Maximum Likelihood Tracking Algorithms for MIMO-OFDM
Christian Oberli
Babak Daneshrad
Department of Electrical Engineering
University of California, Los Angeles
Los Angeles, CA 90095
Email: obelix@ee.ucla.edu
Department of Electrical Engineering
University of California, Los Angeles
Los Angeles, CA 90095
Email: babak@ee.ucla.edu
Abstract— Tracking algorithms for carrier and sampling clock
frequency offset are proposed for pilot assisted MIMO-OFDM
in frequency selective channels. The methods are based on
maximum likelihood estimation theory applied to observations
of received pilot subcarriers at the output of the FFTs of
the receiver. Simulation results show that larger MIMO configurations benefit from lower estimator variances, providing
increased synchronization accuracy at low SNRs, or allowing
for a reduction of the number of pilot subcarriers.
I. I NTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) [1]
is a leading modulation technique for wideband wireless communications. Combining it with the high bandwidth efficiency
offered by the use of multiple antennas in Multiple-Input
Multiple-Output (MIMO) transmission systems [2] promises
a significant increase in the practically achievable throughput
over wireless media. The bit error rate performance of OFDM
systems, however, is sensitive to timing and frequency mismatches between transmitter and receiver oscillators. Precise
synchronization is therefore necessary in order to realize the
potential performance [3]–[7].
In this paper we focus on two particular synchronization
problems for MIMO-OFDM systems, namely tracking the
carrier frequency offset (CFO) and timing frequency offset
(TFO) of the A/D converter in bursty transmissions. We
derive Maximum Likelihood (ML) estimators for the CFO
and TFO, and propose the corresponding tracking algorithms.
Our underlying assumption is that adequate channel estimates,
OFDM symbol timing, and a coarse carrier frequency estimate
within one-half intercarrier spacing are provided by an earlier
acquisition stage.
CFO and TFO are both caused by small differences in
oscillator frequencies between the transmitter and receiver. In
OFDM, a CFO causes an ongoing phase drift of all subcarriers over time, while TFO manifests itself as a phase drift
that grows linearly with frequency, affecting each subcarrier
differently [5]. Tracking these phenomena and compensating
for them in frequency selective channels requires robust algorithms. When the transmission is also bursty, then the algorithms must be fast too, otherwise the overhead incurred while
waiting for their convergence spoils the increased throughput.
Blind and decision directed synchronization methods tend
to be slow, and therefore pilot symbol assisted methods are
necessary [8].
The use of pilot symbols for tracking the CFO and TFO in
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Single-Input Single-Output (SISO) OFDM systems has been
explored by many authors (see, for instance, [3], [8]–[10]).
Direct extension of these algorithms to MIMO-OFDM follows
if all transmitter branches use identical pilot sequences. In
such cases, the pilot subcarriers are perceived by all receiver
branches as SISO transmissions, and the SISO-OFDM tracking
techniques can still be used. However, these methods are
inconvenient for MIMO-OFDM because they render the pilot
symbols useless for other “true MIMO” applications, such as
tracking the state of the MIMO channel.
Our approach in this paper considers the use of a different pilot symbol sequence for every transmitter branch. For
this general case we derive the ML estimators of CFO and
TFO, and propose the corresponding tracking algorithms. The
method uses observations of pilot subcarriers at the output
of the FFTs of the receiver branches. At this stage, before
the MIMO decoder, each subcarrier contains pilot symbols
linearly combined by the channel. Therefore, the step of
actually decoding individual pilot symbols in order to retrieve
their phase information is entirely avoided. Results along this
line of work have not yet been reported in the literature.
Application of our techniques to SISO and SIMO-OFDM
systems follows as a special case.
The paper is organized as follows. In Section II we establish
the notation and introduce an expression for the phase drift
of idividual MIMO-OFDM subcarriers as a function of time,
and in the presence of CFO and TFO. We then use this
result in Section III to derive ML estimators for CFO and
TFO, and propose the corresponding tracking algorithms.
Section IV presents simulation results of our algorithms in
an IEEE 802.11a-like setting. Conclusions are presented in
Section V.
II. BASEBAND MIMO-OFDM M ODEL
Fig. 1 illustrates a MIMO-OFDM system for the case with
two transmit and two receive antennas. Only the fundamental
building blocks relevant to the synchronization problem at
hand are shown. We model the output of the FFT of receiver
branch j and subcarrier k as follows:
Yj (k) = ejφ(m,k,∆f,∆T )
Nt
Xi (k) Hji (k) + Wj (k)
i=1
Xi (k) represents the modulation of subcarrier k on transmitter
branch i, Hji (k) is the channel frequency response at subcar-
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rier k between transmitter i and receiver j, and Nt is the numNs
s
ber of transmit antennas. The range of k is { −N
2 , . . . , 2 −1},
where Ns is the number of OFDM subcarriers. The phase
term φ(m, k, ∆f, ∆T ) represents the phase drift of subcarriers
when the received signal is downconverted with a CFO of
∆f (in Hertz) and sampled with clock period offset ∆T (in
seconds). φ(·) is given by:
k ∆T
∆T
+ ∆f T 1 +
φ(m, k, ∆f, ∆T ) = 2πmNb
Ns T
T
expression:
where T is the sampling period, m is an integer time index
that distinguishes successive MIMO-OFDM symbols and Nb
is equal to Ns plus the number of samples used for a cyclic
prefix. Finally, Wj (k) represents the joint effect of Additive
White Gaussian Noise (AWGN) and Intercarrier Interference
(ICI) caused by CFO. We assume the ICI to be additive and
Gaussian distributed [5], [7], and consider Wj (k) to have a
power spectral density of NW .
By grouping all signals Yj (k) into a column vector, we can
use a more compact vector notation:

