Finite Precision Analysis for Space-Time Decoding

Abstract

Low complexity optimal (or nearly optimal) decoders for space-time codes have recently been under intensive investigation. For example, recent works by Sirianunpiboon and others show that the Silver code and the Golden code can be decoded optimally (or nearly optimally) with quadratic decoding complexity. Fast decodability makes them very attractive in practice. In implementing these decoders, floating-point to fixed-point conversion (FFC) needs to be carefully undertaken to minimize hardware cost while retaining decoding performance. The process of quantization for fixed-point representations is often ignored by research community and lacks investigation, and so FFC is often conducted heuristically based on simulations. This paper studies the effects of quantization to space-time coded systems from an information theoretic perspective. It shows the analytical relationship between quantization error and decoding performance deterioration. This paper also proposes a general finite precision implementation methodology including two FFC criteria for space-time coded systems within an integer optimization framework. As a particular example, this paper examines the finite precision implementation of the quadratic optimal decoding algorithm of the Silver code. However, our methodology and techniques can be applied to general space-time codes.

Full text

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 20094861
Finite Precision Analysis for Space-Time Decoding
Yiyue Wu, Student Member, IEEE, Linda M. Davis, Senior Member, IEEE, and A. Robert Calderbank, Fellow, IEEE
Abstract—Low complexity optimal (or nearly optimal) decoders
for space-time codes have recently been under intensive investi-
gation. For example, recent works by Sirianunpiboon and others
show that the Silver code and the Golden code can be decoded op-
timally (or nearly optimally) with quadratic decoding complexity.
Fast decodability makes them very attractive in practice. In imple-
menting these decoders, floating-point to fixed-point conversion
(FFC) needs to be carefully undertaken to minimize hardware
cost while retaining decoding performance. The process of quanti-
zation for fixed-point representations is often ignored by research
community and lacks investigation, and so FFC is often conducted
heuristically based on simulations. This paper studies the effects
of quantization to space-time coded systems from an information
theoretic perspective. It shows the analytical relationship between
quantization error and decoding performance deterioration. This
paper also proposes a general finite precision implementation
methodology including two FFC criteria for space-time coded
systems within anintegeroptimizationframework. As aparticular
example, this paper examines the finite precision implementation
of the quadratic optimal decoding algorithm of the Silver code.
However, our methodology and techniques can be applied to
general space-time codes.
Index Terms—Finite precision implementation, fixed-point rep-
resentation, quantization, space-time codes.
I. INTRODUCTION
S
antennas. Orthogonal space-time code designs such as the
Alamouti code are able to separate transmit streams with only
linear processing at the receiver. Some higher rate space-time
codes, such as the Golden code and the Silver code, can also
be decoded optimally or nearly optimally with low decoding
complexity (see [1] and [2]). Fast decodability makes these
space-time codes very attractive in practice.
The implementation of space-time encoders and decoders in-
troduces quantization error into the receiver. There are three
main sources of quantization error that can arise: input quan-
tization, coefficient quantization and quantization in arithmetic
PACE-TIME codes are designed to combat fading over
channels by correlating signals across different transmit
Manuscript received November 03, 2008; accepted May 24, 2009. First pub-
lished June 30, 2009; current version published November 18, 2009. The as-
sociate editor coordinating the review of this manuscript and approving it for
publication was Prof. Erik G. Larsson. The work of R. Calderbank and Y. Wu
is supported in part by the NSF under Frant DMS 0701226 and by the AFOSR
under Grant FA9550-05-1-0443.
Y. Wu and A. R. Calderbank are with the Department of Electrical
Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail:
yiyuewu@princeton.edu; calderbk@math.princeton.edu).
L. M. Davis is with Institute of Telecommunication Research, University of
South Australia, Adelaide, SA, 5095 Australia (e-mail: linda.davis@unisa.edu.
au).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2009.2026068
operations [3].Thispaper focusesmainlyoninput quantization,
where input refers to channel gains and received signals which
are inputs to the decoding system. The signals are derived from
the output of the analog to digital converter (ADC) at the re-
ceiver. The sampling and conversion processing will introduce
quantization error to the receive decoding system.
