1.0, 0.2 , 0.4 , -0.6 , 0.3
-0.1, -1.8 , 3.8, 1.8 ,
-0.2, 1.5 , 0.6 , -0.1 , 0.0
hQ
Imaginary
Mismatch parameters for 8 interleaved ADCs
I-component
Gain
gI
Timing
δI
Voltage offset
µI
33
Relative mismatch in %
3.3, -1.4, -9.5, -7.1,
5.7, 9.3, -4.9, 1.1
4.6, -9.6, -1.3, -0.5,
2.9, 6.9, -0.4, -5.1
3.6, 5.2, 4.9, -2.2,
3.1, -6.6, 4.1, -9.4
Chapter 2. Frequency domain Mismatch Compensation
2.6
Illustrations
In this section, we consider example scenarios that illustrate the structure of the
mismatch-induced interference in an OFDM system. We provide simulation and experimental results to evaluate the performance of our mismatch estimation and compensation schemes. The relevant system parameters are given in Table 2.1. We consider
OFDM-based transmission with 128 subcarriers, each of which is modulated by symbols drawn from the 16-QAM constellation. The channel taps are obtained from an
instance of CM 1, the LOS channel model defined in the UWB standardization process
[10]. In Table 2.1, we give mismatch parameters corresponding to a level of 10%, for
a TI-ADC with eight interleaved ADCs. By 10% mismatches, we mean that the values of gain, timing and voltage offset mismatches are chosen uniformly in [−0.1, 0.1],
A A
, 10 ] , respectively. Here, To and A indicate the sampling period and
[− T10o , T10o ] and [− 10
the root-mean-square (rms) value of the ADC input signal’s amplitude. The mismatch
parameters given in Table 2.1 correspond to the TI-ADC for the I-channel, and we
obtain the parameters for the Q-channel TI-ADC by circularly shifting the I-channel
parameters by one element. We now provide simulation results to understand the effect,
structure and suppression of the mismatch-induced interference.
34
Chapter 2. Frequency domain Mismatch Compensation
2.6.1
Mismatch-induced Error Floors
The signal-dependent interference seen at any subcarrier due to mismatch, left uncorrected, is a sum of scaled versions of the constellation symbols from other subcarriers, modeled as independent random variables. Applying the central limit theorem,
we can model the total mismatch-induced interference at any subcarrier as Gaussian,
where the mean is determined by the voltage offset mismatches and the variance can be
obtained in terms of gain and timing mismatch parameters using (2.10). This Gaussian
approximation is validated by simulation results for the 16-QAM constellation in Fig.
2.4, where we show the performance for eight interleaved ADCs in three different mismatch settings of 10%, 5% and 1%. For the 1% and 5% mismatch levels, the mismatch
parameters are the scaled versions (by 0.1 and 0.5 , respectively) of the parameters for
the 10% level given in Table 2.1. The analysis predicts an error floor due to mismatch
as Eb /No gets large, and this is borne out by simulations. For mismatch levels of 10%
and 5%, we observe error floors of 10−2 and 10−3 , respectively. For 1% mismatch, we
do not observe an error floor for BER up to 10−4 . For a given mismatch level, larger
constellations see more severe error floors: while 1% mismatch does not give an error
floor for 16-QAM, there is an error floor at 10−3 for a 256-QAM constellation.
35
Chapter 2. Frequency domain Mismatch Compensation
ï1
10
ï2
BER
10
10% mismatch
ï3
10
5% mismatch
1% mismatch
No mismatch
ï4
10
0
5
10
15
Eb/No (in dB)
20
25
30
Figure 2.4: Effect of mismatches in a TI-ADC (with 8 sub-ADCs) for OFDM transmission
employing 16-QAM constellation. Perfect channel knowledge is assumed. Reprinted from our
c
journal submission [1] with permission, [2010]
IEEE.
2.6.2
Structure of Mismatch-induced Interference
We consider two settings to illustrate how crucially the structure of the interference
depends on whether the number of interleaved sub-ADCs L divides the number of
subcarriers M (we have M = 128): L = 6 (in which L does not divide M ) and L = 8
(in which L divides M ). The mismatch parameters for L = 6 are taken as the first 6
entries of the parameters given for L = 8 in Table 2.1. Consider the real part of the first
subcarrier’s constellation symbol as the signal of interest. The SNR without mismatch
36
Chapter 2. Frequency domain Mismatch Compensation
is fixed at 26 dB, which corresponds to Eb /No of 20dB for 16-QAM. From Fig. 2.5, we
observe that the gain and timing mismatches are the dominant sources of interference
in both settings. For L = 8, only constellation symbols corresponding to seven other
subcarriers and the imaginary part of the first subcarrier’s constellation symbol interfere
with the signal of interest. In contrast, for L = 6, for which L does not divide M , all
other subcarriers interfere with the signal of interest.
10
Signal Level
0
L=8
L=6
Relative interference power (in dB)
ï10
ï20
ï30
ï40
ï50
ï60
ï70
ï80
Signal
Real symbols
Imag. symbols
Thermal noise Vol. ïoffset
Interferers
Figure 2.5: Structure of mismatch-induced interference: the relative interference for the real
part of the constellation symbol of first subcarrier (signal) from all the subcarriers is shown.
The signal level is normalized to 0dB, and the values of “L” in the legend indicate the number
c
of sub-ADCs interleaved. Reprinted from our submission [1] with permission, [2010]
IEEE.
37
Chapter 2. Frequency domain Mismatch Compensation
2.6.3
Suppression of Mismatch-induced interference
We now illustrate the mismatch compensation and estimation algorithms proposed
in Sections 2.4 and 2.5 for eight interleaved ADCs with a mismatch level of 10%.
For training, we use six repetitions of a Pseudo Noise (PN) sequence with 128 QPSK
symbols.
ï1
10
ï2
10
ï3
10
BER
No correction
1 iteration
2 iterations
3 iterations
4 iterations
No mismatch
ï4
10
ï5
10
ï6
10
0
5
10
15
20
23
Eb/No (in dB)
Figure 2.6: BER after estimation and correction of 10% mismatches in a TI-ADC (with 8 subADCs) used for OFDM signal reception employing 16 QAM constellation. The number of
iterations of the estimation algorithm is shown in the legend. Reprinted from our submission
c
[1] with permission, [2010]
IEEE.
38
Chapter 2. Frequency domain Mismatch Compensation
Fig. 2.6 shows that the BER falls rapidly with an increasing number of iterations,
and that we can obtain performance comparable to that without mismatch in only three
iterations. The proposed mismatch compensation scheme has a complexity of 32 realvalued multiplications per sample.
In Fig. 2.7, we show the performance of the proposed algorithms for a TI-ADC
with L = 6 sub-ADCs. From the discussion in Section 2.4.1, the complexity can be as
large as 2M = 256 real-valued multiplications per sample. Fig. 2.7 illustrates the performance for the 256-tap, zero-forcing equalizer, as well as for suboptimal equalizers
with fewer taps, chosen by keeping coefficients of the zero-forcing equalizer with large
absolute values. We refer to these sub-optimal equalizers as truncated zero-forcing
equalizers. While the full-complexity equalizer does eliminate mismatch-induced interference, suboptimal equalizers with fewer than 256 taps incur error floors due to
residual interference. We also optimized the coefficients of the sub-optimal equalizer
(with 48 taps) for minimizing the residual interference power using the MMSE criterion, but obtained an insignificant performance improvement. We conclude that L dividing M is a superior design choice from the point of view of performance-complexity
tradeoffs.
39
Chapter 2. Frequency domain Mismatch Compensation
ï1
10
ï2
BER
10
No correction
Truncated ZF solution (24 taps)
Truncated ZF solution (48 taps)
MMSE solution (48 taps)
Full ZF solution (256 taps)
No mismatch
ï3
10
ï4
10
0
2
4
6
8
10
12
14
16
18
20
Eb/No (in dB)
Figure 2.7: BER after estimation and correction of 10% mismatches in a TI-ADC (with L=6
sub-ADCs) used for OFDM signal reception employing 16 QAM constellation. The complexity
(in terms of number of real-valued multiplications per sample) is indicated in the legend and the
number of iterations of the estimation algorithm is fixed at 3. Reprinted from our submission
c
[1] with permission, [2010]
IEEE.
2.6.4
Experimental results using hardware TI-ADC prototype
The results from computer simulation are supplemented with experimental results
from a hardware prototype that is initially developed by Dr. Munkyo Seo [19], and later
modified to handle communication signals. The prototype (shown in Fig. 2.8) employs
a TI-ADC sampling at 400MSa/s. The TI-ADC was assembled by interleaving four
commercially available ADCs from Analog Devices, Inc., each sampling at 100 MSa/s
40
Chapter 2. Frequency domain Mismatch Compensation
with 14 bits of resolution. We note the low sampling rate of the prototype (compared
to GHz sampling rates required for the multi-Gigabit systems in [11, 10]), chosen on
account of the ease of availability of the commercial ICs, and we refer the reader to
[15, 21, 22], where TI-ADCs with GHz sampling rates are realized. A 100MHz clock,
after bandpass filtering to eliminate wide-band white noise, feeds a clock distribution
board that provides timing for the interleaved ADCs. The distribution board uses a 1to-4 power splitter and delay lines to create four 100MHz clocks with nominal phase
offsets of 0o , 90o , 180o and 270o . Each clock path has a voltage-controlled phase shifter
using a varactor. For our experiments, however, this fine-tuning knob is disabled. Sampling time mismatches of the prototype TI-ADC are mainly due to relative phase errors
in the 1-to-4 power splitter and the voltage-controlled phase shifters.
We now explain the signal flow through the prototype. We assume the baseband
signal to be limited to [−200, 200] MHz. The digital version of the received (baseband)
signal, for both the I and Q channels, is generated using MATLAB at twice the Nyquist
rate (800 MSamples/sec). The signals corresponding to the I and Q channels are separated by a fixed white space so that they can be sampled by the same TI-ADC; the data
for the individual channels can then be obtained by de-serializing the TI-ADC output.
Using an Arbitrary Waveform Generator (AWG520 from Sony/Tektronix) as a D/A
converter, the MATLAB output is converted into an analog signal. The signal is then
lowpass filtered to 200 MHz and fed to the TI-ADC. The Nyquist sampled (400 Msam-
41
Chapter 2. Frequency domain Mismatch Compensation
4 x 100 MSa/s sub-ADCs
Received Baseband (2x)
t = 4 d T0
ADC
MATLAB
D/A
800 MSa/s
LPF
200 MHz
t = (4d+1)T
0
Analog
Input
ADC
4-to-1
MUX
t = (4d+2)T
0
ADC
t = (4d+3)T
0
ADC
Clock
100 MHz
14-bit sampled data
Digital
Ouput
Logic
Analyzer
Figure 2.8: TI-ADC experimental prototype with four interleaved ADCs. D/A and LPF refer
to digital-to-analog converter and low pass filter, respectively. We have To = 2.5ns and that d
c
takes integral values. Reprinted from our submission [1] with permission, [2010]
IEEE.
ples/sec) digital output from the ADC is collected at the Logic Analyzer (Tektronix),
which has a routine for subtracting the running-average mean from the sub-ADC output
to eliminate voltage offsets. We can therefore ignore voltage offset mismatch. We use
MATLAB to process the data obtained from the Logic Analyzer.
Using the joint estimation algorithm of Section 2.5, we obtained gain mismatch estimates of {0.19, 0.08, 0.40, 0.25}% and timing mismatch estimates of {−8.0, 5.0, −5.7, 8.5}%.
