Hindawi Publishing Corporation
EURASIP Journal on Advances in Signal Processing
Volume 2008, Article ID 594928, 15 pages
doi:10.1155/2008/594928
Research Article
A Diversity Guarantee and SNR Performance for
Unitary Limited Feedback MIMO Systems
Bishwarup Mondal and Robert W. Heath Jr.
Department of Electrical and Computer Eng ineering, The University of Texas at Austin, University Station C0803,
Austin, TX 78712, USA
Correspondence should be addressed to Robert W. Heath Jr., [email protected]
Received 16 June 2007; Accepted 26 October 2007
Recommended by David Gesbert
A multiple-input multiple-output (MIMO) wireless channel formed by antenna arrays at the transmitter and at the receiver offers
high capacity and significant diversity. Linear precoding may be used, along with spatial multiplexing (SM) or space-time block
coding (STBC), to realize these gains with low-complexity receivers. In the absence of perfect channel knowledge at the transmitter,
the precoding matrices may be quantized at the receiver and informed to the transmitter using a feedback channel, constituting
a limited feedback system. This can possibly lead to a performance degradation, both in terms of diversity and array gain, due
to the mismatch between the quantized precoder and the downlink channel. In this paper, it is proven that if the feedback per
channel realization is greater than a threshold, then there is no loss of diversity due to quantization. The threshold is completely
determined by the number of transmit antennas and the number of transmitted symbol streams. This result applies to both SM
and STBC with unitary precoding and confirms some conjectures made about antenna subset selection with linear receivers. A
closed form characterization of the loss in SNR (transmit array gain) due to precoder quantization is presented that applies to a
precoded orthogonal STBC system and generalizes earlier results for single-stream beamforming.
Copyright © 2008 B. Mondal and R. W. Heath Jr. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
1. INTRODUCTION
Linear precoding uses channel state information (CSI) at the
transmitter to provide high data rates and improved diversity
with low complexity receivers in multiple-input multiple-

capacity metric were considered in [29] and shown to achieve
full diversity. In the case of a spatial multiplexing system
2 EURASIP Journal on Advances in Signal Processing
Tr an sm it te r
Downlink
channel Receiver
.
.
.
.
.
.
Precoder
set F
Precoder
update
Precoder
selection
Precoder
set F
Index of precoder set F
Low-rate feedback channel
Figure 1: A quantized precoded MIMO system.
employing transmit antenna selection, conjectures on diver-
sity order based on experimental evidence were presented in
[30]. These conjectures were subsequently proved and gener-
alized in [31]. In the special case of single-stream beamform-
ing, the diversity order with limited feedback precoding was
studied in [32]andanecessaryandsufficient condition on
the feedback rate for preserving full diversity is presented. A

Detailed discussion of contributions
A pictorial description of a limited feedback system as con-
sidered in this paper is provided in Figure 1.Afixed,pre-
determined set of unitary precoding matrices is known to
the transmitter and to the receiver. The receiver, for ev-
ery instance of estimated downlink channel information,
selects an element of the set and sends the index of the
selected precoder to the transmitter using B bits of feed-
back. This precoder element is subsequently used by the
transmitter for precoding. For analytic tractability we con-
sider an uncorrelated Rayleigh flat-fading MIMO channel
and we let M
t
, M
r
,andM
s
denote the number of trans-
mit antennas, receive antennas, and symbol streams trans-
mitted, respectively. The uncorrelated Rayleigh channel is
commonly used in rate distortion analysis for limited feed-
back systems [9, 12, 16, 35], including correlation along the
lines of recent work is an interesting topic for future re-
search [32]. Because discussions of diversity and array gain
depend on transmitter and receiver structure, in this pa-
per we consider explicitly two classes of systems—quantized
precoded spatial-multiplexing (QPSM) and quantized pre-
coded full-rank space-time block coding (QPSTBC) sys-
tems. A subclass of QPSTBC systems is due to orthogonal
STBCs and is termed as QPOSTBC systems. The diversity

is an upper bound to log
2
(M
s
(M
t
−M
s
) + 1). This also
means that for sufficiently large feedback, the design
of the set of quantized precoders is irrelevant from the
point of view of diversity.
(ii) SNR analysis: for a QPOSTBC system, the loss in SNR
due to quantization reduces as
∼2
−B/M
s
(M
t
−M
s
)
with
increasing feedback bits B. Thus, most of the chan-
nel gain is obtained at low values of feedback rate
(bits per channel realization) and increasing feedback
further leads to insignificant gains. Our characteriza-
tion also shows that increasing M
r
provides robustness

