Low-complexity prediction techniques of K-best sphere decoding for MIMO systems

Low-complexity Prediction

Techniques of K-best

Sphere

Decoding for MIMO Systems

Hsiu-Chi Chang, Yen-Chin Liao,

and Hsie-Chia

Chang

Department

of Electronics

Engineering

National Chiao

Tung University,

1001

Ta-Hsueh

Road,

Hsinchu,

Taiwan,

R.O.C.

Tel: +886-3-5712121 ext.54246 email:

jasper.ee94ggnctu.edu.tw

Abstract- In multiple-input multiple output (MIMO) systems, search

approach.

Thus the value ofK should be

large

enough,

maximum likelihood (ML) detection can provide good perfor- and the value K dominates the performance and computation

mance, however, exhaustively searching for the ML solution complexity.

becomes infeasible as the number ofantenna and constellation

points increases. ThusML detection is often realized byK-best In thispaper, two modified K-best SD algorithms are

pro-spheredecoding algorithm. posed for

reducing

the

computation

complexity

while

remain-In this paper, two techniques to reduce the complexity of ing the performance similar to

ML

detection. The K-best

algo-K-best algorithm while remaining an error probability similar rithm with

predicted

candidates,oneofour

proposed

methods,

to that of the ML detection is proposed. By the proposed r . .

K-best with predicted candidates approach, the computation r t

complexity can be reduced. Moreover, the proposed adaptive paths before selecting the K best candidates. Moreover, an

K-best algorithm provides a means to determine the value K adaptiveK-best algorithm is proposed,providingan adaptive

according the received signals. The simulation result shows that selection of K by observing the ratio of the second minimum the reduction inthecomplexity of 64-best algorithm rangesfrom and minimum of all paths at the first decoding layer. According

48% to 85%, whereas the corresponding SNR degradation is

maintained within 0.13dB and 1.1dB for a 64-QAM4x4MIMO to oursimulation results, theproposed techniquescan achieve

system. at most 85%

complexity

reduction when

comparing

to

con-ventional 64-best SD

algorithm.

I. INTRODUCTION

The rest of this paper is organized as the

following.

The

Recently,

multiple-input

multiple-out

(MIMO)

systems

are

system

model, SD

algorithm,

and K-best SD

algorithm

are

applied in

many

wireless

applications

for better

transmission

briefly described in SectionII.In SectionIII,the two proposed efficiency and signal quality due to the inherent

diversity

gain detection schemes are presented. The bit error probabilities of

provided by

the multi-path environment.

Maximum-likelihood the proposed schemes are simulated in a 4 x 4 MIMO system

(ME)

sequence

detection

is one of the

detection schemes

for of uncorrelated flat-fading channels, and the simulation results

detecting

thereceived

signals

in MIMOsystems By searching and comparisons are given in Section IV. Finally, Section VI

for the constellation point nearest to the received signal, concludes this

work.

ML detection is optimized for

minimizing

the

symbol

error

probabilities, but exhaustive search becomes infeasible since

the computation

complexity

grows as the number ofantenna 1. SPHERE DECODINGFORMIMO SYSTEM or the

constellation

points

increases.

Sphere

decoding

(SD)

algorithmcanreduce the

computation

complexity by

confining

the number of constellation

points

to be

searched,

Fincke- For a MIMO

system

with NT transmit antennas and

NR

Pohst [1] and Schnorr-Euchner [2] aretwo of the most com- receive

antennas,

the transmitted and received

signals

can be

mon

computationally

efficient search

strategies

for

realizing represented by

the ML detection.

Nevertheless,

the difficulties in hardware

implementation arise because of thenon-constant

computation

=Hs

+

ii,

(1)

complexity and

decoding

throughput.

Alternatively,

K-best

SD algorithm [3], [4]

simplifies

the hardware

implementation

where

yi

is the

NR

x 1 received

complex

signals,

H is an

of SD algorithm

by

keeping

at most K best

paths

in each

NR

x

NT

matrix of

independent

and identical distributed layer, leadingto

fixed-throughput

and

predictable

complexity.

