group detection single short

8/9/2019 Group Detection Single Short

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Reduced Complexity Demodulation of MIMO

Bit-Interleaved Coded Modulation using IQ Group

Detection

Zak Levi and Dan Raphaeli, Senior Member

Abstract

In this paper we propose a novel reduced complexity technique for the decoding of multiple-input multiple-

output bit interleaved coded modulation (MIMO-BICM) using IQ Group Detection (GD). BICM is a well known

transmission technique widely used in practical single-input single-output (SISO) systems and used in the proposed

IEEE standard 802.11n for high speed wireless LAN. It is well known that the decoding complexity of the MAP

detector for MIMO-BICM increases exponentially in the product of the number of transmit antennas and number of

bits per modulation symbol, and becomes prohibitive even for simple schemes. We propose to reduce complexity

by partitioning the signal into disjoint groups at the receiver and then detecting each group using a MAP detector.

Complexity and performance can be traded off by the selection of the group size. Group separation and partitioning

is performed such as to maximize the mutual information between the transmitted and received signal. Simulation

results under both fast Rayleigh fading and Quasi static Rayleigh fading channels show that large SNR gains are

achievable with respect to conventional MMSE per antenna detection schemes. We further propose an iterative

group cancelation scheme using hard decision feedback to enhance performance.

Index Terms: Mutual Information, Group Detection, Maximum A posteriori, Minimum Mean Square Error, Log

Likelihood Ratio, Interference Suppression, Interference Cancelation.

I. INTRODUCTION

Bit interleaved coded modulation (BICM) is a well known transmission technique widely used in

practical single-input single-output (SISO) systems and used in the proposed IEEE standard 802.11n for

high speed wireless LAN. In SISO systems the BICM approach received a lot of attention due to its ability

to exploit diversity under fading channels in a simple way [1]. Inspired by such an approach BICM was

proposed as a transmission technique for multi carrier MIMO systems [2], [3]. MAP decoding of BICM


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transmission involves the computation of the Log Likelihood Ratio (LLR) for each transmitted bit. The

LLR computation is performed using a detector, the complexity of the detector is exponential in the

product of the number of transmit antennas and number of bits per modulation symbol. Even for simple

scenarios this complexity becomes overwhelming. Various techniques have been suggested to reduce the

computational burden of the MAP detector. Most of them can be classified into either list sphere detector

based techniques, or Interference Suppression and Cancelation based techniques.

The list sphere detector [4], [5] was reported to closely approach capacity of the multi antenna fast

Rayleigh fading channel [4], however the list sphere detectors complexity depends on the MIMO channel

and is generally much higher then that of decoding techniques employing Interference Suppression (IS) and

Interference Cancelation (IC). Decoding techniques employing interference suppression and cancelation

were developed in [6], [7], [8], [9]. Such schemes approximate the MAP detector by linear processing

of the MIMO channel outputs followed a per antenna LLR computer . In this paper we propose a Group

Detection (GD) interference suppression based technique. GD was widely studied in the context of Multi

User Detection (MUD) in CDMA systems [10]. The idea is to jointly detect a subset of the transmitted

information while treating the rest of the transmission as noise. Many existing detection techniques can be

regarded as GD based techniques, namely the per antenna detection techniques where each antenna can

be identified as a single group. The authors in [11] used GD in the context of V-BLAST decoding [12]

as a remade for error propagation. In [13] a GD scheme was proposed as a trade off between diversity

gain and spatial multiplexing gain by partitioning the signal at the transmitter into groups. Each group

was encoded separately and per group rate adaptation was performed.

In our work group detection was employed only at the receiver side with no special treatment at the

transmitter. Unlike [7], [8], [9], [11], [13], where a group was defined as a collection of antennas/sub-

channels, we define a group as a collection of In Phase and Quadrature (I or Q) components of the

transmitted symbols, possibly from different antennas. The smallest group is defined as a single I or Q

component of the a transmitted symbol. The GD scheme consists of group partitioning, group separation

and detection. In the proposed GD scheme group separation is performed using a linear operation.

Both separation matrix optimization and group partitioning were derived using an information theoretic


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approach. Under a Gaussian assumption on the transmitted signal, the MMSE detector was identified as

a canonical (information lossless) detector for group detection. A group partitioning scheme was derived

such as to maximize the sum rate. The selection of the group size allows us to tradeoff performance with

complexity. At one end when the number of groups is set to one, the entire transmission is jointly detected

and the scheme coincides with full MAP, while at the other when each dimension is decoded separately,

we show that the scheme coincides with conventional MMSE detection as in [7]. An iterative group

interference canceling technique using hard outputs from the decoder similar to [9] was also investigated.

Finally, performance was evaluated via simulations using a rate 1/2 64-state convolutional code with octal

generators (133,171) and random interleaving. The proposed GD scheme was compared to the full MAP

detection scheme and the conventional MMSE scheme [7], [9] for both fast Rayleigh fading and quasi

static Rayleigh fading channels.

The organization of this paper is as follows. In Section II the system model is presented along with

a review of MIMO-BICM MAP detection. Section III introduces the concept of GD and deals with

group separation and detection. Group partitioning is addressed in Section IV. Iterative group interference

cancelation is discussed in Section V. Simulation results for fast and quasi static Rayleigh fading are

presented in Section VI, and Section VII concludes the paper.

I I . SYSTEM MODEL AND NOTATIONS

We consider a MIMO-BICM system with NT X transmit and NRX receive antennas as illustrated

in Fig. 1. The information bit sequence b = [b1,...,bNb ] is encoded into coded bits which are then

interleaved by a random interleaver. The interleaved bits, denoted by c = [c1,...,cNc ], are mapped onto

an 2m QAM signal constellation using independent I&Q gray mapping. The block of Nc/m symbols

is split into sub-blocks of length NT X . At each instant n a sub-block ac(n) = [ac1(n),...,a

cNT X

(n)]T

is transmitted simultaneously by the NT X antennas. We assume that the transmitted symbols are

independent with a covariance matrix Raa = 2a INT XNT X . The NRX 1 received signal is denoted by

yc(n) = [yc1(n),...,ycNRX

(n)]T and is given by

yc(n) = Hc(n) ac(n) + zc(n) (1)


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Where Hc(n) is the NRX NT X complex channel matrix [hci,j (n)]i=1..NRX ,j=1..NT X and is assumed to

be perfectly known at the receiver (full Channel State Information at the receiver). z(n) is an NRX 1

additive white complex Gaussian noise vector zc(n) = [zc1(n),...,zcNRX

(n)]T with a covariance matrix of

Rzz = 2z INRXNRX . For simplicity we consider the case where NT X =NRX NT. The extension to an

arbitrary number of transmit and receive antennas satisfying NT X NRX is straight forward. Throughout

this paper a vector or matrix with a superscript c means that the vector or matrix is a complex one. The

above system model can be used to describe a multi carrier MIMO system by interpreting the instant index

n as a frequency (subcarrier) index. Each block of Nc/m symbols would correspond to a single multi

carrier symbol with NC/(mNT X ) sub carriers, and H(n) would correspond to MIMO channel experienced

by the nth subcarrier. For the reminder of this paper we omit the instant index n for clarity of notation.

