EE359 – Wireless Communications
Term Project – Autumn 2003
Simple Transmit Diversity Techniques
for Wireless Communications
By Minh-Anh Vuong
February 12, 2016
Table of Contents
1
1.1
1.2
2
2.1
2.2
2.3
2.4
3
4
4.1
4.2
5
5.1
5.2
5.3
5.4
6
Project Outline .................................................................................................. 1
Mitigating Rayleigh Fading .............................................................................. 1
Project Sections ................................................................................................. 1
Alamouti’s Simple Transmit Diversity Technique ........................................... 2
Flat Frequency Fading Channel ........................................................................ 2
Two Branch Transmit Diversity with One Receiver ........................................ 3
Comparison with Maximal Ratio Combining ................................................... 5
Extension to M Receive Antennas .................................................................... 6
Winters’ Transmit Diversity Technique ........................................................... 6
Extension of Alamouti’s to Wideband Fading.................................................. 7
OFDM Overview .............................................................................................. 7
Space-Frequency Coding .................................................................................. 8
Discussions ....................................................................................................... 9
Power Requirements ......................................................................................... 9
Sensitivity to Channel Estimation Errors ........................................................ 10
Decoding Latency ........................................................................................... 10
Soft Failures .................................................................................................... 10
References ....................................................................................................... 11
1
Project Outline
Multipath fading encountered in time-varying channel renders wireless communications highly non-reliable. To obtain an average bit-error rate of 10 3 using BPSK
modulation with coherent detection, performance degradation due to Rayleigh fading
can account for a SNR of 15dB higher than in AWGN [3][7].
1.1
Mitigating Rayleigh Fading
Theoretically, transmitter power control technique known as “water-filling” is the
most effective way to mitigate multipath fading [3][7]. However, the transmitted
power adaptation requires knowledge of channel SNR to be estimated at receiver and
sent back to transmitter that inevitably results in throughput reduction and higher
complexity in both transmitter and receiver. Dynamic range of transmitter amplifier
necessary to accommodate power backoff represents another disadvantage in using
“water-filling” technique.
Diversity is a powerful technique that is more practical and, therefore, widely used in
combination with space-time coding to combat signal fading. Diversity is characterized by the number of independently fading subchannels being created by multiple
antennas at transmitter and/or at receiver. Depending on antenna configurations,
space-time wireless systems can be categorized into SISO (single input single output)
being the traditional channel, SIMO (single input multiple output) having one single
transmit antenna and multiple receive antenna, MISO (multiple input single output)
using multiple transmit antennas and a single receive antenna and MIMO (multiple
input multiple output) having multiple transmit antennas and multiple receive antennas [5][6].
1.2
Project Sections
This project is limited to the review of several study cases of MISO configuration
where suitable coding or signal processing techniques are exploited to allow the extraction of transmit diversity without channel knowledge at the receiver. If the subchannels associated to transmit antennas have independent fades, the order of diversity is proven to be equal to the number of transmit antennas. This approach is attractive in public broadcasting systems such as cellular (for voice) or broadband wireless
access (for data communications) to keep the subscriber side equipment cost down
with simpler hardware requirement and more compact form factor by avoiding the
implementation of several receive antennas.
This report is organized as follows
 Alamouti’s simple but efficient transmit diversity technique [2] is introduced.
This approach uses space-time coding to achieve diversity on one single receive antenna in flat-fading wireless channels. Extension to multiple receive
antennas is also reviewed.
 Winters’ approach [9] using signal processing to create diversity from multiple transmit antennas is briefly reviewed.
1