Φ = 
Y(k) = ejφ(m,k,∆f,∆T ) H(k) · X(k) + W(k)
D/A
RF
RF
A/D



H = 

X
D/A
RF
Fig. 1.
RF
A/D
Nr Np ×1
ejφ(k1 ,Θ) INr
0
..
.
0
H(k1 )
0
..
0
H(kNp )
X(k1 )


..
= 

.
X(kNp ) N



ejφ(kNp ,Θ) INr

.

Nr Np ×Nr Np


Nr Np ×Nt Np

W(k1 )


..
W=

.
W(kNp ) N
×1

r Np ×1
Note that (2) includes all pilot data jointly observed by the
Nr receiver branches on any transmission of a single MIMOOFDM symbol. Also note that the effect of ∆f and ∆T on Y
is captured completely by the matrix Φ, and that we chose a
more compact notation φ(kp , Θ) for φ(m, kp , ∆f, ∆T ). The
1
vector Θ = [∆f T, ∆T
T ] is the parameter we are to estimate .
Since all matrices H(kp ) and all vectors X(kp ) are assumed
to be known, and if Θ is given, then Y is a Gaussian distributed random vector because of the Gaussian AWGN+ICI
term W. For a case like this, maximum likelihood estimation
theory provides well-known methods for estimating Θ [11].
The technique requires maximizing the log-likelihood function
over Θ, which in our case is:
1
FFT
Λ(Y|Θ) =
2NW fδ
[Y∗ ΦHX + X∗ H∗ Φ∗ Y] + K
(3)
In (3), K is a constant term, independent of Θ, and fδ is
the frequency spacing between subcarriers (in Hertz). Careful
inspection of (3) in the light of the diagonal nature of matrices
Φ and H allows us to rewrite it in the following form:
FFT
2 × 2 MIMO-OFDM system.
Λ(Y|Θ) =
III. ML S YNCHRONIZATION PARAMETER E STIMATION
In this Section we derive the ML estimators of ∆f and
∆T for MIMO-OFDM, and propose corresponding tracking
algorithms. We assume that Np predefined subcarriers are
modulated with pilot symbols on each transmitter branch.
To simplify the notation, we enumerate pilot subcarriers as
kp = k1 , . . . , kNp , and consider kp = k(p) to be a mapping
function that relates the pilot index p = 1, . . . , Np to the
actual subcarrier frequency k. We also assume that the channel
matrices H(kp ) are known.
We start our derivation by using (1) and collecting all
Np received pilot subcarriers into one single block-matrix
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Y(kNp )
t Np
MIMO
Decoder
IFFT
(2)
with Y, Φ, H, X and W given by:


Y(k1 )


