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Comparative Study of Efﬁcient Decision-Feedback Equalization Schemes for

MIMO Systems

Christoph M. Degen, Christoph M. Walke, and Bernhard Rembold

Institute of High Frequency Technology, RWTH Aachen, Germany

degen@ihf.rwth-aachen.de

ABSTRACT

Decision-feedback (DF) equalization schemes, well-

known from multi-user detection, offer the opportunity of

an additional performance gain compared to their corre-

sponding linear equalizers by feeding back past decisions

on already detected symbols, based on a ﬁxed symbol

detection order. For MIMO systems, an optimum sym-

bol stream detection order has been found within the V-

BLAST algorithm, which is determined adaptively during

the detection process. The main focus of this paper is, to

show the analogy of these two approaches, where the V-

BLAST algorithm applied to frequency-selective channels

just denotes an extension of the conventional DF system

towards the optimum symbol stream detection order, but

resulting in additional matrix inversions. Monte-Carlo

simulations have shown, that using DF with a less de-

manding detection order approximately achieves the per-

formance of a DF system with the optimum detection or-

der. Finally, an extended DF receiver structure is pre-

sented, comprising as many DF blocks as concurrently

transmitted symbol streams are to be detected. In order

to obtain reasonable computational complexity, equaliza-

tion is performed in the frequency domain for all pre-

sented detection schemes.

1. INTRODUCTION

For MIMO systems as well as for multi-user detection

systems employing receive antenna diversity, generally,

several symbol streams transmitted concurrently at the

same frequency band are impinging on a receive antenna

array, being separable in the code and/or the space do-

main.

Block Decision Feedback Equalizers (BDFE) and the cor-

responding Block Linear Equalizers (BLE) have been

published for multi-user detection in TD-CDMA systems

[1]. For MIMO systems, the iterative V-BLAST algo-

rithm for ﬂat fading channels [2] and its extension to

frequency-selective channels [3] have earned recent sci-

entiﬁc interest. The high complexity of these linear and

iterative equalizers can be enormously reduced by cary-

ing out the equalization in the frequency domain [4, 5],

which is based on extending the blocked block-Sylvester

system matrix to a blocked block-circulant structure. Fur-

ther research is currently being done on extensions of

the V-BLAST algorithm towards systems, in which the

V-BLAST algorithm is carried out -times [6], where

denotes the number of concurrently transmitted sym-

bol streams.

In this paper, the DF and the V-BLAST based equaliza-

tion schemes for frequency-selective channels are pre-

sented, where equalization is carried out in the frequency

domain, based on a particular symbol stream detection

order. We elaborate on the analogy of these schemes.

Finally, an improved receiver structure for frequency-

selective channels based on the extended V-BLAST algo-

rithm is presented. For all presented detection schemes,

the Zero Forcing (ZF) and the Minimum Mean Square Er-

ror (MMSE) optimization criteria can be applied.

This paper is organized as follows: In Sec. 2, the MIMO

system model for a single-user MIMO link is introduced,

describing a system matrix in the time domain as well as

in the frequency domain. Linear detection schemes are

presented brieﬂy in Sec. 3. In Sec. 4, iterative detec-

tion schemes based on either a ﬁxed- or an adaptively de-

termined detection order, and their analogy is presented.

The extension towards sytems with an higher iteration

depth is discussed in Sec. 5. Simulation results are pre-

sented in Sec. 6.

In the sequel, lower-case letters are used for complex-

valued scalars, lower-case bold-face letters for complex-

valued vectors and upper-case bold-face letters for

complex-valued matrices. Conjugation is denoted by

, transposition by and the Hermitian operation

by . denotes the Kronecker matrix product. The

unit matrix of dimension is deﬁned by the sym-

bol . The notation refers to the element in the

-th row and -th column of a matrix .

2. THE MIMO SYSTEM MODEL

The considerations in this paper are based on a single-

user MIMO system with transmit and receive

antennas and concurrently transmitted PSK symbol

streams

(1)

of length , , stacked into the symbol

vector

(2)

In the following, is assumed.

The receive signal sampled at antenna element

is gathered in the

vector

(3)

and stacked into the total receive

vector

(4)

with a corresponding stacked noise vector

. Noise is assumed temporally and spa-

tially white, with zero mean and covariance matrix

.

The total system model then follows as

(5)

where denotes the system matrix and is described in

the following.

2.1 The MIMO System Matrix in the Time Domain

The system matrix is deﬁned

as (6)

is composed of block-Sylvester sub-matrices

of dimension .

Therein, a channel matrix

.

.

.....

.

