BER Criterion and Codebook Construction for Finite-Rate Precoded Spatial Multiplexing With Linear Receivers
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006 1653
BER Criterion and Codebook Construction for
Finite-Rate Precoded Spatial Multiplexing With
Linear Receivers
Shengli Zhou, Member, IEEE, and Baosheng Li, Student Member, IEEE
Abstract—Precoded spatial multiplexing systems with rate-lim- imperfections originating from various sources, such as estima-
ited feedback have been studied recently based on various pre- tion errors, feedback delay, and feedback errors. These consid-
coder selection criteria. Instead of those based on indirect per- erations have sparked recent research interests toward quanti-
formance indicators, we in this paper propose a new criterion di-
rectly based on the exact bit error rate (BER) that is applicable fying and exploiting imperfect (or partial) CSI in multiantenna
to systems with linear receivers and rectangular/square quadra- systems; see, e.g., [9], [30], and references therein.
ture-amplitude-modulation constellations. The BER criterion out- Partial CSI can be modeled in different ways [30]. One class
performs any other alternative in terms of optimizing the BER per- of CSI models imposes a bandwidth constraint on the feedback
formance for an uncoded system with linear receivers. We then de-
velop a precoder codebook construction method based on the gen- channel which is only able to communicate a finite number of
eralized Lloyd algorithm from the vector quantization literature. bits per block. Power control based on finite-rate feedback is in-
This construction is not directly based on the BER criterion. Hence, vestigated in [2] to reduce the outage probability that the mutual
it is suboptimal in the BER sense. However, relative to those cur- information falls below a certain rate. Finite-rate transmit beam-
rently available, our newfound codebooks improve considerably forming has been investigated based on various criteria such as
various minimum distances between any pair of codewords of the
codebook. Finally, we analyze the BER-optimal precoder in the the average signal-to-noise ratio (SNR) [16], [19], the outage
asymptotic case with infinite-rate feedback that amounts to per- probability [18], and the symbol error rate [31], respectively.
fect channel knowledge at the transmitter. The infinite-rate optimal Subject to finite-rate feedback, optimal transmission is also pur-
precoder based on the BER criterion is drastically different from sued in [3], [11], and [22] to maximize the average channel ca-
the counterparts with other criteria, and it leads to a benchmark pacity, while adaptive modulation together with transmit beam-
performance for finite-rate precoded spatial multiplexing systems.
We observe from numerical results that the BER performance of fi- forming has been pursued in [27] to enhance the transmission
nite-rate feedback with suboptimal codebooks approaches quickly rate. Recently, the application of finite-rate feedback in a pre-
the benchmark performance of infinite-rate feedback. This sug- coded spatial multiplexing system has been addressed in [13]
gests that i) the number of feedback bits in practical systems need and [14], where various criteria on precoder selection and code-
not be large and ii) the room for performance improvement via fur- book construction have been proposed.
ther codebook optimization shrinks quickly as the codebook size
increases. As in [13] and [14], we in this paper investigate precoded spa-
tial multiplexing with finite-rate feedback. We focus on linear
Index Terms—Finite-rate feedback, Lloyd algorithm, precoding,
spatial multiplexing. zero-forcing (ZF) and minimum mean-square-error (MMSE)
receivers and rectangular/square quadrature-amplitude-modula-
tion (QAM) constellations. Our contributions are as follows.
I. INTRODUCTION
• Instead of those criteria based on indirect performance
M ULTIANTENNA diversity is by now well established as
an effective fading countermeasure for wireless commu-
nications. To further improve system performance, the receiver
indicators [13], [14], we propose a new precoder selec-
tion criterion directly based on the exact bit error rate
(BER) of the system. The new criterion hence outper-
can feedback channel state information (CSI) back to the trans- forms any other competing alternative in terms of im-
mitter, so that the transmission parameters such as power and proving the BER performance of an uncoded system with
modulation type can be adapted to the channel. In practical wire- linear receivers.
less systems, however, the CSI at the transmitter suffers from • We develop a precoder codebook construction method
based on the generalized Lloyd algorithm from the vector
Manuscript received November 26, 2004; revised July 12, 2005. This work quantization literature [8]. Since this construction is not
was supported by the UConn Research Foundation under Internal Grant 445157. directly based on the BER criterion, it is clearly subop-
This paper was presented in part at the 6th IEEE International Workshop on timal in the BER sense. However, relative to those cur-
Signal Processing Advances in Wireless Communications, New York, June 6–9,
2005. The associate editor coordinating the review of this paper and approving rently available, our new found codebooks improve con-
it for publication was Dr. Javier Garcia-Frias. siderably various minimum distances between any pair of
The authors are with the Department of Electrical and Computer En- codewords of the codebook.
gineering, University of Connecticut, Storrs, CT 06269 USA (e-mail:
shengli@engr.uconn.edu; baosheng@engr.uconn.edu). • We characterize the BER-optimal precoder in the asymp-
Digital Object Identifier 10.1109/TSP.2006.872554 totic infinite-rate feedback case that amounts to perfect
1053-587X/$20.00 © 2006 IEEE1654 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
channel knowledge at that transmitter.1 The optimal
precoder based on the BER criterion is drastically dif-
ferent from the counterparts based on all other criteria
[13], [14]. The system performance with infinite-rate
BER-optimizing precoders serves a performance bound
for finite-rate precoded spatial multiplexing systems.