We note that these parameters correspond to both I and Q channels since the same
TI-ADC is used to sample both channels. We can compare the complexity of the proposed scheme with a standard time-domain compensation scheme in [19], where four
42
Chapter 2. Frequency domain Mismatch Compensation
ï1
BER
10
ï2
10
No correction
After correction
No mismatch (Theoretical)
ï3
10
0
2
4
6
8
10
Eb/No (in dB)
12
14
16
18
Figure 2.9: Experimental results from the prototype confirm the efficacy of the proposed mismatch estimation and equalization scheme. Reprinted from our submission [1] with permission,
c
[2010]
IEEE.
sub-ADCs are interleaved and only the I-channel is present. The scheme in [19] uses
21 multiplications per sample for a required resolution of 14 bits, while the proposed
scheme, independent of the required resolution, has a significantly smaller complexity
of 8 multiplications per sample.
Fig. 2.9 depicts the BER performance. We observe an error floor when the mismatches are left uncorrected, but we obtain a performance close to that without mismatch after zero-forcing equalization, thus verifying the efficacy of the proposed algorithms in a realistic setting. For the Spurious-Free-Dynamic-Range (SFDR) or the
The complexity of the proposed scheme is smaller than the upper bound (of 4L = 16 multiplications/sample) given in Section 2.4.1, because the I and Q channel TI-ADCs have the same mismatch
parameters in the experimental set-up.
43
Chapter 2. Frequency domain Mismatch Compensation
Effective-Number-Of-Bits (ENOB) metrics, we refer the reader to [19], where the same
experimental set-up is used for realizing a general-purpose TI-ADC.
44
Chapter 3
Scalable Mismatch Compensation by
Oversampling
The complexity of the frequency domain compensation scheme presented in the
previous chapter scales with the number of sub-ADCs, and the complexity can become excessive when the number of interleaved sub-ADCs increases beyond 16. In
this chapter, we consider linear mismatch compensation in the time domain instead,
and show that oversampling provides a scalable (in the number of sub-ADCs and in
the desired resolution) approach to mismatch compensation, allowing elimination of
mismatch-induced error floors at reasonable complexity. Intuitively speaking, oversampling increases the dimensions of the signal space containing the received signal
and the mismatch-induced interferers (See Fig. 2.3), and hence eases the design of
zero-forcing equalizers [39]. The results in this chapter are a natural generalization of
such known results on fractionally spaced equalization.
c
Parts of this chapter are reprinted from our conference submission [2] with permission, [2010]
IEEE
45
Chapter 3. Scalable Mismatch Compensation by Oversampling
3.1
Organization
In Section 3.2, we describe a z-domain discrete-time model for TI-ADC with mismatch and for zero-forcing mismatch compensation. Section 3.3 presents the use of
oversampling to reduce the number of taps required for mismatch compensation, including a proof that mismatch-induced interference can be eliminated using finitelength filters when we oversample by a factor of two. These results do not require
the specific structure of an underlying OFDM signal and are valid in general whenever
a TI-ADC with mismatch is used. However, in Section 3.4, we specifically illustrate
the performance-complexity tradeoffs for an OFDM receiver that employs the proposed
scheme, comparing the time domain strategy in this chapter to the frequency domain
approach in Chapter 2.
3.2
System Model
We first describe a linear mismatch model for the TI-ADC and then describe linear
schemes (based on zero-forcing equalization) for mismatch compensation.
3.2.1
TI-ADC model
We consider the problem of sampling an analog signal x(t) with the sampling period
To−1 . The desired digital samples, denoted by x[n] = x(nTo ), are referred to as symbols,
46
Chapter 3. Scalable Mismatch Compensation by Oversampling
t = k L To
t = k L To
G(z)
H(z)
0
0
t = ( k L + 1 ) To
t = ( k L + 1 ) To
X(z)
H(z)
Y(z)
1
G(z)
1
Y’(z)
t = ( k L + L - 1 ) To
t = ( k L + L - 1 ) To
G (z)
H (z)
L-1
L-1
Figure 3.1: Linear model for mismatch in a TI-ADC and zero-forcing based mismatch compensation (k=integer, To =sampling period). All symbols indicate the z-transforms of discrete
streams at the symbol rate To−1 . Reprinted from our conference submission [2] with permission,
c
[2010]
IEEE.
so that the term symbol rate sampling refers to sampling at rate To−1 . We assume that
the values of the continuous signal x(t) can be obtained by interpolating the symbols
as
x(t) =
∞
X
x[n]h(t − nTo ),
(3.1)
n=−∞
where h(t) represents the interpolating function. The class of signals in (3.1) is fairly
general: for example, x(t) could denote a general bandlimited signal, with h(t) taken
as the sinc function, or {x[n]} could be interpreted as symbols transmitted in a linearly
modulated communication system, with h(t) taken as the impulse response of a cascade
of the transmit, channel and receive filters.
47
Chapter 3. Scalable Mismatch Compensation by Oversampling
The ADC has a time-interleaved architecture as in Fig. 1.2, with L sub-ADCs
indexed by integers between 0 and L − 1. We model the lth sub-ADC by a linear,
time-invariant channel response e
hl (t): gain, timing and bandwidth mismatches are
special settings of this model [19]. Thus, the variation of e
hl (t) with l captures the
mismatch among the sub-ADCs [19]. The lth sub-ADC outputs nontrivial samples at
times (kL + l)To for integer k, and outputs zeros at all other times. The samples from
all the sub-ADCs are multiplexed to obtain the TI-ADC output y[m]. Assuming high
enough output resolution, we ignore quantization noise. The digital output of the lth
sub-ADC, yl [m], can then be written as a discrete convolution of the symbols x[n] with
the discrete response function of each sub-ADC:
yl [m] =
∞
X
x[n]hl [m − n],
m mod L = l
n=−∞
= 0,
otherwise,
(3.2)
where mod denotes the modulo operation and the function hl (t), henceforth termed
the sub-ADC response, is the convolution of the sub-ADC response hel (t) with the interpolating function h(t).
In order to find a discrete-time model for the TI-ADC, we need to relate the ztransform of the TI-ADC output y[m] with that of its input x[m]. Consider yel [m],
defined as the convolution of x[m] with hl [m], whose z-transform is given by Yel (z) =
48
Chapter 3. Scalable Mismatch Compensation by Oversampling
X(z)Hl (z). We first relate the transforms of yl [m] and yel [m] by realizing that the former
is non-zero only for samples with indices of the form l + kL (for integer k). Further, for
these samples, yl [m] = yel [m]. Equivalently, expressed in the z-domain, we can collect
terms with degrees l + kL from the polynomial Yel (z) to obtain Yl (z):
L−1
Yl (z) =
1 X −li e i
w Yl (wL z),
L i=0 L
(3.3)
where wL = ej2π/L is an Lth root of unity. We refer readers to [42], where equations
similar to (3.3) appear in the context of multi-rate signal processing. We now add the
outputs of all sub-ADCs and use the linearity of the z-transform to obtain the transform
of the TI-ADC output y[m]:
Y (z) =
L−1
X
X(wL i z)Fi (z),
(3.4)
i=0
where the terms Fi (z) are given in terms of Hl (z) as
L−1
1 X −li
Fi (z) =
wL Hl (wLi z).
L l=0
(3.5)
If Hl (z) = 1 for all l (no mismatch, ideal transfer functions for all sub-ADCs), then
Y (z) = X(z). In general, the expression for Y (z) in (3.4) has a signal term F0 (z)X(z),
49
Chapter 3. Scalable Mismatch Compensation by Oversampling
and interference terms {Fi (z)X(wL i z)} for i 6= 0. We now discuss conventional zeroforcing mismatch compensation for eliminating the interference terms.
3.2.2
Zero-forcing mismatch compensation
First consider a single sub-ADC (L = 1) with non-ideal response, for which we
have Y (z) = H0 (z)X(z). In this setting, the zero-forcing equalizer is given by G0 (z) =
[H0 (z)]−1 . For L interleaved sub-ADCs, zero-forcing mismatch compensation (which
also addresses non-idealities in the sub-ADC transfer functions) can be achieved using
L equalizers in parallel, {Gl (z)}, as shown in Fig. 3.1. These equalizers operate on
the TI-ADC output y[m] such that the lth equalizer output is calculated only for discrete
time indices of the form kL + l for integer k. Thus, in practice, the L parallel equalizers can be implemented as a single filter with periodically time-varying coefficients
with period L. Owing to the similarity between the structures of the TI-ADC and the
0
equalizer, we can use (3.4) for relating the equalizer output Y (z) to the equalizer input
Y (z) as
0
Y (z) =
L−1
X
φk (z)Y (wLk z),
(3.6)
k=0
where φk (z) is defined in terms of the equalizer filters {Gl (z)} as
L−1
1 X −lk
wL Gl (wL k z).
φk (z) =
L l=0
50
(3.7)
Chapter 3. Scalable Mismatch Compensation by Oversampling
0
We now substitute the expression for Y (z) from (3.4) in (3.6) to simplify Y (z) as
0
Y (z) =
L−1 X
L−1
X
Fi (wL k z)φk (z)X(wL i+k z).
(3.8)
i=0 k=0
We now collect the terms of the form X(wL α z) in (3.8). Since, wL α = wLα mod L , we
can restrict the range of α to integers in [0, L − 1]. The right-hand side of (3.8) can now
be rearranged as
0
Y (z) =
L−1
X
X
X(wL α z)
α=0
Fi (wL k z)φk (z),
(3.9)
(i,k)∈Sα
where the set Sα includes all 0 ≤ i, k ≤ L − 1 that satisfy (i + k) mod L = α. Given
k, each Sα has only one element, which is ((α − k) mod L, k). Now, we can replace
the second summation in (3.9) by a single summation over k to obtain
0
Y (z) =
L−1
X
X(wL α z)
X
α=0
Fα−k (wL k z)φk (z),
(3.10)
k
where we have also used the fact that Fi = Fi mod L . For zero-forcing the interference
0
terms {X(wL α z)} (for α 6= 0) in the expression for Y (z) given in (3.10) and to consequently obtain an undistorted signal term X(z) (except for an integer delay d), we need
51
Chapter 3. Scalable Mismatch Compensation by Oversampling
to satisfy the following system of equations in φ(z) = {φ0 (z), · · · , φL−1 (z)}T :
A(z)φ(z) = z −d (1, 0, · · · , 0)TL×1 ,
(3.11)
where the (α, k)th entry of the L × L matrix A(z) of (3.11) is given by
Aα,k = Fα−k (wL k z).
(3.12)
The matrix A(z) contains information about the mismatch responses {Hl (z)}, which
completely characterize the TI-ADC. We therefore refer to A(z) as the system matrix.
Once we obtain {φk (z)} from (3.11), we can find Gl (z) by obtaining the inverse
relation to (3.7) as
Gl (z) =
L−1
X
wL lk φk (wL −k z).
(3.13)
k=0
We now illustrate, through a running example, how linear equalizers can be obtained
when there is timing mismatch among the sub-ADCs.
52
Chapter 3. Scalable Mismatch Compensation by Oversampling
Figure 3.2: Discrete responses for two sub-ADCs with timing mismatch as 10% (solid arrows)
and −10% (dashed arrows), respectively. Popular bandlimited functions like Sinc and Raised
cosine resemble the shape of h(t) (red curve) to the first sidelobe. Reprinted from our conferc
ence submission [2] with permission, [2010]
IEEE.
3.2.3
Running Example
We assume To = 1 and take the sub-ADC response, hl (t), as h(t + δl ), where the
function h(t) is chosen to be
h(t) =




10(1 − |t|),




10(|t| − 2),







0,
|t| ≤ 23 ,
3
2
≤ |t| ≤ 2,
(3.14)
otherwise.
We consider L = 2 (two sub-ADCs) and take δ0 = 1/10, δ1 = −1/10. Hence, the
timing mismatch (relative to To ) is ±10%. The function h(t) and the discrete responses
for the two sub-ADCs are shown in Fig. 3.2. The z-domain responses of the sub-ADCs
53
Chapter 3. Scalable Mismatch Compensation by Oversampling
can be written as
H0 (z) = 9 − z −1 + z − z 2 , H1 (z) = 9 + z −1 − z − z −2 .
