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134 MIMO System Technology for Wireless Communications

the European Union. The IST METRA model generates the channel H accord-

ing to [96]

(5.32)

where is a random matrix with i.i.d. CN(0,1) entries, is the receive

antenna correlation matrix, and is the transmit antenna correlation

matrix. The correlation matrices and can be measured or computed.

Measured correlation matrices were discussed in [44], while computational

methods for obtaining the correlations were studied in [19,83,96].

Note that the IST METRA model is not a fully correlated model [47] and

can be thought of as a transmit-receive correlated model. The correlation is

isolated to only transmit and receive correlation. This kind of correlation

simpliﬁcation was experimentally veriﬁed in [42–44,82,89,98]. Recently, this

model has been adopted by the IEEE 802.11n wireless local area network

working group as a MIMO channel model [96] and has been studied for use

in IEEE 802.20 working group on mobile broadband wireless access [85].

Because of its wide use, this model is one of the cornerstones by which most

MIMO algorithms are tested before implementation.

System performance on transmit-receive correlated models is highly

dependent on the correlation matrices. It is therefore imperative to under-

stand the requirements that and must maintain to guarantee full

diversity order. We will characterize the diversity order of beamforming and

precoded OSTBCing in transmit-receive correlated fading assuming optimal

combining.

FIGURE 5.4

Comparison of the probability of symbol vector errors for 16-bit direct channel quantization,

unquantized MMSE, and 6-bit limited feedback precoding on a 2 substream 4 × 2 system using

16QAM.

−2 0 2 4 6 8 10 12 14 16

Prob. of symbol vector error

−3

−2

−1

(dB)

MMSE(max SV)

Quant. chann. 16 bit MMSE

Grass 6 bit(MSE det)

HRGR=

12 12

4190_book.fm Page 134 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 135

THEOREM 5.1

[60] Beamforming and combining over memoryless, correlated Rayleigh fading chan-

nels provide a diversity order of if and only if (i) the vectors in the beamformer

codebook F span (ii) the rank of is M

, and (iii) the rank of R

Theorem 5.1 provides the intuition that the limited feedback codebook

must always form a basis for the vector space of possible dominant singular

value directions. Thus, it is impossible for the optimal direction to be orthog-

onal to all of the codebook vectors. The correlation matrices change the

distributions of left and right singular vectors. It is important, however, that

the correlations allow for the full number of degrees-of-freedom.

THEOREM 5.2

[52] Precoded OSTBCing provides a diversity of order if and only if (i) the

matrix has a rank of M

, (ii) the rank of is M

, and (iii) the rank

of is

If given a sufﬁcient rate of feedback, limited feedback algorithms can

obtain identical diversity orders to the diversity order from perfect channel

knowledge closed-loop schemes. To obtain full diversity order, Theorem 5.2

tells us that precoded OSTBCing requires log

/M) bits of feedback

because the dimension of for all i is This differs from the result

in Theorem 5.1, which says that limited feedback beamforming requires

log

) bits of feedback. Thus, precoded OSTBCing can provide full diver-

sity order with lower rate feedback. This beneﬁt, however, comes at the

expense of a performance loss for higher feedback amounts.

5.6 Feedback in Broadband Channels with MIMO-OFDM

Previous sections addressed the theory of limited feedback MIMO commu-

nication based on precoder quantization and studied their practical perfor-

mance in narrowband channels. Most developing wireless standards though

are broadband, typically employing some form of OFDM modulation. In

this section we describe how the aforementioned concept of limited feedback

can be extended to the MIMO-OFDM setting. The key idea is to reduce feed-

back by exploiting the correlation of adjacent subcarriers. We describe the

application to beamforming in detail. Extensions to precoding are more

brieﬂy described. Limited feedback for MIMO-OFDM has recently been

adopted as an option in the IEEE 802.16 standard.

5.6.1 MIMO-OFDM System Model and Limited Feedback

OFDM modulation uses a cyclic preﬁx and discrete Fourier transform (DFT)

operations to convert the linear convolution in Equation 5.2 to a multiplication.



, R

EFF F= []



×.

4190_book.fm Page 135 Tuesday, February 21, 2006 9:14 AM

136 MIMO System Technology for Wireless Communications

By sending information in the frequency domain, OFDM converts a linear

convolution into a set of parallel narrowband channels. Precoding can thus

be applied independently per subcarrier.

Consider a MIMO-OFDM system with an M-point DFT, and cyclic preﬁx

that is longer than L, the memory of the discrete-time channel. For illustration

in this section we will discuss precoded spatial multiplexing systems. Exten-

sions to other forms of space–time coding follow in a similar fashion. Let

be the multivariate response of the channel in the frequency domain. Math-

ematically the input-output relationship of a MIMO-OFDM system with

precoding is

(5.33)

where is the precoding matrix for subcarrier k and k = 0, 1, …, M – 1.