..
Y = 

.
(1)
It is important to note that we assume that all transmitter
branches are driven by a common carrier oscillator and a
common sampling clock, and a similar situation for all receiver
branches (Fig. 1). These timing devices produce signals of
fixed frequency, and we seek to solve the synchronization
problem in the digital domain. We also assume that any initial
phase offset of the carrier frequency and the sampling clock are
absorbed by the channel frequency response H(k), estimated
during acquisition (later phase offsets are tracked).
IFFT
Y = ΦHX + W
Np
1
2NW fδ
Y∗ (kp )H(kp )X(kp ) ejφ(kp ,Θ)
p=1
def
= α(kp )
∗
∗
−jφ(kp ,Θ)
+ X (kp )H (kp )Y(kp ) e
+K
(4)
def
= α∗ (kp )
To find Θ̂ that maximizes Λ(Y|Θ) in (4) we set the gradient
of Λ(Y|Θ) with respect to ∆f T and ∆T
T equal to zero. After
some algebraic manipulation we obtain the following system
1 By estimating the normalized CFO and TFO, i.e., ∆f T and ∆T resp.,
T
our procedure becomes independent of the transmission bandwidth.
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of two equations for ∆f T and
e−j2πmNb ∆f T (1+
∆T
T
)
Np
∆T
T
:
kp ∆T
T
α∗ (kp ) e−j2πmNb Ns
(5)
p=1
= ej2πmNb ∆f T (1+
∆T
T
)
Np
kp ∆T
T
phase drift between subcarriers of a given OFDM symbol, as
measured by α(kp )α∗ (kq ). It also shows that the ML estimator
for ∆T
T is given by an averaging operation of such differences
among all possible pairs of pilot subcarriers. We thus propose
the following approximate estimator for ∆T
T :
α(kp ) ej2πmNb Ns
Np Np
ˆ
∆T
−Ns arg{α(kp )α∗ (kq )}
=
T
2πmNb p=1 q=1
kp − kq
p=1
∆T
T
)
Np
kp ∆T
T
kp α∗ (kp ) e−j2πmNb Ns
(6)
p=1
= ej2πmNb ∆f T (1+
∆T
T
)
Np
kp ∆T
T
kp α(kp ) ej2πmNb Ns
p=1
This system does not have a closed form solution for ∆T
T and
∆f T , and we proceed with the following approximations.
A. Solving for the carrier frequency offset ∆f T
Consider, for instance, the case of an OFDM system with
Ns = 1024 subcarriers driven by a sampling clock oscillator
that is accurate to 50 ppm. Then, assuming that during the
acquisition stage the CFO has been reduced to less
than one
< 5 · 10−5
half of the intercarrier spacing (fδ ), we have ∆T
T
0.5
−4
and |∆f T | < Ns ≈ 5 · 10 . Hence, even in this unfavorable
case, the range of ∆T
T is an order of magnitude
smaller than
the range of ∆f T . Therefore, ∆f T 1 + ∆T
is generally
T
dominated by ∆f T , and we approximate it by ∆f T . Eq. (5)
then becomes only a function of ∆f T and not of the subcarrier
index kp nor ∆T
T . Since the LHS of (5) is conjugate to its RHS,
we formulate the ML condition for ∆f T as follows:


Np


α(kp ) ej2π∆f T mNb = 0
Im
(7)


C. Proposed Tracking Algoritms
The CFO tracking mechanism uses observations of the
current FFT output to calculate ∆fˆ T as given by (8). The
result is fed back into the time domain through an accumulator
−z
), whose output controls the rotation speed
(Fig. 2, block z−1
of a phasor that compensates for the CFO at the FFT input in
subsequent OFDM symbols. Even though this controller stops
the subcarriers from rotating any further, it does not return
their phase to zero. However, it is simple to show that the
ML estimator for the steady state phase error, φf , is given by
Np
Y∗ (kp )H(kp )X(kp )} (compare with (8)), and
− arg{ p=1
we use it in feedforward fashion at the FFT output to de-rotate
all subcarriers back to the correct phase (Fig. 2).
ϕ̂ T
− kc
RF
A/D
z + 0.9
z −1
−z
z −1
Interpolator
∆fˆT
α (k p )
ϕ̂ f
FFT
Pilot Subcarriers
e−j2πmNb ∆f T (1+
(10)
RF
A/D
Interpolator
MIMO
Decoder
FFT
p=1
From (7), it suffices to choose ∆f T so that the
phase of the
exponential term is conjugate to the phase of
α(kp ). Thus,
the MLE of ∆f T is given by:
Np
∗
− arg
p=1 Y (kp )H(kp )X(kp )
(8)
∆fˆ T =
2πmNb
Note that this solution is exact for the case in which ∆T = 0.
B. Solving for the sampling clock frequency offset
MIMO-OFDM receiver with CFO and TFO tracking loops.
For tracking the TFO we propose a feedback loop based on
2π
mNb ∆T
the quantity φT = N
T (from (10)), which is a measure
s
of the cumulative effect over time of an uncompensated TFO.
Its desired steady state value is zero, which can be attained
with a standard digital PD controller whose transfer function
−3
).
is −kc z+0.9
z−1 (Fig. 2, kc = 10
∆T
T
IV. S IMULATION
As shown above, ∆f T tends to have a much stronger
effect on the phase drift of subcarriers than ∆T
T . Solving for
∆T
therefore
requires
eliminating
∆f
T
in
a
“clean” fashion
T
from the equations before resorting to approximations. That
elimination is attained by dividing (5) by (6), yielding the
following ML condition for ∆T
T :