This benchmark performance is not available before.
• Our extensive numerical results compare the BER perfor-
mance with different codebooks and with different selec-
tion criteria. We obtain practical guidelines for finite-rate Fig. 1. Precoded spatial multiplexing system with finite-rate feedback.
precoded spatial multiplexing systems, as will be summa-
rized in the conclusions.
matrices as and collect them into a codebook
The rest of this paper is organized as follows. We present the as
system model in Section II and propose the BER-based selection
criterion in Section III. We develop the Lloyd algorithm based (2)
codebook construction method in Section IV. We characterize
the optimal precoder with infinite-rate feedback in Section V. Based on the current channel realization, the receiver will de-
We collect numerical results in Section VI and conclude in Sec- cide which codeword (precoder) from the codebook is the
tion VII. most favorable and inform the transmitter to switch to that pre-
Notation: Bold upper and lower letters denote matrices coder by feeding back its -bits codeword index. Based on the
and column vectors, respectively; , , and denote block fading channel model, channel feedback and transmitter
transpose, conjugate, and Hermitian transpose, respectively; adaptation are done on a per block basis. For such a precoded
stands for the absolute value of a scalar; and stand spatial multiplexing system with finite-rate feedback, the fol-
for the trace and the determinant of a matrix, respectively. lowing two important questions need to be addressed.
denotes the two-norm of a vector or a matrix, while is the i) How does the receiver select a favorable precoder from
Frobenius norm of a matrix. is the identity matrix; the codebook ?
denotes an all-zero matrix of size ; stands ii) How does the system construct a good codebook ?
for the entry of a matrix . These two design issues have been well addressed in [13] and
[14]. We next summarize their results.
II. SYSTEM MODEL AND EXISTING RESULTS
A. Brief Summary of Existing Results
As depicted in Fig. 1, we consider a precoded spatial multi-
plexing system where the transmitter and the receiver have First, the precoding matrices have been constrained to have
and antennas, respectively. The information symbol block orthonormal columns [13], [14]
is precoded by a matrix to obtain
(3)
the precoded block , whose entries are then transmitted
through antennas simultaneously. We assume a block fading With precoders satisfying (3), various precoder selec-
channel model, where the channels remain invariant within a tion criteria have been proposed in [13] and [14].
block but can change independently from block to block. De- • For a maximum likelihood (ML) receiver, the precoder is
note as the channel coefficient between the th receive and chosen to either maximize the minimum receiver symbol
the th transmit antenna, and collect the channel coeffi- vector distance (MD Selection), or, maximize the instan-
cients into the channel matrix with . taneous capacity (Capacity Selection).
The received samples on receive antennas, collected in the • For a linear zero-forcing (ZF) receiver, the precoder is
vector , can then be expressed as chosen to maximize the minimum singular value of
(SV Selection).
(1) • For a linear MMSE receiver, the precoder is chosen to
either minimize the trace of the mean square error matrix
where is the additive white Gaussian noise (AWGN) with each (MMSE-trace Selection) or minimize the determinant of
entry having variance . the mean square error matrix (MMSE-det Selection).
As in [13] and [14], we assume that the receiver is able to
On the codebook design, the following results are available [13],
feedback a finite number of (say, ) bits back to the transmitter,
[14].
and that the feedback link is error-free and delay-free. Under the
constraint of feedback bits, the system only needs to prepare • In the asymptotic case with infinite-rate feedback where
a total of precoding matrices. Let us denote these , the optimal precoder consists of the eigenvec-
tors of corresponding to the largest eigenvalue,
for all selection criteria in [13], [14]. To be more specific,
1Precoder design with perfect channel knowledge has been investigated ex- denote the eigendecomposition of as
tensively in the literature; see, e.g. [5], [7], [20], [23], and references therein.