Input quantization error can largely be ignored for single
input and single output systems (SISO), where fine granularity
of quantization makes quantization error typically less signif-
icant than channel noise. Yet, for multiple input and multiple
output systems (MIMO), the ADC is no longer able to provide
enough resolution and the quantization error incurred can no
longer be neglected [4], [5]. In [4], the authors showed that
for 64-QAM modulation transmitting over channels with four
multipath components, the ADC resolution needed for fine
granularity is 12 bits and 18 bits for one and two transmit
antennas respectively, and it increases rapidly as the number of
transmit antennas increases. Unfortunately, most of the current
MIMO literature ignores the effects of input quantization,
which in fact alters the decoding problem at the receiver.
Coefficient quantization and quantization in arithmetic op-
erations also affect decoder performance in finite precision
implementation. Typically, the approach is one of conversion
from floating-point representation to fixed-point representation
(FFC) and this is normally left to hardware engineers who
are familiar with VLSI constraints, but not necessarily the en-
coding and decoding algorithms. Therefore, bit assignments for
channel gains and receive signals are often made heuristically
based on simulations without theoretical analysis, and the gap
between algorithms and hardware designs is exacerbated. A
transfer function method was proposed by Shi [6], yet it does
not give a clear relationship between quantization and BER
performance. The authors in [7] presented maximum likeli-
hood decoding for quantized MIMO systems by considering
the diversity and outage probability. Automatic gain control
(AGC) for quantization is adopted in [7] and [8], but these work
lack of a systematic FFC framework for space-time decoding.
Some other methods such as [9], [10] are based on simulations
for specific systems and do not treat FFC as an optimization
problem.
The purpose of this paper is to give an information theo-
retic analysis of quantization error and propose a systematic
method of bit assignments for channel gains and received sig-
nals, as well as all the derived signals involved in the decoding
process. This paper focuses on two’s complement representa-
tion and uniform quantization. Nonuniform quantization is be-
yond the scope of this paper. We assume that channel gains
and channel noise are symmetric complex Gaussian and we ap-
proximate all signals involved in the decoding process as com-
plex Gaussian, where high precision can be verified by Monte
Carlo simulations. Based on work by Sripad and Snyder [3],
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4862 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009
we show that the input quantization error converges to a uni-
form distribution when the quantization step size goes to zero.
Using these probabilistic models, we further propose two FFC
criteria, specifically a range criterion and a resolution criterion,
and then formulate the FFC problem as an integer optimiza-
tionproblem.Weemphasizethatourapproachjointlyoptimizes
range and quantization resolution, in contrast to the approaches
in [6], [9], [10] where the whole part and fractional part of the
fixed-point representations are allocated separately. This paper
also reports the relationship between quantization and channel
capacity loss and shows the effects of quantization on BER per-
formance deterioration.
For 2
2 MIMO systems, the Silver code was recently
discovered and intensively studied in [20]–[25]; It is a rate
two code with nonvanishing minimum determinant 1/7 and
can be efficiently decoded with quadratic decoding complexity
where is size of QAM constellation. Fast decod-
ability of the Silver code makes it indeed attractive in practical
use and pushes it into multiple research directions. Y. Wu
and R. Calderbank [12] proposed higher rate superorthogonal
space-time codes based on the Silver code. Biglieri et al. [13]
concatenated the Silver code with an outer trellis code using
the method by Alamouti et al. [14].
This paper studies the fixed-point implementation of the
quadratic decoding algorithm of the Silver code using the
proposed finite precision methodology. The bit assignments for
the quadratic decoding system are given within an optimiza-
tion framework and the proposed FFC criteria are verified by
simulations. Though the analysis in this paper focuses on the
Silver code, the proposed methodology and techniques can be
applied to other space-time coded systems and we note that
different decoding algorithms may lead to different specific
analysis. In our more recent work, we have integrated our finite
precision space-time decoding scheme into the MIMO OFDM
framework.