(i.i.d.)

circular Gaussian random variables

(flat

fading

is

as-Note that the term layer refers to the signal constellations sumed),sis an NTx1 complexvectorrepresentingthe signals of an transmit antenna. However, K-best SD algorithm can transmitted by each transmit antenna, and

ni

is the NR X 1 not guarantee

ME

performance since the ML path might i.i.d. complex Gaussian noise vector. Moreover, the complex be eliminated due to the breadth-first nature of K-best SD model in (1) is often described by the equivalent real-valued

representation, which is PED and the accumulated Euclidean distance corresponding

F

Re{i}

1 to

s(i+±),

denoted

by

T(s'+1)),

that is

[ Im{ } J

T(s('))

=

T(s(+))

+

e(S)(7)

= F

Re{}

I

{}

lHj

Re{}

l +F

Re{fi}

The detection process starts from

i=NT, resulting

to a

tree-[

Im{H}

Re{H}

I [

Im{s}

J [

Imfn}

i structure, or called depth-first, search strategy. However,

ex-- Hs+ n. (2) haustively searching for the ML solution becomes infeasible

This is also referred to as the real value decomposition. For [5] since the computation complexity grows exponentially QAM signals, real value decomposition transforms the com- with Nt or the number of constellation points. Thus, sphere plex constellation into two real-valued PAM

constellations,

decoding (SD) algorithm has been proposed and recognized

which can result to fewer computation. as a powerful means to solve the

ML

detection problems [6] For detecting the received signals, maximum likelihood [4]. SD algorithm reduces the computation by restricting the

(ML)sequencedetectionis one of the MIMO system detection search range. Instead of searching all candidates in Q, SD

technique that optimizesthe symbol error probability. Accord- algorithm constrains a much smaller search range

QSD

=

ing to the system model described in, Fig.1 ML detection

{s

s R

Rs

< r2}; only the candidates in QSD Will be is equivalent to searching for the vector s that minimizes compared. By the aforementioned procedure, the candidate

IY-

Hsl12.

That

is,

of the smallest

T(s(1))

is always the

ML

solution as long

as r is properly defined. However, not only the value r, s =argmin

IY- Hsll2,

(3) but the computation varies with

SNR,

leading to a

non-SGE2 constant decoding throughput. Hardware implementation of

where Q is the setconsisting ofall possible

2Nt-dimensional

SD algorithm becomes complicated.

signal constellation points. Fig.1 shows the simplified block K-best SDalgorithm is an alternative method that improves

diagram ofa MIMO receiver. The channel estimator provides the decoding throughput. It simplified the original SD

algo-the required channel state informationH. By QR decomposi- rithm and maintains a constant throughput by keeping only

tion, the channel matrix H is decomposedby H = QR, and the K smallest accumulated PED at each layer. However,

(3)can be rewritten as K-best SD algorithm can not guarantee the performance of

IlY- Hsl2

=

(s

-

sZf)HHHH(s

-

szf)

ML detection since the

ML

solution may be eliminated when

+ y-ff

(I

-

H(HHH)-'HT)Y

it is not of the K best accumulated PEDs. Thus, larger K

is required and the value K becomes a tradeoff between

and complexity and error performance.

-

argmin(s-szf)HHTH(s

- szf) Transmit Channei

arg min

sHRHRS.

(4) Symbols (H)

Notethat the matrixR derivedfrom QRdecompositionis an Estimation Decomposition st

uppertriangular matrix with non-negative diagonal elements,

and

HHH=RHR.

Moreover, Szf is the zero-forcing (ZF)

solution that can be derived by Szf =

H+y

for H+ is the _{Detect MaxmumLikdelihood}

pseudo-inverse

of H. It is

perceived

that s - szf is the Symbols Aithm

distance from the candidates ofsignal tothe ZF solution.

Due to the triangular form ofR, we can rewrite (4) as _{Fig. 1. Block diagram of MIMo detection}

2

NR NT

argmin E -

1

Rijs>

(5) Fig.2illustrates the bit error rate of a4 x 4 MIMOdetector

i=1 j=i ofdifferent values ofK, and there isperformance degradation where

Ri

and

sj

denote the i-th row,j-th column ofRand when K is chosen too small.

the j-th element ofs. Moreover, we can define

e(s(')),

the III. PROPOSED K-BEST SD ALGORITHM

WITH

PREDICTED

partial square Euclideandistance(PED) of the i-th layer, by CANDIDATES

NT 2 AlthoughK-best SDalgorithmremainsconstant

throughput

e

(s()

=

Yi-E

Rij

,

(6)

and computation, itscomputationcomplexityis notnecessarily lower than the conventionalSD algorithm since all the PEDs of each layer still need to be calculated. However, only the where

s(i)

- [sis

).*s.)]

andsKi) isthe j-th element of K PEDs resulting to the K best accumulated PEDs can affect s() Then the accumulated Euclidean distance corresponding the PED calculation in the next decoding layer. That is, part to the candidate

s(i)

can be derived recursively from the of computations of the PEDs are unnecessary. A method to

I---64-best SD

complexity

and error

probability.