We will detail distinctions of our result with [5], [7], [20], and [23] in Section V. (4)ZHOU AND LI: BER CRITERION AND CODEBOOK CONSTRUCTION FOR FINITE-RATE PRECODED SPATIAL MULTIPLEXING 1655
where contains on its diagonal when the channel realization is and the precoder is . The
the eigenvalues arranged in a nonincreasing order: proposed BER-based selection rule is then
. The optimal precoder is then
(9)
(5)
whose codeword index is fed back to the transmitter.
where consists of the first columns of . The BER expression for an ML receiver is not available up
• In the finite-rate feedback case, if MMSE-det or ca- to date. Hence, the BER-based criterion is not applicable to an
pacity selection is used, the codebook shall be ML receiver. However, the BER for linear receivers can be easily
designed to maximize , where computed thanks to the recent results in [4] and [24]. We thus
is the Fubini–Study distance defined for focus our attention on linear receivers.
two subspaces spanned by and [1]
A. BER Expression for AWGN Channel [4]
(6) Let denote the relationship between BER and SNR
in an AWGN channel. We consider rectangular or square QAM
• In the finite-rate feedback case, if MMSE-trace, SV, constellations with size that can be decomposed into two in-
or, MD Selection is used, the codebook shall be de- dependent pulse-amplitude modulations (PAMs), one with size
signed to maximize , where and the other with size such that [4].
is the projection two-norm subspace dis- Define the Gaussian-Q function as
tance defined as [1]
. The closed-form expression for
is [4]
(7)
Precoder codebooks have been constructed using modified (10)
versions of the algorithm from [10], and sample codebooks are
provided in [12]. In a different scenario, finite-rate precoding
has been applied in an orthogonal space time block coded where
system [15], where the codebook was proposed to maximize
, where is the chordal
subspace distance defined as [1]
(8)
(11)
Although a variety of criteria on precoder selection and code-
book design have been offered, the following practical questions
have not been addressed.
1) All the selection criteria in [13] and [14] are based on
metrics that are indirect (though good) performance in-
dicators, e.g., the trace or the determinant of the MSE
matrix. However, the performance evaluation is eventu- (12)
ally done based on the uncoded BER for the considered
system. Why not directly adopt BER as the selection cri-
terion? The BER criterion will certainly outperform any Equations (10)–(12) reveal that is a finite sum of
other alternative for an uncoded system. Gaussian-Q functions. We can write it compactly as
2) Various codebooks based on optimization of different
subspace distances are available. Which codebook shall (13)
one adopt for a practical system? What is the optimal pre-
coder codebook with the BER-based selection criterion?
We will address these questions in our following developments. where the constants need to be figured out for each con-
stellation in use. The simple examples are 2-QAM and 4-QAM,
where we only have one term in the summation: ,
III. BER-BASED SELECTION CRITERION FOR
for 2-QAM, and , for 4-QAM. Alter-
LINEAR RECEIVERS
natively, one can compute easily using a simple recursive
Different from all selection criteria in [13] and [14], we pro- algorithm in [24].
pose to directly use the exact BER as the selection criterion. We next present the average BER for linear ZF and MMSE
Denote as the BER averaged over data streams receivers.1656 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
B. Linear ZF Receiver of optimizing the BER performance for an uncoded system with
The linear ZF receiver is . linear receivers.
Applying the ZF receiver on (1), we obtain
IV. CODEBOOK CONSTRUCTION WITH FINITE-RATE FEEDBACK
(14) The codebook eventually dictates the overall system per-
formance. Thus it shall be carefully designed. The codebooks in
where the processed noise has variance
[12] are obtained via modified versions of the algorithm in [10].
. With denoting the average symbol
Interestingly, the codebook design for finite-rate beam-
energy for each symbol , the SNR for the th data stream is
forming and finite-rate precoding can be linked to a vector
quantization problem [8]. Using the generalized Lloyd algo-
(15) rithm to search for a finite-rate beamforming codebook was
first used in [19] with . It is later used in [26] to
design finite-rate beamforming codebook for both independent
where for notational brevity we define
identically distributed fading channels and correlated fading
channels, with an arbitrary ; the codebooks are listed in
(16) [25]. We next show how the Lloyd algorithm can be utilized
to search for good precoder codebooks. The Lloyd algorithm
The average BER over data streams is then based codebook construction also provides an alternative sys-
tematic approach for the subspace packing problem in [6]. We
underscore that the Lloyd algorithm and other existing methods
(17) on codebook construction do not use the system BER directly
as the optimization criterion.