Therest ofthispaper isorganized asfollows.SectionII intro-
duces the system model and formulates the problem. Section III
presents our finite precision methodology and FFC criteria. Im-
plementation of the quadratic decoding algorithm for the Silver
code using our proposed methodology is detailed in Section IV.
Section V provides some further discussion about our method-
ology and this paper concludes in Section VI.
The following notations are used. Boldface upper-case letters
denote matrices, nonboldface upper-case letters denote scalars,
boldface lower-case letters denote vectors and nonboldface
lower-case letters denote scalar variables. Operations
anddenote complex conjugate, Hermitian, Frobenius
norm, and expectation with respect to
,,
respectively.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System Model
We consider a space-time coded MIMO system with
transmit antennas and receive antennas. The received signal
is given by
(1)
where
resenting the channel gain between the
the
receive antenna,
codeword with entries in the constellation
of symbol periods in the transmission of one space-time code-
word and
is the additive noise with entries as sym-
metric complex Gaussian with zero mean and variance
they follow the distribution
and
tion with mean zero and variance
Forspace time codes, weusually assumea quasi-static model
for the wireless channel which means that channel gains remain
constant during one coherence block and change independently
from one block to the next. Throughout this paper, we also as-
sume channel gains are symmetric complex Gaussian with vari-
ance
. The average signal to noise ratio
is the channel matrix with entryrep-
transmit antenna and
is a space-time block
, is the number
, i.e.
where
denotes Gaussian distribu-
.
is given by
(2)
where
lation power.
For a real decoding system, the received signals are sam-
pled and quantized by the analog to digital converter, which
willintroducequantizationerrortothedecodingsystemthrough
quantization errors on the received signal and also the derived
channel estimate. By taking quantization error into considera-
tion, the quantized received signal
be written as
is the transmission rate andis the average constel-
and channel matrix can
(3)
(4)
where
model represents the channel in the decoder by the true channel
plus a single quantization error,
only a function of the finite precision representation, but also
the estimation algorithm, and it is dependent on the data, , and
indeedthequantizationofthereceivedsignal, , fromwhichthe
estimate is derived. The channel estimation error is not taken
into consideration in this paper.
Using (3) and (4), (1) can be rewritten as
and are corresponding quantization errors. This
. In practice, this error is not
(5)
where
(5) is the system model on which the receiver operates. In this
paper, we consider that receivers treat
as independent noise. If we suppose that
and variance
and is independent of channel noise, then the
average signal to noise ratio for the quantized system can be
written as
is the sum quantization error. Equation
as the true channel and
has zero mean
(6)
We define the
deterioration factor as
(7)
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WU et al.: FINITE PRECISION ANALYSIS FOR SPACE-TIME DECODING4863
B. Channel Capacity
The channel capacity of the system without quantization is
derived by Telatar [15]
(8)
where
is the expectation with respect to
and is the
We consider that the additive quantization error
pendent of channel noise
unknown complex distribution denoted as
,
identity matrix.
is inde-
and each of its entries follows an
, where
and denotes the
. In this
unknown distribution with mean zero and variance
case, an inequality for the capacity of the quantized system
given by Ihara [16]
is
(9)
where
(10)
and
is the divergence defined as
(11)
Note that
if and only if follows complex Gaussian
independently of.
The capacity loss due to quantization is
(12)
Thecapacitylossisameasureforperformancedeteriorationdue
to quantization.
C. Problem Formulation
With fine granular quantization resolution, the quantization
error
may be far less significant than channel noise
can be neglected. Yet, it has been pointed out that for MIMO
systems, the ADC resolution for fine granular quantization be-
comesinfeasibleinpractice(see[4]and[5]).Furthermore,once
the quantization resolution of the input signals is fixed, all inter-
mediate signals involved in decoding arithmetic operations can
be assigned with certain word lengths to preserve decoding ac-
curacy. In a sense, for a fixed decoding procedure, the quantiza-
tion resolution of the input signals determines the complexityof
decoding hardware. Therefore, hardware system design should
reduce quantization resolution of channel gains and received
signals to the minimum such that transmitted data can still be
recovered with high precision.
