A Family of Suboptimum Detectors for Coherent MUD

8/11/2019 A Family of Suboptimum Detectors for Coherent MUD

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683

E E E

J O U R N A L ON

S E L E C T E D

A R E A S

IN COMMUNICATIONS.

V O L . 8.

NO. 4. MA Y 199

A Family of Suboptimum Detectors

for

Coherent

Multiuser Communications

Z H E N H U A X I E , R O B E R T T . S H O R T . A N D C R AIG K . R USHFOR TH

Abstract-In this paper we consider a class of suboptimum detectors

for data transmitted asynchronously by K users employing direct se-

quence spread spectrum multiple access (DS/SSMA) on the additive

white Gaussian noise (AWGN) channel. The general structure of these

detectors consists

of

a bank of matched filters, a linear transformation

that operates on the matched-filter outputs, a nd a set of threshold de-

vices. The linear transforma tions are chosen

to

minimize either a mean

squared error or a weighted squared error performance criterion. Each

detector can be implemented using a tapped delay line. The number of

computations performed per detected bit is linear in K in each case ,

and the resulting detectors are thus much simpler than the optimum

detector. Under typical operatin g conditions, these d etectors will per-

form much better than the conventional receiver and often nearly as

well as the optimum detector.

I . I N T R O D U C T I O N

E consider

in

this paper a coherent asynchronous

W

K-user DS/SSM A system for transmitting signals

on the additive white Gaussian noise channel. The re-

ceiver makes symbol estimates using a detector that con-

sists of a bank

of

matched filters followed by a decision

algorithm. Tw o well-known detectors of this type are the

conventional receiver, which simply compares each

matched-filter output to a threshold, and the maximum-

likelihood

( M L )

sequence detector whose decision algo-

rithm is the Viterbi algorithm [

1

1-[4]. The complexity of

the M L detector grows exponentially with K, and is thus

impractical unless the num ber of users is quite sma ll. At

the same time, the performance of the conventional re-

ceiver deteriorates rapidly as t he number

of

users in-

creases. We are therefore led to study suboptimum detec-

tors that are easier to implement than the M L receiver but

that perform much better than the conventional receiver.

Some examples of suboptimum detectors that have been

previously studied are a sequential decoding algorithm

[ 5 ] ,

a decision-feedback equalizer [6], 171 for asynchronous

DS/SS MA system s, and an optimized linear multiuser de-

tector for synchronous DS/SSMA systems

[SI.

In this paper, we consider a class of suboptimum de-

tectors for the asynchronous DS/SSM A system whose de-

Manuscript received February 2 1 , 1989; revised September 18. 1989.

This work was supported by the Unisys Corporation under the University

of Utah Contract 5-20516. This paper was presented in part at MILCOM

'89, Boston, MA, October 15-18, 1989.

Z . Xie and C.

K .

Rushforth are with the Department of Electrical En-

gineering, University

of

Utah, Salt Lake City, UT 841 1 2 .

R .

T. Short is with Communication Syste ms Division. Unisys Corpo-

ration, Salt Lake City,

UT

841 16.

IEEE Log Number 9034793.

cision algorithms consist of a linear transformation

T

fol-

lowed by a set of threshold devices. This approach is

motivated by equalization tec hniqu es developed f or chan -

nels with intersymbol interference [ 111 and by the ap-

proaches taken in [6] and [SI. Because of the sim ilarities

between the two problems,

it

should not be surprising that

techniques which are effective in combatting intersymbol

interference can also be used to advantage in the multi-

access problem. In addition to the obvious similarities,

however, there are some important differences. For ex-

ample, the near-far problem (i. e., the problem of signif-

icantly different user powers) encountered in multiple-ac-

cess communications has no counterpart in the

intersymbol interference context. Moreover, the system

designcr can typically exercise more control over the level

of

interference in designing a multiple-access system

through his or her choice of signature sequences than

would be possible in the intersymbol interference prob-

lem. Thu s, we might reasonably expect suboptimum mul-

tiple-access receivers to perform well under a broader

range of conditions than their intersymbol-interference

counterparts .

We obtain two different versions of T using two differ-

ent performance criteria: minimum mean squared error

and weighted least squares. These choices, though not

op-

t imum in terms of bit-error probability, are mathemati-

cally tractable and lead to elegant and simple detection

structures that can be implemented using tapped delay

lines. These structures have computational complexities

per detected bit which are linear

in

K ,

and are thus much

simp ler than the optimu m detector . This result rests on

the assumption that the parameters

of

the system have

been computed and are fixed. The practical problems as-

sociated with estimating the parameters of the system and

adapting the structure of the detector must be faced re-

gardless of the decision rule adopted and thus do not ap-

preciably affect the comparisons we make.

Our initial results are based on the assumption that the

linear transformation operates on the entire received

waveform to estimate the entire transmitted sequence.

This approach requires too much m emory and en tails too

much decoding delay to be practical, so we next consider

some modified algorithms that have small fixed decoding

delays. We also discuss appropriately modified versions

of two schemes that have been widely used to combat in-

tersymbol interference: the linear equalizer and the deci-

sion-feedback equalizer.

0733-87 16/90/0500-0683$01

OO

@ 1990 IEEE


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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. VOL. 8.

NO.