.
.
.
.
.
.
Demod.
decode
bits
Precoder set
Precoder
update
Precoder index
Quantizer Precoder set
Low-rate feedback channel
Figure 2: Discrete-time quantized precoded MIMO spatial multiplexing system.
This paper is organized as follows. The system model is
described and the assumptions are mentioned in Section 2.
The diversity of such systems and the effective channel gain
are analyzed in Sections 3 and 4, respectively, before the re-
sults are summarized in Section 5.
Notation. Matrices are in bold capitals, vectors are in bold
lower case. We use H to denote conjugate transpose,
·
F
to denote the Frobenius norm, ·
2
to denote matrix 2-
norm, [A]
ij

) denote a complex normal distri-
bution with zero mean and N
0
variance with i.i.d. real and
imaginary parts.
2. SYSTEM OVERVIEW
In this section, a precoded spatial multiplexing system and
a precoded space-time block coding system, both with pre-
coder quantization and feedback, are described. Then a brief
motivation is provided for unitary precoding assuming per-
fect CSI at the transmitter. Subsequently limited feedback
precoding is introduced and formulated as a quantization
problem. Finally the main assumptions of the paper are sum-
marized.
2.1. Quantized precoded spatial multiplexing
system (QPSM)
As shown in Figure 2, in a spatial multiplexing system a sin-
gle data stream is modulated before being demultiplexed
into m
s
symbol streams. This produces a symbol vector s of
length m
s
for a symbol period. We assume that E{ss
H
}=I.
The symbol vector s is spread over M
t
antennas by mul-
tiplying it with an M

s
is the energy for one symbol period, H is a ma-
trix with complex entries that represents the channel transfer
function, and n represents an additive white Gaussian noise
(AWGN) vector. For a QPSM system we assume M
t
>M
s
,
M
r
≥ M
s
. In this paper we only concentrate on ZF and
MMSE receivers that enable low-complexity implementa-
tion.
We also consider a fixed predetermined set of precod-
ing matrices F
={F
1
, F
2
, , F
N
} that is known to both the
transmitter and the receiver. Depending on the channel real-
ization H, the receiver selects an element of F and informs
the transmitter of the selection through a feedback link. Note
that
log

be constant for the T symbol periods and changes randomly
in the next symbol period. The discrete-time baseband signal
model for T symbol periods may be written as
Y
=

E
s
M
s
HFC + N,
(2)
where Y is the received signal at the M
r
receive antennas over
T symbol periods, E
s
is the energy over one symbol period,
and N is the AWGN at the receiver for T symbol periods.
We a ss um e M
t
>M
s
, but there is no restriction on M
r
.As
before, we consider a set of precoding matrices F known to
both the transmitter and the receiver. The receiver chooses an
4 EURASIP Journal on Advances in Signal Processing
bits

Precoder
update
Precoder index
Quantizer Precoder set
Low-rate feedback channel
Figure 3: Discrete-time MIMO system with quantized precoded STBC.
element of F depending on H and sends this information to
the transmitter using a feedback link. As mentioned before,
we restrict ourselves to full-rank STBCs for which,
λ
min

E
ij
E
H
ij

> 0 ∀i
/
= j,(3)
where E
ij
= C
i
− C
j
is the codeword difference matrix be-
tween the ith and the jth block code. Full-rank STBCs en-
compass a wide variety of codes differing in rate and com-

forming where a single symbol is spread over M
t
antennas by
the beamforming vector. The ML receiver, in this case, be-
comes a maximum-ratio combiner (MRC).
2.3. Limited feedback unitary precoding
In the following sections, it will be of interest to define a
perfect-CSI precoding matrix (or a precoding matrix with
infinite feedback bits) as
F
∞
= U,
(5)
where H
H
H = UΣU
H
denote the SVD of H
H
H, Σ =
diag(λ
1
, λ
2
, , λ
M
t
), λ
1
≥ λ