We formulate our general optimization problem for quantiza-
tion as
and
(13)
where
sponding to quantization resolution),
cost evaluation function,
for the decoding performance, and
coding performance.
Throughout this paper, the performance benchmark func-
tion
is defined by the
in (7). We note that average
directly to error probability performance in general, but will
show this gives very reasonable results. An alternative would
be to consider outage probability as in [7]. Another option for
the performance benchmark function could be the capacity loss
function in (12).
The hardware cost evaluation function
tonically increasing with
and
mentation technologies. For the purpose of analysis, we define
a generic cost function
are word lengths of the input signals (corre-
is a hardware
is the benchmark function
represents required de-
deterioration function
does not translate
is mono-
and it varies for imple-
where
signals, respectively.
andare numbers of channel gains and received
III. FINITE PRECISION IMPLEMENTATION METHODOLOGY
We consider uniform quantization and two’s complement fi-
nite precision system with a general data type representation as
where is binary with
denoting unsigned, is the word length and
step size represented by the least significant bit. When
also denotes the fractional bit assignment. For example, [1,8,3]
represents the data type of a signed variable with 8 bit word
length and resolution
. The range of data type
denoting signed and
is the
,
is
.
In order to represent a signal more precisely with limited bit
resource, we need to capture its range and required resolution.
We propose two criteria, specifically a range criterion and a res-
olution criterion, where the range criterion is based on Gaussian
approximationofsignalsandtheresolutioncriterionis basedon
system quantization error analysis.
A. Gaussian Approximation of Signals
As shown in the linear (1), provided that the variances for
channel gains and noise are fixed and the symbol constellation
is chosen, we can easily derive the statistics of
involved signals in a given decoding procedure.
For each entry of the received signal
mean as zero and its variance as
approximate the entries of
as Gaussian distribution
and any other
, we can calculate its
. We
Fig. 1 shows the empirical distribution of the received signal
(real part or imaginary part of one entry) and its Gaussian
approximation and it verifies the precision of Gaussian approx-
imationofthereceivedsignal.Wenotethatthedistributionof
is not exactly Gaussian due to the multiplication of a Gaussian
distributed channel matrix
with a uniformly distributed code
matrix
.
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4864 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009
Fig. 1. Gaussian approximationof the received signals with ?? ? ???,? ?
?, ?? ?, ? ? ?,
? ?? ?? and symbols on 4-QAM.
Similarly, we can approximate the distribution for the entries
ofalltheothervectorsandmatricesinvolvedinagivendecoding
procedure. In this paper, by saying a vector or a matrix follows
certain distribution, we mean the entries of the vector or the
matrix follow that distribution.
B. Approximation of Quantization Error Distribution
A uniform roundoff quantization with step size
zation resolution) is a nonlinear process converting continuous
signals into discrete signals in a staircase-type relation (see
[17]). We represent the quantized output of a general real signal
as
(quanti-
(14)
where
variable with probability density function
function
is the quantization error. We take to be a random
and characteristic
The probability density function of
Snyder [3]
is given by Sripad and
otherwise
(15)
and they also proved that a necessary and sufficient condition
for
to be uniform is
If
is Gaussian, then
and we can calculate the mean and variance of the quantization
error [3] as
for all.
Wenowprovethedistributionofquantizationerrorconverges
to the uniform distribution as the quantization step size goes to
zero in certain cases.
Lemma 1: If there exists
, then
such that
Proof:
Since
Proposition1: If
such that
of the quantization error
tion when the quantization step size goes to zero, i.e.
converges, the limiting value is zero.
is symmetricwith
, then the distribution
converges to the uniform distribu-
and thereexists
otherwise.
Proof: Since
, we have
Then
The proposition is proved by applying Lemma 1 as follows:
Corollary 1: If
verges to the uniform distribution when the quantization step
size goes to zero.
