Tsoulos George (ред.) MIMO System Technology for Wireless Communications

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124 MIMO System Technology for Wireless Communications

transmit antennas. The performance of OSTBCs, however, can be improved

by leveraging channel knowledge at the transmitter [47].

At the kth transmission, an OSTBC encoder maps a block of n

symbols s

, …, s

taken from a constellation S to an (with space–time

code matrix The power is constrained so that [40]

(5.9)

for all In addition, we will assume that the constellation has been normal-

ized so that

The space–time block codeword is precoded by an matrix F.

As in the limited feedback beamforming case, this matrix will be restricted

to lie in a codebook F = {F

, F

, …, F

}. The matrix is restricted according

to a peak power constraint for all such that The matrix will

be further restricted, but this restriction will be developed later.

We will assume a narrowband MIMO system with a spatially uncorrelated,

memoryless linear channel that is constant over several codeword transmis-

sions before independently changing. These assumptions allow the system

input-output expression to be written as

(5.10)

FIGURE 5.2

Performance of limited feedback beamformer for a four-transmit and ﬁve-receive antenna

system using 16-QAM. This plot demonstrates the beneﬁts of using quantized signal adaptation

rather than direct channel quantization.

−8 −6 −4 −2 0 2 4

−4

−3

−2

−1

(dB)

Probability of symbol error

Optimal beamforming

Quantized channel(40 bit)

Grassmannian BF(6 bit)

MT× MM

< )

Ccc c() [ () () ()]kkkk

=….

CC I() ()kk s





C()k

F F Q

1{}F f . F

YHFCN=+

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Feedback Techniques for MIMO Channels 125

where is the channel matrix with independent entries distributed

as CN(0,1), W is the signal-to-noise ratio (SNR), and is an M

× T noise

matrix with independent entries distributed according to CN(0,1). The

receiver has perfect knowledge of and and it decodes using optimal

ML decoding.

To implement a limited feedback precoded OSTBCing system, two main

problems must be addressed. First, we must develop a selection criterion to

choose from F. Second, we must determine how to design the codebook

F to maximize some performance criterion.

We will deﬁne performance with respect to the symbol error rate (SER)

given denoted by Pr(ERROR冨H). Using the orthogonality properties of

OSTBCs, it can be shown that

(5.11)

where L is a function that depends on W, M, and S [46,47]. Because the SNR,

constellation, and signaling architectures are ﬁxed during transmission, max-

imizing corresponds to minimizing the SER bound. This yields a nat-

ural precoder selection criterion. The receiver chooses the linear precoder F

according to

(5.12)

Consider the optimal precoder chosen to maximize over the set

of maximum singular value constrained matrices L(M

, M). This matrix F

opt

will not be unique over L(M

, M) because for arbitrary U  U(M,M), 㥋HF

opt

㥋

㥋HF

opt

U㥋

. Denote singular value decomposition (SVD) of by

(5.13)

where and 8 is an M

× M

diagonal matrix

with Q

{H} at entry Let denote the matrix taken by selecting the ﬁrst

M columns of The following lemma summarizes

LEMMA 5.1

[57] The precoder matrix maximizes over L(M

, M).

Lemma 5.1 tells us that not only should but Q

opt

} = 1 for 1 f

j f M. Intuitively, this means that we should perform waterﬁlling to maxi-

mize compound channel when transmitting with a peak power con-

straint. When the transmitter has no a priori knowledge of the transmitter

will have no knowledge of Therefore, we will use a limited feedback

path to convey a suboptimal precoder. The following lemma shows that any

H MM

HF, Y

Pr ERROR exp

HHF

()

f

FHF



argmax



opt

HVV=



RttLrr

MM MM ,  ,UU(), ( ),

()jj,.

. F

opt

opt R

= HF

opt

1{}F

opt

4190_book.fm Page 125 Tuesday, February 21, 2006 9:14 AM

126 MIMO System Technology for Wireless Communications

suboptimal linear precoder should always have unit singular values under

a maximum singular value constraint in order to minimize the SER.