2
An extension to frequency selective fading for Alamouti’s technique in the
OFDM framework as proposed by N. Ahmed et al [1].
The project then concludes with a discussion on implementation issues.
Alamouti’s Simple Transmit Diversity Technique
This technique [2] is described to be applicable to two transmit antennas and one receive antenna configuration.
2.1
Flat Frequency Fading Channel
The author presents the received signals following separate propagation paths as identical except for a complex scalar. This implies flat frequency fading channel, i.e.,
Equation 2-1:
BTm  1
where B is the signal bandwidth and Tm the channel delay spread. In addition, it is
necessary that
Equation 2-2:
BTZ  1
where TZ is transit time of the wave front across antenna array or difference in propagation delay from transmit antennas to receive antenna. For distance between
transmit antennas in the order of 10-20 wavelengths and for carrier frequencies in the
range 2-5GHz, the transit time TZ  5ns , i.e. much smaller than practical values of
delay spread or symbol period.
TX Ant 0
RX Ant
TX Ant 1
TZ
Figure 2-1. Transit time across antenna array.
Under the aforementioned conditions, the channels between N transmit antennas and
the receive antenna can be modeled as a row vector h  [h0 h1 ... hN 1 ] and the
received signal at time k as
Equation 2-3:
y[k ] 
Es
hx[k ]  n[k ]
N
where x[k ]  [x 0 [k ] x1 [k ] ... x N 1 [k ]]T is the transmitted signal vector at time k
and n[k ] additive noise [5].
2
Two Branch Transmit Diversity with One Receiver
The scheme is defined by the following three functions:
 The encoding and transmission sequence of information symbols at the
transmitter;
 The combining scheme at the receiver;
 The decision rule for maximum likelihood detection.
h0
sn
TX Ant 0
Combiner
~s
n
ML Detector
x0 [k ]
y[k ]
Space-Time
Coding
n[k ]
x1[k ]
h1
Channel
Estimator
Noise and
interference
TX Ant 1
ŝn
h0,1
Figure 2-2. Alamouti’s simple transmit diversity scheme.
-s*n+1
k
TX Ant 0
Coding for
antenna 0
Coding for
antenna 1
s*n
Sequence
of symbols
to transmit
x0 [k ]
sn+1
sn+1
sn
The Encoding and Transmission Sequence
The space-time coding scheme is based on pairs of symbols that are transmitted in
two consecutive symbol intervals. During the first (even-numbered) interval, the two
symbols are transmitted unaltered and simultaneously by two antennas. During the
following (odd-numbered) interval, the complex conjugates are transmitted as shown
in Figure 2-3.
sn
2.2
x1 [k ]
k
TX Ant 1
Figure 2-3. Alamouti’s space-time coding for TX antennas.
Let s n be the sequence of symbols to be transmitted. Let x0 [k ] and x1[k ] be the signals transmitted by antennas zero and one respectively at time k. Please note that the
symbol time k is introduced to highlight the sequential aspect in transmitted and received signals. However, the symbol index n is retained to avoid eventual confusion
that apparent non-causality in the following equations may cause.
Two consecutive symbols are transmitted simultaneously over two antennas during
the symbol interval k, where k is even, as
3
Equation 2-4:
 x0 [k ]  sn

 x1[k ]  sn1
During the odd-numbered symbol interval, the same symbols are re-transmitted, in
complex conjugate form and inverted polarity on antenna zero, as
Equation 2-5:
 x0 [k  1]   sn1

 x1[k  1]  sn
While the fading is assumed to remain unchanged across two consecutive symbol intervals (Equation 2-2), the channel can be modeled by complex multiplicative distortions for transmit antenna zero and one as
Equation 2-6:
h0 [k  1]  h0 [k ]  h0   0 e j0

h1[k  1]  h1[k ]  h1  1e j1
And since the time delay between receptions of x0 [k ] and x1[k ] caused by spatial
distance between transmit antenna locations (order of several wavelengths) is assumed negligible compared to the symbol interval, the received signal can be expressed as
Equation 2-7:
y[k ]  h0 x0 [k ]  h1 x1[k ]  n[k ]
The Combining Scheme
y[k  1]
h1
Conj
~
sn

