Chase Decoding for Space-Time Codes
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Chase Decoding for Space-Time Codes
David J. Love Srinath Hosur and Anuj Batra Robert W. Heath, Jr.
School of Electrical and Computer Engr. DSPS R&D Center Dept. of Electrical and Computer Engr.
Purdue University Texas Instruments The University of Texas at Austin
West Lafayette, IN 47907 Dallas, TX Austin, TX 78712
djlove@ecn.purdue.edu {hosur, batra}@ti.com rheath@ece.utexas.edu
Abstract— Multiple antenna wireless systems are known performance of successive detection schemes is inferior
to provide a higher capacity than traditional single antenna to that provided by ML decoding. Sphere decoding [6]–
systems. Over the past few years, space-time signaling [8] is a low complexity symbol decoding technique
schemes that make use of this increased capacity have been
studied. Because of the large capacity of multiple-input that provides performance close to that of ML symbol
multiple-output channels, the multidimensional constella- decoding by performing minimum distance decoding
tions used by these space-time techniques are large in size over a small number of symbol vectors that fall within
making it impractical to perform optimal maximum like- a metric ball around the received signal vector. The
lihood decoding even for a moderate number of transmit sphere decoder, however, has several serious problems
antennas. In this paper, we propose a space-time version of
the binary Chase decoder. The decoder generates an initial that make its implementation challenging. First, the
estimate of the transmitted bit sequence from successive optimal ML vector within the metric ball might be
detection and then uses this bit estimate to generate a the first or last vector searched meaning that the worst
reduced search space (or list) to perform minimum distance case complexity of the sphere decoder is always equal
decoding. Three algorithms for constructing the space-time to that of ML decoding. Second, there is no simple
reduced search space are overviewed.
algorithm for choosing the sphere radius. As well, a list-
based sphere decoder must be used to allow the sphere
I. I NTRODUCTION
decoding algorithm to generate the log-likelihood, or soft
Multiple-input multiple-output (MIMO) wireless sys- bit, information that is critical to error control codes [8],
tems, which use multiple antennas at the transmitter and [9].
receiver, provide substantial gains in capacity compared In this paper, we propose space-time Chase (ST
to single antenna systems. As well, MIMO systems Chase) decoding for MIMO wireless systems as a solu-
can offer an increase in diversity, the rolloff factor tion to the MIMO decoding problem. ST Chase decoding
of the probability of error curve on a log-log scale. is a modification of existing successive detection de-
To achieve maximum diversity advantage, space-time coders that operates in two-stages: successive detection
transmission techniques must be optimally decoded [1], and reduced search space ML decoding. We design our
[2]. Unfortunately, the optimal decoder complexity of decoder to attempt to minimize the raw (uncoded) bit
most space-time signaling methods grows exponentially error rate by refining, in some sense, the initial bit
with the number of transmit antennas. Thus maximum estimates returned from the successive detection decoder.
likelihood (ML) decoding is often impractical even for The initial bit estimate is used to generate a list of
a modest number of transmit antennas. candidate symbol vectors over which minimum distance
Several reduced complexity decoding schemes have decoding can be performed. We present three differ-
been proposed to solve this implementation problem. ent algorithms for constructing these candidate symbol
Because most space-time transmission is based on the vectors based on the binary block decoding algorithms
idea of transmitting multiple substreams (see for example proposed in Chase’s famous work [10]. The ST Chase-
spatial multiplexing [3], [4]), multiuser communication 1 decoding algorithm performs maximum likelihood de-
receivers can be employed by thinking of each substream coding on symbol vectors corresponding to bit sequences
as a user and the intereference between substreams as lying in a predetermined radius Hamming ball around
interference between users. Successive detection tech- an initial estimate of the transmitted bit sequence. The
niques (see [3]–[5], etc.) avoid computationally expen- second decoding algorithm, known as the ST Chase-
sive joint detection by detecting and then cancelling the 2 decoder, uses soft bit information from a successive
effect of each substream. Unfortunately, the bit error rate detection stage to construct a reduced search spacey H
using all combinations of bit patterns in the indices
with the weakest soft bits. The final decoding algorithm, y
Successive
^
c Create Search L ML ^
b
ST Chase-3 decoding, functions similarly to ST Chase- Detection Space Decoding
H
2 decoding but only considers the most-likely error
patterns for each possible error weight. Fig. 1. Block diagram of a ST Chase decoder.
The ST Chase-1 decoder can actually be thought of
as a variant of the list sphere decoder proposed in [8].