The codebook design of is linked to a vector quantization
C. Linear MMSE Receiver problem as follows. Suppose that we have a random ma-
trix , which is isotropically distributed. We now want to quan-
Now let us consider the linear MMSE receiver tize to a finite number of codewords that form . Conceptu-
ally, one can adopt any subspace distance as in (6)–(8)
(18) and set up an objective to minimize an average distortion as
The signal-to-(interference plus noise) ratio (SINR) after
MMSE equalization is
(21)
(19)
where stands for expectation and is the prob-
The residual interference-plus-noise can be well approximated ability of a random V belonging to the region , which is de-
by a Gaussian random variable [21]. Although this Gaussian ap- fined as
proximation is derived under various asymptotic conditions, it
does lead to accurate system performance evaluation [21] (see (22)
also some numerical verification in [32]). Therefore, we can
compute the average BER as The cost function in (21) can then be iteratively reduced via the
Lloyd algorithm [8].
It turns out that the Lloyd algorithm is feasible only when the
(20) chordal distance is chosen as the distance measure. The reason
is that the chordal distance can be reexpressed as
It is a well-known fact that the MMSE receiver strikes a
balance in suppressing the additive noise and the interference (23)
among parallel data streams. At high SNR when , the
interference is dominant and the MMSE receiver reduces to the that will render simple analytical solution inside the iterations
ZF receiver. On the other hand, at low SNR when , the of the Lloyd algorithm. This cannot be achieved with other dis-
additive noise is dominant. The MMSE receiver boils down to tance measures.
a matched filter with , as can be seen from (18). Based on the chordal distance, the codebook design steps are
This SNR-dependent characteristic of the MMSE receiver will as follows.
dictate the optimal precoder design with infinite-rate feedback. S1) To avoid the expectation operation in (21), we use the
With any given codebook , the BER-based selection crite- Monte Carlo approach, as in [26]. We generate a training
rion outperforms any other alternative in [13] and [14], in terms set with samples .ZHOU AND LI: BER CRITERION AND CODEBOOK CONSTRUCTION FOR FINITE-RATE PRECODED SPATIAL MULTIPLEXING 1657
S2) Starting with an initial codebook (obtained via random TABLE I
computer search or using the currently best codebook THE MINIMUM DISTANCES OF THE CODEBOOKS WITH N = 4, K=2
if available), we carry out the following two substeps
iteratively.
— Nearest neighbor rule [8]: assign to one of the
regions using the rule
(24)
— Centroid condition [8]: For each region , find the
optimal codebook as
(25)
TABLE II
where is defined as THE MINIMUM DISTANCES OF THE CODEBOOKS WITH N = 6, K=3
(26)
Performing the eigendecomposition of as
(27)
it is easy to show that shall be taken as the
eigenvectors of corresponding to the largest
eigenvalues.
Notice that the Lloyd algorithm converges with
monotonically decreasing. But this does not mean that
the minimum distance of the codebook is monotonically
block coding, codebooks with optimized minimum chordal dis-
improving, as observed in [26] and [28]. During each it-
tance are proposed [15]. We have the following observations.
eration, we examine the tentative codebook, and record
it if its minimum distance is larger than the currently • Our newfound codebooks have better distance properties,
best. This is done for each distance , , and . no matter what distance definition is preferred. Hence, our
S3) Go back to S1) to generate another training set and rerun new codebooks can be used in the scenarios of [13]–[15]
the Lloyd algorithm in S2). We stop until no further im- to improve the system performance relative to existing
provement on the minimum distance is observed. codebooks.
• We have multiple codebooks at hand with different dis-
Example 1) (Codebooks Obtained by the Lloyd Algo-
tances optimized, but none of them was optimized based
rithm): We collect some codebooks obtained via the Lloyd
on BER criterion. Which one should we use for our BER
algorithm in Table I for and in Table II for
based selection criterion? Or, should we adopt a totally
, respectively. In Tables I and II, we use bold-
new distance measure to construct the codebook for the
face fonts to highlight the maximized minimum distances ,
BER criterion? We are not able to answer these questions
, or , if multiple codebooks are listed for one configura-
analytically, but will get some practical guidelines based
tion. The new codebooks (collected in [29]) have much larger
on numerical study in Section VI.
minimum distances than the codebooks currently available
in [12]. This demonstrates the effectiveness of our proposed
V. CODEBOOK CONSTRUCTION WITH INFINITE-RATE
codebook construction method based on the generalize Lloyd
algorithm. FEEDBACK
Notice that with precoded spatial multiplexing, codebooks We now consider BER-optimal codebook construction in the
with optimized minimum Fubini–Study distance are advocated limiting case with . The importance for analyzing this
for MMSE-det and Capacity-based precoder selection criteria, limiting case is twofold. First, it will provide much insight into
while codebooks with optimized minimum project two-norm this problem. Indeed, the optimal precoders with the BER cri-
distance are advocated for MMSE-trace, SV, and MD-based se- terion are drastically different from the counterparts based on
lection criteria [13], [14]. With precoded orthogonal space time other criteria. Second, with the BER-optimizing precoders in1658 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
the limiting case, we provide a benchmark performance for fi- or Hadamard, matrix. As long as these two conditions are sat-
nite-rate precoded spatial multiplexing. This benchmark perfor- isfied, a specific choice of does not affect our following
mance is not available before. Notice that existing work [14] conclusions for linear receivers.
usually compares with the asymptotic system performance with Proposition 1: We consider a linear ZF receiver and a rectan-
MMSE-optimal precoders, which has performance difference gular/square QAM constellation with size . We define a con-
with BER-optimal precoders. stellation-specific threshold as shown at the bottom of the page.