.(7)

is deﬁned, where the Toeplitz

sub-blocks contain the complex-valued Chan-

nel Impulse Responses (CIR) between

transmit station antenna element and re-

ceive station antenna element with an

assumed channel memory .

Finally, an spreading matrix

blockdiag (8)

being composed of the matrices

(9)

denotes spreading of each symbol stream with the

spreading code of length .

2.2 The MIMO System Matrix in the Frequency Do-

main

The computational complexity of symbol estimation can

be reduced by accomplishing equalization in the fre-

quency domain, using Fourier techniques [4]. For this

purpose, the system sub-matrices are trans-

formed into a block-circulant structure [7] with

a block size , resulting in the modiﬁed system ma-

trix . is chosen here to be . Thus,

a slightly lengthened receive vector is supposed, being

composed of stacked -dimensional

vectors per receive antenna element as

otherwise (10)

A lengthened noise vector is determined accordingly.

Symbol sub-vectors are deﬁned according to (2),

comprising the transmitted symbol streams and

appended zeros. In the sequel, the estimated

symbol sub-vectors contain

the desired symbols plus appended symbols

that need to be skipped. Note, that can also be set to

if a cyclic preﬁx is added to the symbol streams

before transmission. For notational convenience

the choice of is assumed in the following, but

keeping in mind that the transmission of a cyclic preﬁx

reduces capacity.

Now, pre-multiplying with a discrete Fourier transform

matrix and post-multiplying with

yields the expression

.

.

.....

.

.

(11)

where the sub-matrices contain

blocks on its diagonal. denotes the DFT matrix

.

.

..

.

.....

.

.

(12)

and is the corresponding IDFT matrix, with

. For implementation, the DFT and the

IDFT matrices can be replaced by efﬁcient FFT and

IFFT algorithms, respectively.

The block-diagonal matrices contain the

eigen-values of the block-circulant channel sub-matrices

. The sub-matrices can be deter-

mined using the relation [7, 4]

blockdiag (13)

where denotes the ﬁrst column of

and the vector blockdiag contains

blocks of its block-diagonal argument.

With an row permutation matrix

and an column permutation matrix ,

(11) can be block-diagonalized as

...

(14)

yielding the ( )-dimensional sub-matrices

. Combining the Fourier- and the

permutation matrices according to and

, we ﬁnally obtain the block diagonaliza-

tion of as (15)

which can be rewritten as

(16)

Depending on whether the Zero Forcing (ZF) or Mini-

mum Mean-Square-Error (MMSE) equalization criterion

is chosen in the sequel, the following Cholesky decom-

positions are applied:

ZF:

MMSE: (17)

is an upper triangular matrix with ones along the di-

agonal and is a diagonal matrix with the real diago-

nal entries of . Each of these Cholesky decompositions

comprises Cholesky decompositions of ( )-

dimensional matrices. Note, that in the following and

, respectively, are different for ZF and MMSE consid-

erations.

3. LINEAR DETECTION SCHEMES

The well-known linear equalization schemes either use

the ZF or the MMSE criterion in order to minimize:

ZF:

MMSE: (18)

The solution to (18) using (16) is:

ZF:

MMSE: (19)

With the Cholesky decomposition in (17), (19) becomes:

diag

diag

(20)

Here, diag( ) denotes a diagonal matrix containing only

the diagonal elements of its argument and diag denotes

a matrix containing all but the diagonal elements of its

argument.

The Signal-to-Noise-and-Interference Ratio (SNIR) after

symbol detection at the receiver can be expressed as:

(21)

(22)

An estimation of computational complexity of the ZF-

BLE and the in the sequel presented ZF-based iterative

detection schemes is given in Tab. 1. For using the

MMSE criterion, negligible additional computational ef-

fort on estimating the noise power density needs to be

spent.

ZF-BLE Computational complexity

computation of

Cholesky decomposition

Forward-back substitutions

ﬁxed ZF-BDFE additional calculations

in feedback loop

Channel energy ordering

linear SNIR ordering

adaptive SNIR ordering

adaptive ZF-BDFE Computational complexity

computation of

Cholesky decomposition

Forward-back substitutions

in feedback loop

in feedback loop

Table 1 : Analysis of computational complexity for

ZF-BLE and ZF-BDFE

4. ITERATIVE DETECTION SCHEMES

From multi-user detection systems, iterative detection

schemes have been published as Decision-Feedback (DF)

equalization in TD-CDMA systems [1, 5]. Within these

approaches, a Cholesky decomposition of is com-

puted either in the time- or in the frequency-domain. The

iteration procedure is carried out either without any par-

ticular detection order or with an order that has been de-

termined before symbol detection and is kept ﬁxed for

the whole iterative detection process. The advantage of

these techniques is that no additional matrix inversion or

Cholesky decomposition needs to be computed compared

to conventional linear detection systems [1, 5], except the

variable computational effort that needs to be spent in or-

der to obtain the detection order.