Np Np


kp −kq ∆T
Im
kq α(kp ) α∗ (kq ) ej2πmNb Ns T
= 0 (9)


p=1 q=1
Eq. (9) cannot be solved in closed form for ∆T
T . However, it
reveals that the effect of TFO is embedded in the differences of
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Fig. 2.
A. Channel Model
We tested our algorithms with simulations using the exponentially decaying Rayleigh fading channel model, whose
discrete impulse response sequence is given by:
h(n) = Ae−γn [x(n) + jy(n)],
n = 1, 2, . . .
(11)
Above, all x(n) and y(n) are independent and identically
distributed Gaussian random variables with zero mean and unit
variance, and A and γ are given by:
1
Tsample
(1 − e−2γ ),
γ=
A=
2
2τRM S
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B. Simulation Parameters
We chose to run our simulations under conditions that
mimic the main parameters of the IEEE 802.11a standard, i.e.,
a 20 MHz wideband transmission with Ns = 64 subcarriers,
of which 6 are left unmodulated on each band edge and other
4 (unless noted otherwise) are used for pilot symbols. The
cyclic prefix is 16 samples long.
For the channel model we chose τRMS = 50 ns, representing
an indoor office environment.
Next, we considered uncoded 4-QAM modulation and independent random data and random pilot symbols for each
transmitter branch. We assumed perfect OFDM symbol timing at the receiver and used the ML criterion for decoding
the MIMO signals. The ML decoder consists of minimizing
(individually for each subcarrier k) the following metric over
all possible transmission vectors X(k):
X̂(k) = min Y(k) − H(k) · X(k)2
X(k)
Note that using ML decoding is an arbitrary choice, since
our tracking algorithms are compatible with any other MIMO
decoding method.
Finally, for generating each single data point of the results
presented next, we collected statistics over 2000 MIMO channels, transmitting 120 MIMO-OFDM symbols for each one of
them. In all cases, the total transmitted power was kept the
same.
1x1
2x2
3x3
4x4
5x5
6x6
7x7
8x8
−2
var(∆f T)
For generating MIMO channels we used Nt × Nr independent
realizations of the above model.
Finally, to test our algorithms with imperfect channel
estimates, we added independent complex-valued Gaussian
distributed random perturbations to each element of the exact
channel matrices H(k) (these matrices result from taking
Ns -point discrete Fourier transforms of the channel impulse
responses hji (n)). The perturbances have zero mean and
a variance 15 dB smaller than the expected energy of the
elements of the matrices H(k) [5], [12].
4 × 4 system at 0 dB SNR (dotted line). 5 × 5 systems and
larger provide estimator variances at 0 dB SNR that cannot be
attained by a SISO system even operating at 20 dB SNR.
10
−3
10
0
2
4
6
8
10
Es/No
12
14
16
18
20
ˆ T of (8) as a function of SNR with 4 pilot subcarriers
Fig. 3. Variance of ∆f
and ∆f
=
0.3
(m
=
1 and ∆T = 0).
f
δ
In Fig. 4 we present the dependency of the CFO estimator
variance on the number of pilot subcarriers, and observe that
as the number of antennas grows, fewer pilot subcarriers are
necessary to achieve similar estimator variances. For instance,
our estimator used in a 2×2 MIMO-OFDM configuration with
only 2 pilot subcarriers provides estimates of equivalent reliability as if it was used in a SISO-OFDM system with 4 pilot
subcarriers, such as one that complies with the IEEE 802.11a
standard.
1x1, 10 dB
2x2, 5 dB
4x4, 0 dB
2x2, 10 dB
4x4, 10 dB
8x8, 10 dB
−2
10
var(∆f T)
For practical purposes we limited the impulse responses to a
length equal to eight time constants of the decaying exponential in (11), i.e.,
8
τRMS
nmax =
= 16
γ
Tsample
−3
10
2
4
6
8
10
12
14
16
18
20
N
pilots
ˆ T as a function of the number of pilot subcarriers
Fig. 4. Variance of ∆f
( ∆f
= 0.3, m = 1 and ∆T = 0).
f
δ
C. Simulation Results
First we verified by simulation that (8) provides unbiased
estimates in cases in which the CFO is smaller than one half of