(3.15)
We now obtain F0 and F1 using (3.15) in (3.5) and then find the system matrix A.
Solving for φ in (3.11) and using the values of φ0 and φ1 thus obtained in (3.13), we
get the zero-forcing equalizers as
G0 (z) =
z 4 − z 3 − 9z 2 + z
z 3 − 9z 2 − z + 1
,
G
(z)
=
.
1
8(z 4 − 10z 2 + 1)
8(z 4 − 10z 2 + 1)
(3.16)
Note that G0 (z) and G1 (z) possess an infinite power series expansion, so that the corresponding time-domain functions g0 [n] and g1 [n] cannot be implemented as FIR filters.
This observation holds in more generality. Whenever the mismatch responses {hl [n]}
are of finite length, the entries of the matrix A(z) are finite length polynomials. The solution φ to (3.11), when it exists, is, in general, a rational function. Consequently, the
zero-forcing choices of Gl (z) are rational functions with infinite-length time domain
responses in general. In the next section, we show that, under certain conditions, we
can obtain finite-impulse response (FIR) equalizers for mismatch compensation by the
use of oversampling.
54
Chapter 3. Scalable Mismatch Compensation by Oversampling
3.3
Oversampling for Scalable Mismatch Compensation
For ease of exposition, we first consider oversampling with L = 1. We consider a
rational oversampling ratio of p/q, where p and q are relatively prime positive integers
(without loss of generality) such that p ≥ q. From (3.1), the mth output sample of the
p/q-oversampling TI-ADC can be obtained as
y[m] =
X
x[n]h
n
mq
− n To .
p
(3.17)
In order to find the output z-transform, we first consider the following discrete signals:
x
e[m] =



 x[m/p], m mod p = 0



0,
mT
o
e
,
h[m] = h
p
otherwise,
ye[m] = x
e[m] e
h[m],
(3.18)
where represents the convolution operation. Hence, the corresponding z-transforms
e X(z).
e
are related as Ye (z) = H(z)
It can be shown from (3.17) and (3.18) that y[m] =
ye[qm]. Now, we use the z-transform properties related to up/down sampling (e.g., see
[42]) to obtain
q−1
Y (z) =
1X
e k z 1/q ),
X(wqpk z p/q )H(w
q
q k=0
55
(3.19)
Chapter 3. Scalable Mismatch Compensation by Oversampling
where wq = ej2π/q . Note that when p = q = 1, the expression for Y (z) in (3.19)
reduces to X(z)H(z), which agrees with the discussion in (3.2.2).
We now consider the general setting of L interleaved sub-ADCs. As in (3.18), we
sample the corresponding sub-ADC response hl (t) at p times the symbol rate to obtain
a discrete signal e
hl [m] for each l. If the lth sub-ADC were to obtain all the samples, that
is at the rate of pTo−1 /q, the output z-transform is obtained from (3.19) by replacing
e
e l (z). In the time-interleaved architecture, we use (3.19) in (3.3) and (3.4) to
H(z)
by H
obtain
Y (z) =
q−1 L−1
X
X
X(wpkL+pi z p/q )Fi,k (z 1/q ),
(3.20)
k=0 i=0
j2π
where w = e qL . Compared to the expression obtained for the p = q = 1 setting in
(3.4), the coefficients Fi,k (z 1/q ) vary over two variables (i, k) and are defined in terms
e l (z)} as
of the sub-ADC responses {H
L−1
Fi,k (z) =
1 X −qli e kL+i
w Hl (w
z).
qL l=0
(3.21)
We now analyze the special setting of oversampling at twice the symbol rate in order to
obtain useful insights regarding the length of the zero-forcing equalizers. The analysis
also applies to other integer oversampling factors, but in practice, we would probably
be interested in rational oversampling factors between 1 and 2.
56
Chapter 3. Scalable Mismatch Compensation by Oversampling
3.3.1
Oversampling factor = 2
Substituting p = 2, q = 1 in (3.20), we obtain the following expression for the
TI-ADC output:
Y (z) =
L−1
X
X(w2i z 2 )Fi (z),
(3.22)
i=0
where Fi (z) is now defined as
L−1
1 X −li e i
w Hl (w z).
Fi (z) =
L l=0
(3.23)
For zero-forcing equalization, we consider L filters {Gl (z)} as in (3.6) such that successive outputs are obtained from different filters operating in succession. Using (3.22)
in (3.6), the output of the equalizer can be written in the z-domain as
0
Y (z) =
L−1 X
L−1
X
Fi (wk z)φk (z)X(w2i+k z 2 ),
(3.24)
i=0 k=0
where we used the fact wL = w for q = 1. The equalizer output in (3.24) refers
to a discrete signal at twice the symbol rate. In order to obtain the “symbols”, we
0
first downsample (by 2) the signal represented by Y (z) in (3.24) and later, we give
conditions for zero-forcing the interference terms. The transform of the downsampled
57
Chapter 3. Scalable Mismatch Compensation by Oversampling
version is given by [42]
L−1 L−1
1 XX
Yd (z) =
(Fi (wk u)φk (u) + Fi (−wk u)φk (−u))
2 i=0 k=0
0
X(w2i+k z),
where u =
√
(3.25)
z. We realize that the functions φk (u) and φk (−u) are dependent on each
other. To obtain an unconstrained zero-forcing problem formulation, we define two
transformed variables φk,e (u) and φk,o (u) as
2φk,e (u) = φk (u) + φk (−u), 2φk,o (u) = u−1 (φk (u) − φk (−u)).
(3.26)
Using the power series expansion for φk (u), we can infer that φk,e (u) and φk,o (u) contain different coefficients of the expansion and hence, we can choose them independent
of each other. Now, the zero-forcing conditions (with a delay d) for the 2-times oversampling setting are given by (3.11) with φ = {φ0,e , · · · , φL−1,e , φ0,o , · · · , φL−1,o }T
and in this setting, the L × 2L system matrix A(z) has its entries as
Aα,k =


P


i∈Sα−k
Fi,e (wk u),
0 ≤ k ≤ L − 1,
(3.27)

P

 u2 i∈S Fi,o (wk u), L ≤ k ≤ 2L − 1,
α−k
58
Chapter 3. Scalable Mismatch Compensation by Oversampling
where u =
√
z. The functions Fi,e and Fi,o are defined as in (3.26) by replacing φk
by Fi . The set Sa , for an integer a, is defined as Sa = {i : (2i) mod L = a}. After
solving the equation (3.11) using (3.27), the solution φ can be used in (3.13) to obtain
the equalizers {Gl (z)}. We now revisit the running example to show how oversampling
can help to simplify the equalizer design.
3.3.2
Running Example
We consider the setting of L = 2 sub-ADCs described in Section 3.2.3, but assume that the net sampling rate is two times the symbol rate. The sub-ADC responses,
sampled at twice the symbol rate, are given by
e 0 (z) = −z 8 − 4z 7 + z 6 + 6z 5 + 9z 4 + 4z 3 − z 2 − 4z,
z4H
e 1 (z) = −4z 7 − z 6 + 4z 5 + 9z 4 + 6z 3 + z 2 − 4z − 1.
z4H
(3.28)
To determine the system matrix A from (3.27), we calculate F0 and F1 using (3.23)
and determine Sa of (3.27) for the allowed values of a = {0, 1}. We obtain S0 = {0, 1}
and S1 as empty. We can now find the zero-forcing conditions by using the value of A
in (3.11) as
b(z)φ0,e + zc(z)φ0,o = z −d ,
b(z)φ1,e + zc(z)φ1,o = 0,
59
(3.29)
Chapter 3. Scalable Mismatch Compensation by Oversampling
where we replaced u with
√
z. The functions b(z) and c(z) in (3.29) are given as
b(z) = −z 4 + z 3 + 9z 2 − z and c(z) = −4z 3 + 4z 2 + 6z − 4. Due to the greater number
of variables than equations, the system of equations in (3.29) may have several solutions
but we are particularly interested in polynomial solutions. For the second equation, we
have a trivial solution: φ1,e (u) = φ1,o (u) = 0. For the first equation, the application
of the standard Bezout’s identity to polynomials with no common zeros, b(z) and c(z),
implies the existence of polynomial solutions for both φ0,e and φ0,o . These solutions
can be found by using the extended Euclidean algorithm [43]. These solutions are then
used in (3.13) to obtain finite length equalizers G0 (z) and G1 (z).
The existence of finite length equalizers can be generalized for a two-times oversampling TI-ADC with L sub-ADCs. From (3.27), we can decompose the L × 2L
matrix A into two L × L matrices B and C, such that B consists of the first L columns
of A and C has the next L columns. Now, we can rewrite (3.11) (used with A obtained
from (3.27)) as
B(z)φb (z) + C(z)φc (z) = z −d (1, 0, · · · , 0)TL×1 ,
(3.30)
where φb (z) = {φ0,e (z), · · · , φL−1,e (z)}T and φc (z) = {φ0,o (z), · · · , φL−1,o (z)}T . We
now state the following lemma regarding the existence of finite length zero-forcing
60
Chapter 3. Scalable Mismatch Compensation by Oversampling
equalizers expressed in terms of the determinants (denoted by det) of the matrices B(z)
and C(z).
Lemma 1. Finite length zero-forcing equalizers exist for mismatch compensation with
two-times oversampling when the polynomials det B(z) and det C(z) have no nontrivial zeros in common.
Proof. We first note that the factors of z k in detB(z) or detC(z) can be absorbed into
φb (z) and φc (z) in (30), so that we only need to consider nontrivial zeros (i.e., zeros at z 6= 0). Next, we note that (3.30) is a system of linear equations in φb (z) and
1
φc (z). Also, by definition Fi,e and Fi,o contain only even powers of u = z 2 or equivalently, they are polynomials in z. Hence, the coefficient matrix U = [B(z) C(z)] is
a L × 2L matrix with polynomial (in z) entries. We can form the augmented matrix
Ua by appending the column vector on the R. H. S of (3.30) to the matrix U . From
[44], polynomial solutions exist for all the entries of φb (z) and φc (z), when the greatest
common divisor (GCD) of all the L×L determinants is same for both U and Ua . (Actually, [44] provides results for when the variables and coefficients in the linear system of
equations are integers, but this result extends to polynomials). By hypothesis, det B(z)
and det C(z) have no common zeros, hence their GCD is 1. These two determinants
constitute two of all the L × L determinants calculated for both the matrices U and Ua .
Since the GCD of any other polynomial with respect to the trivial polynomial p(z) = 1
61
Chapter 3. Scalable Mismatch Compensation by Oversampling
is the trivial polynomial, we conclude the required GCDs are same (equal to p(z)) for
both U and Ua , implying the existence of a polynomial solution to (3.30).
Referring to the running example, we have, from (3.29), that det B(z) = b2 (z) and
det C(z) = z 2 c2 (z) and the determinants can be verified to have no common zeros,
except at z = 0. We give further illustration of the relation between the zeros of
det B(z) and det C(z) in Fig. 3.3, where we consider L = 4 sub-ADCs and assume
different levels of timing mismatch. For a given relative mismatch level δ, the mismatch
parameters for all the sub-ADCs {δl } are generated uniformly in [−δ, δ]. We observed
no common nontrivial zeros between the determinants and hence, the existence of a
finite-length equalizer is guaranteed by Lemma 1.
3.4
Application to an OFDM receiver employing a TIADC
We now illustrate the use of oversampling for mismatch compensation for a communication link using 128-subcarrier OFDM with 64-QAM signaling on each subcarrier, transmitted (with no excess bandwidth) over a frequency selective communication
channel. In our numerical results, we use a channel impulse response obtained as a
realization of the near Line-of-Sight (LOS) channel model defined in the UWB standardization process [10]. For the TI-ADC, we consider a 10% relative timing mismatch
for each sub-ADC.