Now we consider statistics and distributions of quantization
errors of received signals and channel gains for the case where
the channel gains are exactly Gaussian and the received signals
are approximated as Gaussian. If the received signal
type
(for both real part and imaginary part), then its
stepsizeis
andthequantizationerror
uniform distribution with variance
the channel gains (with data type
and the quantization error is approximately uniform with
variance
.
is Gaussian with zero mean, thencon-
has data
isapproximately
. Similarly,
) have step size
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WU et al.: FINITE PRECISION ANALYSIS FOR SPACE-TIME DECODING4865
C. System Quantization Error Analysis
Now, we consider the system quantization error
linear decoding model in (5). The
expanded as
this paper is quantization effects rather than channel estimation,
we assume the best case with
of each other, thus
of the
can beentry of
. Since the focus of
, andindependent
(16)
Then the
written as
deterioration factor defined in (7) can be
(17)
The approximate system performance loss due to quantization
is then
.
Remark: The
deterioration due to quantization mea-
sured here is for exhaustive search (maximum likelihood) de-
coding and may be different for other schemes including dif-
ferent forms of maximum likelihood decoding as we will see in
our example. In other words,
zation also depends on the signal processing in addition to the
analog to digital conversion at the receiver.
We also note that since the receiver assumes the quantized
channel to be the true channel, the impact of channel quantiza-
tion error is amplified by the fact
very sensitive to channel quantization.
deterioration due to quanti-
making the receiver
D. Bit Assignment Criteria
1) Range Criterion: The range criterion ensures that the
fixed-point representation adequately represents the range of
data values. For a general complex variable
and data type , when
the word length
should be the smallest integer that satisfies
with variance
are predetermined,and
(18)
where the left hand side of the inequality is the range of two’s
complementdatapresentation,
ment and in the case of
as least 99.7% confidence if
.
Remark: Note that, AGC (see [7], [8]) can be integrated with
the range criterion. The function of the AGC is to ensure that
the allocated bits are fully utilized which translates to equality
in (18).
2) Resolution Criterion: Given the noise variance
channel gain variance and average constellation power
the resolution criterion is stated as
isadjustabletosystemrequire-
, the range is covered with
follows Gaussian distribution
,
,
(19)
where
threshold according to system requirements.
is definedin (17) and (in dB) is thedeterioration
3) Optimization Formulation: Using these two criteria, we
can formulate a bit assignment integer optimization problem
using the formulation in (13) as
(20)
This optimization problem is nonconvex due to the non-
convex constraints
. By relaxing the range criterion in-
equality (18) into
(21)
(20) is tailored to be convex, specifically
(22)
Then, we can apply integer programming techniques (see [18])
and general convexoptimization techniques (see [19])tohandle
this problem.
4) Pragmatic Bit Assignment Operation: When data types
(i.e. word lengths and quantization resolutions) for the channel
gains and received signals are fixed, it is straightforward to de-
cide the data types for other intermediate decoding signals. Pre-
servingexactresultsfortheintermediatesignalsgeneratedfrom
arithmetic operations such as addition and multiplication may
require a large amount of storage. For example, the word length
required for exact multiplication of one channel gain and one
received signal is
. Considering the fact that channel
gains and received signals have been contaminated with quan-
tization errors, it is not worth preserving the exact arithmetic
results. In order to reduce signal storage while retaining accept-
able accuracy, we provide a feasible low storage cost method
which is verified by simulations as the following.
• Signals which are generated from channel gains are as-
signed with the same word ength as that for the channel
gains.
• Signals generated from received signals are assigned with
the same word length as that for the received signals.
• Signals generated from both of them are assigned with the
maximum word lengths of the channel gains and the re-
ceived signals.
Then, the resolution levels of these intermediate signals can be
calculated by selecting the largest integer that satisfies (18).
In practice, the channel variance
statistics of other decoding variables may also differ dramati-
cally due to different channel variances and different constel-
lation power levels. In order to avoid tedious recalculation, we
can decide the data types of all decoding signals by comparing
the system with a reference system.
may vary widely and
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