With , we can equivalently assume that the transmitter Using the BER-based selection criterion, the optimal column-
has full knowledge of the channel and selects directly orthonormal precoder shall be chosen as follows according
based on . Precoder design with perfect channel knowledge to the channel .
has been investigated extensively in the literature—see, e.g. [5], 1) When , we have .
[7], [20], [23], and references therein. Our results in this sec- 2) When , we have .
tion are distinct from those in [5], [20], and [23] in that i) we 3) When conditions in 1) or 2) do not hold, the optimal
use exact BER as the performance criterion for linear receivers cannot be found analytically. Suppose that within the set
and ii) we focus on column-orthonormal precoder . Notice there are entries larger than . Then one
that various MSE-related performance criteria have been used in suboptimal precoder that is better than can be con-
[20] and [23] for linear receivers, while optimal precoders based structed as
on a minimum distance criterion are derived in [5] for systems
with 2-QAM and 4-QAM constellations and maximum likeli- (29)
hood receivers. Also with BER as criterion, [20] has used an
approximate BER expression for linear MMSE receivers, while
the results in [7] are limited to ZF receivers for 2-QAM and The quantities , , are defined in (4), (5),
4-QAM. Also, both [20] and [7] did not characterize the op- and (16), respectively.
timal design at very low SNR. Proposition 2: We consider a linear MMSE receiver and a
For all the selection criteria presented in [13] and [14], it is rectangular/square QAM constellation with size . We define
concluded that the optimal subject to the column-orthonor- two constellation-specific thresholds as shown at the bottom of
mality constraint shall be chosen as . When , we the page. Using the BER-based selection criterion, the optimal
have and column-orthonormal precoder shall be chosen as follows ac-
cording to the channel .
(28) 1) When , we have
Essentially, the channel is diagonalized to provide parallel .
subchannels. Each information symbol in goes through one 2) When or , we have
subchannel with distinct SNR , . .
Is the optimal choice for the BER selection cri- 3) When conditions in 1) or 2) do not hold, the optimal
terion? We next show that it is not. We list our main results in cannot be found analytically. Suppose that within the set
Section V-A, provide the proofs in Section V-B and -C, and , there are entries larger than and
then give some intuitive explanations in Section V-D. Readers entries smaller than . Then one suboptimal precoder
not interested in the proofs can jump from Section V-A to V-D. that is better than can be constructed as
A. Main Results
(30)
With signifying the dimensionality, we denote as a
matrix that possesses two properties: i) it is unitary, i.e.,
; and ii) each entry has the same modulus . The quantities , , are defined in (4), (5),
For example, could be a normalized fast Fourier transform, and (16), respectively.ZHOU AND LI: BER CRITERION AND CODEBOOK CONSTRUCTION FOR FINITE-RATE PRECODED SPATIAL MULTIPLEXING 1659
Since in Proposition 2 when and ,
we have the following corollary.
Corollary 1: For 2-QAM and 4-QAM, the optimal precoder
is always for linear MMSE receivers, with the
BER-based selection criterion.
B. Proof of Proposition 1
We perform the eigen decomposition of as
(31)
where the unitary matrix contains eigenvectors. Define the
matrix such that and is
diagonal. Therefore, must contain any columns of .
With any given , choosing always improves the
system performance relative to any other columns of , as Fig. 2. Second-order derivative of (1=x) for M -QAM where M =
16; 32; 64.
it selects the largest eigenvalues from . We then have
The conditions in (37) suggests that the entries of are less
(32) spread out than those of . A real valued function is said to
be Schur-convex over the region [17, p. 54] if
Now, let us define a matrix as
where (38)
(33) On the other hand, is Schur-concave if the inequality in (38)
is reversed.