LEMMA 5.2

[57] Let FeL(M

, M) have a singular value decomposition such that

Given this, when

Proof: Let be deﬁned as in the lemma. Using (i) the invariance of the

Frobenius norm to unitary transformation, (ii) the singular value bound for

Fe, and (iii) the lemma’s deﬁnition of

Lemma 5.2 is important because it tells us that the precoder should always

be designed to have singular values that are as large as possible. Because of

the maximum singular value restriction, this singular value maximization

result corresponds to designing precoders in the set U(M

, M) rather than

L(M

, M). For this reason, we will restrict our codebook precoders to lie in

U(M

, M) (i.e.,

We would like the quantized equivalent channel to provide SER per-

formance close to that provided by the optimal precoded equivalent channel

opt

when F is restricted to F. One method for doing this was proposed in

[57] by using the average of the loss in received channel power as a measure

of distortion. Using this deﬁnition of performance loss, the average distortion

of the codebook is given by

(5.14)

It was shown in [54,55,57] that the average distortion can be bounded as

(5.15)

This bound on the average distortion can be thought of as the product of

two different terms. The ﬁrst term relates to the distribution of the maximum

channel singular value, which is not under the control of the system designer.

The second term represents the “quality” of the codebook F and can there-

fore be optimized.

Before dealing further with the codebook design, we must review some

properties and notations dealing with ﬁnite subsets of U(M

, M). The column

space of each matrix in U(M

, M) corresponds to an M-dimensional subspace

of The set of all subspaces in is known as the complex Grassmann



FUU

{}

<.F 1 HF HF



FUI=.

[]0



HF HU HU I HF

= f =

, []0



FF I



)

opt

HF HFmin







opt

HF HFmin





f

{} min

HVVFF











. 

4190_book.fm Page 126 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 127

manifold [105]. Just as in the traditional Euclidean space, the Grass-

mann manifold can have distances deﬁned on it. The chordal distance between

subspaces and where denotes the subspace generated by F, is

given by

(5.16)

Consider the set of subspaces generated by the code-

book matrices This ﬁnite set of subspaces can be thought

of as a subspace code or subspace packing in Just as in binary

coding theory, we can deﬁne the minimum distance of a packing:

(5.17)

In applied mathematics, the Grassmannian subspace packing problem is

the problem of choosing N subspaces (i.e., designing an N element subspace

code) in order to maximize the minimum distance I.

Using a metric ball approach, the “codebook quality” term in Equation

5.15 can be approximately bounded as [57]

(5.18)

This bound is a decreasing function of I when which

can be easily proven by differentiating the bound and noting that

Thus maximizing I approximately minimizes Equation 5.18. This establishes

that a low-distortion, high-performance codebook can be designed by pack-

ing subspaces in the Grassmann manifold using the chordal distance metric.

Note that when M = 1, this bound corresponds to the beamforming scenario

discussed in the previous section.

The experiment shown in Figure 5.3 compares the performance of antenna

subset selection [23] with 8-bit chordal distance precoding on an 8 × 1

wireless system transmitting an Alamouti code constructed from

16-quadrature amplitude modulation (QAM). For comparison, the SER per-

formance for an Alamouti code transmitted on a system is also shown.

For example, at an error rate of 10

–3

, antenna subset selection gives an 8 dB

gain over the two antenna open-loop system. An additional gain of approxi-

mately 1.4 dB results from using 8-bit feedback chordal distance precoding.

5.4.3 Precoded Spatial Multiplexing

In spatial multiplexing, a bit stream is demultiplexed into different bit

streams that are independently modulated using the same constellation S.

G()MM

, P

()FF FF FF

12 11 22

,= 



VPP P=,,…,{}

FF F

12 N

=,,…, .{}FF F

G()MM

I =,

f < f

min ( )

1 klN

d FF

EMN

MM o

VV FFmin

(







 <+

)



223MM oM

+>/,()

I <.M

M = 2

21×

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128 MIMO System Technology for Wireless Communications

At time instance K, the M symbols can be written as a vector s

k,1

k,2

… s

k,M

]

with E

]= I

. In precoded spatial multiplexing, the sym-

bol vector is multiplied by an matrix F before transmission pro-

ducing a length M

dimensional vector

where is the total transmit energy, is the number of transmit antennas,

and

This formulation allows the baseband, discrete-time input-output relation-

ship to be written as

(5.19)

where is an channel matrix and is a length noise vector.

The receiver, which is assumed to have perfect knowledge of H and F,

decodes the received vector using a vector decoder to produce a hard

decoded symbol vector

. We assume the channel matrix H to have inde-

pendent CN(0,1) entries, and the noise vector to have independent

CN(0, N

) entries.

FIGURE 5.3

Symbol error rate comparison of two-antenna open-loop Alamouti space–time coding, eight

transmit antenna subset selection combined with an Alamouti code, and eight antenna 8-bit

feedback precoding combined with an Alamouti code. The constellation was assumed to be

16-QAM, and the receiver was assumed to have a single antenna.