Due to

fading,

the

signals

0-2 ~~~~~~~~~~sufferfrom low SNR when they are in deep fades, and K

...:.when the

signal strength

is

high. Dynamic

K

implies

an

signal

...q lity...in tor...is...re ire d.nd to ir

10-3____~~~~~~

...adaptive.... ...K -best...algorithm,...provides.. ...a...m eans...to...observe...the..

i

~

~~~~~~~~~antennas,

this indicatorcan be

acquired by

the ratio

... ... ... .. ... .. .. .... ... ... ... .. ... .. .. ... .. .. .. .... ... .... .... ... ...

Fi.2. Cmprsoso M ndKbetS

aloihmowhee MLndM

path

tbein selim

natd

duingmu

the

K-bestm

SD

-7

-5

3 5 7

processing

increases.

o Q - 0

(1± 1)

-th

Fig.4

is an illustrative

example

ofa4x4

64-QAM system,

4:::-: _zz-

~~layer

which shows the relation between T and the

symbol

error

probability

conditioned onthe value T. The curve stands for

i -t the

probability

Pr(R

<

T),

and the

histogram

shows the

----

~~~~~~~~~ayr

the conditional

symbol

error

probability.

It is

perceived

that

symbol

error

probability

is small as T increases.

Thus,

the

Fig. 3. K-best with predicted candidates value K can be determined

by

first

computing

ft in

(9),

then

K

K,

ifR<T; I

K2

otherwise. (0

predict

the more

likely

PEDs is

presented

in the

following.

Only

a fraction of the PEDs are

computed,

and

thus,

the

computation

can be

greatly

reduced.07

At

decoding layer

i,

the

point k'j resulting

in the smallest

Pr(symbolerror

occurs

IR

=T)

PED fora

given s('±l)

can be derived

by

0.6-~(±)-Yi NTP+1

Rijs05'

k

j=i~~f

(8)

.-and

only

the L- 1

points

nearest to

s~i±l)will

be

computed

for

e(s(')).

That

is,

the

s$')

of the vector

s(')

will be

ki(i±1)

and its L- 1 nearest constellation

points. Only

L PEDs from

0.3-e(s(i±1))

should be calculated instead.

Accordingly,

we can

always

have the PEDvalues

computed

in an

ascending

order,

0.2-and the first L smallest PEDs will contribute to more

likely

candidates.

Fig.3

is a

64-QAM

example

with L =3. The 0.1

constellation

corresponds

to the i-th

layer

is denotedas

Si,

as

the

figure

shows,

the

points

with ofcross mark is the 0(±1 and

only

the three constellation

points (linked by

solid

lines)T

willbe

cmputd.

Tus,

he

ompuatio

com

lexiy

ca be Fig. 4. The probability of R < T and the conditional symbol error

reduced,

especially

when

NT

is

large.

probability.

IV. PROPOSEDADAPTIVE K-BEST SPHERE DECODING The value R can be

regarded

as a

signal quality

indicator

layers

can be reduced if K

=1K2

is chosen.

However,

if reduce

computation

effort, however,

the

performance

will also R is determined

earlier,

there are chances that ft cannnot

degrade

since some

computation

is

ignored.

provide

sufficient informationto

report

the

signal quality

and

the

performance

will

degrade.

120.00%

V. SIMULATIONRESULTS 0.%

80.00%-Inthis

section,

a 4x4 MIMO

system

is simulated for

com-paring

the

proposed

schemes and the conventionalSDandK- 60.00%

best

algorithms (K =64),

whereas theMEI detection

provides

40.00%

a

performance

baseline. The

signal

is modulated

by

64-QAM

and the MIMOchannel is assumedto fade

uncorrelatedly

and 20.00%

independently. Totally

106

bits are simulated when the

SNR.