A. Review of MIMO-BICM MAP detection

The decoding scheme is shown in Fig. 1. The MAP detector performs soft de-mapping by computing

the conditional LLR for each coded bit. The conditional LLR for the kth coded bit is given by

LLR(ck| yc, Hc) = log

Pr

ck = 1| yc, Hc

Pr

ck = 0| yc, Hc

(2)

For clarity and ease of notation from here-on we omit the conditioning on the channel matrix Hc

from our

notation. Using Bayes rule and the ideal interleaving assumption, the conditional LLR of the kth coded

bit is given by

LLR

ck| yc

= log

acSk,r1

e

12z

ycHcac2

acSk,r0

e

12z

ycHcac2

(3)

Where Sk,rl CNT is the set of all complex QAM symbol vectors whose kth bit in the rth symbol is

l {0, 1}. The complexity of the LLR computation is 2mNT and is thus exponential in the product of the

M-QAM constellation size and the number of antennas.

III. GROUP DETECTION

Before presenting the group detection scheme let us reformulate the system model using real signals

only. Eq. (1) is transformed into


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yc

Ryc

I

=

HcR H

cI

HcI HcR

acRacI

+

zcRzcI

(4)

The subscripts R or I imply taking the real or imaginary part of the vector or matrix it is associated with.

For clarity of notation for the rest of this paper all real vectors and matrices derived from the complex

ones described in Sec. II will inherit the names of their complex versions without the superscript c. For

example from here-on the vector a refers to

a =real {ac1} ,...,real

acNT

, imag {ac1} , ..., imag

acNT

T

To reduce the complexity of Eq. (3) we propose to use a group detection approach. Instead of jointly

detecting the entire transmission, namely a, we prose to partition a into Ng disjoint groups. For simplicity

we assume that all groups are of equal size M, where M = N/Ng, were N denotes the number of

real dimensions (N = 2NT). The extension to non equal sized groups is trivial. Each group is detected

separately using a MAP detector. Let = {1, 2, . . . N } be the set of indexes of entries in the transmitted

vector a. Define the group partitioning of into disjoint groups Gi such that:Gi ,|Gi| = M andN g

i=1Gi = . For any a

N, the group aGi |Gi| is a subgroup of a that is made up of ak, k Gi.

The channel experienced by the group Gi, namely HGi , is a sub matrix of H and is made up of the

columns of H namely hk such that k Gi. At the receiver the groups are separated by an N N

separation matrix denoted as W. The sub-matrix WGi of size M N, that filters out the ith group out

of the received vector y, is made up of the rows of the matrix W, namely wTk such that k Gi. The

separate detection of real and imaginary parts of the transmitted symbols is possible due to the use of

independent I&Q mapping. The separation scheme is depicted in Fig. 2

A. Group Separation Matrix

Group separation is performed by linear processing of the received samples, namely using a separation

matrix. Given a group partitioning scheme, we propose to optimize the separation matrix W such as to

maximize the sum rate RGD

RGD =N gi=i

I

WGi y; aGi

(5)


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Denote the output of the separation matrix corresponding to group Gi by

aGi WGi y (6)

Eq. (5) is maximized by choosing the group separation sub matrix WGi such as to maximize the mutual

information between each transmitted group and the output of the separation matrixMN

WoptGi = arg maxWGi{I(aGi; aGi)} (7)

From the data processing inequality [18] it follows that

I

y; aGi

I

WGi y; aGi

= I

aGi; aGi

(8)

Thus, if exists a separation matrix WGi that achieves the equality in Eq. (8), then it clearly maximizes

the mutual information in Eq. (7) and in Eq. (5). It is well known [14], [15] that the sub-matrix of the

MMSE separation matrix Wmmse

Wmmse = RayR1yy = H

T

HHT +

2z2a

INN

1(9)

corresponding to the group Gi achieves the equality in Eq. (8). The sub-matrix for group Gi is given by

WmmseGi = HTGi

HHT +

2z2a

INN

1(10)

The separation error covariance matrix is given by

Ree = Raa RayR1yy Rya (11)

and the separation error covariance matrix for the group Gi is a sub-matrix of Ree and is denoted by

ReGi eGi

B. Group LLR Computation

The output of the MMSE separation matrix for group G is given by

aG = Wmmse

G HGaG + Wmmse

G (HGaG + z) vG

(12)

where HG is the N |G| sub matrix of H corresponding to group G and HG is the NG sub matrix

of H corresponding to group G, and G G = . Denote by vG the noise experienced by group G. The

covariance matrix of vG is given by


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RvGvG = Wmmse

G

2a2

HGHTG

+ 2z2

INN

(WmmseG )T

(13)

The conditional LLR for the kth coded bit, where k belongs to the set of coded bits mapped into one of

the symbols belonging to group G is given by

LLR(ck| aG) = log Pr {ck = 1| aG}Pr {ck = 0| aG} (14)To compute Eq. (14) we need the conditional pdf f( aG| aG). From Eq. (12) and under the Gaussian

assumption on the inter group interference [16]

aG| aG N

WmmseG HGaG, RvGvG

(15)

The fact the the noise term in Eq. (12) is colored complicates the evaluation of Eq. (15). We propose to

whiten the noise in Eq. (12). The noise covariance matrix is symmetric positive semi-definite and thus

has the following eigen value decomposition

RvGvG = UGGUTG (16)

where UG is a |G| |G| unitary matrix and G is a |G| |G| diagonal matrix of the eigen values of

RvGvG . The noise whitening matrix is given by

FG = 12

G UTG (17)

The output of the group whitening separation matrix for group G is given by

aG = FGWmmse

G HGaG + vG (18)

where RvGvG = I|G||G|. The conditional LLR is then derived by using Bayes law and the ideal interleaving

assumption. The conditional LLR is given by

LLR( ck| aG) = log

aGSk,rG,1e

12aGFGWmmseG HGaG

2

aGS

k,rG,0

e12aGFGWmmseG HGaG

2 (19)

Sk,rG,l R|G| is the set of all real |G| dimensional PAM symbol vectors whose kth bit in the rth symbol

is l {0, 1}. The complexity of the LLR computation for all groups is Ng2m2|G|

and is exponential in

the product of the group size and the number of bits per real dimension. We are then able to trade off

performance with complexity by the selection of group size.