62
Chapter 3. Scalable Mismatch Compensation by Oversampling
Relative timing
mismatch (in %)
10
5
2.5
1
Zeros of det B
Zeros of det C
ï2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
Real part of the zeros
Figure 3.3: Zeros of the determinants of the matrices B(z) and C(z) (obtained by the decomposition of the system matrix A(z)), plotted as a function of the relative timing mismatch.
Since the imaginary parts are insignificant, only the real parts of the zeros are shown. There are
some zeros of det B(z), but not of det C(z), beyond the depicted range. Reprinted from our
c
conference submission [2] with permission, [2010]
IEEE.
Following the discussion in the previous sections, ideal zero-forcing equalizers for
mismatch compensation can have an infinite number of taps. In this setting, we can
employ Minimum Mean-Squared Error (MMSE) mismatch compensation, minimizing
the total residual interference power with a finite number of taps. When there exists a finite length zero-forcing (ZF) solution (for example, when Lemma 1 holds for two times
oversampling), a ZF equalizer would be obtained as the MMSE solution, in the absence
of noise, for a sufficient number of taps. However, when a finite length ZF equalizer
does not exist, our numerical results illustrate that the equalizer length must increase
63
Chapter 3. Scalable Mismatch Compensation by Oversampling
(a)
(b)
ï1
ï1
10
10
ï2
ï2
BER
10
BER
10
ï3
ï3
10
10
No compensation
PreïFFT (5 taps)
PreïFFT (21 taps)
PostïFFT
No mismatch
ï4
10
0
ï4
10
20
10
30
Eb/No (in dB)
0
No compensation
PreïFFT (5 taps)
No mismatch
10
20
30
Eb/No (in dB)
Figure 3.4: BER in a 64-QAM, 128-subcarrier OFDM system employing a sloppy TI-ADC
with 10% timing mismatch. For the left subfigure (a), Nyquist rate sampling is performed and
the TI-ADC interleaving factor L is 8. On the other hand, we assume sampling at twice the
Nyquist rate and L = 32 in the right subfigure (b). Reprinted from our conference submission
c
[2] with permission, [2010]
IEEE.
with the desired resolution in order to limit the residual interference to an acceptable
level.
Zero-forcing time domain mismatch compensation is of general applicability, but
given that our focus is on OFDM in this section, we also evaluate the performance of
the frequency-domain mismatch compensation scheme that we proposed in Chapter 2,
which is specifically designed for OFDM receivers. It was shown in Chapter 2 that,
regardless of the desired resolution, we can compensate for mismatch after the FFT
64
Chapter 3. Scalable Mismatch Compensation by Oversampling
using L-tap frequency domain equalizers operating on groups of subcarriers of size L,
when the number of subcarriers is a multiple of the number of sub-ADCs L. We refer
to this scheme as Post-FFT compensation, and to the general zero-forcing mismatch
compensation solution proposed in this chapter as Pre-FFT compensation. For large
constellations, we desire a high ADC resolution: in this setting, post-FFT compensation works well for small L, but the pre-FFT compensation with oversampling (to limit
complexity as the desired resolution increases) becomes attractive for large L.
We first consider a Nyquist sampling TI-ADC with a moderate interleaving factor
of L = 8. BER results, depicted in Fig. 3.4 (a), indicate that the mismatch, when left
uncorrected, leads to significant error floors. Also, pre-FFT compensation, even with
as many as 21 taps, could not completely eliminate the mismatch-induced interference.
On the other hand, post-FFT compensation approaches the ideal performance without
mismatch at a much smaller complexity of L = 8 taps.
When we consider increasing the interleaving factor of the TI-ADC to increase the
net sampling rate, the complexity of the post-FFT scheme increases and beyond a point,
we resort to oversampling to enable low-complexity mismatch compensation. For illustration, we consider a TI-ADC with L = 32, for which the post-FFT compensation is
less attractive. We consider sampling at twice the Nyquist rate. For a given technology,
the absolute mismatch remains fairly constant. Assuming 10% relative mismatch for
Nyquist sampling, we have 20% relative mismatch for 2x oversampling. From Fig. 3.4
65
Chapter 3. Scalable Mismatch Compensation by Oversampling
(b), we observe that the Pre-FFT scheme requires only 5 taps to achieve the ideal performance without mismatch for BERs as low as 10−4 . For the same range of BERs, when
the oversampling factor is decreased to 5/4 (corresponds to oversampling by 25%), the
number of taps increased to 9 to approach the performance without mismatch.
66
Chapter 4
Scalability of Joint Channel and
Mismatch Estimation
The previous chapter focuses on compensating mismatches in a scalable fashion as
the number of sub-ADCs increases, assuming that the mismatch parameters have been
accurately estimated. However, any such mismatch compensation scheme requires sufficiently accurate estimates of the mismatch parameters. In this chapter, we ask how
well the joint channel and mismatch estimation procedures presented in Chapter 2 scale,
in terms of the length of training required, as we increase the number of sub-ADCs.
Our starting point is Chapter 2, where we present an iterative algorithm for joint
estimation of the channel coefficients and TI-ADC mismatch parameters (modeled as
gain, timing and voltage-offset mismatches). A least squares cost function is alternately minimized over the channel and mismatch parameters, keeping the other set of
parameters fixed. In this scheme, channel estimation with fixed mismatch parameters
c
Parts of this chapter are reprinted from our conference submission [3, 4] with permission, [2010,
2011] IEEE respectively.
67
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
has a closed form solution, while mismatch estimation with fixed channel coefficients
requires a linear search to find the timing mismatch estimate for each sub-ADC. In this
chapter, we consider a linear approximation in modelling the timing mismatch, so that
closed-form solutions exist for both channel and mismatch estimates.
4.1
Organization
We organize the rest of the chapter as follows. In Sections 4.2 and 4.3, we restrict
attention to “gain mismatch” in order to obtain detailed insight into the progress and
scalability of the joint channel and mismatch estimation. Section 4.4 extends the scalability results to include both gain and timing mismatches by linearizing the model
with respect to timing mismatch. For the simpler setting of gain mismatch, Section 4.5
presents an interesting observation on the progression of channel estimates, and gives
closed-form expressions for the convergence rate of the joint channel and mismatch
estimation algorithm. Section 4.6 deals with the effects of noise on the convergence of
joint estimation. Finally, while we have hitherto considered pseudorandom training sequences, we demonstrate in Section 4.7 that well-designed periodic training sequences
can achieve much higher convergence rates.
68
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
t = 4 d T0
ADC
t = (4d+1)T
0
b[m]
h(t)
r(t)
Training
sequence
Dispersive
channel
Analog
Input
Transmit
filter
Receive
filter
r[m]
ADC
4-to-1
MUX
t = (4d+2)T
0
Digital
Ouput
ADC
t = (4d+3)T
0
ADC
Figure 4.1: Base-band model for transmission over dispersive channel using a time-interleaved
ADC with 4 sub-ADCs at the receiver (d =integer, T0 = sampling period). Reprinted from our
c
conference submission [3] with permission, [2010]
IEEE.
4.2
Thursday, May 26, 2011
Dispersive channel and Gain mismatch
For transmission over a dispersive communication channel (See Fig. 4.1), the received signal is given by
r(t) =
M
−1
X
b[k]h(t − kT ) + n(t),
(4.1)
k=0
where {b[k]} is the training sequence of length M , 1/T is the symbol rate, and the
“channel impulse response” h(t) includes the effect of transmit, channel and receive
filters. We assume Nyquist sampling using the TI-ADC, so that the sampling interval
for each sub-ADC is LT , where L is the number of sub-ADCs. Throughout this chapter,
we assume that the sub-ADCs used to sample the I-channel and the Q-channel have the
69
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
same mismatch parameters. A generalization, however, is possible using ideas similar
to those in Chapter 2.
The received samples in the presence of gain mismatch among the interleaved
ADCs are given by
r[m] = (1 + gi )
m
X
b[k]h[m − k] + n[m],
m ∈ {0, 1, · · · , M − 1}
(4.2)
k=m−(N −1)
where i = m mod L (corresponding to the samples collected by sub-ADC i, i =
0, 1, ..., L − 1), gi denotes the gain mismatch for the ith sub-ADC (with respect to
unity gain), and {h[m]} denotes the channel coefficients expressed at symbol rate, assumed to be non-zero only when m ∈ {0, · · · , N −1}. We also assume that the training
sequence has a cyclic prefix of length N − 1 so that b[k] is defined for negative k.
We now express (4.2) in vector notation:
ri = (1 + gi )Ai h + ni ,
i = 0, 1, · · · , L − 1
(4.3)
where ri , h and ni represent the vectors of received samples, channel and noise.
Using (4.2), the matrix Ai has columns aj given by
70
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
aj =
t
M
b[x], b[x + L], · · · , b x +
−1 L
,
L
x=i−j
(4.4)
Given the vectors of received samples in (4.3), the goal is to estimate the channel
vector h, and the L mismatch parameters, {gi }. Neglecting correlations (if any) among
the noise samples n, the maximum likelihood (ML) estimates are obtained by solving
the following minimization:
(ĥ, {ĝ0 , · · · , ĝL−1 }) = arg min
L−1
X
||ri − (1 + gi )Ai h||2
(4.5)
i=0
Our numerical experiments show that the objective function in (4.5) is non-convex,
because its Hessian matrix (with respect to channel and mismatch parameters) has positive as well as negative eigenvalues. Thus, the powerful tools of convex optimization
cannot be applied (except as an approximate solution to the problem). Direct search is
also not a valid option, due to the increase in the dimensionality of the problem (4.5)
when either L or N become large.
However, we observe from (4.5) that the objective function is quadratic in the channel parameters given the mismatch parameters. This implies a closed form solution
for the channel estimate in (4.5) given the mismatch parameters. Similarly, closedform solutions exist for the mismatch estimates when the channel parameters are given.
71
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
We now illustrate this alternating minimization strategy (a similar solution is given for
OFDM transmission in Chapter 2) to solve the joint estimation problem in (4.5).
(n)
Given the mismatch estimates gi
at the start of nth iteration, the ML channel
estimate for (4.5) can be obtained by setting the partial derivative with respect to h
zero:
L−1
X
!
(1 +
(n)
gi )2 Ati Ai
(n)
h
=
L−1
X
(n)
(1 + gi )Ati ri ,
(4.6)
i=0
i=0
where n = 1, 2, · · · . On the other hand, given the channel estimates, the ML estimate
(n)
for the mismatch parameters gi
progress as follows:
(n)
1 + gi
=
(ri , Ai h(n−1) )
,
||Ai h(n−1) ||2
(4.7)
(1)
The algorithm is initialized by setting the initial mismatch estimates to zero: gi
= 0.
We note that there is a scale ambiguity in (4.5): if (h, {1 + gi }) are solutions, then so
are (xh, {(1 + gi )/x}). Without loss of generality, we therefore scale the gain estimates
(n)
(1 + gi ) by a constant to set the mean mismatch to zero.
72
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
4.3
Joint channel and mismatch estimation: convergence
behaviour
In this section, we use a pseudorandom training sequence to estimate the channel
and the TI-ADC mismatches. Specifically, we consider an m-sequence with generator
polynomial z 8 +z 6 +z 5 +z 4 +1. We append a zero to this length 255 m-sequence to obtain a training sequence of length 256 (an integer number of bytes). Shorter training sequences of length, say M , are obtained by truncating the sequence to retain the first M
bits. The transmitted training sequence is comprised of BPSK symbols b[n] = (−1)t[n] ,
where t[n] ∈ {0, 1} are the elements of the m-sequence. The gain mismatches are
generated uniformly and independently in the range [−0.1, 0.1]. We generate channel
coefficients as independent zero mean Gaussian random variables, and normalize the
channel norm squared,
PN −1
q=0
|h[q]|2 , to 1.