H
A = I where A can be either tall or square.)
At the receiver, corresponding to a channel realization
H, a precoding matrix is chosen from the set F . This selec-
tion may be described by a map Q such that Q(F
∞
) ∈ F ,
where F
∞
is obtained from H using (5). The map Q may also
be visualized as a quantization process applied to the set of
all perfect-CSI precoding matrices. Then borrowing vector
quantization terminology, the map Q is a quantization func-
tion, F
∞
is the source random matrix, F is a codebook, ele-
ments of F are codewords (or quantization levels), and the
cardinality of F is the number of quantization levels or the
quantization rate. This justifies the “Q” in QPSM and QP-
STBC systems. The quantization function Q is also referred
to as the precoder selection criterion in the literature and we
will use these terms interchangeably in this paper. It may be
noted that assuming a feedback of
log
2
N per channel re-
alization H, the precoding matrix F in (1)and(2)becomes
an element of F chosen by a precoder selection criterion de-
scribed by F
= Q(F

the elements of H are distributed as i.i.d. CN (0, 1). The i.i.d.
assumption is typically used for the analysis of limited feed-
back systems [9, 12, 27, 31] mainly due to the tractable nature
of the eigenvalues and eigenvectors in this case. The elements
of n, N represents AWGN, are distributed as i.i.d. CN (0, N
0
).
The feedback link is assumed to be error-free and having
zero-delay, and we assume perfect channel knowledge at the
receiver.
3. SUFFICIENT CONDITION FOR NO DIVERSITY LOSS
A concern for QPSM and QPSTBC systems is whether the
diversity order is reduced due to quantization. The objective
of this section is to provide a sufficient condition that will
guarantee no loss in diversity due to precoder quantization
for such systems.
As evidenced by simulation results it turns out (this will
be proved in the following) that the diversity order of QPSM
and QPSTBC systems does not change if, corresponding to
a given channel realization H, the precoding matrix F is
substituted by FQ,whereQ is an arbitrary unitary matrix.
B. Mondal and R. W. Heath Jr. 5
This motivates the representation of the precoding matrix F
as a point on the complex Grassmann manifold which is in-
troduced in the next subsection. In the following we outline
a strategy for the proof and introduce the projection 2-norm
distance and the chordal distance as analysis tools. In the
course of the analysis, a special class of codebooks called cov-
ering codebooks is defined that satisfies a certain condition on
its covering radius (measured in terms of projection 2-norm

, we can associate an element ω ∈ G
M
t
,M
s
such that ω is the column space of F. We can explicitly write
this relation as ω(F)
∈ G
M
t
,M
s
. Also since a rotation of the
basis does not change its span, ω(FQ) is the same element in
G
M
t
,M
s
for all M
s
-by-M
s
unitary matrices Q. This models the
fact that the precoding matrices F and FQ provide the same
diversity irrespective of any Q.
3.2. Proof strategy
This subsection provides an intuitive sketch of the proof
ideas and not a rigorous treatment. In order to implement
a limited feedback system, a precoder selection criterion Q

where d
P
(·, ·) is the projection 2-norm distance and is de-
fined as [44]
d
P

F
1
, F
2

=


F
1
F
H
1
−F
2
F
H
2


2
,
(7)

1
)
and ω(F
2
)onG
M
t
,M
s
. It turns out that the d
P
(·, ·) is a distance
measure in G
M
t
,M
s
. The proofs leading up to the diversity re-
sults follow in two steps: (i) first, we assume that Q
P
is used as
the precoder selection criterion and prove that the diversity
result is true for such a system; (ii) second, if Q
∗
as defined
in (14), (16), (17) is used instead of Q
P
, the diversity perfor-
mance of the system is identical or better, thus the result is
true for systems using Q

,
(8)
where d
C
(·, ·) is the chordal distance [44]
d
C

F
1
, F
2

=


F
1
F
H
1
−F
2
F
H
2


F
,

P
(·, ·)andd
C
(·, ·) simplifies the proofs for diversity and
SNR respectively but we were unable to discover any funda-
mental reason behind this. It is mentioned in passing that
the distance measures d
P
(·, ·)andd
C
(·, ·) coincide in G
M
t
,1
,
d
P
(·, ·) ≤ d
C
(·, ·)andd
P
(·, ·) ≈ d
C
(·, ·) when either is close
to zero.)
3.3. Covering codebook
The notion of a covering codebook is another mathemati-
cal aid. Covering codebooks define a subset of all possible
codebooks and we show later that a covering codebook along
with a precoder selection criterion Q