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C. Simplified LLR Computation for group size of 2

For a group size of Ng = 2 it is possible to derive a simplified closed form approximation for the LLR

without computing the noise whitening matrix in Eq. (17). The derivation is in the spirit of [7] and can

be done for an arbitrary group size. The LLR will be derived given a zero forcing separation matrix. The

MMSE structure will then emerge from the derivation. Consider the group aG of size two, and let the

group G = {i, j}, thus aG = [aiaj]. As in [7] by using the log max approximation [1] the conditional

LLR can be expressed as

LLR

ck| y

maxdSk,rG,1

{logf( aZF| aG = d)}

maxdSk,rG,0

{log f( aZF| aG = d)}(20)

where

aZF = H#y =

HTH

1HTy = a + w (21)

H# is the ZF separation matrix given by the Moore Penrose pseudo inverse of H. The noise covariance

matrix is given by

Rww =12

2z

HTH

1(22)

The output of the zero forcing matrix can be written as

aZF = aiei + aj ej +

k /{i,j}

akek + w (23)

where ei is the N 1 ith unit vector. Under the Gaussian assumption on the inter group interference [16]

follows that

f( aZF| aG) =1

(2)M |G|exp

1

2Q (aG)

(24)

where

Q (aG) =

aZF G

T1G

aZF G

(25)

The mean and covariance in Eq. (24) can be found by using Eq. (23) and are are given by

G

= E{ aZF| aG} = aiei + ajej (26)

G = E

aZF G

aZF G

T

aG

= 1

22a

INN VGV

TG

+ Rww (27)


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where VG = [eiej]. Substituting Eq. (26,27) into Eq. (24) gives

Q (aG) = aTZF

1G aZF 2a

TZF

1G

aiei + ajej

+ a2i e

Ti

1G ei + a

2j e

Tj

1G ej + aiaj e

Ti

1G ej + aiaje

Tj

1G ei(2

Substituting Eq. (28) and Eq. (23) into Eq. (20) and noting that the first term in Eq. (28) is not a function

of aG and thus cancels out we arrive at

LLR

ck| y

mindSk,rG,0

Q (d)

min

dSk,rG,1

Q (d)

(29)

where

Q (aG) = 2aTZF

1G

aiei + ajej

+ a2i e

Ti

1G ei + a

2j e

Tj

1G ej + aiaje

Ti

1G ej + aiaj e

Tj

1G ei (30)

In Appendix. (A) we show that

Q (aG) = 122a (1pjj )pii

(1pii)(1pjj )p2ij

aMMSEipii ai

2 + 122a (1pii)pjj(1pii)(1pjj )p2ij aMMSEj

pjj aj2 +

12

2a

1(1pii)(1pjj )p2ij

aMMSEi pij aj

2+ 1

22a

1(1pii)(1pjj )p2ij

aMMSEj pijai

2+

2apij

(1pii)(1pjj )p2ijaiaj + C

(31)

where C is not a function of aG and will cancel out in Eq. (29) and pi,j is the i,jth element of the matrix

P, and P = WmmseH. In Appendix. (B) we show that 1/pii is the MMSE bias compensation factor [17]

and that the best unbiased linear estimate of aMMSEi from aj is pijaj . Thus in Eq. (31) we identify four

mean square error terms and a correlation term coupling between the two group elements.

D. Simple Antenna Partitioning (group size of 1)

For i, j taken as the I and Q of the same transmit antenna it is easy to show that at the output of the

MMSE separation matrix

E

ammsei ammse

j

= E{ammsei aj} = E

ammsej ai

= 0 (32)

. From Eq. (32) follows that the detection of the ith and jth element can be done separately. Thus simple

antenna partitioning and per antenna detection is equivalent to GD scheme with group size of |G| = 1.

Using Eq. (32) it is easy to show that for i, j taken as the I and Q of the same transmit antenna follows

that pij = 0 and pii = pjj . In Appendix. (B) we show thatpii

1pii= SN RMMSE U,i

By denoting the unbiased mean square error by

ij =

aMMSEi

pii d1

+j

aMMSEj

pii d2

(33)


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Eq. (31) coincides with the LLR in [7] namely:

LLR

ck| y

SN RMMSE U,i

min

dSk,r{i,j},0

2ij (d) mindSk,r

{i,j},1

2ij (d)

(34)

IV. GROUP PARTITIONING

The number of ways to partition the transmitted signal into groups is a function of the transmitted

signal size N and the groups size M. For example when both N and M are powers of 2 the number of

partitioning possibilities is given by Eq. (35)

NP =12

N

N/2

12

N/2N/4

. . . 12

2MM

=12

log2(N/M) N!M!

log2(N/M)i=1

(N2i)!(35)

Thus for example there are three ways to partition a two antenna scheme into groups of size 2, and

105 ways to partition a four antenna scheme into groups of size 2. Eq. (35) suggests that the number of

partitioning schemes grown combinatorially in number of transmit antenna. We are faced with the problem

of choosing a partitioning scheme from amongst all the partitioning possibilities. A natural selection would

be to choose the partitioning scheme that minimizes some probability of error measure, this although very

intuitive is very difficult to trace analytically. Instead we propose to select the partitioning scheme that

maximizes the sum rate of all the groups. By using the chain rule of mutual information the mutual

information of the MIMO channel can be written as Eq. (36)

I

y; a

= I

y; aG1

+ I

y; aG2

aG1 + Iy; aGNg aG1 , . . . , aGNg1 (36)When using the GD scheme information is not exchanged between groups, and so the mutual information

of Eq. (36) cannot in general be realized. The mutual information or sum rate RGD given the GD scheme

is given by Eq. (37)

RGD =

N gi=1 Iy; aGi Iy; a (37)The sum rate is simply the sum of mutual information of the individual groups since the groups are disjoint.