4 6 8 10 12 14 16 18 20 22 24 26

Probability of symbol error

No precoding(2 × 1)

Ant select

Grass 8 Bit

−3

−2

−1

(dB)

xFs

v .

yHFsn

H MM

4190_book.fm Page 128 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 129

The precoding matrix is chosen at the receiver from a ﬁnite set of possible

precoding matrices F = {F

, F

, …, F

} known to both the transmitter and

receiver. The receiver sends the binary index of the chosen precoding matrix

back to the transmitter over a limited capacity, zero-delay feedback link. As

in the precoded OSTBC scenario, each has unit column vectors that

are orthogonal. This orthogonality assumption is not especially restrictive

since it follows from the form of the optimal spatial multiplexing precoders

derived in [81] that consist of a matrix with orthogonal columns pre-multi-

plied by a diagonal matrix.

As with beamforming and precoded OSTBCing, we need to (i) determine

how to choose the precoded F given a codebook F and (ii) design the

codebook F. Unlike the other algorithms, spatial multiplexing is often imple-

mented with a variety of different decoder algorithms. We will brieﬂy discuss

the different decoders and their accompanying precoder selection criterion.

5.4.3.1 Precoder Selection for Spatial Multiplexing

Maximum Likelihood Receiver Inspired Selection

The ML receiver decodes to

by solving

(5.20)

To improve probability of error performance, it is of interest to ﬁnd a

closed-form expression for the probability of symbol vector error. Unfortu-

nately, a closed-form expression for the probability of symbol vector error

has yet to be derived in the literature. One solution is to upper bound the

probability of symbol vector error for high signal-to-noise ratios (SNR) using

the vector Union Bound [30]. Since we assume that the SNR J

is held

constant, the Union Bound is solely a function of the receive minimum

distance of the multidimensional constellation S

[30]. This motivates the

following criterion for ML receivers.

SC-ML: Pick such that

(5.21)

Linear Receiver Inspired Selection

Unlike ML decoding, linear receivers can be designed in different ways

depending on the desired decoding matrix A linear decoder decodes to

the vector

= Q(Gy

) where G is an matrix and is a function

that performs single-dimension maximum likelihood decoding for each

entry of a vector.

F F

yHFs

= 



argmin

FHFss

ss s s

= 



,  : |

argmax



min

12 1 2

¹¹

× Q()

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130 MIMO System Technology for Wireless Communications

Minimum Singular Value

It was shown in [29] that the performance of a spatial multiplexing system

using a linear receiver is characterized by the minimum substream SNR.

Maximizing the minimum singular value provides a close approximation

to maximizing the minimum substream SNR [29]. The reason for this is that

as card(W) grows large, the probability of an error vector lying collinear to

the minimum singular value direction goes to one. This motivates the fol-

lowing criterion

SC-MSV: Pick such that

(5.22)

Minimum Mean Squared Error Receiver: Minimizing some function of the

mean squared error matrix has been studied in [81] to improve the overall

system performance. The average mean squared error when decoding with

a minimum mean squared error (MMSE) linear decoder is

(5.23)

We can use Equation 5.23 to derive a selection criterion for choosing F

from F.

SC-MSE: Choose such that

(5.24)

where is either (the trace of a matrix) or (the determinant of a

matrix).

Capacity Selection

The ﬁnal criterion can be used for both ML and linear receivers. When the

transmitter precodes with before transmission, we can view the system as

experiencing an effective channel of Thus the mutual information

assuming independent and identically distributed Gaussian signaling for a

ﬁxed realization of and a ﬁxed is given by

(5.25)

Therefore, a capacity selection can be derived from Equation 5.25.

FHF



argmax



min

{}

MSE( )FI FHHF=+





MMN

()



argmin MSE



m ()

m() tr()

det()

HF.

Idet

() logFIFHHF=+



4190_book.fm Page 130 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 131

SC-Capacity: Choose such that

(5.26)

5.4.3.2 Codebook Design for Spatial Multiplexing

As with precoding for OSTBCing we will think of the codebook as a set of

subspaces in Thus, the limited feedback precoded spatial multiplexing

codebook F represents a set, or packing, of subspaces in the Grassmann

manifold. We will use the subspace coding interpretation to simplify notation

and analysis.

The measure on induces a normalized invariant measure µ on

G(M

, M). Volumes can be computed within G(M

, M) using this induced

measure. In addition to the chordal distance, a number of other distances

can be deﬁned on G(M

, M) [5]. The projection two-norm, and Fubini-Study

distances between the two subspaces and are given by

Let denote the set of subspaces obtained from the col-

umn space of the matrices in The minimum distance of this packing is

(5.27)

where is a distance function on

Understanding the optimal unquantized matrix will allow us to under-

stand how to design F. The ﬁrst important point to notice is contained in

the following lemma.