Now we characterize the Schur-concave or Schur-convex re-
and denote its diagonal entries as . Collect the diagonal gions for in (35). We first evaluate the second-order
entries and the eigenvalues of in the vectors derivative of . Using the compact expression for
as in (13), the second-order derivative of is found as
(34)
respectively. With , the average BER in (17) is then
(39)
(35) Numerical testing reveals that there exists a constellation-spe-
cific threshold , such that
Our problem at hand is to
(40)
minimize subject to It is easy to verify that for 2-QAM and
(36) for 4-QAM (notice that the second-order deriva-
To proceed, we need the concept of vector majorization [17]. tive is provided in [7] for 2-QAM). For other QAMs,
For any real vector , let we resort to numerical testing to find the threshold as
denote the components of in a decreasing order. Then for two when
vectors and , vector is majorized by vector (denotes as . For example, the second-order
) [17, p. 7] if derivatives of are shown in Fig. 2 for 16-QAM,
32-QAM, and 64-QAM.
Equation (40) reveals that 1 is convex when
and is concave when . Over the region
, the sum of convex functions
is Schur-convex, according to [17, Theorem 3.C.1]. Similarly,
(37) over the region , is
Schur-concave.1660 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
We will rely on the fact the diagonal entries of any Hermitian
symmetric matrix is majorized by the eigenvalues [17, p. 218].
Specifically, we have
(41)
where is the vector with all entries to be one. Define the con-
stant , whose values are listed in Proposition 1. The
results in Proposition 1 correspond to the following scenarios.
• When , or equivalently , we
have . Subsequently as . As
is Schur-concave over this region, we conclude
, which dictates that .
• When , or equivalently , we
have . As is Schur-convex over this
region, . This can be achieved
only when all entries of have the same amplitude Fig. 3. Second-order derivative of (1=x01) for M -QAM, where
1 . M = 2; 4; 16; 64.
• For all other cases, is neither Schur-concave nor
Schur-convex. The optimal precoder thus cannot be ob- In general, we will have two constellation-specific constants
tained analytically. We can, however, pursue separate op- and such that
timization for different subsets of subchannels belonging
to either the convex region or the concave region. That when or
(46)
leads to the suboptimal precoder in (29), which obvious when
outperforms .
The 2-QAM and 4-QAM constellations turn out to be spe-
C. Proof of Proposition 2 cial since we always have .
To put them into the same framework as other con-
Similar to the proof of Proposition 1, we have ,
stellations, we arbitrarily assign .
with defined in (31). We now define the matrix
For , we find the constants
as and
. The deriva-
(42) tives are plotted in Fig. 3 for .
The diagonal entries and eigenvalues of are collected in Define new constants as and
(see the values in Proposition 2).
Using the same arguments for Proposition 1, we obtain in
Proposition 2 results suitable for different scenarios depending
(43) on whether can be identified to be Schur-convex
Based on (41), the diagonal entries fall in the interval or Schur-concave. If is neither Schur-convex nor
, which belongs to (0, 1). The BER Schur-concave, we can have a suboptimal precoder through
performance is now separate optimizations over different subsets of subchannels.
D. Intuitive Explanation
(44)
When , we have parallel subchannels with dis-
tinct SNR’s as in (28). But when , we have
Now we proceed to determines Schur-convex or Schur-con-
cave regions for . Similar to (39), we find the second- (47)
order derivatives of as
(48)
Therefore, the SNRs are balanced for all information symbols
when . Then when should the SNRs be balanced
and when should not?
We first discuss the ZF receiver. Intuitively, if all subchannel
SNRs for data streams are sufficiently high, then the worst
subchannel dominates the overall performance. Hence, the op-
(45) timal precoder shall balance the SNRs for all data streams toZHOU AND LI: BER CRITERION AND CODEBOOK CONSTRUCTION FOR FINITE-RATE PRECODED SPATIAL MULTIPLEXING 1661
achieve the best performance in this scenario. This confirms
Proposition 1 for the high SNR case, where . This
high SNR behavior for the ZF receiver was observed in [7] only
for 2-QAM and 4-QAM. Notice that the balanced SNR in (47)
for ZF receiver is proportional to the harmonic mean of the
largest eigenvalues of . On the other hand, when all sub-
channel SNRs are sufficiently low, they shall not be balanced,
as in (28).
We next discuss the MMSE receiver. It is well known that at
high SNR, the MMSE receiver reduces to a ZF receiver. Hence,
the behavior of the MMSE receiver shall be similar to the ZF re-
ceiver at high SNR. This is reflected in Proposition 2, where the
subchannel SNRs shall be balanced when they exceed a certain
threshold.
In contrast to the ZF receiver, however, the subchannel SNRs
shall also be balanced when they are extremely low. The intu-
ition is that the MMSE receiver reduces to a matched filter at
Fig. 4. Performance comparison with infinite-rate feedback: (N ; N ; K) =
extreme low SNR. Following the same steps in Sections V-B (6; 3; 3), 4-QAM.
and -C, it is easy to verify that the corresponding to the
matched filter at low SNR (neglecting the interference among
symbols) is always convex, hence balancing the SNRs is the op-
timal thing to do at low SNR. In short, at low SNR, the MMSE
receiver behaves like a matched filter, while at high SNR, the
MMSE receiver behaves like a ZF receiver. At both ends, the
subchannel SNRs should be balanced, but for different reasons.