By writing the mutual information as a difference in entropies and using the fact that the covariance of

the MMSE estimator of aGi from y equals the covariance of the estimation error, namely R aGi |y aGi |y=

ReGi eGi[15] and by the assumption that the Inter Group Interference is Gaussian [16], and thus aGi

y isalso Gaussian, follows that


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RGD =N gi=1

h

aGi

1

2

Ngi=1

logReGi eGi

(38)Under the assumption that transmitted symbols are i.i.d the first term in Eq. (38) does not depend on the

partitioning scheme, the determinants of the error covariance matricesReGi eGi

, i = 1 . . . N g are a function

the partitioning scheme. Given a partitioning scheme {G1, G2, . . . GN g} the error covariance matrix for

the ith group ReGi eGiis obtained from the covariance matrix Ree by deleting the kth rows and the kth

columns k / Gi. Thus finding the partitioning scheme that maximizes the mutual information in Eq. (38)

is equivalent to finding the partitioning scheme that minimizes the product of the the determinants of the

group error covariance matrices. Thus we need to solve the following optimization problem

{Gopt1 , Gopt2 , . . . GoptNg } = arg min{G1, G2, . . . GNg }s.t : Gi , |Gi| = MGi Gj = i = j

N gi=1

ReGi eGi (39)

The complexity of solving Eq. (39) using brute a force search quickly becomes overwhelming (Eq. (35)).

To reduce the complexity of Eq. (39) we turn to the structure of Ree. Since Ree is a real covariance matrix

it is obviously symmetric positive semi-definite, however its structure goes deeper due to the symmetry

in the real channel matrix H (Eq. 4). This structure can be utilized to greatly simplify Eq. (39) for the

2 2 scheme. Intuition form the simplified expressions for the 2 2 scheme will then lead us to develop

simple suboptimal ad hoc approximations to (Eq. (39).

A. Simplified Partitioning for 2 2 scheme

For the 2 2 antenna scenario the MMSE error covariance matrix (Eq. (11)) is given by:

Ree =12

2a I44 H

T

HHT +

2z

2aI44

1H (40)

In Appendix. (C) we prove that

Ree =2

2a

1

ad b2 c2

d b

b a0 cc 0

0 cc 0

d bb a

(41)

where hi denotes the ith column of H and the scalars ,a,b,c,d are given by

= 2z

2aa = 1 + hT1 h1 d = 1 + h

T2 h2 b = h

T1 h2 c = h

T1 h4 (42)


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There are three possible partitioning schemes, the first is such that G1 = {1, 2}, G2 = {3, 4}, the second

is such that G1 = {1, 3}, G2 = {2, 4} and the third is such that G1 = {1, 4}, G2 = {2, 3}. By substituting

Eq. (41) into Eq. (39) it is easy to show ifhT1 h2 > hT1 h4 choose scheme 1 else chose scheme 3. Scheme

2 will never be better then scheme 1 or 2, this is no surprise since in Sec. III-D we showed that per

antenna detection is equivalent to group detection of group size |G| = 1. The above result is simple to

compute and very intuitive. Each one of the columns of the channel matrix H can be regarded as the

channel experienced by a single element of the transmitted signal. Elements are thus grouped together

based on the correlation between their corresponding channels. This is intuitive since grouping correlated

elements will result in less noise enhancement in the group separation process.

B. Ad hoc Simplified Partitioning

In general for NT > 2 obtaining a closed form expression for the noise covariance matrix as was done

for NT = 2 is hard to tackle. Instead we draw intuition from the 2 2 scenario and propose ad hoc greedy

algorithms for the partitioning of a general NT NT system into groups of size M. Denote the nth group

by Gn = {in1 , in2 ...,i

nM} We propose to partition the groups according to the heuristical correlation like

measure in Eq. (43) iGn1 , i

Gn2

= arg maxk,l

hTk hliGn3 = arg maxk

hTiGn1 hk +

hTiGn2 hk

...

iGnM = arg maxk

hTiGn1 hk +

hTiGn2 hk + . . . +

hTiGnM1

hk

(43)

The correlation like measure is built in the following fashion. Since the algorithm is based on correlations

between pairs of elements a candidate list of all possible pairs is formed. The list is denoted by n.

At the beginning of the algorithm, due to the symmetry in the HTH matrix, the list 1 is initialized

to 1 = {(k, l) : k < l}. First a pair of elements with maximal correlation is added to the group,

the third element is selected such that maximizes the sum of correlations with the already selected

pair. The fourth element is selected such as to maximize the sum of correlations with respect to

the pre selected triplet. The same procedure is repeated in the same fashion until the adding of the

Mth element. After the partitioning of a group is done the candidate list is updated by excluding all


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pairs that involve elements that have already been grouped. The partitioning algorithm is given by:

Simplified partitioning algorithm for a group size of M

1) n = 1, n = {(k, l) : k < l}

2) hk,l = hTk hl , (k, l) n

3) [iGn1 , iGn2 ] = arg max(k,l)n (hk,l)

iGn3 = arg maxk:(iGn1 ,k)&(iGn2 ,k)n&k={i

Gn1 ,i

Gn2 }

hiGn1 ,k

+ hiGn2 ,k

iGn4 = arg maxk:(iGn1 ,k)&(i

Gn2 ,k)&(i

Gn3 ,k)n&k={i

Gn1 ,i

Gn2 ,i

Gn3 }

hiGn1 ,k

+ hiGn2 ,k+ hiGn3 ,k

...

iGnM = arg maxk:(iGn1 ,k)&(iGn2 ,k)&&(i

GnM1

,k)n&k={iGn1 ,i

Gn2 ,i

GnM1}

hiGn1 ,k

+ hiGn2 ,k+ . . . + hiGn

M1,k

4) Gn = iGn1 , iGn2 , . . . , iGnM 5) n = {(k, i

Gnm )m=1...M(i

Gnm , k)m=1...M : k : (k, i

Gnm )m=1...M, (i

Gnm , k)m=1...M n}

6) n+1 = n \ n

7) if + + n Ng goto 3 else end The matrix HTH is a byproduct from the computation of Wmmse

(see Appendix. (A)) and needs not to be recomputed in stage 2. The above algorithm is very simple and

its complexity is that of finding the maximum entry from a list. At each one of the Ng iterations of the

algorithm the list size decreases drastically. This is a tremendous reduction compared the combinatorial

complexity in Eq. (12). In Section VI the performance degradation of the above simplified partitioning

was measured with respect to optimal partitioning via simulations under a fast Rayleigh fading channel,

and was found to be less than 0.35[dB] for partitioning a 4 4 into groups of size 2 and 4.