0
h
z 1
 h0
y[k ]
h1
~
sn 1
Figure 2-4. Alamouti’s RX combining scheme.
The received signal is combined across two consecutive symbol intervals in the following manner
Equation 2-8:
 ~
sn  h0 y[k ]  h1 y  [k  1]
~
 sn1  h1 y[k ]  h0 y  [k  1]
where the channel complex multiplicative factors h0,1 are estimated and made available by channel estimator. Substituting Equation 2-4 through Equation 2-7 into Equation 2-8 and simplifying produce
Equation 2-9:
 ~
sn  ( 02  12 ) sn  h0 n[k ]  h1n  [k  1]
~
 sn1  ( 02  12 ) sn1  h1 n[k ]  h0 n  [k  1]
4
Equation 2-9 shows the combined signals are in fact transmitted symbols being scaled
and received in additive noise.
The Maximum Likelihood Decision Rule
The detector makes decision of which symbol being transmitted based on the Euclidean distance to all possible symbols
s s |
Equation 2-10: sˆ  s ; j  arg min | ~
n
i
n
i
Comparison with Maximal Ratio Combining
To illustrate the comparable performance between the simple transmit diversity
scheme presented in previous sections and maximal ratio combining diversity
scheme, let us consider the one transmit antenna and two receive antennas diversity
system as shown by Figure 2-5.
RX Ant 0
h0
y0 [ k ]
sn+1
TX Ant
sn
2.3
j
x[k ]  sn
h0
n0 [k ]
k
~
sn
h1
y1[k ]
RX Ant 1
n1 [k ]
h1
Figure 2-5. Maximal ratio combining diversity.
The received signals are
Equation 2-11:
 y 0 [k ]  h0 x[k ]  n0 [k ]

 y1[k ]  h1 x[k ]  n1[k ]
j
where x[k ]  s n is the transmitted signal during symbol interval k, h0,1   0,1e 0 ,1
channel response assumed to remain unchanged across consecutive symbol intervals
and n0,1 [k ] additive noise terms. The MRC produces
Equation 2-12:
~
sn  h0 y 0 [k ]  h1 y1[k ]
 ( 02   12 ) s n  h0 n0 [k ]  h1 n1[k ]
that is indeed very similar to the expressions in Equation 2-9.
5
2.4
Extension to M Receive Antennas
Alamouti’s technique can be applicable to MIMO configuration of two transmit antennas and M receive antennas to provide a diversity order of 2M. While the transmission coding remain as specified by Equation 2-4, the channel matrix becomes
 h0, 0

Equation 2-13: H   
hM 1, 0
h0,1 

 
hM 1,1 
where hl , m is the complex scalar associated to transmission path between transmit
antenna l, l  {0,1} , and receive antenna m, m  {0,1,..., M  1} . The received signals
are
 y 0 [k ] 
 n0 [ k ] 
 x 0 [k ] 



Equation 2-14: y[k ] 
    H  x [k ]    
 1  n [k ]
 y M 1 [k ]
 M 1 
And the combining scheme is


~

 sn  h0,0  hM 1,0  y[k ]  h0,1  hM 1,1 y [k  1]
Equation 2-15: 


~

sn1  h0,1  hM 1,1  y[k ]  h0,0  hM 1,0 y [k  1]
Replacing Equation 2-13 and Equation 2-14 into Equation 2-15 yields
M 1
M 1
 ~
2
2
s

s




hm , 0 nm [k ]  hm ,1 nm [k  1]

n
m,0
m ,1
 n

m 0
m 0
Equation 2-16: 
M 1
M 1
~
s n 1  s n 1   m2 , 0   m2 ,1   hm ,1 nm [k ]  hm , 0 nm [k  1]