We study the convergence of the algorithm using the square of Euclidean distance
between the estimate and the truth as the metric. Thus, the channel estimation error is
given by
PN −1
q=0
|h[q] − ĥ[q]|2 . For mismatch estimation, we average the metrics over all
sub-ADCs to obtain
PL−1
i=0
|gi − gˆi |2 /L as the estimation error for the gain mismatch.
73
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
−20
Channel
Gain mismatch
Mean−square error (in dB)
−40
−60
−80
−100
−120
−140
−160
0
2
4
6
8
10
Iterations
Figure 4.2: Progress of iterations: Multiple iterations of joint channel and mismatch algorithm
decrease the MSE linearly in the dB scale (or exponentially in the absolute scale). Here M =
256, N = 20 and L = 64.
4.3.1
Progress of iterations
We first consider the progress of the algorithm in the absence of noise. Fig. 4.2
depicts the decrease in estimation error as the iterations progress for L = 64 sub-ADCs,
with channel length N = 20 and training length M = 256. We say that convergence
is achieved when the estimation error falls below -100 dB. Fig. 4.2 shows that the
algorithm converges in as few as 6 iterations. Since the graphs in Fig. 4.2 (with error
expressed on a log scale) can be closely approximated by straight lines, we infer that the
estimation errors decrease exponentially with iterations, with rate given by the slopes of
74
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
the corresponding linear fits. We observe similar trends for the mismatch and channel
estimation errors, and hence restrict our attention to the latter hereafter.
Rate of convergence (in dB/iteration)
25
M = 32
M = 64
M = 128
M = 256
20
15
10
5
0
0
20
40
60
80
100
120
140
Number of sub−ADCs (L)
Figure 4.3: Mean convergence rates versus number of sub-ADCs: Increasing the training length
for a given number of sub-ADCs increases the convergence rate of joint channel and mismatch
estimation algorithm. Here N = 20 channel taps and the convergence rate is defined with
respect to the channel estimate error.
4.3.2
Rate of Convergence
We now consider values of M given by powers of 2 ranging between 32 and 256.
For each M , we choose the number of sub-ADCs L as powers of 2 between 2 and
M/2. We study the convergence behaviour after a few iterations in order to avoid
any initial transient behaviour. So, we take the convergence rate to be the decrease in
the channel estimate error (in dB) from the 9th to the 10th iteration. From Fig. 4.3,
75
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
we observe that the convergence rate (averaged over 50 random instances of channel
and mismatches) decreases with L (for M fixed) and increases with M (for L fixed).
Thus, when convergence is desired with fewer iterations, the training sequence length
is observed to scale with the number of sub-ADCs.
4.3.3
Geometry of estimate progression
The estimate for the channel vector can move along any path in N -dimensional
space as iterations progress. However, our simulation studies indicate that the channel
estimates in successive iterations are well approximated as proceeding along a straight
line (in N -dimensional space, where N is the number of channel taps), see Figure 4.4,
which shows that the straight line approximation improves as the training sequence
length M increases. Also, the projection of the estimates onto the orthogonal complement of the depicted 2-D space is observed to be negligible throughout all the iterations.
4.4
Joint estimation with gain and timing mismatches
In this section, we extend the modelling in Section 4.2 to include timing mismatches. Unlike the gain mismatches, the model for timing mismatches becomes nonlinear. Since the mismatch is small, we linearise the model.
Including the effect of timing mismatch, the received samples are given by
76
Second coeff. error
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
M =64
0.01
0
−0.01
−0.05
0
0.05
Second coeff. error
First coeff. error
−3
M =128
x 10
5
0
−5
−0.03
−0.02
−0.01
0
0.01
0.02
Second coeff. error
First coeff. error
−3
x 10
M =256
4
2
0
−2
−4
−0.025−0.02−0.015−0.01−0.005
0
0.005 0.01 0.015
First coeff. error
Figure 4.4: Approximately linear trajectories of the channel estimate with iterations of joint
estimation algorithm: Trajectories for several randomly generated channels (length N = 20
taps) are plotted in the two-dimensional plane containing the true channel, h (indicated by
black diamond at origin), the initial channel estimate, h0 , and the channel estimate after the
first iteration, h1 . The “coefficients” represent the projections onto the orthogonal singular
vectors for the matrix, that is obtained for each realization, with columns h − h0 and h − h1 .
M denotes the training sequence length and L = 8 ADCs are interleaved. Reprinted from our
c
conference submission [4] with permission, [2011]
IEEE.
77
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
r[m] = (1 + gi )
m
X
b[k]h((m − k + δi )T0 ) + n[m],
(4.8)
k=m−(N −1)
where δi represents the (normalized) timing mismatch for the ith sub-ADC. Equation (4.8) contains non-uniformly sampled channel taps. We now try to express these
taps in terms of the channel taps sampled at the symbol rate.
We assume that there is no excess bandwidth in the transmission (a good approximation for OFDM, for example), so that the transmit filter is band-limited to [− 2T1 0 , 2T1 0 ].
This implies that h(t) is also band limited to the same range. We can therefore use the
sampling theorem to write h(t) in terms of (the symbol-rate samples) h[q] as follows:
h(t) =
N
−1
X
h[q] sinc (
q=0
t
− q),
T0
(4.9)
where we have used the assumption that h[q] = h(qT0 ) is zero unless q lies between
0 and N − 1. We now substitute h(t) from (4.9) in (4.8) and collect all samples corresponding to the ith sub-ADC (i.e., {r[m]} for m = i + pL) into a vector ri , modeled
as follows:
ri = Ci (gi , δi )h + ni
78
(4.10)
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
where h and ni denote the vector of N channel coefficients {h[q]} and the vector
of noise samples, respectively. The matrix Ci is a function of mismatch parameters,
with (p, q)th element given by
i+pL
X
[Ci ](p,q) = (1 + gi )
b[k] sinc (i + pL − k − q + δi )
(4.11)
k=i+pL−(N −1)
We now describe a linear approximation to model timing mismatch, where the samples
of the sinc function are approximated as follows:
sinc (k + δ) =




1
k=0
(4.12)


 δ · sinc 0 (k) k 6= 0, k ∈ Z
0
where sinc denotes the derivative of the sinc function. As shown in our simulations later, this is a good approximation (in the least squares sense) as long as the
timing mismatches (relative to T ) are small (< 10%). Using (4.12) in (4.11), we can
decompose the matrix Ci as
Ci = (1 + gi )Ai + δ̃i Bi
(4.13)
where δ̃i = (1 + gi )δi . The elements of Ai are as given in (4.4) while the elements
of Bi are given by
79
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
i+pL
[Bi ](p,q) =
0
X
b[k] sinc (i + pL − q − k)
(4.14)
k = i + pL − (N − 1)
k 6= i + pL − q
Our aim now is to jointly estimate the unknown channel coefficients h and the gain
and timing mismatch parameters corresponding to all sub-ADCs. Using (4.10) and
(4.13), we can write the maximum-likelihood (ML) joint estimation problem as,
(ĥ, {ĝ0 , · · · , ĝL−1 }, {δ̂0 , · · · , δ̂L−1 }) = arg min
L−1
X
||ri − (1 + gi )Ai h − δ̃i Bi h||2
i=0
(4.15)
where we assume that the noise samples are i.i.d. and hence, the ML estimate coincides
with the least-squares estimate given in (4.15). A discussion regarding the correlation
structure of noise is given in Section 2.4.2 in Chapter 2. However, simulations suggest
that it is a good approximation to assume i.i.d noise samples as long as the relative
mismatches are small (less than 10%).
80
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
4.4.1
Iterative algorithm for joint estimation
We can now derive iterations similar to (4.6) -(4.7). It is worth noting that the
mismatch estimation problem (with channel fixed) is quadratic in δ̃i , rather than in the
actual timing mismatches δi . Thus, we first estimate gi and δ̃i in closed form, and then
estimate δi . We can now specify each step of the iteration explicitly as follows:
−20
−30
Channel estimation
Gain mismatch estimation
Timing mismatch estimation
Estimate error (in dB)
−40
−50
−60
−70
−80
−90
−100
0
1
2
3
4
5
6
7
Iterations
Figure 4.5: Progress of the joint estimation algorithm: The estimate errors for channel and
mismatches decrease exponentially. Here, M = 256, N = 20 and L = 32. Reprinted from our
c
conference submission [3] with permission, [2010]
IEEE.
• Channel estimation given mismatches:
(n)
Given the gain mismatch estimates gi
start of nth iteration, the channel estimate can be obtained by solving:
81
(n)
and timing mismatch estimates δi
at the
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
L−1
X
!
Cit Ci
i=0
(n)
h
=
L−1
X
(n)
Cit ri ,
(n)
Ci = g̃i Ai + δ̃i Bi
(4.16)
i=0
• Mismatch estimation given channel:
On the other hand, given the channel estimates, the ML estimates for g̃i = 1 + gi
and δ̃i = g̃i δi can be obtained by solving the following pair of linear equations:
(n)
(n)
g̃i ||Ai h(n−1) ||2 + δ̃i
(n)
g̃i
4.4.2
t
h(n−1) Ati Bi h(n−1)
= rit Ai h(n−1)
t
(n)
h(n−1) Ati Bi h(n−1) + δ̃i ||Bi h(n−1) ||2 = rit Bi h(n−1) (4.17)
Convergence behaviour
We now repeat the simulations in Section 4.3 to understand the effect of timing
mismatches in joint estimation. Fig. 4.5 depicts the decrease in estimation error as the
iterations progress for L = 32 sub-ADCs, with channel length N = 20 and training
length M = 256. From Fig. 4.6, we observe that the convergence rate (averaged over
1000 random instances of channel and mismatches) is inversely proportional to L (with
M fixed) and proportional to M (with L fixed). Thus, similar to setting with only gain
mismatch, when convergence is desired with fewer iterations, the training sequence
length must scale with the number of sub-ADCs.
82
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
25
Rate of convergence (in dB/iteration)
M = 32
20
M = 64
M = 128
M = 256
15
10
5
0
0
20
40
60
80
100
120
140
Number of sub−ADCs (L)
Figure 4.6: Variation of mean convergence rate with L for different values of M (N = 20).
c
Reprinted from our conference submission [3] with permission, [2010]
IEEE.
For fixed training length M , it is of interest to determine the largest number of
sub-ADCs, L, for which the algorithm converges. When M/L = 1, the equations in
(4.17) become degenerate. This is because the matrices Ai and Bi have only M/L = 1
row. This implies that there is no unique solution for the mismatch parameters (gi , δ̃i ).
Hence, we at least need M/L ≥ 2. In Fig. 4.7, we demonstrate for M = 32 and L = 16
that the iterative algorithm can get stuck (albeit in fewer than 2 out of 104 instances)
away from the true channel and mismatch parameters. We also observe from Fig. 4.7
that the graph of the least-squares cost (i.e., the right-hand side of (4.15)) settles near
83
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
−5
Channel estimate
Gain estimate
Timing estimate
Least squares cost
Estimate error (in dB)
−10
−15
−20
−25
−30
−35
−40
0
5
10
15
20
25
30
Iterations
Figure 4.7: Progress with iterations for the joint estimation algorithm with (M, L, N ) =
c
(32, 16, 4). Reprinted from our conference submission [3] with permission, [2010]
IEEE.
-39 dB, or about 10−4 , suggesting that the solution is not a global minimum (otherwise
the least squares cost would be 0). This cost floor is not observed for M ≥ 4L when
we simulate 105 instances of channel and mismatch parameters, for values of M given
by powers of 2 between 32 and 256.