V. ITERATIVE GROUP INTERFERENCE CANCELATION

The detector in Eq. (3) does not exploit dependencies between coded bits which leads to degraded

performance. The detector in Eq. (19) is an approximation to Eq. (3) and is even more information lossy

since information is not exchanged between groups. An optimal decoder would regard the channel code

and MIMO channel as serially concatenated codes and would decode them jointly, such a decoder would

have extraordinary complexity. Many authors [4], [9], [8], [19], [20] propose to use iterative schemes

since it has been shown that such schemes are very effective and computationally efficient in other


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joint detection/decoding problems [21], [22]. The iterative scheme proposed here uses hard decisions

from the decoder. The motivation for the use of hard decisions comes from the fact that hard output

decoders are commonly implemented in many practical systems and are much less complex then soft

output decoders. For each group namely group G, hard decoded bits from the decoder are re-encoded,

re-interleaved and used to reconstruct a version of the transmitted MIMO symbol from all symbols but

the ones corresponding to group G. This reconstructed signal is then passed through the effective MIMO

channel. Group interference canceling is performed by subtracting the filtered reconstructed signal from

the true received signa. The signal after Interference Cancelation is given by:

ylG

= HGaG + HG

aG alG

e G

+z (44)

The superscript l in Eq. (44) denotes the iteration number. Assuming correct decisions alG = aG the

above expression is further simplified.

ylG

= HGaG + z (45)

A. Group Partitioning and Iterative Group Detection

The partitioning into groups for the iterative stage introduces a new trade off with respect to the

original group partitioning. In the first part of the decoding process we traded off decoding complexity

with performance, where larger groups resulted in better performance and higher complexity. After the

first decoding pass we have hard estimates for all bits. If one partitions the signal into large groups then

one is using less new information and at the extreme not using any new information when no partitioning

is done thus only one group (MAP decoding). On the other hand if one partitions the signal into very

small groups one may be more susceptible to error propagation since one only has hard estimates of the

decoded bits with no reliability measure. Nevertheless we propose to partition into groups of size |G| = 1

(per antenna partitioning). Note that the partitioning scheme in Sec. IV is no longer relevant since it does

not take into account the new information from the initial stage. The signal after IC Eq. (45) can be

detected using Eq. (34). Denote the matrix P (See Eq. (B-1)) for the Gth group by PG

PG = HTG

HGH

TG +

2z2a

INN1

HG (46)


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15

Since antenna partitioning is performed it is more convenient to use the complex signal model in Eq. (1).

Eq. (46) can be simplified by using complex signal notation and by using the matrix inversion lemma 1

and thus it is easy to show assuming that group G corresponds to the symbol transmitted from antenna

i, that

PG =

2a2z hcG2

1 + 2a

2zhcG

2

I22 (47)

where hcG is the ith column of the complex channel matrix Hc. Again using the matrix inversion lemma

it is easy to show that the complex MMSE separation matrix WMMSEc

G coincides with MF or Maximum

Ratio Combiner (MRC) and is given by

WMMSEc

G = (hcG)

H hcG (hcG)H + 2z

2a

INTNT1

=

2a2z

(hcG)H

1 +2

a2z hcG2

(48)

The MMSE estimate after IC is then given by

acmmse

G =

2a2z

(hcG)H

1 + 2a

2zhcG

2

yc HcGa

cG

(49)

The LLR for the group G can then by computed by using Eq. (34) where

SN RMMSE U,i =pG11

1 pG11=

2a

2zhcG

2(50)

were and pG11 is the 1, 1 element of the matrix PG ,and

i,j (dc) =

acmmse

G

pG11 dc

=(h

cG)

H

hcG2

yc HcGa

cG

dc

(51)At the end of each iteration one obtains hard decoded bits that can be used by the next iteration.

Simulation results in chapter VI suggest that performing two iterations achieves most of the performance

gain.

V I. SIMULATION RESULTS

The performance of the GD scheme for MIMO-BICM was evaluated via Monte-Carlo simulations.

At the transmitter blocks (packets) of 2000 information bits were encoded and interleaved using a rate

1/2 64 state convolutional encoder with octal generators (133, 171) followed by a random per packet

interleaver. Two antenna configurations were considered, a 2x2 configuration and a 4x4 configuration. For

1

A + uv

H

1

= A1

1+vHA1uA1

uvH

A1


16/24

16

the 2x2 configuration the detection schemes considered were full MAP detection, Per Antenna detection

(|G| = 1) and optimal search GD with |G| = 2 all with zero, one and two hard iterations. Most of

the performance gain due to iterations was achieved after two iterations. For the 4x4 configuration we

considered the partitioning into Ng = 8 groups of size |G| = 1 each (PA GD), the partitioning into Ng = 4

groups of size |G| = 2 each and the partitioning into Ng = 2 groups of size |G| = 4 each. The detection

schemes considered were full MAP detection (only for fast fading), Per Antenna group detection (PA GD

- conventional MMSE), Optimal Search GD (OS GD), Simplified Search GD (SS GD) all with zero, one

and two iterations. The complex MIMO channel matrix entries were drawn from a zero mean complex

Gaussian distribution with variance 1/NT in an iid fashion. Simulation results were summarized via average

Bit Error Rate (BER) and average Packet Error Rate (PER) versus SNR2 plots.

A. Fast Fading

For fast fading the MIMO channel was independently generated at each instant. Fig 3 presents simulation

results for the 2x2 configuration for both 16 and 64QAM. Table I summarizes the gain of the MAP scheme

over GD, the gain of GD over PA and the gain due to iterations for each one of the schemes. The gain was

measured at a BER of 104 - 105. The gains in Table I correspond to both 16 and 64QAM since they

were found to be similar. Performing more than two iterations did not show much gain. Fig 3 suggests

that the GD gain over PA increases with the SNR. Without iterations GD shows a substantial gain over

PA. Performing iterations closes the gap between GD and full MAP as well as the gain of GD over PA.