m 0
m 0



3





Winters’ Transmit Diversity Technique
TX M 1
D
h
MLSE
TX1
n[k ]
1
Ts
D
sn
TX 0
Figure 3-1. Winters’ transmit diversity.
6
ŝn
Unlike Alamouti’s, the technique [9] being briefly reviewed here operates based on
artificial introduction of intersymbol interference achieved by multiple antennas sending delayed versions of transmit symbols as shown in Figure 3-1.
The channels between transmit antennas and receive antenna are assumed to be independent Rayleigh. In order to obtain uncorrelated signals over transmit antennas, the
delay D must be larger than max( Ts , Tm ) where Ts is the symbol interval and Tm the
multipath delay spread. Aggregate noise is assumed additive white Gaussian. At the
receiver side, the signal is sampled at the symbol rate before maximum likelihood sequence estimation (MLSE) with perfect channel knowledge is employed to determine
the transmitted symbols.
The author uses a tree pruning algorithm, modified to handle multilevel signal in
complex channel, to determine the minimum Euclidean distance in MLSE implementation and to produce the performance curves presented in his paper. The paper reports simulated results that are identical to the theoretical matched filter bound performance for the case of M  2 and have negligible degradation, i.e. less than 0.1dB,
for larger M.
One significant difference Winters’ approach has from Alamouti’s resides in the
matched filter bound reference. In contrast to MRC receive diversity, the matched
filter has only one noise source that is unable to provide array gain. This results in a
gain reduction by M as discussed in this paper.
4
Extension of Alamouti’s to Wideband Fading
Alamouti’s proposed technique is based on narrowband fading assumption where the
channel is determined by a set of complex scalars that remains unchanged across several symbols interval. This section summarizes an extension to wideband fading
channel, studied by Nadeem Ahmed et al, in the framework of OFDM [1].
4.1
OFDM Overview
The basic idea being exploited here is to divide the symbol stream into substreams to
be transmitted over a set of orthogonal subcarriers where resulted symbol period in
each substream is much larger than the multipath channel delay spread to efficiently
mitigate the ISI.
In practice, OFDM can be efficiently implemented using the well-known FFT and
IFFT functions as shown in Figure 4-1.
7
D2A
CP
Insertion
P2S
IFFT
S2P
sn
Channel
A2D
CP
Removal
S2P
ŝ n
FFT
P2S
n[k ]
Figure 4-1. OFDM modem block diagram.
In the context of OFDM where each subchannel centered on a subcarrier experiences
flat fading, Alamouti’s technique can be applied using a coding scheme spanning
multiple antennas and adjacent OFDM sub-carriers, instead of timeslots, to obtain
diversity over a single receive antenna.
Space-Frequency Coding
Let s n be the symbols to be transmitted. The transmit encoding scheme consists of
mapping two consecutive symbols on to same subcarriers f to be transmitted over
antennas zero and one as
Equation 4-1:
 X 0 ( f )  sn

 X 1 ( f )  s n1
and complex conjugates of same symbols on to adjacent subcarrier f  f as
Equation 4-2:
 X 0 ( f  f )   s n1

 X 1 ( f  f )  s n
H0( f )
-s*N-1
sN-2
-s*3
s2
-s*1
s0
Figure 4-2 shows the formation of the OFDM frames, from N  N FFT / 2 symbols, to
be transmitted over two antennas.
X0( f )
Ant 0
f
X1( f )
s*N-2
sN-1
s*2
s3
s*0
RX Ant
s1
4.2
H1 ( f )
Ant 1
f
Figure 4-2. Space-frequency coding for OFDM.
8
Assuming fading is similar across adjacent subchannels that can then be represented
j
by complex factors H 0,1 ( f )  H 0,1 ( f  f )  H 0,1   0,1e 0 ,1 , the received signals,
for the transmitted symbols s n and s n 1 , are
Equation 4-3:
Y ( f )  H 0 X 0 ( f )  H 1 X 1 ( f )  n( f )