LEMMA 5.3

[54,55] Let where and 8 has diagonal entries

The optimal unquantized precoding matrix (i.e., F  U(M

M)) for each of the criteria in Section 5.4 is the ﬁrst M columns of

For SC-ML it was shown in [30] that the receive minimum distance is

approximately maximized by maximizing the minimum singular value of

HF. SC-MSE using the trace cost function chooses that minimizes

Equation 5.23. For high SNR with scaled to unity, the trace of Equation

5.23 can be approximated by tr [(F



HF)

–1

]. However,



arg max ( )





U()MM

proj

()

( ) arccos

FF FF FF

12 11 22

,= 



eet FF



SPP P=,,…,{}

FF F

12 N

I =,

f < f

min ( )

1 ijN

d FF

d(), G()MM

HH V V



8 V

Rtt

MMU(,)

QQ Q

0vv… vv.

, V

F F

4190_book.fm Page 131 Tuesday, February 21, 2006 9:14 AM

132 MIMO System Technology for Wireless Communications

(5.28)

Therefore this bound requires the maximization of

Thus the SC-MSV and the bounds on SC-ML and SC-MSE using the trace

cost function all lead to maximizing We will therefore use our

codebook to minimize

(5.29)

for SC-ML, SC-MSE using the trace cost function, and SC-MSV where once

again is the Mth largest singular value of Using the subspace pack-

ing properties of F, we will show that Equation 5.29 can be bounded by a

decreasing function of the minimum projection two-norm distance of F,

denoted by

Codebook Design Criterion 1 [58]: Codebooks for systems using SC-ML,

SC-MSV, and SC-MSE using the trace cost function to select F from

should be designed by using an N subspace packing in the Grassmann

manifold that maximizes the minimum projection two-norm distance.

It was shown in [54,55] that selecting F F using SC-MSE with the deter-

minant cost function is equivalent to maximizing

I(F) where

Thus the codebook design criterion for SC-Capacity and SC-MSE with the

determinant cost function must be identical.

Note that

(5.30)

Using the above analysis and observing that

we will use the cost function

min min

{}

min

FHHF









()

min

{}HF .

min

{}HF .

opt MH

HF HFQQQ

min min

{}max{} {

222





F

HHHF

}max { }

min





F

{}HH.

proj



Idet

()FI FHHF=+





I det det

()

FVFIv

()



Idet

opt M

()FI=+

4190_book.fm Page 132 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 133

(5.31)

where Equation 5.31 follows from the independence of and

Using the subspace packing interpretation of F, this distortion cost func-

tion can be bounded by a decreasing function of the minimum Fubini-Study

distance of F, denoted by This yields the following design criterion.

Codebook Design Criterion 2 [58]: Codebooks for systems using SC-

MSE with the determinant cost function or SC-Capacity to select from

F should be designed by using an N subspace packing in the Grassmann

manifold that maximizes the minimum Fubini-Study distance.

Finding good Grassmannian packings for arbitrary and is actu-

ally quite difﬁcult [10]. The codebook design difﬁculty is only increased for

the non-standard projective two-norm and Fubini-Study distances [5]. One

practical technique for designing good packings is to use the non-coherent

constellation designs from [32]. This design algorithm usually yields code-

books with large minimum distances and can be easily modiﬁed to work

with any distance function on the Grassmann manifold.

Figure 5.4 demonstrates the problems associated with directly quantizing

the matrix channel and the beneﬁts of using a precoder codebook. A 4 × 2

wireless system using M = 2 and 16QAM was simulated. For a probability

of symbol vector error of 10

–2

, directly quantizing the channel with 16 bits

of feedback performs approximately 4.7 dB worse than a 6-bit limited feed-

back precoder. With only 6 feedback bits, the limited feedback precoder

obtains performance almost identical to the unquantized MMSE precoder with

the maximum singular value power constraint.

5.5 Effect of Spatial Correlation

In the previous sections, the channel was modeled as a spatially uncorrelated

Rayleigh fading channel. This assumption was used to derive codebook

design criteria. Real channels will experience spatial correlation. In this sec-

tion, we will present some results that show that diversity performance is

not compromised for our codebook designs.

There are several models for spatial correlation. One of the most popular

models was studied in the Information Society Technologies (IST) Multi-

Element Transmit and Receive Antennas (METRA) Project [18] funded by

Edet

det det

IVFI+



()



88 max

Edet



I 88





()





Edet

VFmax

¹¹

8 V

,, N

4190_book.fm Page 133 Tuesday, February 21, 2006 9:14 AM