Among the different publications on MIMO systems, the

V-BLAST algorithm has earned scientiﬁc interest. This

algorithm is based on an iterative detection of concur-

rently transmitted symbol streams as well. However,

within this approach an optimum detection order has been

found [2], which is based on an adaptive determination of

the symbol stream to be detected and subtracted in each

iteration step. This results in an additional matrix inver-

sion within each iteration step, and thus, in an increased

computational complexity. It will be shown, that using

this approach equals symbol stream-wise DF equaliza-

tion for which the optimum detection order has been de-

termined before symbol detection.

In the sequel, all iterative detection schemes are termed

Block Decision Feedback Equalizer (BDFE), but making

the distinction of using either a ﬁxed or an adaptive detec-

tion order. As already shown in [5, 3], detection and feed-

ing back of past decisions are based on complete symbol

streams rather than single symbols, so that no past de-

cisions on single symbols can be used to remove trail-

ing ISI; past decisions on complete concurrent symbol

streams can only be used to improve MAI cancellation.

4.1 Fixed Detection Order

The input vector is transformed into the frequency do-

main, where matched ﬁltering is performed subsequently,

resulting in

(23)

Thus, in contrast to [5] matched ﬁltering is performed in

the frequency domain, in general resulting in less compu-

tational complexity (cf. Tab. 1).

Using (15), (16) and (17), and additionally multiplying

(23) with the whitening ﬁlter and the scaling ma-

trix , the following expression holds

(24)

and can be rewritten as

(25)

Rewriting (25) and using (17) yields the symbol esti-

mates:

(26)

The operations and

in (26) constitute the feedback operator. Taking into ac-

count the fact that is an upper triangular matrix, the

decision on the symbol vector is now obtained by

feeding back and subtracting past decisions on previously

detected symbol vectors , starting

with the lowest symbol sub-vector of the stacked

symbol vector or . Up to now, the previ-

ously described receiver structure turns into a linear block

equalizer. However, with quantization of the already de-

tected sub-vectors before feeding them back, we ﬁnally

can obtain a performance improvement compared to the

linear block equalizer. The structure of this BDFE with a

ﬁxed detection order is depicted in Fig.1.

In order to determine the SNIR we rewrite the equations

H

ΣH

RT

−1

T

xd

L−I

−1

∆L

−1

Figure 1: Receiver structure for a BDFE with a

ﬁxed detection order

(26) into the following form:

(27)

Note, that the symbol estimate in

(27) still contains the ISI and MAI components

diag , where diag equals its

argument but having zeroed its diagonal elements.

Assuming all past decisions in the iteration process to

be correct and quantized, and thus and

in (27), respectively, the SNIR is equal

to

(28)

(29)

If a past decision is incorrect, error propagation may oc-

cur, whose impairing effect can be reduced by reordering

the sub-vectors in the vector and the correspond-

ing columns in , so that symbols which are most likely

to be detected correctly are arranged at the bottom of the

symbol vector and thus are detected ﬁrst. Three possi-

ble reordering criteria are:

1. Channel energy:

with .

2. SNIR from linear detection:

according to (21) and (22). Thus, the SNIR of the

ﬁrst detected symbol stream equals the SNIR of the

corresponding linear equalizer. This order yields

at least for the ﬁrst detection step the most reliable

symbols to be detected. However, using the SNIR as

a detection criterion requires an additional Cholesky

decomposition.

3. SNIR from adaptive detection:

according to (33) and (34), yielding the opti-

mum detection order [2], that will be used in

Sec. 4.2 as well. However, just for providing the

detection order, matrix inversions but with

decreasing matrix dimensions need to be performed.

4.2 Adaptive Detection Order

Iterative MIMO detection systems with an adaptive de-

tection order are based on a re-design of the receiver ﬁlter

structure within each iteration step. Therefore, additional

variables etc. are introduced in the following,

denoting reduced vectors and matrices, in which already

detected symbols and data belonging to these symbols

are zeroed. For the complementary vectors and matri-

ces, containing the set of already detected symbols and

having zeroed all other entries, and their corresponding

matrices the index is appended.