Fig 4 summarizes simulation results for the 4x4 configuration for 16QAM. Table II presents a

comparison between the various GD schemes and the MAP scheme as well as a comparison between

GD with group size of |G| = 2 to that of GD with a group size of |G| = 4, the gain due to iteration is

also included. The results show that performing iterations reduces the gap between MAP, the various GD

schemes and PA.Fig 5 presents simulation results for the 16QAM 4x4 configuration for the various GD

schemes with the simplified group partitioning (SS GD) algorithms. Results were compared to those of

the Optimal Search partitioning (OS GD). The simplified partitioning into groups of size |G| = 2 (See

IV-A) showed a loss of no more then 0.2[dB] with respect to optimal partitioning, the loss after one

2The SNR is defined as E

Ha2

E

z2

= 1

2z


17/24

17

iteration dropped to 0.1[dB]. The simplified partitioning into groups of size 4 (See IV-A) showed a loss

of no more then 0.35[dB] with respect to the optimal partitioning, the loss after one iteration remained

around 0.35[dB].

B. Quasi Static Fading

For quasi static fading the MIMO channel remained constat over a duration of a block and changed

independently from block to block. Fig 6 presents simulation results for 16QAM 2x2 configuration with

simplified partitioning. Table III presents a comparison between the gain of MAP scheme over GD, the

gain of GD scheme with respect to PA scheme as well as the gain due to iterations for each one of the

schemes all at a PER of 102-103 all with simplified partitioning (See IV-A).

Table IV summarizes results for 16QAM 4x4 configuration with simplified partitioning as well as a

comparison between the various GD schemes at a PER of 102-103.

V II. CONCLUSIONS

In this paper we proposed a scalable reduced complexity detection algorithm for MIMO-BICM.

Complexity reduction was achieved by performing detection in groups instead of joint detection of

the entire MIMO signal. A simple group partitioning algorithm was derived as well as a approximate

expression for the LLR for group size of |G| = 2. Performance and complexity were shown to be traded

off by the selection of the group size, and the detection complexity was shown to be roughly Ng2m2|G|

.

Computer simulations showed that group detection with a group size of |G| = 2 achieves gains of 1-4[dB]

with respect to Per Antenna detection. Per Antenna detection was identified as a special case of group

detection where |G| = 1. Gains of up to 10[dB] were achieved by using a group size of |G| = 4. A simple

hard iterative interference canceling scheme was further proposed to enhance performance. Performing

hard iterations improved performance of all the schemes as well as reduced the gaps between them.

APPENDIX

APPENDIX A - CALCULATION OF Q (aG)

To compute Q (aG) we first derive a closed form expression for 1G . Define

P =

INN +2z2a

HTH

11(A-1)


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18

And note that

G =12

2a

INN +2z2a

HTH

1

P1

+VG (I22) VT

G

(A-2)

Then make use of the matrix inversion lemma 3

1G =12

2aPINN + VG I22 VTG P VG1

T

VTG P (A-3)

Noting that

T =

I22 VT

G P VG1

=

1 pjj pij

pij 1 pii

(1 pii) (1 pjj ) p2ij(A-4)

where pij is the i,jth element of P in Eq. (A-1). To compute the first term in Eq. (30) we evaluate

aTZF1G ei =

12

2aaTZFP

INN + VGT VT

G Pei (A-5)

Noting that P converts ZF estimation into MMSE estimation

aTZFP =

PTaZFT

=

INN +

2z2a

HTH

11 HTH

1HTy

T

=

HTH+ 2z

2aIN

1HTy

T=

WmmseyT

= aTMMSE

(A-6)

The forth equality in Eq. (A-6) follows from the following Eq. (A-7)HTH+

2z

2aI1

HT = 2a

2z

2a2z

HTH+ I1

HT = 2a

2z

I HT

HHT +

2z

2aI1

H

HT

= 2a

2zHT

I

HHT +

2z

2aI1

HHT

= 2a

2zHT

2a2z

HHT + I1

= HT

HHT + 2z

2aI1

= Wmmse(A-7)

The second and forth equalities in Eq. (A-7) follow from the matrix inversion lemma while the rest are

trivial. Substituting Eq. (A-6) into Eq. (A-5) yields

aTZF1G eiai =

12

2aaTMMSE

ei + VGT V

TG P ei

ai =

12

2a(1pjj )a

MMSEi +pij a

MMSEj

(1pii)(1pjj )p2ijai

aTZF1G ej aj =

12

2aa

TMMSE

ej + VGT V

TG P ej

aj =

12

2a

pij aMMSEi +(1pii)aMMSEj

(1pii)(1pjj )p2ij

aj

(A-8)

The last four terms in Eq. (30) evaluate to

a2i eTi

1G ei =

12

2aa2i e

Ti P ei +

12

2aa2i e

Ti P VGT V

TG P ei =

12

2apii(1pjj)+p

2ij

(1pii)(1pjj )p2ija2i

a2j eT

j 1G ej =

12

2aa

2j e

Tj P ej +

12

2aa

2j e

Tj P VGT V

TG P ej =

12

2a

pjj (1pii)+p2ij

(1pii)(1pjj )p2ija2j

aiajeTi 1G ej = 122aaiajeTi P ej + 122aaiaj eTi P VGT VTG P ej = 122a pij

(1pii)(1pjj)p2ijaiaj

aiajeT

j 1G ei =

12

2aaiaj e

Tj P ei +

12

2aaiaj e

Tj P VGT V

TG P ei =

12

2a

pij(1pii)(1pjj)p2ij

aiaj

(A-9)

Substituting Eq. (A-6,A-8) into Eq. (A-3) yields

Q (aG) =12

2a(1pjj )pii

(1pii)(1pjj )p2ij

aMMSEi

pii ai

2+ 1

22a

(1pii)pjj(1pii)(1pjj )p2ij

aMMSEj

pjj aj

2+

12

2a

1(1pii)(1pjj )p2ij

aMMSEi pij aj

2+ 12

2a

1(1pii)(1pjj )p2ij

aMMSEj pijai

2+

2apij

(1pii)(1pjj )p2ijaiaj + C

(A-10)

3A1 A1B

D1

+ CT

A1

B

1

CT

A1

=

A + BDC

T

1


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19

APPENDIX B - CONNECTING P WITH MS E

From the matrix inversion lemma follows that

P =

INN +2z2a

HTH

11= HT

HHT +

2z

2aINN

1H (B-1)