List sphere decoding can be thought of as decoding over we are only concerned with vector-by-vector detection
a reduced search space of candidate symbol vectors by of s.
searching a sphere around the zero-forcing symbol vector We assume that each space-time signal corresponds
estimate using a modified, channel-dependent distance to the transmission of B bits. Each realization of s is
function. In contrast, the ST Chase-1 algorithm uses a obtained from an invertible map M : FB 2 → C
M
(i.e.
bit vector obtained from successive detection decoding the function M(·) is a modulation function). Thus the
as the center of a binary sphere using the Hamming purpose of the decoder is to find an estimate of the vector
distance. The symbol vectors corresponding to binary b ∈ FB2 with s = M(b) that generated y in (1).
vectors within the sphere are used to construct a reduced Note that we make no assumptions about the vector
search space. Thus the ST Chase-1 decoder can be constellation C being written as the product of single-
viewed as a binary list sphere decoder. dimensional real constellations as in [6], [7]. Thus, we
will not require that each entry of s have real and
II. S YSTEM OVERVIEW imaginary parts chosen from a real lattice. This allows
Over the past eight years, researchers have developed common constellations such as m-PSK where m > 4 to
a wide variety of spatio-temporal signaling techniques be decoded without any modification to the decoder. We
(see the discussion and references in [1], [2]). This also make no assumptions about the constellation being
paper focuses on systems that use some form of multidi- decomposable into subsets of PSK constellations as in
mensional modulation. Multidimensional constellations, [8]. The only constellation assumption is that C is a finite
unlike single-dimensional constellations, are subsets of1 subset of C.
Cn where n > 1. In this paper, we will restrict ourselves
to multidimensional modulations that can be written in III. S PACE -T IME C HASE D ECODING
the framework of linear dispersion codes [1], [11]. The fundamental difference between a ST Chase
Multidimensional constellations based on linear dis- decoder and previously published decoders is that the
persion codes possess the important property of having ST Chase decoder concentrates on correcting bit errors
a simple bijective map to a set C M where C is a com- instead of symbol errors. In this section, we will combine
plex constellation (ex. 8-PSK, 16-QAM, etc.). Assuming ideas from sphere decoding and binary block decoding
perfect pulse-shaping and sampling, linear dispersion to create a low complexity MIMO decoder.
codes have a matrix input/output relationship that can Fig. 1 presents a block diagram of a ST Chase
be modeled as decoder. The decoder’s first stage consists of a successive
y = Hs + v (1) detection decoder. Possible first stage decoders could
where y, v ∈ CN , s ∈ C M , H ∈ CN ×M , and N ≥ M. use ZF detection, ordered ZF detection, ordered mini-
The noise vector v is normalized so that each entry of mum mean squared error (MMSE) detection, etc. The
v is independent and distributed according to CN (0, 1). received vector y and effective channel H are fed into
ρ a successive detection decoder [3], [4]. The first-stage
We will assume that Es [ss∗ ] = M IM where ρ is the
average signal-to-noise ratio (SNR). Note also that any decoder generates a length B log-likelihood vector ĉ
kind of temporal indices in (1) have been removed since with the entries of ĉ corresponding to soft estimates of
the transmitted bit sequence.
1 We use F to denote the two element finite field F = {0, 1},
2 2 Let h(·) denote a function that returns hard decisions
FB2 to denote B-dimensional vector space over F2 ,
T to denote the
and c = h(ĉ) (i.e. ci = 12 (1 + ĉi /|ĉi |)). The ST Chase
the transpose of a matrix, ∗ to denote the conjugate transpose of a
matrix, C M to denote the M -fold Cartesian product of the set C with decoder uses this vector c as an initial estimate of b in
itself, IM for the M × M identity matrix, · 2 to denote the vector FB2 . This serves as a starting point for constructing a
two-norm, † to denote the matrix pseudo-inverse, | · | to denote the reduced complexity search.
absolute value, b ⊕ e to denote entry by entry exclusive-or of b with
e (i.e. b ⊕ e = [b1 ⊕ e1 b2 ⊕ e2 · · · bB ⊕ eB ]T ), card(·) to denote Let w(·, ·) denote the Hamming distance where
the cardinality of a set, and Ey [·] to denote expectation with respect
to y. w(b , b ) = card ({ i | bi = bi }) . (2)The set FB2 with the Hamming distance is a metric space. where Ci,1M M
(Ci,0 ) is the set of vectors in C M with bit i
Therefore, we can construct closed metric balls in FB 2 set to one (zero).