Transforming the receive vector into the frequency do-

main, we initially obtain

(30)

Subtracting the inﬂuence of already estimated symbols

from (30), multiplying with the adaptively de-

termined ﬁlter structure , and trans-

forming the result back into the time domain yields

(31)

The symbols in (31) are quantized before feeding

them back as well as the already estimated symbols in

Sec. 4.1. For ZF and MMSE, we ﬁnally obtain:

(32)

The structure of the BDFE with an adaptive detection or-

der and the resulting adaptively determined ﬁlter struc-

ture is depicted in Fig. 2. The SNIR is now obtained

under the same assumption as in Section 4.1, where all

quantized past decisions are assumed to be correct, mean-

ing that and

in (32):

(33)

Σ

HH

RT

−1

T

x

adaptive determination

d

red red

red

red

L L

−1 −1

d

Σ

Figure 2: Receiver structure for a BDFE with an

adaptive detection order

(34)

For each symbol stream , the reduced matrices in

(33) and (34) denote those matrices, that have been used

to estimate that particular symbol stream.

As has been shown in [2], the optimum detection order

consists of selecting the symbol stream with the highest

SNIR according to (33) or (34) in each iteration step,

resulting in altogether Cholesky decompositions

according to (17) with decreasing matrix dimensions.

Analogy to the BDFE in Sec. 4.1:

If the BDFE in Sec. 4.1 uses a ﬁxed but optimum symbol

stream-wise detection order that has been determined

before symbol detection, then the detection scheme

originating from the V-BLAST algorithm turns out to

be equivalent to the conventional decision-feedback

equalizer.

Proof:

Without loss of generality, the same entity that deter-

mines the optimum detection order in Sec. 4.1 and the

corresponding vector and matrix rearrangements can be

used here before starting the iterative detection process as

well. Now, in each iteration step the matrix in (33)

and (34), respectively, equals the original block-diagonal

matrix but containing 0s in the last columns in each of

the sub-matrices . These zeroed columns corre-

spond to those in the matrix and belong to already

detected symbol streams . This correspondence be-

tween zeroed columns in and can be illustrated,

rewriting the Cholesky decomposition in (17) for the ZF

case [8]:

(35)

where denotes a unitary transformation matrix with

(cf. Fig. 3) (36)

The matrix can now be determined just by zeroing

the last columns corresponding to already detected sym-

bol streams as well. This statement also holds for the

MMSE case. Thus, the expression

(37)

Lred

X

red

Σ

MT

MQ

RMT

MT

MT

MQ

R

Figure 3: Deduction of the reduced matrices

and from the original matrices and for

a ﬁxed detection order (the light grey parts are ze-

roed)

in (33) and (34) holds for the particular symbol stream

of interest which has just not been detected yet, and

therefore the SNIR in (33) and (34) equals (28)and (29),

accordingly.

In order to proof the overall equivalence between the two

presented iterative equalization schemes, also the error

propagation terms in (27) and (32), which have been ne-

glected for the SNIR considerations, need to be taken into

account. Therefore, the respective term in (32) for the ZF

case can be written as

=

=

=

= .

In the above equation, the expression

(38)

holds, and thus,

(39)

with for the particular symbol

stream which is to be detected. For the MMSE case,

the error propagation terms in (27) and (32) are equal as

well.

5. ENHANCED ITERATIVE DETECTION

In [6], an iterative detection algorithm has been proposed,

in which the estimated symbols after a V-BLAST detec-

tion stage are further processed. Among different strate-

gies, the best performance was obtained by feeding back

after one V-BLAST stage the symbol stream that has

been detected at last in the preceding V-BLAST process.

Subsequently, a further V-BLAST detection process on

the remaining symbols is started, etc. Altogether, the

V-BLAST algorithm is performed -times. If no er-

ror propagation has occurred, and thus all past decisions

have been correct, we ﬁnally obtain estimated sym-

bol streams that are not affected by MAI.

An extension of this receiver architecture for frequency-

selective channels is depicted in Fig. 4, employing suc-

cessive BDFE with an overall (BDFE) com-

plexity. Within the ﬁrst BDFE stage, a certain detection

order needs to be chosen and after the iterative detection

process the symbol stream is fed out, that has been

detected at last. This symbol stream is subtracted from

the receive vector, followed by a further BDFE stage for

which a new detection order for the remaining symbol

streams needs to be determined and from which again the

last detected symbol stream is fed out, and so on. Here,

the lower index of the estimated symbol streams de-

notes the number of the current BDFE detection stage and

the upper index denotes that in this BDFE stage only the

last detected symbol stream is fed out.