From Eq. (11) and Eq. (A-1) then follows that Ree =2a2 (INN P) Since Ree is the MMSE error

covariance matrix its diagonal elements are the MSEMMSE in the estimation of each element of a. The

unbiased SNR (See [17]) of the ith element ai is given by SN RMMSE U,i =pii

1pii. The bias compensation

scaling factor is given by 1pii

. We next prove that the best unbiased linear estimate of aMMSEi from aj is

pijaj . It is easy to show that follows that

pij = 2

2aEai a

MMSEi aj a

MMSEj =

22a

EaMMSE

j ai aMMSEi

22a

E{aiaj} +

22a E

aMMSEi aj (B-2)The first term in the second equality is zero from the orthogonality principle and the second term

in the second equality is zero since transmitted symbols are statistically independent. Thus pij =

22a

E

aMMSEi aj

= 22a

E

aMMSEj ai

From linear estimation theory follows that aMMSEi (aj) =

E{aMMSEi aj}E{a2j }

aj =2

2aE

aMMSEi aj

aj = pij aj . Since both aj and aMMSEi are zero mean follows that

the best unbiased linear estimate of aMMSEi from aj is pij aj . The same proof can be repeated for aMMSE

j .

APPENDIX C - COMPUTING Ree FOR THE 2 2 SCHEME

Using the matrix inversion lemma follows that

Ree =12

2a

I44 H

T

HHT + 2z

2aI44

1H

= 122a

I44 +

2a2z

HTH1

(C-1)

Denoting the ith row of H by hTi follows that

HTH =

hT1 h1 h

T1 h2 0 h

T1 h4

hT1 h2 hT2 h2 h

T1 h4 0

0 h

T

1 h4 h

T

1 h1 h

T

1 h2hT1 h4 0 hT1 h2 h

T2 h2

(C-2)

Denote = 2z

2aand denote the scalars a,b,c,d

a = 1 + hT1 h1 d = 1 + hT2 h2 b = h

T1 h2 c = h

T1 h4 (C-3)

The inverse of the matrix in Eq. (C-1) can be computed in closed form by noting that the matrix in

Eq. (C-1) has the following block symmetry Ree =12

2a

A B

BT A

1And then using the block matrix


20/24

20

inversion lemma 4 follows

Ree =2

2a

1

ad b2 c2

d b

b a0 cc 0

0 cc 0

d bb a

(C-4)

REFERENCES

[1] G.Caire, G. Taricco and E. Biglieri, Bit-interlived coded modulation , IEEE Trans. Inf Theory, vol. 44, pp. 927-946,

May 1998.

[2] J. Boutrus, F. Boixadera, and C. Lamy, Bit-Interleaved coded modulations for multiple-input multiple-output channels,

Proc. of the IEEE 6th ISSSTA00, New Jersey, Sept. 2000.

[3] S. H. Muller-Weinfurtner, Coding approaches for multiple antenna transmission in fast fading and OFDM, IEEE Trans.

Signal Proces., vol.50, pp 2442-2450, Oct. 2002.

[4] B. Hochwald and S. ten Brink, Achieving near-capacity on a multipleantenna channel, submitted to the IEEE Transactions

on Communications, July 2001.

[5] J. Boutros, N. Gresset, L. Brunel and M. Fossorier Soft-input, soft-output lattice sphere decoder for linear channels

IEEE GLOBECOM 2003, San Francisco, USA

[6] Michael R.G Butler, and Iain B. Collings, A Zero-Forcing Approximate Log-Likelihood Receiver for MIMO Bit-

Interleaved Coded Modulation, IEEE Commun Letters. Vol 8, no 2, pp 105-107, Feb 2004.[7] D. Seethaler, G. Matz, and F. Hlawatch, An Efficient MMSE-Based Demodulator for MIMO Bit-Interleaved Coded

Modulation, IEEE Globecom 2004 Dallas, Texas ,Nov/Dec 2004.

[8] M. Witzke, S. Baro, F. Schreckenbach, and J. Hagenauer, Iterative. detection of MIMO signals with linear detectors, in

Proc. Asilomar Conference on Signals, Systems and Computers (ACSSC), Pacific Grove, CA USE, IEEE, Nov. 2002, pp.

289 -293.

[9] K.B Song, and S.A Mujtaba,A Low Complexity Space-Frequency BICM MIMO-OFDM System for Next-Generation

WLANs, Globecom 2003.

[10] M. K. Varanasi, Group detection for synchronous Gaussian code-division multiple-access channels, IEEE Trans. Inf.

Theory, vol. 41, no4, pp 1083-1096, Jul. 1995.

[11] W.J Choi, R. Negi and J. Cioffi, Combined ML and DFE Decoding for the V-BLAST Systems, IEEE International

Conference on Communications 2000.

[12] G.J. Foschini, Layered space-time architecture for wireless communications in a fading enviroment when using multi-

element antennas, Bell Labs. J, vol. 1, no 2, pp 41-59, Autumn 1996.[13] Sana Sfar, Lin Dai, Khaled B.Letaief, Optimal Diversity-Multiplexing Tradeoff With Group Detection for MIMO Systems,

IEEE TRans Commun, Vol. 53, No 7, July 2005.

[14] G.D. Forney, Jr. Shannon meets Wiener II: On MMSE estimation in successive decoding schemes, To appear in Proc.

2004 Allerton Conf. (Monticello, IL), Sept. 2004, Vol 43, No 3, May. 1997.

[15] W. Yu and J. Cioffi, The sum capacity of a Gaussian vectorbroadcast channel, IEEE Trans. Inform. Theory, submitted

for publication

[16] H. Vincent Poor and Sergio VerdU Probability of Error in MMSE Multiuser Detection, IEEE Trans Infomation Theory,

Vol 43, No 3, May. 1997.

[17] J.M. Cioffi, G.P. Dudevoir, M.V. Eyuboglu and G.D. Forney, , MMSE decision-feedback equalizers and coding - Part

I:Equalization results IEEE Trans. Commun, vol 43, no 10, pp. 2582-2594, Oct 1995.

[18] Thomas M. Cover and Joy A. Thomas , Elements of Information Theory. John Wiley, 1991.

[19] Boronka, A. Rankl, T. Speidel, J.Iterative Nonlinear Detection of MIMO Signals using an MMSE-OSIC Detector and

Adaptive Cancellation, ITG Conference on Source and Channel Coding (SCC), Erlangen, Germany, January 2004, pp.17-24. IEEE Trans Infomation Theory, Vol 43, No 3, May. 1997.