We now discuss the errors due to the linear approximation of timing mismatch. We
employ the formulae based on linear approximation in (4.16) and (4.17), but use the
non-linear model in (4.10) to generate the received samples ri . As the mismatches
increase from 1% to 10%, we observe that the linearisation errors increase from -80
84
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
dB to -40 dB. Henceforth, we modified the joint estimation algorithm by performing
a fixed-point iteration (to fit Quadratic approximation for timing mismatch) to refine
the mismatch estimates. For the modified algorithm, we observe the estimate errors to
settle below -70 dB even for 10 % mismatch.
4.5
Progression of Channel estimates: Analysis
In this section, we try to provide some analytical insights for the convergence behaviour of the joint estimation algorithm. For ease of analysis, we consider the simpler
model of only gain mismatches from Section 4.2.
4.5.1
Progression along the dominant direction
When viewed jointly with respect to the variables h and gi , the estimation problem
in (4.5) is non-linear. Proving that the estimates proceed dominantly along a straight
line seems harder; however, we argue below that such an assumption should only reduce the residual estimate error. Thus, approximating the algorithm’s progression as a
straight line should provide an optimistic estimate of performance. Indeed, we observe
analytical formulae for convergence rates upper bounding the simulation results.
85
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
We ignore noise in (4.3). The progression of channel estimates can be mathematically expressed as:
h(n) = βn h0 + (1 − βn )h + e
(4.18)
where n denotes the iteration number, βn represents the positioning of the channel
estimate h(n) on the line joining h0 and h. Further, e denotes the deviation vector
from the straight line progression and by definition, e is orthogonal to (h − h0 ). We
introduce the more compact notation g̃i = 1 + gi , and refer to g̃i as the gain of the ith
sub-ADC. The gain estimates can be written using (4.7) as follows
g̃ˆi = g̃i fi (βn−1 )
(4.19)
where fi (βn−1 ) is given by
fi (βn−1 ) =
(Ai h, Ai h(n−1) )
(Ai h(n−1) , Ai h(n−1) )
(4.20)
where (x, y) = xt y. Using (4.18), the channel estimate at the nth iteration (given
the gain estimates in (4.19)) depends only on a scalar parameter βn and a vector e. We
can now use (4.18) to recast the channel estimation problem in terms of βn and e:
(βn , e) = arg min
L−1
X
||ri − g̃ˆi Ai (βn h0 + (1 − βn )h + e)||2
i=0
86
(4.21)
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
The function in the right-hand side of (4.21) is quadratic in both βn and e. The
minimizing solution with respect to βn is given by
PL−1
βn =
vit (ui − ei ))
PL−1
2
i=0 ||vi ||
i=0
(4.22)
where the vectors ui , vi and ei are given by
ui = g̃i (1 − fi (βn−1 ))Ai h
(4.23)
vi = g̃i fi (βn−1 )Ai (h0 − h)
(4.24)
ei = g̃i fi (βn−1 )Ai e
(4.25)
We now show that βn depends weakly on the vector e as the training sequence gets
longer. Specifically, we expand the term that depends on e in (4.22):
L−1
X
i=0
!
vit ei = (h0 − h)t
X
g̃i2 fi2 (βn−1 )Ati Ai
e
(4.26)
i
Since the mismatches are small, the gains g̃i can be assumed to be in a small range
around unity. Also, when the algorithm is near convergence (that is, when n is large),
fi (βn−1 ) ≈ 1 because the gain estimates in (4.19) get close to the true gains. Moreover,
87
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
for a pseudorandom training sequence, the matrix
P
i
Ati Ai is very close to a scaled
identity matrix. These observations, coupled with the fact that e is orthogonal to (h0 −
h), imply that the right-hand side of (4.26) is zero. Thus, we have shown that βn is
independent of e under the above assumptions.
After the nth iteration, the residual estimate error is given by βn (h − h0 ) + e.
Since βn is independent of e, a non-zero solution for e could only increase the residual
error, and hence proceeding along a straight-line provides an optimistic performance
estimate.
4.5.2
Convergence rate formulae
In the joint estimation algorithm (4.6)-(4.7), the complexity of each iteration depends dominantly on inverting an N × N matrix (the operator that acts on h(n) in
(4.6)), hence the overall complexity of the algorithm scales with the number of iterations. Thus, a scalable design for a TI-ADC architecture must maintain the convergence
rate as the number of sub-ADCs, L, increases. We therefore focus on characterizing
the convergence rate defined by,
αn =
||h − h(n) ||
,
||h − h(n+1) ||
(expressed on a logarithmic scale as 20 log10 (αn ) dB/iteration).
88
(4.27)
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
Our analysis ignores noise, and relies on the interesting behaviour depicted in Fig.
4.4, where the channel estimates h(n) progress along a straight line joining the initial
channel estimate h(0) and the true channel vector h. Our results (derived in Appendix
C) indicate that the convergence rate depends on the norms and the inner products of
the vectors ui = Ai h and vi = Ai h0 for i = {0, · · · , L − 1}. Further, given the
vectors ui and vi , the asymptotic convergence rate (as iteration number n gets large)
is independent of n and weakly depends on the mismatches. In the limit of small
mismatches (tending to zero), the asymptotic convergence rate is given by
PL−1
σi
α = PL−1 i=0
i=0 ρi (κi − κ)
(4.28)
where the terms ρi , σi and κi are defined as follows:
ρi = (Ai h, Ai (h − h0 )), σi = ||Ai (h − h0 )||2 , κi =
ρi
,
||Ai h||2
(4.29)
and κ is a weighted average of κi :
L−1
κ=
1X
(1 + gi )κi
L i=0
(4.30)
The formula in (4.28) enables us to compute convergence rates as a function of the
training sequence and the initial channel estimate h(0) . A simple geometric interpretation of the terms in (4.29) and (4.30) can be obtained based on the following equality:
89
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
Ai h0
O
Ai h
θi
Figure 4.8: A geometric insight behind the convergence rate formula in (4.28). The convergence
rate is significantly determined by θi (for each i ∈ {0, 1, · · · , L − 1}), and is given by (4.32).
ρi κi
= cos2 (θi )
σi
(4.31)
where θi is the angle between the vectors Ai h and Ai (h−h0 ). If we assume a long
Sunday, May 22, 2011
random training sequence, there is little variation between σi , and the convergence rate
formula in (4.28) can be expressed as follows:
α=
E[cos2 (θ)]
1
− (E[cos(θ)])2
(4.32)
where E denotes the empirical mean taken over all i. We can further simplify the
convergence rate formula by exploiting the random-like properties of long pseudorandom sequences:
90
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
αr =
M
L
(4.33)
The preceding formula indicates that we need to increase the training sequence length
M linearly with the number of sub-ADCs, L, in order to maintain a given convergence
rate.
Next, we present simulation results comparing our analytical results (4.28) and
(4.33) with empirically observed convergence rates. A sketch of the derivations of
the above presented analytical results is included in Appendix C.
25
Convergence rate (dB/iteration)
PN sequence (analytical)
PN sequence (observed)
20
15
10
5
0
0
20
40
60
80
100
120
140
No. of sub−ADCs (L)
Figure 4.9: Analytical results for the mean convergence rate of joint channel and mismatch
estimation: Training sequence length M increases as we move away from origin as 32, 64, 128
c
and 256. Reprinted from our conference submission [4] with permission, [2011]
IEEE.
91
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
45
Simpler formula
PN sequence (analytical)
PN sequence (observed)
Convergence rate (dB/iteration)
40
35
30
25
20
15
10
5
0
20
40
60
80
100
120
140
No. of sub−ADCs (L)
Figure 4.10: Simple formula for convergence rate: Mean convergence rate is well approximated
as 20 log10 (M/L) dB/iteration (Equation 4.33) when L ≥ 32 for a pseudorandom training
sequence of length M = 256. Reprinted from our conference submission [4] with permission,
c
[2011]
IEEE.
4.5.3
Simulation Results
We use the same simulation setting as in Section 4.3. Fig. 4.9 plots the convergence
rate as a function of the number of sub-ADCs, L. The analytical estimates obtained
using the straight-line progression approximation agree closely with the observed convergence rates (averaged over 50 randomly generated channel and mismatch sets). The
analytical results deviate from simulations at large L (i.e., small M/L). This is con-
92
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
sistent with Fig. 4.4, which shows that approximating the algorithm’s progression as a
straight line is less accurate for smaller M/L. Also, as indicated in Section 4.5.1, the
analytical convergence rates upper bound the performance. Next, fixing the training
sequence length M = 256, we see from Fig. 4.10 that the convergence rate agrees
closely with the formula in (4.33) for relatively large L, when M/L ≤ 8.
4.6
Effects of noise on convergence behaviour
MSE in channel estimation (dB)
−30
SNR = 30 dB
SNR = 40 dB
SNR = 50 dB
−35
−40
−45
−50
−55
−60
1
2
3
4
5
6
7
8
9
10
Iterations
Figure 4.11: Performance of the iterative algorithm in the presence of thermal noise with
(M, L, N ) = (256, 64, 20). Reprinted from our conference submission [3] with permission,
c
[2010]
IEEE.
93
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
In the previous sections, we have studied the convergence behaviour of joint channel
and mismatch estimation algorithms in the absence of any noise. In Fig. 4.11, we
consider thermal noise in (4.15) with a non-zero power σ 2 = 1/SNR. The meansquared error (MSE) refers to the estimate error averaged over 50 algorithm runs with
a randomly generated instance of thermal noise in each run. First, we observe that
the iterative algorithm converges even in the presence of noise. We observe from Fig.
4.11 that the convergence rate, before the estimate error settles, ranges between 6 − 8
dB/iteration, which is close to the no-noise, mean-convergence rate for M = 256 and
L = 64 in Fig. 4.6. Thus, we observe that the convergence rate, before the algorithm
settles, is fairly independent of the noise power, and the significant effect of varying the
noise power is to change the MSE floor for the estimates.
4.6.1
Comparison with CRLB
We now evaluate the Cramer-Rao lower bound (CRLB) for the joint estimation
problem, which serves as a lower bound on the level of MSE at which the algorithm
settles in Fig. 4.11. Assuming i.i.d gaussian noise samples, the likelihood function is
a scaled (by 1/σ 2 ) version of the least-squares cost function given in (4.15). We first
evaluate the Fisher information matrix, F , which can be inverted to obtain the CRLB:
94
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
MSE in estimation (in dB)
−10
CRLB (channel)
CRLB (gain)
CRLB (timing)
Alt. min. (channel)
Alt. min. (gain)
Alt. min. (timing)
−20
−20
−40
−50
−60
0
5
10
15
20
25
30
35
40
Input SNR (in dB)
Figure 4.12: Comparison of the algorithm’s MSE with Cramer-Rao lower bounds for
(M, L, N ) = (256, 64, 20). We considered 10 iterations for the alternating minimization alc
gorithm. Reprinted from our conference submission [3] with permission, [2010]
IEEE.
1
F = 4E
σ
"
∂l(x)
∂x
∂l(x)
∂x
t #
(4.34)
where x denotes the concatenation of channel vectors {<[h[0]], · · · , <[h[N − 1]]}
and {=[h[0]], · · · , =[h[N − 1]]}, and the mismatch vector {g0 , · · · , gL−1 , δ0 , · · · , δL−1 }.
We evaluate the derivatives required in (4.34) numerically at x. Further, the expectation
in (4.34) is approximated by an empirical average over 104 instances of noise samples.
95
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
From Fig. 4.12, we observe that the channel MSE is close to the corresponding
CRLB for all SNR (= 1/σ 2 ) values considered. For example, the CRLB for channel
estimation for SNR levels 30 dB and 40 dB are -40 dB and -50 dB, respectively, which
are very close to the values the MSE settles in Fig. 4.11. However, the mismatch
MSEs are approximately 3 dB higher than the corresponding CRLB, indicating that the
iterative algorithm is slightly sub-optimal with respect to mismatch estimation.