A straightforward application of precoded spatial multiplexing entails that

the precoder for each subcarrier is quantized and the appropriate index

is sent back to the transmitter. For a codebook F of cardinality N = 2

, this

requires feedback of Mb bits to relay the precoders for the whole OFDM

symbol. The feedback requirements grow in proportion to M, the number

of subcarriers. The channels for adjacent subcarriers, however, are correlated.

Based on this, it can also be shown that the adjacent precoders, in some

sense, are also correlated [9]. Feedback can thus be reduced by exploiting

the correlation and sending back less information per subcarrier.

5.6.2 Interpolated Beamforming in MIMO-OFDM

First we consider the special case of transmit beamforming. This allows us

to present the essence of the extension to MIMO-OFDM before getting into

the full complexity of precoding. With transmit beamforming, the receiver

processing consists of matched ﬁltering and scalar detection for each sub-

carrier. The combined signal at subcarrier k can be expressed as

(5.34)

where is the receive combining vector, chosen to be H(k)f(k), which in

this case corresponds to maximum ratio combining at the receiver.

A simple beamforming method for using the correlation of the beamforming

vectors is to combine the neighboring subcarriers into a cluster. The idea is to

choose the cluster size to be less than the coherence bandwidth of the chan-

nel, thus all subcarriers see the same effective channel. Then a representative

H() ()kle

jklM

yFsn() ()()() ()kkkkk=+H

F()k

rk k k ksk k k N

() (){ ()()() ()}=+,ffzfnH1

z()k

4190_book.fm Page 136 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 137

beamforming vector is chosen for the whole cluster. A simple approach,

which we call clustering, is to use the beamforming vector for the middle

subcarrier of the cluster for all the subcarriers in the cluster [8]. A better

approach, which we call modiﬁed clustering, chooses the best vector accord-

ing to some criteria [65] motivated by error rate or squared error.

An alternative to clustering is to use interpolation to reconstruct the trans-

mit beamforming vectors given a fraction of the beamformers f(lK + 1) where

K is the cluster size (which divides N) and Conventional

interpolation, however, does not preserve the structure in the beamforming

problem. First, all beamforming vectors are unit norm due to the transmit

power constraint. This means that the optimal beamforming vector is a point

on a sphere. The interpolator should give outputs that are also on the sphere.

Second the optimal beamforming vector is not unique. The reason is that if

one examines the effective channel gain both f(k) and e

f(k)

have exactly the same value. Thus the best beamforming vector is not unique.

To solve this problem, we proposed a modiﬁcation of the spherical inter-

polator in [99] that includes a phase rotation parameter [8]. This interpolator

performs a weighted average of the beamforming vectors and renormalizes

the result to place it on the unit sphere. Given {f(lK + 1), 0 f l f N/K – 1},

the proposed interpolator is expressed as

(5.35)

where c

= (k – 1)/K is the linear weight value, are the quan-

tized transmit beamforming vectors, 1 f k f K, and is a parameter for

phase rotation with 0 f l f N/K – 1. The role of is to remove the distortion

caused by the arbitrary phase rotation of the optimal beamforming vectors.

To maximize the performance of the interpolator, the receiver evaluates

the optimal phase {/

, 0 f l f N/K – 1} and sends this information back to

the transmitter along with the quantized beamforming vectors. In practice,

the set of possible phase values is quantized to and

P is a desired number of quantization levels (we have found that P = 4 is

sufﬁcient). The feedback of slightly increases the amount of feedback

information compared to clustering but results in better performance than

the simple clustering approach.

The phase can be optimized in different ways. We ﬁnd that maximizing

the minimum effective channel gain where 0 f l f N/K – 1 and

(5.36)

works well.

lNK=,,…, 01 1.

,() ()() ,kkk= H f

()

()( ){(()

lK k

clK ce lK

kQ k

+; =

 ++ +

11 1

 ++ + +

11 11

)}

()( ){(())}clK ce lK

kQ k

ff()()N +=11

6 =,,,,

{}





()

{}/

lK k lK k k K=++;,ff



argmax min{ ( ) ( ) }

H f

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138 MIMO System Technology for Wireless Communications

5.6.3 Interpolated Precoding in MIMO-OFDM

As with beamforming, a variety of methods can also be used for the more

general case of precoding in MIMO-OFDM including clustering, modiﬁed

clustering, and interpolation. Because the extension of the interpolator is not

trivial, we discuss it here.

The challenge with precoder interpolation is that the unitary property

and the invariance to right multiplication by a unitary

matrix, i.e., and where is a square unitary matrix give the same

performance, must be respected. The latter property implies that the optimal

precoder is not unique. The matrix Q, however, has a substantial inﬂuence

on the performance of an interpolator. Based on this observation, we propose

the following interpolation algorithm, which utilizes a unitary matrix Q

and the quantized precoders for l = 0, 1, …, N/K – 1

(5.37)

(5.38)

The proposed algorithm in Equation 5.37 and Equation 5.38 is a non-

obvious extension of the beamformer interpolator in Equation 5.35. Note

that is the projection of into with respect to the Frobe-

nius norm (see Section III-F of [94]). Notice that of Equation 5.35 is

replaced by in Equation 5.37. In beamformer interpolation, removes

the distortion caused by the arbitrary phase rotation of beamforming vectors.