F, F
k

< 1,
(10)
where F
k
∈ F and F ∈ G
M
t
,M
s
.
6 EURASIP Journal on Advances in Signal Processing
(ii) The complements of the elements of F provide a cov-
ering for G
M
t
,M
t
−M
s
. This may be written as c(F
1
) ∪
c(F
2
) ∪ ··· ∪ c(F
N
) = G

P
(F, F
k
) takes values in [0, 1], it is intuitive that
a codebook, chosen at random, will be a covering codebook
with probability 1. This is proved in the work by Clark and
Shekhtman [37]. They have studied the problem of covering-
by-complementsforvectorspacesoveralgebraicallyclosed
fields. Since
C is algebraically closed, it follows from [37]
that the least cardinality of F to be a covering codebook is
M
s
(M
t
− M
s
) + 1. It also follows from [37] that almost all
(in probability sense) codebooks of cardinality larger than
M
s
(M
t
−M
s
) + 1 are covering codebooks.
3.4. Diversity of QPSM with linear receivers
The diversity of a QPSM or a QPSTBC system is the slope
of the symbol-error-rate curve for asymptotically large SNRs
defined as a limit expressed by

F
H
H
H
(12)
is applied to the received signal vector y in (1) and the re-
sulting M
s
data streams (corresponding to Gy) are indepen-
dently detected. The postprocessing SNR for the ith data
stream after receiving ZF filtering is given by [30]
SNR
(ZF)
i
(F) =
E
s
M
s
N
0

F
H
H
H
HF

−1
ii

(14)
Then if F is a covering codebook, the precoder Q
∗
(F
∞
) pro-
vides the same diversity as provided by F
∞
.
Proof. The proof of Corollary 1 proceeds in two stages as de-
scribed in Section 3.2. We prove that a precoder chosen from
F according to Q
P
as in (6) provides the same diversity as
F
∞
. Then we show that Q
∗
given by (14)providesabet-
ter diversity performance than Q
P
; for a detailed proof see
Appendix B.
Corollary 1 states that a covering codebook preserves the
diversity order of a precoded spatial multiplexing system
with a ZF receiver. (It is worth mentioning that the diver-
sity order of a precoded spatial multiplexing system (using
F
∞
as the precoder) with a ZF receiver is not available. As a

Recall that in a QPSTBC system (2) the difference codewords
E
ij
= C
i
−C
j
, i
/
= j are full rank. It is known that these systems
provide a diversity order of M
t
M
r
. QPOSTBC systems are a
subset of QPSTBC systems where E
ij
= αI, i
/
= j,andα ∈ C.
The Chernoff bound for pairwise error probability (PEP) for
a QPSTBC system may be expressed as [45]
P

C
i
−→ C
j
| H


∗

F
∞

=
arg max
F
k
∈F
min
i,j


HF
k
E
ij


2
F
.
(16)
Then if F is a covering codebook, the precoder Q
∗
(F
∞
) pro-
vides the same diversity as provided by F

k


2
F
,
(17)
and from Corollary 2 it follows that a covering codebook
provides full diversity. The special case of QPOSTBC has also
been studied in [33]andasufficient condition for preserv-
ing full diversity was derived. It follows from Corollary 2 and
Lemma 1that a full-rank STBC system with transmit antenna
subset selection is guaranteed to achieve full diversity.
3.6. Observations
It is proven that precoder selection criteria motivated by
postprocessing SNR and the Chernoff bound on PEP pre-
serve diversity order. This is a pleasing result for system de-
signers. Diversity can be guaranteed by a codebook chosen
at random of size determined only by M
t
and M
s
. The struc-
ture in the codebook or a particular element of a codebook
is irrelevant and thus codebook design algorithms need not
consider diversity as a criterion. It is also interesting to note
that diversity can be preserved with less feedback than that
for antenna subset selection.
4. CHARACTERIZATION OF SNR LOSS
The objective of this section is to quantify the loss in ex-

k
∈F


HF
k


2
F
.
(18)
Notice that the expected SNR of a system using a precoder F
does not change if F is substituted by FQ,whereQ is an ar-
bitrary square unitary matrix (of dimension M
s
× M
s
). This
fact, similar to the case of diversity, justifies the representa-
tion of a precoding matrix on a complex Grassmann mani-
fold. Recall from Section 3.2 that the first step in the proof
is to consider a precoder selection criterion based on d
C
(·, ·)
given by (8). Then we have the following result.
Theorem 2. Assumeaprecoderselectioncriteriongivenby
Q
C