4.7
Training sequence design for fast convergence
The previous sections have focussed on the convergence of joint channel and mismatch estimation algorithm using pseudorandom training sequences. In this section, we
show, for a training length of M = N L, that well-designed periodic training sequences
require much fewer iterations for convergence than a pseudorandom training sequence.
Our design requires that the training sequence is periodic with period N samples
(recall that N is the nominal channel length). We refer to the subsequence {b0 , · · · , bN −1 }
as the training frame. Thus, the actual training sequence is a repetition of L training
frames. Also, we need that the training frame does not equal any i-circular shift of itself
for 0 ≤ i ≤ N − 1.
We now simplify the structure of the matrices Ai and Bi in (4.13) for the above
periodic training sequences.
96
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
Lemma 2. Structure of Ai : The collection of the rows of Ai is the collection of all
possible circular shifts of the training frame {b0 , · · · , bN −1 }.
Proof. We first prove by contradiction that no two rows of Ai are equal. From the
definition in (4.4), we observe that pth row of Ai is obtained by circularly shifting the
th
first row by pL. Suppose the pth
1 and the p2 rows of Ai are equal. Using property (c)
of the training sequence, this can only happen when (p1 − p2 )L mod N = 0. Since
N and L have no common factors, this readily implies that p1 = p2 , since for 0 ≤
p1 , p2 ≤ N − 1, we have −(N − 1) ≤ p1 − p2 ≤ (N − 1). Then, the equation
(p1 − p2 )L mod N = 0 has only one solution p1 = p2 .
The lemma is true for any Ai . Thus, the rows of Ai could be obtained by permuting
the rows of A0 in some order. Equivalently, Ai = Pi A0 , where Pi represents the matrix
obtained by performing the same row permutations on an Identity matrix.
Lemma 3. Structure of Bi : The matrix Bi can be expressed as Pi B0 , where Pi is the
same row-permutation matrix as in Ai = Pi A0 .
Proof. From (4.14), we can rewrite the elements of Bi as,
N
−1
X
Bi (p, q) =
0
0
0
0
b(i + pL − q ) sinc (q − q)
0
q =0,q 6=q
97
(4.35)
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
Using (4.35) and the definition of Ai from (4.4), it can be observed that the columns of
Bi are linear combinations of the columns of Ai , and hence we can write Bi = Ai Q.
It is noteworthy that Q does not depend on the choice of i. Since Ai = Pi A0 , it implies
that Bi = Pi A0 Q. Thus, we infer that Bi = Pi B0 .
Lemma 4. Fast convergence: For the proposed periodic training sequence, the channel
estimate can be directly obtained from the received samples without any iterations (to
first order in mismatch terms).
Proof. We use Lemmas 1 and 2 in (4.10) to obtain
ri = Pi [(1 + gi )A0 + δ̃i B0 ]h
(4.36)
Since row permutation matrices are invertible, we can premultiply (4.36) by Pi−1 .
Further, we sum over i to obtain
L−1
X
Pi−1 ri = A0 h + gi δi B0 h
(4.37)
i=0
where we used the fact that the gain and timing mismatches have zero-mean without
loss of generality. From (4.37), we note that the second term on the right-hand side includes a product of mismatch terms, which represents a second-order term in mismatch
magnitude. Assuming mismatches to be small, we can approximate the estimate of h
from (4.37) as
98
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
h≈
A−1
0
L−1
X
Pi−1 ri
(4.38)
i=0
We note that when we initialize the values of mismatches to zero in (4.37), the leastsquares estimate for h is as given in (4.38). Thus, we can jointly estimate the channel
and mismatch parameters, accurate to first order in the relative mismatch parameters,
within the first iteration.
Channel estimate error (in dB)
−40
Pseudo−random training
N−periodic training
−60
−80
−100
−120
−140
−160
−180
−200
0
2
4
6
8
10
12
Iterations
Figure 4.13: Comparison of the progress of the iterative algorithm with the pseudorandom and
the proposed training sequences. (M, L, N ) = (1216, 64, 19). Reprinted from our conference
c
submission [3] with permission, [2010]
IEEE.
In Fig. 4.13, we plot the channel estimation error for pseudorandom training and
the proposed periodic training. We choose N = 19 and L = 64 (no common factors)
99
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
for a training length of M = N L = 1216. For the proposed training, we first randomly
generate a vector with entries as ±1 of length 19 and take its FFT, which is repeated
64 times to obtain the overall training sequence. This sequence has the advantage that
any matrix Ai (see (4.4)) has N eigenvectors with all eigenvalues equal in magnitude.
Thus, we see from (4.38) that when we invert Ai , there is no noise enhancement in
the estimation of the channel vector. For pseudorandom training, we use the same msequence (from section 4.3) of length 255, repeated five times, but with the last 59 bits
deleted to obtain a total length of 1216. We observe that the proposed sequence indeed
demonstrates much better convergence rates, over several random instances of channel
and mismatch parameters, compared to pseudorandom training.
4.8
Rules of Thumb
We now summarize some rules of thumb for training sequence selection for the
purpose of joint channel and mismatch estimation. Firstly, the training sequence needs
to be longer than two times the number of interleaved ADCs; that is, M > 2L. A
conservative choice is to set M ≥ 4L so that the algorithm does not get stuck at a
local optimum. Using a randomized assumption, the convergence rate of pseudorandom sequence can be approximated as 20 log10 (M/L) dB/iteration. This formula holds
reasonably accurate for M ≤ 8L, while it becomes optimistic for smaller values of
100
Chapter 4. Scalability of Joint Channel and Mismatch Estimation
L. On the other hand, when longer training sequences are allowed, well-designed periodic training sequences of length M = N L achieve much faster convergence than a
pseudorandom sequence with the same length.
101
Chapter 5
Conclusions
This thesis was motivated by the recognition that the analog-to-digital converter
is a critical bottleneck in scaling mostly digital implementations of communication
transceivers to multi-Gigabit speeds. The TI-ADC architecture investigated here is an
attractive approach for relieving this bottleneck, and the compensation techniques developed in this thesis provide a means of alleviating performance floors due to mismatch. To get a sense of how these ideas could impact practice, let us now consider some potential applications to multi-Gigabit wireless transceiver design in the
millimeter-wave band [11].
Consider OFDM transmission (with no excess bandwidth) for a 4 Gbps indoor
millimeter-wave link (Refer Fig. 5.1). For 16-QAM modulation on each of the 128
subcarriers, we can obtain an uncoded data rate of 4 Gbps using 1 GHz of spectrum.
Thus, we can use two TI-ADCs (each sampling at a rate of 1 GHz), one for each of
the I and Q channels. Numerical results indicate that 8 bits of resolution at the dig-
102
Chapter 5. Conclusions
Data
on
16-QAM
data symbols
I
IFFT
D/A
I
I+jQ
Q
Pseudorandom
training (M =32)
Q
Data
off
[a]
Joint compensation
1
I
1 GHz
bandwidth
D/A
I-channel TI-ADC
RF
section Q
RF
section
Equalize
Group 1
Data
on
2
Data
symbol
estimates
FFT
(128)
j
Equalize
Group 8
8
Friday, May 20, 2011
Data
off
Q-channel TI-ADC
Estimate
channel
Estimate
mismatch
Iter = 3
Joint estimation
[b]
Figure 5.1: Proposed system architecture for multi-Gigabit OFDM transceiver (4 Gbps uncoded
data rate) employing a time-interleaved ADC: [a] transmitter [b] receiver
Sunday, May 22, 2011
103
Chapter 5. Conclusions
ital output from each ADC suffice to keep the quantization noise to negligible levels.
When we use a time-interleaved architecture with L = 8 sub-ADCs, the sampling
rate is 125 MHz for each sub-ADC, which is slow enough that we can use power efficient successive-approximation or pipelined architectures to implement the sub-ADCs
[13, 14]. The complexity of the frequency-domain equalizer is 32 real-valued multiplications per sample, and this complexity is independent of the desired resolution.
We now consider a more demanding scenario where the transmission is over 2 GHz
bandwidth, and a larger constellation (64-QAM) is used, to obtain an uncoded bit rate
of 12 Gbps (Refer Fig. 5.2). In this setting, we potentially need a higher resolution
of 8-12 bits from the TI-ADC. When we restrict the sub-ADC sampling rate to 62.5
MHz for increased power efficiency, we need 32 ADCs to be interleaved for each of
the I and Q channels. To keep the complexity of mismatch compensation tractable,
we use two times oversampling. Each of the I and Q components therefore requires
a TI-ADC operating at an aggregate sampling rate of 4 GHz, and we need L = 64
sub-ADCs, each sampling at 62.5 MHz with 12 bits of resolution. Again, such ADCs
can be obtained from the pipelined or successive-approximation architectures [13, 14] .
From our simulations in Chapter 3, we see that as few as 5 taps per sample are required
for mismatch compensation. Thus, we expect a decreased overall power consumption
compared to using a Flash ADC [45], even taking into account the inefficiency due to
oversampling.
104
Chapter 5. Conclusions
[a]
Data
on
64-QAM
data symbols
I
IFFT
D/A
I+jQ
I
Q
Pseudorandom
training (M =256)
Q
Data
off
RF
section
2 GHz
bandwidth
D/A
[b]
I-channel TI-ADC
1
I
RF
section Q
4 GSa/s
12 bit
2
Data
on
Compensate
mismatch
Zeroforcing
2 GSa/s
12 bit
4 GSa/s IN
j
Friday, May 20, 2011
FFT
and scalar
correlation
Data
bits
2 GSa/s
12 bit
64
Data
off
Q-channel TI-ADC
Estimate
channel
Estimate
mismatch
Iter = 3
Joint estimation
Figure 5.2: Proposed system architecture for a more demanding multi-Gigabit OFDM
transceiver (12 Gbps uncoded data rate) employing a time-interleaved ADC: [a] transmitter
[b] receiver
Friday, May 20, 2011
105
Chapter 5. Conclusions
We now discuss the implications of joint mismatch and channel estimation on the
transceiver. For 1 GHz symbol rate (in Fig. 5.1), assuming a delay spread of 20 ns
(for indoor millimeter wave communication), we have N = 20 channel coefficients to
estimate, while this number increases to N = 40 for the 2 GHz symbol rate (in Fig.
5.2). The training sequence needs to be at least of this length to provide the necessary dimensions for channel estimation. Since the complexity of joint estimation is
dominated by the number of iterations, we limit the number of iterations to 3. For the
setting in Fig. 5.1, our simulations suggest that we can use a pseudorandom sequence
of length M = 128 with 3 iterations of the joint estimation algorithm without any significant error floors for BERs as low as 10−4 . On the other hand, for the setting in
Fig. 5.2, we increase the length to M = 256, so that the convergence rate is approximately 20 log10 (M/L) = 12 dB/iteration. This implies that 3 iterations of the algorithm
could lower the (relative) mean-squared error in the channel estimates by 36 dB, which
should be adequate to support constellations as large as 64QAM without significant
error floors. Further, the complexity of estimation is tractable, owing to closed-form
solutions for each step of the iteration.
Alternatively, we can use the periodic training sequence from Chapter 4 with length
M = 672, so that one iteration suffices to obtain the same estimation error (e.g., we set
the number of channel coefficients being estimated to N = 21, in order to ensure that
N and L have no common factors).
106
Chapter 5. Conclusions
Overall, the results of this thesis demonstrate that mismatch compensation for timeinterleaved ADCs, when considered in the light of the underlying communication system, yields significant reduction in complexity. Moreover, estimating the mismatches
jointly with the communication channel could eliminate the need for dedicated on-chip
training to estimate the mismatches. Finally, the thesis presents scalable solutions for
mismatch compensation and estimation in a communication receiver as the number of
sub-ADCs increases.