just as was done in the sphere decoding literature. Define Discussion
the metric ball as Note that algorithms decrease in complexity when mov-
ing from ST Chase-1 to ST Chase 3. The ST Chase-
BH (P, c) = {b ∈ FB
2 | w(b , c) ≤ P }. 1 decoder is the most complex of the three algorithms
Note that BH (P, c) can be constructed easily by com- P B it has the largest list size with card(L) =
because
puting c ⊕ e for all e such that w(e, o) ≤ P where i=0 P . The ST Chase-2 and ST Chase-3 decoders
o = [1 1 · · · 1]T . have smaller lists of cardinality 2P and P + 1, respec-
A list of candidate symbol vectors in C M can be tively.
constructed from the bit vectors in BH (P, c). Therefore, The ST Chase integer parameter P must satisfy 0 ≤
P ≤ B. Interestingly, varying parameter P allows a
Ltotal = {M(b ) | b ∈ BH (P, c)} complexity vs. performance tradeoff. When P = 0, all
is a list of candidate symbol vectors that can be used as three ST Chase decoders use only a successive detection
a reduced search space for minimum distance decoding. decoder (i.e. the list size is one). When P = B, ST
This list can be efficiently constructed because BH (P, c) Chase-1 and ST Chase-2 become true ML decoders that
is easily constructed via bit manipulations and Ltotal compute the soft bits over the entire constellation with
can be found by simply remodulating the bit vectors in L = C.
BH (P, c). Outer Codes
The final stage of ST Chase decoding performs min- The algorithms discussed in Section III are discussed
imum distance decoding over a set L ⊆ Ltotal . We in the case where they return hard bit decisions. These
propose three different algorithms for constructing the algorithms, however, can be easily modified for log-
reduced search set L based on algorithms presented in likelihood information using the unsliced portions of (3).
[10]. It is interesting to compare the symbol vector lists
Algorithm 1 (ST Chase-1): Set L = Ltotal . L constructed by the ST Chase-1 and list sphere de-
Algorithm 2 (ST Chase-2): Let I be the set of indices of coder. The list sphere decoder creates the list using a
the P weakest bits in ĉ. This means that if i ∈ I then modification of the real (or complex) sphere decoder.
|ĉi | ≤ |ĉj | for all j ∈
/ I. Construct L by choosing Thus, there will exist situations where L will have
reduced cardinality because of the lack of candidate
L = {M(b ) | b ∈ BH (P, c) and ∀j ∈
/ I, bj = cj }. vectors within the metric ball around H† y. The fixed-
For example if P = 2 and ĉ = [0.2 5.3 −0.3 1.3]T , then complexity ST Chase-1 decoder will always return soft
c = [1 1 0 1]T and bit information computed with a maximum cardinality
list L. This implies that the ST Chase-1 decoder always
L = M [1 1 0 1]T , M [1 1 1 1]T , returns a high quality soft bit estimate regardless of the
M [0 1 0 1]T , M [0 1 1 1]T . channel response.
Algorithm 3 (ST Chase-3): Let Ij be a set that contains IV. S IMULATION
the indices of the j weakest bit locations where 0 ≤ j ≤ We performed Monte Carlo simulations to compare
P. Define I0 to be the empty set. Construct L as the performance of ST Chase decoding with other de-
L = {M(b0 ), M(b1 ), . . . , M(bP )} coders. The probability of bit errors are shown as a
function of Eb /N0 = ρ/B.
where bj is c with the bits in the j The first experiment compares the ST Chase (using
indices in Ij complemented. Once again, P = 3 bits), ML, and ZF decoding on a 16-QAM 2 × 2
T
ĉ = [0.2 5.3 −0.3 1.3]
= 2 yield L
and P = spatial multiplexing system. The probability of bit error
M [1 1 0 1] , M [0 1 0 1]T , M [0 1 1 1]T . results are shown in Fig. 2. The ST Chase decoders used
T
After the list of candidate symbol vectors has been a ZF decoding first stage to generate initial bit estimates.
constructed, minimum distance decoding can then be ST Chase-3 and ST-Chase-2 decoding provide 0.5dB
performed over L. The decoder would compute the and 1dB gains, respectively, over ZF decoding. The ST
detected bit vector with each entry given by Chase-1 decoder performs within 0.5dB of an optimal
2
ML decoder at a bit error rate of 10−2 . The ST Chase-1
M ∩L exp − y − Hs 2
s∈Ci,1
b̂i = h ln 2) (3) decoder, thus, provides most of the gain available from
M
s∈Ci,0 ∩L exp(− y − Hs 2 ML decoding with a reduced complexity.0
10 Chase−1 (OIMMSE) 6bit
Chase−1 3bit Chase−1 (ZF) 6bit
Chase−2 3bit ML
Chase−3 3bit OIMMSE
ML ZF
ZF
−1 −1
10 10
BER
BER
−2
10
−2
10
−3
10 −2 0 2 4 6 8 10 12 14
−5 0 5 10 15 20
Eb/N0
Eb/N0
Fig. 2. Bit error rate comparisons for a 2 × 2 spatial multiplexing Fig. 3. Bit error rate comparisons for a 2 × 2 number theoretic
system using 16-QAM with various decoders. space-time code using 16-QAM with various decoders.