−
E



HQ

F
∞



2
F

=
(Λ−Λ)E

d
2
C

F
∞
, Q

F
∞

,

M
t
≥ 0, are the ordered eigenvalues of
H
H
H.
Proof. See Appendix F.
An intuitive understanding of the final SNR result fol-
lows directly from Theorem 2. It follows from a result
in [46–48] that (8) defines a quantization problem with
a distortion function as d
2
C
(·, ·) and the expected distor-
tion, E
{d
2
C
(F
∞
, Q(F
∞
))}∼N
−1/M
s
(M
t
−M
s
)

chordal distance between F
∞
and its quantized version as-
sumingaprecoderselectioncriteriongivenby(8). Note that
E
{d
2
C
(F
∞
, Q(F
∞
))} is the expected distortion for the quanti-
zation function Q defined by (8). This class of quantization
problems with chordal distortion has been studied in [46–
48]. In the particular case of an uncorrelated Rayleigh fading
channel the probability distribution of F
∞
is known [49]. A
lower bound on the expected distortion E
{d
2
(F
∞
, Q(F
∞
))}
is derived in [36] for large N which takes the form
E



c

M
t
, M
s

N

−1/M
s
(M
t
−M
s
)
,
(21)
where c(M
t
, M
s
) is a constant and may be expressed as
c(M
t
, M
s
) = (1/(M
t

)!)

M
s
i=1
((M
t
−
i)!/(M
t
−M
s
−i)!) otherwise. Thus for large N and with pre-
coder selection criterion given by (8)wecanwrite
E



HQ

F
∞



2
F

≤
E

2
C

F
∞
, Q

F
∞

=
0 =⇒ lim
N→∞
E



HQ

F
∞



2
F

=
E


=
arg max
F
k
∈F


HF
k


2
F
(24)
maximizes E
{HQ(F
∞
)
2
F
}.Itiseasytoseethatforanygiven
codebook F ,wehave
E



HQ

F
∞


;
(25)
and using the same sequence of codebooks as before, we have
from (23)and(25)
lim
N→∞
E



HQ

F
∞




=
E



HF
∞



.

−1/M
s
(M
t
−M
s
)
,
(27)
where the approximation in (27) means that the left-hand
side and the right-hand side can be contained in a ball of
radius
 > 0.
4.3. Special case of MRT-MRC
In the special case of single-stream beamforming with M
s
=
1, F
∞
reduces to maximum-ratio transmission (MRT). Con-
sidering a maximum-ratio combining (MRC) receiver, the
loss in expected SNR of the received symbol due to quan-
tization of the beamformer F
∞
may be expressed as ΔSNR =
E{HF
∞

2
F

ulations. A 4
× 4 QPOSTBC MIMO system is considered
and precoding with M
s
= 1,2issimulated.Inbothcases,
the codebooks are designed using the FFT-based search al-
gorithm proposed in [51]. The precoder selection criterion
is given by (24)andE
{HQ(F
∞
)
2
F
} is plotted in dB as a
function of log
2
N in Figure 4. The experimental results show
that the approximation in (27) is reasonably accurate even at
small values of N and provides a practical characterization of
performance.
4.5. Observations
To better understand the result in (27), we provide an anal-
ogous result from vector quantization theory [52, 53]. Con-
sider a D-dimensional (complex dimension) random vector
23 4567
log
2
N
10.5
11

∞
)
2
F
}), is plotted
against the number of bits used for quantization. The simulation re-
sults are compared against the closed-form approximation in (27).
The system parameters are M
t
= 4, M
r
= 4, and the perfect CSI case
meaning E
{HF
∞

2
F
} is also plotted for comparison.
and let every instance of the vector be quantized indepen-
dently with B bits. Then the average error due to quantiza-
tion measured in terms of square-Euclidean distance follows
∼2
−B/D
. The loss in expected SNR from (27)maybewritten
as
∼2
−B/M
s
(M

s
), and then equiva-
lently quantized. (The space of dimension M
s
(M
t
−M
s
) is the
complex Grassmann manifold and this equivalent formula-
tion of quantization is available in [47, 54].) The reduction
in dimension (as well as the bounded nature of the space)
implies that we are quantizing a much smaller region (com-
pared to
C
M
t
M
s
) which is the precise reason why the loss in
performance due to quantization is surprisingly small. This
also justifies the quantized precoding matrices being unitary.
The loss in expected SNR reduces exponentially with the
number of feedback bits B. Thus, most of the gains in chan-
nel power is obtained at low values of feedback rates and
increasing feedback further leads to insignificant gains (also
evident from Figure 4). It may be noted from (20) that the
loss in expected SNR depends on the spread of the expected
eigenvalues. The number of receive antennas M
r

of an arbitrary matrix F also by F. The connotations are ob-
vious from context.
Claim 1. Let S
∈ G
M
t
,M
s
be any point and F
k
be any element
of F (both S, F
k
are unitary). Then
d
P