5.1
Future work
There are a number of open issues that deserve further investigation. First, detailed
circuit design and experimental evaluation are required to demonstrate the efficacy of
highly interleaved TI-ADCs. It is also interesting to determine whether the proposed
iterative algorithm for joint mismatch and channel estimation is guaranteed to converge to a good (or, even better, an optimal) solution. It would be useful to devise
low-complexity techniques for tracking the mismatch estimates (rather than estimating
them afresh for each packet). For this purpose, a least-mean-squares (LMS) implementation of the joint channel and mismatch estimation appears to be a natural approach
for further investigation.
107
Chapter 5. Conclusions
Finally, other communication scenarios for which joint mismatch and channel estimation/compensation are of interest include fiber optic and multiple-input, multipleoutput (MIMO) communications.
Application to standard fiber optic processing requires handling the non-linearities
induced by the photodiodes, but emerging trends in linear radio-like processing for optical communication open up even more opportunities for mostly digital architectures
[46, 7, 8, 6, 9]. These architectures need sufficient bits of precision (such as 8 bits)
from the ADC to equalize for the optical fiber dispersion electronically. The main
challenge in the ADC design is the extremely high sampling rates required (around 50
GSa/sec at the time of writing), which increase the interleaving factors to over 200 for
using a power-efficient sub-ADC architecture [15]. It would be interesting to investigate whether it would be possible to reduce the mismatch compensation complexity by
addressing it jointly with the compensation of fiber dispersion.
In the setting of MIMO communication, we must consider trade-offs between analog and digital processing for diversity, multiplexing and beamforming functionalities.
For example, in the beamforming setting, it is possible to implement the spatial equalizers in the analog domain, and use only two TI-ADCs (for the I and Q channels) to
sample the analog signal. When it is desired to avoid analog equalizers for a mostly digital implementation, the spatial processing needs to be performed in the digital domain,
and hence a different TI-ADC has to be used for each receive antenna. It then becomes
108
Chapter 5. Conclusions
important to understand whether it is still possible to compensate for the mismatches
among the sub-ADCs corresponding to all the TI-ADCs in the different receive antenna
chains, using the communication resources and with minimal additional complexity.
109
Appendix A
Invertibility of matrix A
˜ are invertible.
From the definition in (2.12), A is invertible whenever F̃ and ∆
Clearly, F̃ is invertible: F̃ x̃ = 0 implies that F x = 0, so that we can infer the
invertibility of F̃ from that of F . We now assume that ∆I = ∆Q = ∆, when we have
˜ = ∆x, and hence ∆
˜ is invertible whenever ∆ is invertible. Using (2.9), we can
∆x̃
write ∆ as a product of a diagonal matrix D∆ and a Vandermonde-matrix V∆ , which
are defined as,
D∆ (m, m) = 1 + gm ; D∆ (m, y) = 0, for m 6= y; V∆ (m, y) = e
j2πy
(m+δm )
M
(A.1)
where m and y take integral values between 0 and M − 1, and where we dispense with
the subscripts I and Q, since ∆I = ∆Q . The diagonal matrix D∆ is invertible when
all of its diagonal elements 1 + gm , which correspond to the sub-ADC gains, are nonj2π
zero. The Vandermonde matrix V∆ is invertible when all its parameters {e M
110
(m+δm )
}
Appendix A. Invertibility of matrix A
are distinct. Clearly, when m varies over integer values between 0 and M − 1, the Vandermonde parameters are all distinct, as long as |δm | < 1 for all m (i.e., the normalized
timing mismatches are bounded by one). Under these conditions, therefore, the matrix
A is invertible.
111
Appendix B
Structure of the matrix A when L
divides M
We first consider the structure of the matrix F ∆I when L divides M . We write the
(k, y)th element of the matrix F ∆I as follows:
(F ∆I )(k,y)
L−1
X
1 X
=
(1 + gI,l )ej2πyδI,l /M
e−j2πm(k−y)/M
M l=0
m∈M
(B.1)
l
where Ml denotes the set of all m for which m mod L = l. Evaluating the summation
over m in (B.1),
(F ∆I )(k,y) =







1
L
PL−1
l=0
(1 + gI,l )ej2πy(l+δI,l )/M e−j2πkl/M ,
y ∈ Yk
(B.2)
0
112
otherwise
Appendix B. Structure of the matrix A when L divides M
where Yk is a set of L indices given by
Yk = {y : 0 ≤ y ≤ M − 1, y mod (M/L) = k}
(B.3)
From (B.3), we observe that Yy = Yk for any integer y ∈ Yk . This readily implies that
the set of indices ξ = {0, · · · , M − 1} can be partitioned into M/L disjoint groups,
− 1}, such that (F ∆I )(k,y) is nonzero only
which are the sets Yk for k = {0, · · · , M
L
when k and y belong to the same group. Since no information about the specific mismatch parameters is utilized during the proof, the result directly extends for the matrix
∗
F ∆Q . We can repeat the analysis between (B.1)-(B.3) for the matrix F ∆I to understand that its (k, y)th element is non-zero only when y ∈ Y−k .
Now, we understand that all the M × M sub-matrices of A, that appear in the defi∗
nition of A given in (3.12), can be written in terms of F ∆I , F ∆I and their conjugates
∗
∗
∗
F ∆I and F ∆I (Here X ∗ denotes the matrix obtained by the complex conjugation of
all the elements of the matrix X). Hence, we extend the results derived for the structure
of these matrices to infer that the index set {0, · · · , 2M − 1} can be partitioned into
disjoint groups such that A(k, y) is non-zero only when k and y belong to the same
113
Appendix B. Structure of the matrix A when L divides M
group. When M/L is even, the disjoint groups can be obtained as
Gk =



 Yk ∪ Y−k ∪ (M + Yk ) ∪ (M + Y−k ), 1 ≤ k ≤



Yk ∪ (M + Yk ),
M
2L
−1
(B.4)
M
}
k ∈ {0, 2L
where M + X denotes the addition of M to all the elements of the set X. From (B.4),
we have two groups of size 2L and
M
2L
− 1 groups of size 4L. The complexity of
zero-forcing equalization, given in (2.13), for a group of size X is at most X 2 . Hence,
we can obtain the total complexity to be at most 8M L − 8L2 . We can perform a similar
analysis for odd M/L to obtain an equalization complexity of 8M L − 4L2 .
114
Appendix C
Convergence rate formulae:
derivations
Using the definition of convergence rate in (4.27) and of βn in (4.18), we obtain the
convergence rate α as
α=
βn−1
βn
(C.1)
We now use the expression for βn from (4.22) in (C.1) to obtain an expanded expression for α :
PL−1
2
i=0 ||vi ||
α = βn−1 PL−1
H
i=0 <(vi ui )
(C.2)
Using the expressions for vectors u and v from (4.23), we can rewrite α as
PL−1
2 2
i=0 σi g̃i fi (βn−1 )
2
i=0 ρi g̃i fi (βn−1 )[1 − fi (βn−1 )]
α = βn−1 PL−1
where the scalars σi and ρi are defined as in (4.29).
115
(C.3)
Appendix C. Convergence rate formulae: derivations
C.1
Asymptotics of the convergence rate
While the iterative algorithm only takes a few steps to converge in a well-designed
system, it is useful to compute the asymptotic convergence rate (i.e., the limit of α
for large n) in order to provide simplified rules of thumb. From the expression for
convergence rate in (C.3), we see that the iteration number is buried in βn−1 . If the
algorithm converges, βn−1 → 0, hence we evaluate the limit of α as βn−1 → 0. We first
consider the following limits:
lim fi (βn−1 ),
βn−1 →0
1 − fi (βn−1 )
βn−1 →0
βn−1
lim
(C.4)
where fi (βn−1 ) is defined in (4.20) in terms of hn−1 . We can rewrite fi (βn−1 ) as a
function of βn−1 by using the expression for hn−1 from (4.18):
fi (βn−1 ) =
β 2e
βci + (1 − β)e
2
0 + (1 − β) e + 2β(1 − β)ci
(C.5)
where we have dropped the subscript of β for simplicity, and where the scalars ei , e0i , ci
are defined as:
ei = ||Ai h||2 ,
e0i = ||Ai h0 ||2 ,
ci = (Ai h, Ai h0 )
(C.6)
We can now employ expansions in terms of β to evaluate the desired limits as follows
116
Appendix C. Convergence rate formulae: derivations
lim fi (β) = 1
β→0
1 − fi (β)
ci − ei
=
, κi
β→0
β
ei
lim
(C.7)
We now obtain the asymptotic convergence rate by evaluating the limit of (C.3) as βn−1
tends to zero:
PL−1
i=0
α = PL−1
i=0
C.1.1
σi g̃i2
ρi g̃i2 κi
(C.8)
Accounting for gain scaling
We now modify the asymptotic convergence rate estimate to account for the scaling step in our algorithm, where the gain mismatches are scaled such that the mean
mismatch is zero. We use (4.19) to obtain the modified gain mismatch estimates as,
modified g̃ˆi =
1
L
g̃i fi (β)
PL−1
i=0 g̃i fi (β)
(C.9)
Comparing (C.9) with (4.19), we observe that they are identical except that in the
former, fi (β) is scaled by an additional factor, G, given by,
L−1
1X
G=
g̃i fi (β)
L i=0
117
(C.10)
Appendix C. Convergence rate formulae: derivations
We can simplify (C.10) by using fi (β) from (C.5) and the fact that the true mismatches have zero-mean:
L−1
1X ˆ
κi g̃i
G = 1 − βκ + o(β), where κ =
L i=0
(C.11)
We now re-evaluate the limits in (C.7) taking into account that fi (β) are scaled by
G:
lim fi (β) = 1,
β→0
1 − fi (β)
= κi − κ
β→0
β
lim
(C.12)
Hence, the asymptotic convergence rate gets modified to:
PL−1
σi g̃ 2
α = PL−1 i=0 2 i
i=0 ρi g̃i (κi − κ)
C.2
(C.13)
First order approximation
In practice, the mismatches are small, and the gains {g̃i } are in a small range around
unity. From (C.13), we understand that these cause second order variations in the convergence rate. Ignoring these, we can simplify the expression for the convergence rate
as follows:
PL−1
σi
α = PL−1 i=0
i=0 ρi (κi − κ)
118
(C.14)
Appendix C. Convergence rate formulae: derivations
C.3
Long pseudo-random training sequence and channel
We now use the random-like properties of long pseudo-random sequences to simplify (4.28) to (4.33). For large N and M/L N , the matrix Ati Ai has M/L non-zero
eigenvalues all of which lie close to N . Then, we can rewrite ρi in (4.29) as,
M/L
ρi = N
X
(ht ai )(ati (h − h0 ))
(C.15)
i=1
where {ai } represent orthonormal eigenvectors of Ati Ai . The random variables
ht ai (and similarly, ati (h − h0 )) are zero-mean and independent across i. Also, since a
randomly chosen channel vector, say h, looks random with respect to an orthonormal
basis, say {ai }, we have E[(ht ai )2 ] = ||h||2 /N . This implies,
E[ρ2i ] =
M
· ||h||2 · ||h − h0 ||2
L
(C.16)
We now estimate the terms κi and σi in the expression (C.14) for the convergence
rate, using the approximation ||Ai x||2 ≈
κi ≈
ρi
,
M
||h||2
L
M
||x||2 .
L
σi ≈
119
This yields
M
||h − h0 ||2
L
(C.17)
Appendix C. Convergence rate formulae: derivations
For small mismatches, we can approximate κ by the mean of κi , which implies the
convergence rate to be,
α≈
M ||h − h0 ||2
M
||h||2
PL−1
2 ·
PL−1 2 ( k=0 ρi )
L
k=0 ρk −
L
(C.18)
Using (C.16) and neglecting the negative term in the denominator, we obtain the result
in (4.33).
120
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