lower bound on the bit error rate for list decoders of this
The second experiment, displayed in Fig. 3, shows size. Even with the optimality assumption on this bound,
the average bit error rate of ST Chase-1 decoding for a the ST Chase-1 decoder performs identically to the list
2 × 2 number theoretic space-time code [12] using 16- sphere decoder with approximately the same complexity.
QAM transmitting over an uncorrelated Rayleigh fading The ST Chase-1 decoder has a gain of approximately
channel. This space-time code has a multidimensional 3dB over ordered MMSE decoding at a bit error rate of
constellation C of size 164 = 65536 symbol vectors 10−3 .
making ML decoding difficult in a real-time imple-
0
mentation. ST Chase-1 decoding was simulated with 10
OIMMSE
P = 6 bits and two different first stage decoders, one List Sphere
Chase−1
produced by ZF decoding and the other produced by
ordered MMSE decoding. The plot shows ZF, ordered −1
10
MMSE, and ML decoding for performance comparison.
ST Chase-1 decoding with an ordered MMSE first stage
performs within 0.35dB of ML decoding. Similarly, ST
BER
−2
Chase-1 decoding with a ZF decoding initial bit estimate 10
performs within 0.4dB of ML decoding.
The final experiment simulated the ST Chase-1 de-
coder with an outer convolutional code on a 5×5 spatial −3
10
multiplexing system transmitting 4-QAM symbols. A
rate 1/2 convolution code of memory two was simulated
with interleaving over 600 bits. It is assumed that the bits −4
10
are interleaved over ten independent channel realizations. −5 −4 −3 −2 −1
Eb/N0
0 1 2 3 4
The coded bit error rates for an ordered MMSE decoder,
a ST Chase-1 decoder using an ordered MMSE first Fig. 4. Coded bit error rate comparisons for the ordered MMSE, list
10 P = 5 bits, and a list sphere decoder using
stage sphere, and ST Chase-1 decoder.
5 with
i=0 i = 638 candidate symbol vectors are pre-
sented in Fig. 4. The list sphere decoder was simulated
assuming optimal candidates. This means that for every V. C ONCLUSIONS
symbol vector decoded the 638 symbol vectors with We proposed a new, reduced complexity decoder for
minimum values of y − Hs 2 were chosen. Thus the MIMO wireless systems using space-time signaling. The
simulated curve for the list sphere decoder is a strict decoder is based on the classic Chase decoder presentedin [10] and can be easily implemented. The decoder [10] D. Chase, “Class of algorithms for decoding block codes
works by using an initial stage of successive detection with channel measurement information,” IEEE Trans. Info. Th.,
vol. 18, pp. 170 –182, Jan. 1972.
decoding that generates a list for use with minimum dis- [11] B. Hassibi and B. Hochwald, “High-rate codes that are linear in
tance decoding. We presented three different algorithms space and time,” IEEE Trans. Info. Th., vol. 48, pp. 1804–1824,
of varying complexity for constructing the symbol vector July 2002.
[12] M. O. Damen, A. Tewfik, and J. C. Belfiore, “A construction of
list. The decoder does not require that the transmit con- a space-time code based on number theory,” IEEE Trans. Info.
stellation be decomposable into a real lattice. Simulation Th., vol. 48, pp. 753–760, March 2002.
results show that the decoders provides probability of
error improvements over successive detection and can
perform close to optimal ML decoding.
There are a large number of open questions and
extensions to the ST Chase decoding algorithm. In
particular, statistical analysis or extensive simulations
are needed to understand the performance comparisons
to the list sphere decoder. Efficient implementations
of the ST Chase and list sphere decoder must also be
compared. As well, extensions of MIMO ST Chase
decoding can possibly be employed in decoding
space-time trellis coded systems by only computing
distance metrics for the L most likely symbol vectors.
Reduced state or reduced path ML algorithms have
been employed in decoding single-dimensional systems,
and the ST Chase decoder could be modified into the
framework of a MIMO reduced path ML decoder.
VI. ACKNOWLEDGMENT
We would like to thank Dr. John T. Coffey of Texas
Instruments for his helpful comments.
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