During the transition from low to high SNR, there exists an
SNR range where the subchannel SNRs shall not be balanced
for MMSE receivers, as established in Proposition 2. The
2-QAM and 4-QAM are exceptional special cases, where the
subchannels should be always balanced throughout the entire
SNR range. The intuition for these special cases, however, is
unclear to the authors.
Since the optimal precoder is not clear for all cases, and the
suboptimal construction in (29) and (30) requires the knowl-
edge of the constellation-specific thresholds, we propose the fol-
lowing practical solution.
Proposition 3: With linear ZF or MMSE receivers, for each Fig. 5. Performance comparison with infinite-rate feedback: (N ; N ; K) =
channel realization , the transmitter selects the precoder to be (6; 3; 3), 16-QAM.
either or , depending on which one yields better
BER performance. This way, all cases can be treated in a co- Fig. 4 that i) the ZF receiver with the precoder out-
herent fashion, and there is no need to know the constellation performs that with at low SNR, but vice versa at
specific thresholds. high SNR; and ii) the suboptimal precoder in (29) outperforms
the precoder uniformly. These observations confirm
VI. NUMERICAL RESULTS our theoretical analysis in Proposition 1 for ZF receivers.
In Fig. 4 with 4-QAM, we observe that the MMSE receiver
We now present numerical results. We focus on two different
with always outperforms that with .
configurations: and
This is not the case with 16-QAM, where the MMSE receiver
. In all plots, we define the average SNR at each receive
with achieves better performance than that with
antenna as
when the SNR is either at the high end or at the
low end (not distinguishable from the curves at low SNR, so
(49) we numerically checked), but leads to inferior performance in a
moderate SNR range, as shown in Fig. 5. Also, the suboptimal
We use 4-QAM constellation unless specified otherwise. Each choice of (30) always performs better than . Hence,
BER curve is averaged over 10 channel realizations under a Figs. 4 and 5 confirm our theoretical analysis in Proposition 2
block fading channel model. for MMSE receivers.
Test Case 1 (Infinite-Rate Feedback): We first test our theo- Interestingly, by choosing the better one from and
retical analysis in Section V where . We present the case for every channel realization, the proposed solu-
with . With 4-QAM, we observe from tion in Proposition 3 outperforms all other suboptimal choices1662 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006 Fig. 6. Comparing codebooks optimized based on different distances, Fig. 7. BER comparison between the existing and the new codebooks, (N ; N ; K)= (4; 2; 2). (N ; N ; K)= (4; 2; 2), MMSE receiver. aforementioned. Hence, we will adopt this scheme as the bench- mark performance. The performance improvement over the con- ventional choice of in [14] is significant, and the dif- ference gets larger as the SNR increases. We next turn our attention to the finite-rate feedback case. Test Case 2 (BER Comparison for Codebooks Optimized Under Different Distance Criteria): Tables I and II provide various codebooks, which are optimized based on different definitions of subspace distance. Then, which codebook should one use for a better BER performance? With and , Fig. 6 depicts the BER performance for the codebooks with maximized min- imum , , . We observe that the performance for dif- ferent codebooks are not distinguishable. The same observation applies to all other configurations. Hence, sticking to the code- book with any distance optimized will be equally good, in terms of the system BER performance. Although individual codebook has different minimum distance optimized, the impact on the Fig. 8. ZF receiver: SV and BER criteria, (N ; N ; K) = (6; 3; 3). overall system performance is negligible. This has to do with the following two facts: i) when one minimum subspace dis- yields performance quite close to the BER criteria. This is tance is maximized, the other minimum subspace distances of reasonable, as the SV criterion tries to improve the worst SNR the same codebook are also large (intuitively, the subspaces of for data streams, implicitly enforcing some averaging over the codebook are well distributed), as evidenced by Tables I and all subchannel SNRs. On the other hand, Fig. 9 compares II; and ii) more importantly, BER is a quantity that is averaged the MMSE-trace, MMSE-det, and BER criteria for MMSE over different channel realizations, unlike the maximized min- receivers. We observe that MMSE-det performs worse than imum distance. the MMSE-trace criterion, and both of them are inferior to the Test Case 3 (BER Comparison With Existing Code- BER criterion. books): We now compare our codebooks with the existing Therefore, the BER-based selection criterion has the perfor- ones in [12]. Fig. 7 shows that the performance improvement is mance advantage over competing alternatives. Notice that com- only moderate. This is also consistent with Test Case 2, where puting BER is also straightforward through either closed-form BER is not very sensitive to the distance optimization due the expressions [4] or a recursive algorithm [24]. On the other hand, averaging effect. While distance optimization is an important SV and MMSE-trace do not require the knowledge on the signal research topic in subspace packing problem [6], practical constellation, at the cost of performance degradation. systems only need to deploy reasonably good codebooks. Test Case 5 (Performance Improvement With the Number of Test Case 4 (Comparison Among Different Selection Cri- Feedback Bits): We now test the performance improvement as teria): We now compare the proposed BER criteria with a function of the number of feedback bits. Fig. 10 shows the those in [14]. Fig. 8 compares the SV and the BER selection case for and with MMSE receivers. criteria for ZF receivers. We observe that the SV criterion Fig. 11 shows the case for and
ZHOU AND LI: BER CRITERION AND CODEBOOK CONSTRUCTION FOR FINITE-RATE PRECODED SPATIAL MULTIPLEXING 1663
Fig. 9. MMSE receiver: MMSE-det, MMSE-trace, and BER criteria, Fig. 11. Performance improvement with number of feedback bits,
(N ; N ; K ) = (6; 3; 3). (N ; N ; K ) = (6; 3; 3), ZF receiver.