.0

is below

30dB,

and i7bits are simulated for SNR > 30dB. SNR(dB)/Adaptiv Kbest 30 32 34

The

proposed adaptive

K-best

algorithm

can be

applied

w-K2=28,L2=8 35.54% 41.18% 51.37%

with the above mentioned candidate

prediction technique,

*Kl=64,L1=8 64.46% 58.82% 48.63%

whereas the

K,

and

K.2

can have distinct

L,

and

L2

values,

respectively. Fig.5

presents

the error

probabilities

versus SNR Fig. 6. Reduce computation effort inSNR 30, 32,and 34dB for T 30.

for different detection methods. It is

perceived

that for SNR

loweror

equal

to 30

dB,

all the

proposed

schemescan

provide

_______________________

performance

very close to that of the

MEI

detection. When 120.00%

SNR is

greater

than

30d,

a

slight degradation

is

shown,

and 100.00%

the value L dominates the

degradation.

As shown in

Fig.

5,

for K1

K2

=64,

the one with

L,

2 8

outperforms

80.00%-the one with

L,

=

-2

3.

60.00%-...F... ...F. .0 -- K1=64,K2=32,L1=8,L2=3.20.00% -I--Kl=K2=64,₂₍Ll=L2=8. _{~~~~~~~~~~~~~~~~~~~~0.00%} 102 - -..--.K.K2..4.L..2=3.SNR(dB)/Adaptiv K-best 30 32 34 .e--K1=64,K2=28.L1=L2=8__- K23L23 517580%35% .* Kl=64,L1=8 47.83% 41.98% 26.48%

LU ...Fi.7. Reue.o puato efotinSN 0,32 4d frT 5

co

...7 7percentage....of....2..being...selected...also...increases,..

25... 26..27..28...31..32...33...overall com putation...com plexity...Thus,...the...num ber...of. sorting...

.---operations----

are recorded--

--and

aeshownteinrSTABLE0 I2for3comparingSN

.---i--- ---i -i

_thceanume

r,

ofe

srcetinge

operat

bio

n

ga

allmethodaso inorm

alied,

error probability.

Since- --smaller --.2may

leat3prfrmnc showstatte

redpucation

intcomplexity

bof64-este.

algoih

o

drops whentT>t1.oAccordingly,oercomparewthewoocasesuQAM

4he4

MIMO system

64,26.27 28, 30 wit L1 8,33

whereascmpu

3

VI.o

CONCity

TusION

ume f otn

the prameers hose wil resut tosimiar cmputtion twont rehiqesordeducing thew compALExIt fof K-bestiSD

copeitie. Asig5E

coprshows,o

thfernlattertiresultmstoslghl

alortmplfortisiga

dhetetonmaineMoriMO

systmspaext

prefesented

smalervlerro

probailites.a

thus,of ietwcanbhe obseredithatdth oB thet poposhedKovninl6-best

algorithm.wthprdite

S cadiatles

valuerLpaffectlerrorSprobability.

Th maximumt valerfofmaLc reduces thatteneumberof sringth

compleration.

Morbeover

agrthemro

istegrdaimension

ofithePNaM consterTwllation

SmlequrLwloed.O

ragsfo

adativ K-bestS algoriasthm

poroviespaomeans

toR

TABLE I

COMPARISONOFMLANDK-BEST SPHERE DECODINGANDRATIO SPHERE DECODING DESIGN

Method ML K1=K2=64 K1=64,K2=28 K =64,K2 =32 K1 K2 =64 Number of LL = L2=8 L =L2 =8 |L =8,L2 3 L1 L2 =3 Number of 1.19 x 1019 6.59 x 1010 3.43 X 1010 1.9 x 101O 9.39 x 109 Sorting Operations Normalized Sorting 1.8X 108 100% 52.04% 28.83% 14.2% Complexity 2 SNR(dB)for 32.64 32.72 32.85 33.24 33.82 BER- 5 x10-4k 326

determine the value K by observing the received signals. These two schemes can be applied at the same time when

considering

the error

probability

and complexity, providing

flexibility

and tradeoff between system performance and im-plementation cost. According to our simulation results, the reduction in the complexity of 64-best algorithm ranges from 48% to 85%, whereas the corresponding SNR degradation is maintained within 0.13dB and

l.1dB

for a 64-QAM 4 x 4 MIMO system.

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