H

ΣH−1 d

−1

∆L

−1

H

ΣH

R−1

x d

−1

∆L

−1

Σ

ΣH−1 d

∆−1 ∆−1

(1)

MT

2

1

(1)

(1)

T

M

T

M

T

MT

M

T

M

T

M

2 2 2

2

2

L −I

1

L −I

1 1 1

1 1

T

T

T

T

T2

1

1

2

1

−1

Σ −1

T

T

Figure 4: Receiver structure for the enhanced

BDFE

6. SIMULATION RESULTS

In this section, Monte-Carlo simulations are presented in

order to qualify the Bit Error Ratio (BER) performance

of the iterative MIMO techniques proposed in this pa-

per. The link-level simulation results are based on a

frequency-selective propagation environment and on the

physical layer of a TD-CDMA system according to the

UTRA-TDD mode [9].

Transmission of QPSK symbols with a chip rate of

is assumed, where the spreading gain is set

to . Both transmitter and receiver deploy uni-

form linear antenna arrays with inter-element spacings

, where is the wavelength of a narrow-

band signal with center frequency . The frequency-

selective spatial CIRs are generated by a stochastic direc-

tional channel model, which is parameterized to represent

a rich scattering pico-cellular propagation scenario with

channel length , delay-spread , no

LOS component and scatterers being distributed around

both transmitter and receiver. Channel estimation is per-

formed based on a multi-user ZF-BLE and a midamble

length of chips according to the UTRA-TDD

standard [9].

In Fig. 5 and Fig. 6, the uncoded BER is depicted for

all previously described BDFE, using the ZF and MMSE

criterion, respectively. An antenna constellation with

and has been chosen. As can be

seen, for both the ZF and MMSE criterion, an increas-

ing performance improvement can be achieved for the

BDFE compared to the linear block equalizer, depend-

ing on the effort spent on choosing a certain detection

order. However, the curve for the BDFE with a ﬁxed de-

tection order based on the linear SNIR criterion and for

which only the ﬁrst detected and fed back symbol stream

has been chosen optimally, equals the curve of the BDFE

with the optimum adaptive SNIR detection order. The

best result was obtained using the enhanced BDFE that

employs the adaptive SNIR ordering within each of its

BDFE stages. As mentioned already in Sec. 5, without

error propagation the data streams would not be af-

fected by MAI within this approach, and thus, the BER

curves in Fig. 5 and Fig. 6, respectively, would equal the

single-transmitter bound1.

0 5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0 [dB]

BER

ZF−BLE

ZF−BDFE w/o ordering

ZF−BDFE w/ channel energy ord.

ZF−BDFE w/ lin. SNIR ordering

ZF−BDFE w/ adapt. SNIR ordering

Enhanced ZF−BDFE w/ adapt. SNIR ord.

Single−transmitter bound

Figure 5 : Average uncoded BER versus the average

for different ZF-BDFE in a

MIMO system

0 5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0 [dB]

BER

MMSE−BLE

MMSE−BDFE w/o ordering

MMSE−BDFE w/ channel energy ord.

MMSE−BDFE w/ lin. SNIR ordering

MMSE−BDFE w/ adapt. SNIR ord.

Enhanced MMSE−BDFE w/ adapt. SNIR ord.

Single−transmitter bound

Figure 6 : Average uncoded BER versus the average

for different MMSE-BDFE in a

MIMO system

7. CONCLUSIONS

In this paper, we have presented three categories of

decision-feedback equalization schemes that can be used

in MIMO systems. First, a detection order for symbol

1Equivalent to the single-user bound in multi-user systems.

streams that need to be detected is determined, followed

by the iterative detection process based on an efﬁcient

implementation using triangular Cholesky matrices. For

this category, a negligible additional computational ef-

fort needs to be spent for the detection process com-

pared to a linear equalizer, but having a varying compu-

tational complexity for the determination of the detection

order. Second, an optimum adaptive detection order that

is redeﬁned in each iteration step, well-known from the

V-BLAST algorithm, is used, resulting in an additional

matrix inversion within each iteration step. Third, a re-

ceiver structure was presented, comprising succes-

sive BDFE stages.

It was found, that using an optimum detection order

within the ﬁrst category of equalization schemes exactly

equals V-BLAST based detection systems. Addition-

ally, Monte-Carlo simulations have shown approximately

equal results for the DF system with the optimum de-

tection order and a DF system with a less demanding

detection order. The best performance for both the ZF

and the MMSE criterion was obtained using BDFE

successively. A reasonable computational complexity

was achieved for all presented algorithms by perform-

ing equalization in the frequency domain. The system

model and all considerations in this paper can be applied

to multi-user detection in TD-CDMA systems as well.

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