the intercarrier spacing (not shown). The variance of the CFO
estimator of (8) as a function of SNR2 is shown in Fig. 3 for
different MIMO configurations and a CFO (normalized to one
intercarrier spacing) of ∆f
fδ = 0.3. There is a clear SNR gain
as the number of antennas grows. For instance, the estimator
variance of a SISO-OFDM system operating at 10 dB SNR is
attained with a 2 × 2 configuration at 5 dB SNR, and with a
2 We
define SNR as the average signal-to-noise ratio of all receiver branches.
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Next, we illustrate the symbol error rate (SER) performance
of our tracking algorithms using 4 pilot subcarriers. We
first analyzed their transient behavior at the beginning of
transmission, and found that it lasts for approx. 4 OFDM
symbols (not shown). During that period the SER is dominated
by intercarrier interference (ICI) caused by the residual CFO
still present at the FFT input.
The joint performance of our algorithms during steady state
is shown in Figs. 5 and 6 for 1 × 1 SISO-OFDM and 4 × 4
MIMO-OFDM, respectively. Each figure shows simulations
with perfect channel state information and with a channel
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estimation error (CEE), simulated as described in Section IVA. In both cases we compare the SER performance of an ideal
receiver that is perfectly synchronized to the transmitter with
the corresponding curve when both tracking algorithms are
working jointly on transmissions with an initial normalized
CFO of ∆f
fδ = 0.3 and a TFO of 50 ppm. For reference, the
dotted lines show theoretical single-carrier performance of 4QAM in AWGN and Rayleigh fading. We find that the tracking
algorithms follow the ideal performance closely, even at low
SNR and under conditions of imperfect channel estimates.
0
10
Ideal Receiver
∆f/fδ = 0.3 & TFO = 50ppm
Ideal Receiver with CEE
∆f/fδ = 0.3 & TFO = 50ppm with CEE
−1
10
−2
SER
10
−3
10
Single Carrier in
Flat Rayleigh Fading
The algorithms result from applying maximum likelihood
estimation theory to observations of received pilot subcarriers
at the output of the FFTs of the receiver branches.
In the most general case, different pilot symbol sequences
are used on each transmitter branch. In such case, we observed
that our carrier frequency offset estimator benefits from diversity gain as the MIMO system grows in number of antennas.
This may be exploited by larger MIMO-OFDM configurations
either for operating reliably at lower SNR, or using less
subcarriers for pilot symbols. The algorithms, however, require
knowledge of the MIMO channel matrices.
The ML condition we derive for the sampling clock frequency offset is independent of the carrier frequency offset.
Conversely, a sampling clock offset is negligible in the ML
condition for the carrier frequency offset. This important
property allows for tracking algorithms that are decoupled
from each other, thus avoiding mutual destabilization while
one or both algorithms are in a transtient stage.
Even though we have illustrated the performance of our
algorithms in an IEEE 802.11a setting, the results can be
readily applied to MIMO, SIMO and SISO-OFDM systems
of different bandwidth and number of subcarriers.
ACKNOWLEDGMENT
−4
10
This work was supported in part by the ONR AINS Program.
Single Carrier
in AWGN
−5
10
5
Fig. 5.
OFDM.
10
15
Es/No [dB]
20
SER performance of the proposed tracking algorithms in SISO-
0
10
Ideal Receiver
∆f/fδ = 0.3 & TFO = 50ppm
Ideal Receiver with CEE
∆f/fδ = 0.3 & TFO = 50ppm with CEE
−1
10
−2
SER
10
−3
10
Single Carrier in
Flat Rayleigh Fading
−4
10
Single Carrier
in AWGN
−5
10
5
10
15
Es/No [dB]
20
25
Fig. 6. SER performance of the proposed tracking algorithms in 4 × 4
MIMO-OFDM.
V. C ONCLUSIONS
Algorithms for tracking carrier and sampling clock frequency offsets in MIMO-OFDM systems were presented.
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