S, F
k

< 1 ⇐⇒ S
⊥
∈ c

F
k

,
(A.1)
where S

,M
t
−M
s
)
cos θ
i
< 1(A.3)
⇐⇒ F
k
∩S
⊥
={0},
(A.4)
where (A.2) follows from the representation d
P
(S, F
k
) =

F
H
k
S
⊥

2
mentioned in [55], (A.3) follows from the notation
that cos θ
i

⊕ S
⊥
and the claim
follows. From Claim 1, it follows that the following are equiv-
alent.
(i) d
P
(S, F
k
) < 1forsomeF
k
∈ F for all S ∈ G
M
t
,M
s
.
(ii) c(F
1
) ∪c(F
2
) ∪···∪c(F
N
) = G
M
t
,M
t
−M
s

s
. This implies
sup
F∈G
M
t
,M
s
f (F) = δ<1
(A.6)
since f (F) < 1forF
∈ G
M
t
,M
s
and G
M
t
,M
s
is compact.
B. PROOF OF COROLLARY 1
Recall the definition of U, Σ,
U based on the SVD
of H
H
H from Section 2.3.LetΣ = diag(λ
M
t

min(M
t
,M
r
)
}. Since H
H
H is of rank equal to
min (M
t
, M
r
) with probability 1, in the following we consider
Σ
to be full rank. It may be noted, however, that the rank de-
pends on the value of M
t
, M
r
,andM
s
and in case M
r
= M
s
, Σ
and U are not defined and the following derivation remains
valid while ignoring all terms involving Σ
and U.
Claim 2. Consider F

M
s
N
0

F
H
∞
H
H
HF
∞

−1
kk
=
E
s
M
s
N
0
[Σ]
−1
kk
=
E
s
M
s

N
0

(B.3)
≤
M
s

k=1
E

e
−(E
s
d
2
min
/4M
s
N
0
)λ
k

,
(B.4)
where
N
e
is the number of nearest neighbors and Q(·) is the

/2
.Itisstraightfor-
ward to show that Q(x)
≥ η
1
e
−η
2
x
2
for some constants η
1
,
10 EURASIP Journal on Advances in Signal Processing
η
2
and a lower bound to P
e
could be derived using the same
arguments as before. Thus the diversity can be expressed as
d
=−lim
η
3
→∞
log E

e
−η
3

H
H
H
HF

−1
kk
≥
1

F
H
UΣU
H
F

−1
kk
.
(B.7)
Since F is a covering codebook, according to Theorem 1
we have d
P
(F
∞
, F) < 1. Noting F
∞
= U, it follows that F
H
U is

−1
(B.9)
= A
−1
−A
−1
Y

Σ
−1
+ Y
H
A
−1
Y

−1
Y
H
A
−1
(B.10)
= A
−1
−A
−1
YVSV
H
Y
H

change in notation where B
= A
−1
YVS
1/2
. Since BB
H
have
real-positive diagonal entries, it follows from (B.12) that

F
H
H
H
HF

−1
kk
≤

F
H
UΣU
H
F

−1
kk
(B.13)
whichjustifiestheclaim.