Fig. 10. Performance improvement with number of feedback bits, Fig. 12. Comparison of ZF and MMSE receivers, (N ; N ; K) = (4; 2; 2).
(N ; N ; K)= (4; 2; 2), MMSE receiver.
receiver by a small margin, as the former incorporates additional
with ZF receivers. The case corresponds to an SNR knowledge.
MIMO system, where linear receivers can be applied. The Our observations in Test Cases 1–5 provide useful guidelines
case corresponds to antenna subset selection, with antennas on the choices of receiver type, precoder codebook, and pre-
partitioned to two sets, each with elements. The asymptotic coder selection criterion for finite-rate precoded spatial multi-
case with is included as a performance benchmark. We plexing systems.
observe the following.
i) Feedback link improves the system performance drasti- VII. CONCLUSION
cally. In this paper, we considered precoded spatial multiplexing
ii) The performance gain demonstrates diminishing returns transmissions assisted by finite-rate feedback. We proposed a
as the number of feedback bits increases. new precoder selection criterion based on the exact BER for
iii) The large portion of the feedback gain is achieved with linear ZF and MMSE receivers. The proposed BER criterion
only moderate number of bits (e.g., in both cases). outperforms any other competing alternative for an uncoded
Hence, the number of feedback bits in practical systems needs system with linear receivers. We then developed a precoder
not be large. Also, the small gap between and codebook construction method based on the generalized Lloyd
suggests that the codebook for is very good; further algorithm. This construction was not directly based on the
optimization on the codebook may not lead to substantial gain. BER criterion. Hence, it is suboptimal in the BER sense. How-
Finally, we collect the BER performance with ZF and MMSE ever, various minimum distances of our new found codebooks
receivers in Fig. 12. The MMSE receiver outperforms the ZF have been improved considerably relative to those currently1664 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
available. In the asymptotic case of infinite-rate feedback, we [8] A. Gersho and R. M. Gray, Vector Quantization and Signal Compres-
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Process., vol. 51, no. 9, pp. 2410–2423, Sep. 2003. 2004.ZHOU AND LI: BER CRITERION AND CODEBOOK CONSTRUCTION FOR FINITE-RATE PRECODED SPATIAL MULTIPLEXING 1665
Shengli Zhou (M’03) received the B.S. and M.Sc. Baosheng Li (S’05) received the B.S. and the M.Sc.
degrees in electrical engineering and information degrees in electronic and communication engineering
science from the University of Science and Tech- from Harbin Institute of Technology, Harbin, China,
nology of China (USTC), Hefei, in 1995 and 1998, in 2002 and 2004, respectively. He is currently pur-
respectively. He received the Ph.D. degree in elec- suing the Ph.D. degree in the Department of Elec-
trical engineering from the University of Minnesota, trical and Computer Engineering, University of Con-
Minneapolis, in 2002. necticut, Storrs.
He has been an Assistant Professor with the De- His research interests lie in the areas of com-
partment of Electrical and Computer Engineering, munications and signal processing, with emphasis
University of Connecticut, Storrs, since 2003. His on adaptive modulation, multiuser and multicarrier
research interests lie in the areas of communications communications, and space-time coding.
and signal processing, including channel estimation and equalization, multiuser
and multicarrier communications, space-time coding, adaptive modulation, and
cross-layer designs.
Prof. Zhou is an Associate Editor for IEEE TRANSACTIONS ON WIRELESS
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