Y ( f  f )  H 0 X 0 ( f  f )  H 1 X 1 ( f  f )  n( f  f )
A combining scheme similar to Equation 2-8 yields the following estimates
Equation 4-4:
 ~
s n  ( 02   12 ) s n  H 0 n( f )  H 1 n  ( f  f )
~
s n 1  ( 02   12 ) s n 1  H 1 n( f )  H 0 n  ( f  f )
The author reports that simulation results with QPSK-OFDM over multipath channels, respectively 4- and 16-tap long, to be very close to narrowband performance for
FFT length N FFT  512 . For N FFT  256 , the SNR degradation is about 2dB for
BER  10 3 . This reduced performance is explained by the fact as number of FFT
bins decreases, adjacent channel spacing increases and the assumed flat fading across
subchannels ceases to be effective.
5
Discussions
This reports reviewed the performance of two transmit diversity techniques proposed
by Alamouti [2] and Winters [9]. Compared to equivalent receive diversity, the presented techniques observe the advantage of having simpler, smaller and less expensive subscriber equipments while providing efficient remedy to Rayleigh fading
channel.
These techniques are reported based on different benchmarks, MRC and matched filter bound criteria respectively, render direct comparison impractical. However, while
Alamouti’s, and MRC in general, provides an array gain due to different noise
sources, the matched filter bound doesn’t and suffers a gain reduction proportional to
diversity order M. Therefore, considering same TX power equally distributed among
transmit antennas, Alamouti’s technique would be expected to outperform Winters’
due to array gain. Besides, Alamouti’s approach appears to be more popular thanks
to its simple formulation, efficiency and flexible adaptability to other configurations
(MISO and MIMO) and coding scheme (space-time vs. space-frequency).
And finally, following are several issues that need to be properly addressed and made
provisions for in order to take full advantage of the reviewed techniques.
5.1
Power Requirements
As per Alamouti’s paper (Fig. 4), the simulated cases (2-by-1 or 2-by-2 antennas) report 3dB performance degradation from the equivalent MRC (1-by-2 or 1-by-4 antennas). This performance loss is due to the fact only half of total power is radiated
from each transmit antenna. It can also be seen as one receive antenna can pick up
only half the equivalent power compared to the MRC cases. Of course, MRC compa-
9
rable performance can be achieved whenever total power is not a constraint and,
hence, can be doubled to accommodate both transmit antennas.
In Winters’ report, it is unclear how power repartition would affect the overall performance.
5.2
Sensitivity to Channel Estimation Errors
Both techniques are built upon the assumption of perfect knowledge of the channels.
Pilot tones or training symbols are commonly used to assist the receiver in estimating
channel information. In the case of transmit diversity, same pilot symbols need to be
alternately transmitted from every antenna or orthogonal symbols from all antennas
simultaneously, i.e. more overhead than receive diversity is needed for channel estimation purpose. Some constraint is also imposed on received signal sampling rate
that must meet both Nyquist requirement and exceed twice the maximum Doppler
frequency.
5.3
Decoding Latency
Both methods are introducing delayed symbols to allow diversity extraction at receiver side. Detection function must wait until all transmitted symbols are received to
make optimal decision. Instead of space-time coding, Alamouti’s technique can also
be combined with coding across space-frequency dimensions. If so, the decoding delay can be eliminated.
5.4
Soft Failures
If one transmitted signal fails to reach the receive antenna, e.g. due to transmitter failure or deep fade, the diversity gain is reduced but the transmitted symbols can still be
decoded successfully.
10
6
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Ahmed, N.; Baraniuk, R.G; “Asymptotic performance of transmit diversity
via OFDM for multipath channels,” IEEE Conference on Global Telecommunications, Nov 2002, vol. 1, pp. 691-95.
Alamouti, S. M.; “A simple transmit diversity technique for wireless communications,” IEEE Journal on Select Areas in Communications, Oct 1998, vol.
16, no. 8, pp. 1451-58.
Goldsmith, A.; “EE359 Wireless Communications, Course Reader,” 2003.
Lindskog, E.; Paulraj, A. J.; “A transmit diversity scheme for channels with
intersymbol interference,” IEEE Conference on International Communications, Jun 2000, vol. 1, pp. 307-11.
Paulraj, A.; Nabar, R.; Gore, D.; “Introduction to Space-Time Wireless Communications,” Cambridge University Press 2003.
Paulraj, A. J.; Papadias, C. B.; “Space-time processing for wireless communications,” IEEE Signal Processing Magazine, Nov 1997, pp. 49-83.
Stuber, G.; “Principles of Mobile Communications,” Kluwer Academic Publishers, 2nd Edition, 2001.
Tarokh, V.; Alamouti, S. M.; Poon, P.; “New detection schemes for transmit
diversity with no channel estimation,” IEEE International Conference on Universal Personal communications, Oct 1998, vol. 2, pp. 917-20.
Winters, J. H.; “The diversity gain of transmit diversity in wireless systems
with Rayleigh fading,” IEEE Transactions on Vehicular Technology, Feb
1998, vol. 47, no. 1, pp. 119-23.
11