,
(B.14)
where η is a positive constant.
In the following e
k
denotes a vector of unit magnitude
where the kth element is unity:
e
H
k

F
H
UΣU
H
F

−1
e
k
≤


e
H
k

F
H
U



WSV
H


2
F
λ
−1
M
s
(B.17)
≤ max
1≤i≤M
s

M
s
cos
2
θ
i

λ
−1
M
s
(B.18)
=

F
,
(B.17) follows from the SVD decomposition of
(F
H
U)
−1
= WS V
H
,(B.18) holds due to the fact that S =
diag(1/ cos θ
1
, ,1/ cos θ
M
s
), where θ
i
are the critical angles
between the column spaces of F and
U,andfinally(B.19)
holds because the covering radius of the codebook is upper
bounded by δ<1fromTheorem 1. Thus the claim is
justified.
Let us define the selected precoder E as [cf. (14)]
E
= Q
∗

F
∞

kk
(B.21)
≥ min
k
E
s
M
s
N
0

E
H
H
H
HE

−1
kk
(B.22)
≥ min
k
E
s
M
s
N
0

F

(B.25)
where ζ is a constant. In the above (B.23)holdsbecauseF is
chosen according to the criterion F
= arg min
F
i
∈F
d
P
(F
∞
, F
i
)
and is a suboptimal precoder [cf. (6)], (B.24)followsfrom
Claim 3,and(B.25)holdsduetoClaim 4.
From (B.25)andClaim 2 it follows that the diversity is
preserved when F is a covering codebook and the precoder
selection criterion is (B.20).
C. PROOF OF LEMMA 1
An alternate representation for d
P
(F
1
, F
2
) for arbitrary
F
1
, F

. Consider an arbitrary precoder F ∈ G
M
t
,M
s
and the an-
tenna selection codebook F
={F
1
, F
2
, , F
N
},whereN =

M
t
M
s

. Since rank(F) = M
s
, ∃ asetofM
s
linearly indepen-
dent rows in F.Suppose
{i
1
, i
2

<π/2, where θ
i
are the critical angles between
the column spaces of F
∗
and F.Thenmax
1≤i≤M
s
sin θ
i
< 1.
Then from (C.1) it follows that d
P
(F
∗
, F) < 1.
B. Mondal and R. W. Heath Jr. 11
D. PROOF OF COROLLARY 2
Let us define U, Σ,
U, U, Σ, Σ as in Appendix B. Also assume
E
ij
is the codeword difference matrix between the codewords
i and j,isofsizeM
t
× T,andsubsumesi
/
= j. Also assume
E
ij

= Q
P

F
∞

=
arg min
F∈F
d
P

F
∞
, F

.
(D.2)
Note that in an unprecoded OSTBC system with M
t
transmit
and M
r
receive antennas, the Chernoff bound on the pairwise
error probability is of the form e
−η
1
H
2
F

d
P
(F
∞
, F), then


HFE
ij


2
F
≥ ηH
2
F
(D.3)
for some positive constant η.
The left-hand side of (D.3) may be expressed as


HFE
ij


2
F
= tr

E

H
FE
ij

(D.5)
≥ tr

ΣU
H
FE
ij
E
H
ij
F
H
U

(D.6)
= tr

Σ
VSV
H

(D.7)
= tr

SV
H

F
H
UU
H
F

(D.11)
≥ tr(
Σ
)λ
min

E
ij
E
H
ij

λ
min

F
H
UU
H
F

(D.12)
≥ tr(Σ)λ
min

H
FE
ij
is her-
mitian nonnegative definite so its trace is nonnegative,
(D.7) can be explained by the SVD decomposition given by
U
H
FE
ij
E
H
ij
F
H
U = VSV
H
,(D.8) follows from the repeated
use of the property tr(AB)
= tr(BA), in (D.9) we use the
fact tr(V
H
ΣV) = tr(Σ), (D.10)holdsbecauseS is a diago-
nal matrix consisting of the eigenvalues of
U
H
FE
ij
E
H

F and
U along with the fact that the covering radius of F
is δ and by Theorem 1 we have δ<1, and finally (D.14)is
because min
i,j,i
/
=j
λ
min
(E
ij
E
H
ij
) is a positive constant by defini-
tion of a full-rank STBC and tr(
Σ) ≥ λ
1
≥ (1/M
t
)H
2
F
. This
justifies the claim.
Then we have


HEE
ij

≥ ηH
2
F
,
(D.17)
where (D.16) is because of the suboptimality of the precoder
F compared to E and (D.17)isduetoClaim 5.
E. DIVERSITY ORDER FOR QPSM WITH M
R
= M
S
Lemma 2. A precoded spatial multiplexing system with a ZF
receiver, F
∞
is the precoder and M
r
= M
s
can attain a diversity
order of M
t
−M
s
+1.
Recall that since M
s
= M
r
, H
H