Similarly, the role of is to get rid of the distortion caused by the non-

uniqueness of the precoding matrices.

Given the subsampling rate and the codebook F, the quantized precod-

ers {F(lK + 1), 0 f l f N/K – 1} are obtained by using the same criterion as

precoding design, e.g., SC-MSE or SC-Capacity. Since we are quantizing Q

for limited feedback, we can numerically ﬁnd the best Q by selecting a matrix

from a codebook Q of unitary matrices. Given a codebook Q, we can ﬁnd

the best in a similar way as before by optimizing

(5.39)

The performance and complexity are dependent on 冨Q冨, the cardinality of Q.

A reasonable way to design Q is to employ the Frobenius matrix norm

and maximize the minimum norm between the entries of Q. A variety of

other approaches for codebook design are also possible; optimal quantization

of Q is still under investigation.

FF I

kk()()=

F()k FQ()k Q

lK()+ 1

ZFQF()()() ()lK m c lK c l K

mQ mlQ

+= ++ + +

()

11 11

( )()()()FQZ Z ZlK m lK m lK m lK m

+; = + + +

{}



()F k; / Z()k U ()MM

QFQ

MSE lK m=+;.



argmin max (

())

4190_book.fm Page 138 Tuesday, February 21, 2006 9:14 AM

Feedback Techniques for MIMO Channels 139

5.6.4 Performance Example

To illustrate the potential performance of limited feedback precoding for

MIMO-OFDM we provide a performance example in this section. In the

simulation, we consider a MIMO-OFDM system with parameters M

= 4,

= 2, M

= 2, N = 64, and K = 8. We assumed that the discrete-time channel

impulse responses had 6 taps with uniform proﬁle and i.i.d. complex Gaussian

distribution; the channels between different transmit and receive antenna pairs

were independent; was i.i.d. zero mean complex Gaussian; the feedback

channel had no delay and no transmission error; quadrature phase shift

keying (QPSK) modulation and MMSE detection were used for BER simu-

lations; and the transmit power was identically distributed to all subcarriers.

We compared antenna subset selection [29] where the best set of 2 out of

4 antennas is chosen for each subcarrier, simple clustering, the proposed

interpolator, and MMSE detection with the optimal precoder (with and with-

out quantization). To quantize the precoding matrices, we used the codebook

with 冨F 冨 = 64 requiring 6 bits of feedback for each precoding matrix which

was designed according to [54]. Also, we considered the codebook given in

[9]. Consequently, we used 64 feedback bits for the proposed method. In

antenna subset selection, the transmit antenna was selected independently

for each subcarrier, so 166 feedback bits were used. We used 48 feedback bits

for clustering and 384 bits for the optimal precoder with feedback for all

subcarriers. For comparison, we also considered the optimal precoder with-

out quantization.

Figure 5.5 shows the coded BER performance of the proposed and existing

precoding methods. The maximum minimum singular value criterion [29]

was used for antenna subset selection and for the proposed scheme was

determined by Equation 5.39. For channel coding, we used a convolutional

code with generator polynomials g

= 133

and g

= 171

with coding rate

1/2, the interleaver deﬁned in the IEEE 802.11a standard extended for

64 subcarriers, and soft Viterbi decoding. The frame length was 30 OFDM

symbols and each OFDM symbol transmitted 128 bits. The proposed pre-

coder outperformed clustering and antenna subset selection, and had only

0.1 dB loss compared to the optimal MMSE with quantization. The quanti-

zation loss was about 1 dB even with per-subcarrier quantization. Clustering

obtains nearly as good performance with less feedback required.

5.7 Conclusions

In this chapter we presented an overview of closed-loop MIMO communi-

cation systems. We discussed both direct channel quantization and quantized

signal design, choosing to focus on the latter because it typically requires

less feedback. As concrete examples we discussed strategies for quantized

n()k

4190_book.fm Page 139 Tuesday, February 21, 2006 9:14 AM

140 MIMO System Technology for Wireless Communications

beamforming and precoding for both spatial multiplexing and space–time

coding. We also summarized some extensions of quantized beamforming

and precoding for spatial multiplexing with MIMO-OFDM.

Other important aspects of limited feedback MIMO communication were

not considered in this chapter due to a lack of space. Perhaps the most

important omission is multi-mode precoding, where the number of spatial

streams M is also varied in the channel. Results along these lines are available

in [25,56,59]. We also overlooked the combination of adaptive modulation

and limited feedback, though naturally the two will be combined in a real

system. The effect of channel estimation error, delay in the feedback channel,

and, more generally, the effect of time variation of feedback performance

remain to be studied.

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