[20] W. J. Choi, K. W. Cheong, and J. Cioffi Iterative soft interference cancellation for multiple antenna systems, IEEE

Wireless Communications and Networking Conference, 2000

[21] Xiadong Li and J.A. Ritcey, Bit Interleaved Coded Modulation with Iterative Decoding, IEEE Communications Letters,

vol. 1, No. 6, Novermber 1997, pp. 169-171.

[22] S. Ten Brink, J. Speidel and R. H. Yan, Iterative demapping and decoding for multilevel modulation, Proc. of the

IEEEGlobal Telecommunications Conference, vol. 1, pp. 579-584, November 1998

4

A B

CT

D

1=

E1 E1BD1

D1BTE1 D1 + D1CTE1BD1

E = A BD1CT


21/24

21

bi

ciRandom

InterleaverBinarySource

EncoderSymbol

Mapper (Gray)

c

NTX

a

1

ca

S/P

TX

Hc

Detector(Bit LLR)

1

cy

c

NRXy

DeInterleaver Decoder

bi

RX

Fig. 1. MIMO-BICM NRX x NTX System Model.

y

1Ga

LLR

LLR

LLR

P/S Deinterleaver Decoder

b

Group Separation

Group Detection

2Ga

NgG

a

M x 1

M x 1

M x 1

N x 1

N x 1

N x 1

1GW

2GW

NgGW

Fig. 2. Group Detection Scheme.

Gain [dB] @BER MAP/GD GD/PA

104 105

No Iter 0.2-0.3 0.8-1.71 Iter 0.1 0.4-0.8

2 Iter 0.1-0.3 0.35

MAP Iter Gain GD Iter Gain

1 Iter 1-1.3 1-1.5

2 Iter 1 0.8-1

PA Iter Gain

1 Iter 2-2.5

2 Iter 1-1.5

TABLE I

FAS T RAYLEIGH FADING MAP,GD,PA COMPARISON AT BE R 104 - 105 , 2 2 SCENARIO


22/24

22

3[f]

6 8 10 12 14 16 18 20 22 24 26

10

-4

10-3

10-2

10-1

Snr

Ber

Ber(Snr) Mimo (2,2) 64QAM Conv K=7 ,Rate = 1/2

16QAM Coded PA Detection

16QAM Coded GD Detection

16QAM Coded Map Detection

16QAM Coded IPA Detection

16QAM Coded IGD Detection

16QAM Coded IMAP Detection

16QAM Coded I2PA Detection

16QAM Coded I2GD Detection16QAM Coded I2MAP Detection








64QAM Coded I2GD Detection

64QAM Coded I2MAP Detection

16QAM

64QAM

2 Iteration

0 Iteration

1 Iteration

2 Iteration

0 Iteration

1 Iteration

Fig. 3. 2 2 16QAM,64QAM Fast Fading Rayleigh.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2210

-5

10-4

10-3

10-2

10-1

Snr

Ber



16QAM Coded 4G_PA_GD Detection

16QAM Coded 2G_PA_GD Detection16QAM Coded 4G_OS_GD Detection

16QAM Coded 2G_OS_GD Detection


16QAM Coded 4G_PA_IGD Detection

16QAM Coded 2G_PA_IGD Detection16QAM Coded 4G_OS_IGD Detection

16QAM Coded 2G_OS_IGD Detection

16QAM Coded 4G_PA_I2GD Detection

16QAM Coded 2G_PA_I2GD Detection

16QAM Coded 4G_OS_I2GD Detection16QAM Coded 2G_OS_I2GD Detection


0 Iteration

1 Iteration

2 Iteration

Fig. 4. 4 4 16QAM Fast Fading Rayleigh.


23/24

23

5 10 15 2010

-5

10-4

10-3

10-2

10-1

Snr

Ber


16QAM Coded 4G_SS_GD Detection


16QAM Coded 2G_SS_GD Detection


16QAM Coded 4G_SS_IGD Detection


16QAM Coded 2G_SS_IGD Detection


0 Iteration

1 Iteration

Fig. 5. 4 4 16QAM Fast Fading Rayleigh Optimal Search Vs Simple Search.

Gain [dB] @BER |G|=2 MAP/GD |G|=2 GD/PA10

4 105

No Iter 1.5-2 1-2

1 Iter 1 0.5

2 Iter 0.6 0.3

|G|=4 MAP/GD |G|=4 GD/PANo Iter 1 1.5-2

1 Iter 0.4-0.8 0.7

2 Iter 0.3-0.5 0.5

GD,PA |G|=4/2 Iter GainNo Iter 1-2

1 Iter 0.4-0.7 1.5-4.5

2 Iter 0.1-0.3 1-2

TABLE II

FAS T RAYLEIGH FADING MAP,GD,PA COMPARISON AT BE R 104 - 105 , 4 4 SCENARIO

Gain [dB] @PER MAP/GD GD/PA

102 103

No Iter 4-8 3-4

1 Iter 4-5 2-4

2 Iter 2-3 2

MAP Iter Gain GD,PA Iter Gain

1 Iter 1-2 3-4

2 Iter 0.5-1 1

TABLE III

QUASI STATIC RAYLEIGH FADING MAP,GD,PA COMPARISON AT BE R 102 - 103 , 2 2 SCENARIO


24/24

24

Gain [dB] @PER |G|=2 GD/PA |G|=4 GD/PA10

2 103

No Iter 3-4 1.5-2

1 Iter 3.5-4 1.5-2

2 Iter 2.5-3 1-1.5

PA |G|=4/2 SS |G|=4/2No Iter 7-10 5-8

1 Iter 4-7 3-5

2 Iter 3-4 2-3

|G|=2 Iter Gain |G|=4 Iter Gain1 Iter 5-6 2.5-3.5

2 Iter 2.5-3.5 1.5-2

TABLE IV

QUASI STATIC RAYLEIGH FADING GD,PA COMPARISON AT PE R 102 103 , 4 4 SCENARIO

5 7 9 11 13 15 17 19 21 23 25 27 29 31 3310

-3

10-2

10-1

100

Snr

Ber









16QAM Coded I2GD Detection


Fig. 6. 2 2 16QAM Quasi Static Fading Rayleigh.

group detection single short

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