(u)
M
s
} denote the set of
unordered eigenvalues. Then the joint probability density
functions of
{λ
1
, λ
2
, , λ
M
s
} and {λ
(u)
1
, λ
(u)
2
, , λ
(u)
M
s
} differ
only by a scaling factor of M
s
!. The marginal probability den-
sities of λ
(u)
i


2
λ
M
t
−M
s
e
−λ
,
(E.1)
where L
M
t
−M
s
k
(λ) = (1/k!)e
λ
λ
M
t
−M
s
(d
k
/dλ
k
)(e
−λ

tenas(see(B.2))
SNR
(ZF)
k
=
E
s
M
s
N
0
[Σ]
−1
kk
d
=
η

Σ
(u)

−1
kk
= ηλ
(u)
k
,
(E.2)
12 EURASIP Journal on Advances in Signal Processing
where the constant η absorbs the factors E

2
min
/4)
}
(E.4)
=
N
e
M
s

∞
0
e
−ηλ(d
2
min
/4)
M
s
−1

k=0
a
k

L
M
t
−M


k=0
a
k
×

k

p=0
b
p
L
2(M
t
−M
s
)
2p
(2λ)

λ
M
t
−M
s
dλ
(E.6)
=
N
e

c
q
λ
q

λ
M
t
−M
s
dλ
(E.7)
=
N
e
M
s
M
s
−1

k=0
a
k
×

k

p=0
b

s
M
s
−1

k=0
a
k
×

k

p=0
b
p

2p

q=0
(−1)
q
c

q

η
d
2
min
4

·) is the Gaussian
Q-function, λ denotes an unordered eigenvalue, d
2
min
is the
minimum distance of the constellation, (E.4) represents the
Chernoff bound for Q(
·), (E.5) follows from the form of
the pdf of λ in (E.1)anda
k
is a positive constant, (E.6)fol-
lows from an expansion of [L
M
t
−M
s
k
(λ)]
2
given in [59]and
b
p
> 0 are constants, (E.7) follows from an expansion of
L
2(M
t
−M
s
)
2p

F
= Q
C

F
∞

=
arg min
F
k
∈F
d
C

F
∞
, F
k

.
(F.1)
Recall from Section 2.3 that the SVD of H
H
H is given by
H
H
H = UΣU
H
,

λ
M
s
+1
, λ
M
s
+2
, , λ
M
t
. Then we have the following:
E

HF
2
F

=
E

tr

F
H
UΣ
H
U
H
F


F
H
UΣU
H
F

=
ΛE

tr

U
H
FF
H
U

,
(F.5)
where
Λ = (1/M
s
)

M
s
k=1
E{λ
k

U
H
F)}
(F.6)
= E{tr(F
H
UΣ
Π
U
H
F)},
(F.7)
wherewedefine
Σ
Π
= ΠΣΠ
H
.Now,
E

tr(F
H
UΣU
H
F)

=
1
M
s

Σ
Π

U
H
F

(F.9)
= tr

E

U
H
FF
H
U

E

1
M
s
!

Π
Σ
Π

(F.10)

s
!)

Π
Σ
Π
}=ΛI. This justifies Claim 6.
Claim 7.
E

tr

F
H
UΣU
H
F

=
ΛM
s
−ΛE

tr

U
H
FF
H
U

permutation matrix. Then
E

tr

F
H
UΣU
H
F

=
E

tr

F
H
UΠΣΠ
H
U
H
F

=
E

tr

F


U
H
FF
H
U

.
(F.14)
Then we can write
E

tr

F
H
UΣU
H
F

=
ΛE

tr

F
H
U
H
UF

from
UU
H
+ UU
H
= I,and(F. 17 ) holds also due to tr(AB) =
tr(BA). This justifies Claim 7.
Note that
U = F
∞
and E{tr(U
H
FF
H
U)}=M
s
−
E{d
2
C
(F
∞
, F)} by definition (applying the relation A
2
F
=
tr(AA
H
)). Then from (F.4 ), Claims 6, 7 we have
E


d
2

F
∞
, F

.
(F.18)
ACKNOWLEDGMENTS
Bishwarup Mondal is the recipient of a Motorola Partner-
ships in Research Grant. This material is based in part upon
work supported by the National Science Foundation under
Grant CCF-514194. This work has appeared in part in the
Proceedings of IEEE International Workshop on Signal Pro-
cessing Advances, in Wireless Communications pages 1–5,
July 2–5, Cannes, 2006.
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