4. MAY 1990

Matched

Filter

Bank

In Section 11, we review the

DS/SSMA

signaling

scheme and present a discrete model for its analysis. The

minimum mean squared error (MMSE) detector, its per-

formance, its implementation, and various suboptimum

modifications of it are developed and discussed in Section

111. The computational complexity of the MMSE detector

is shown to be

3 K

multiplications/bit. In Section IV, the

linear equalizer is derived and its performance is ana-

lyzed. One common feature of the MMSE detector and

t he

linear equalizer is that the preliminary estimate of the

transmitted bit sequence obtained using the linear trans-

formation T is biased. This com plicates the analysis and

leads to a performance that depe nds on the interfering sig-

nal power. Th e weighted least squares (WLS ) detector de-

scribed in Section V avoids this problem by providing an

unbiased estim ate of the transmitted sequence. T he com-

putational complexity of the W LS detector is once again

3K multiplications /b it, and its bit-error probability and

asymptotic efficiency are independent of the interfering

users powers.

A

natural suboptimu m implementation of the WL S de-

tector yields the well-known decision feedback equalizer

[6], which we discuss in Section VI. In Section VII, we

summarize the results of several computer simulations

conducted to illustrate the performances of the detectors

described in this paper and to compare them to the per-

formances of the conventional receiver and the optimum

(ML) receiver. In many circumstances of interest, the re-

ceivers described here perform much better than the con-

ventional receiver and nearly as well as the more compli-

cated M L receiver.

Dension

Algorithm

11. P R E L I M I N A R I E S

We consider BPSK transmission through a common

AWGN channel shared by

K

asynchronous users employ-

ing

DS/SSMA,

as illustrated in Fig. 1. The received sig-

nal

r ( r )

can be written in the form 131

M K

r t ) =

C

b , ( i ) ~ , ( t - i ~ - 7 , ) + n ( r )

1 = I

= I

M K

= h 1 1 1 ) ( r ( 4 )1 1 l , ( f r ( m ) T

111

=

I

- 7 , ( 1 1 1 ) + n ( r )

( 2 . 1 )

where n t ) s additive white Gaussian noise with two-

sided power spectral density N 0 / 2 , k ( m )

1

= m mod

K , ~ ( m )

1 is the integer part of m/K, and M is the

number of bits transmitted by each user.

b,

i

)

E { -

,

1

} is the ith transmitted bit of the

k t h

user,

S,

r ) =

J ~ w ,

, ( 1 ) COS

a c t

+

0,

),

t

E

[o ,

T )

( 2 . 2 )

is the signature waveform of the k t h user,

a , r )

is the

spreading signal of the kth user normalized to unit power,

Fig.

I .

System model for asynchronous

DSiSSMA

communications.

carrier phase,

a,

s the common ca rrier frequency, and w ,

is the received signal power of the kth user. The powers,

phases, and time delays of all users are assumed to be

known, and without loss of generality the users are or-

dered in such a way that the time delays satisfy 0 ~

~ < T . In practice, these system parameters

must of course be estimated and continually updated, a

problem that we are currently studying.

The spreading signal

a,

( t ) , which is used to reduce

multiuser interference, is a signal of duration

T

seconds

that can be written in the form

L

r ) = u t ~ p T f I

( m

- ( 2 . 3 )

111

=

I

where

p T ,

t ) s the spreading chip waveform whose du-

ration is T ,

=

T / L seconds and { a t , E ( - 1 , is

the spreading sequence of the kth user. The assumption

regarding the duration of the sign ature waveform has been

made almost universally by others working on this prob-

lem [ l ] , [ 2 ] , [ 3 ] , and our results show that

i t

does not

seriously limit multiple-access performance. If additional

protection against jamming or unauthorized interception

were required, the generalization would be analytically

straightforward, although the implementation would be

somewhat more complicated.

The matched-filter output y, (

1,1

)

(

7 ( m

)

associated with

the k( m) th user

in

the r( m )t h bit interval is given by [3]

9 1 1 l ) + l ) T + T i ( l r l I

J h d ~ ~ ) )i7,,),lr+,h

r f ) O , I , ( t - 7 m ) T- Q , I 1 1 , ) dr

111 + K

I

= b m T i ) )

K

, , I 1 ,

1

, r ( m

= t i - K + l

r 4 ) + 1 1 1 ) ( d ~ ) )

(2 .4)

where

m

H1,(m)= s , ( r -

s , ( t

+

mT

7, r

( 2 . 5 )

is the signal crosscorrelation functi on. We see that

y, ( 7 m ) depends upon the transmitted bits, the sig-

-m

rk is the time delay of the k t h user,

0,

is the kth users

nal crosscorrelations, and the noise.


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S U B O P T I M U M

DETECTORS

F O R M U L T I U S E R C O M M U N I C A T I O N S

685

Using vector notation,

it

is not difficult to show that the

MK matched-filter outputs yL(,, q ( t n ) can be expressed

as [ I O ]

y =

PMb +

n ( 2 . 6 )

where

y ( i ) = [ y l ( i > , 2 ( i> , . * Y K ( i ) l T ~

~ ,

b ( i ) = [ b l ( i ) , b 2 ( i ) , * * * b K ( i ) l T E - I , I

I ,

n

=

[ n (

) , n(2) ' ,

*

. n ( M ) ' ] '

E

R M K ,

y = [ y (

)' ,

y(2 ) ' , * * . ,y ( M ) ' ] '

E

R M K ,

b

= [ b ( l ) ' , l ~ ( 2 ) ~ ,

. ~ ( M ) ' ] ' E { - 1 , I } M K ,

n ( i > = [ n l ( i > , 2 ( i ) , .

n K ( i ) ] ' E R ~ ,

and

PM =

H ( 0 )

H 1f

0

H ( l ) H ( 0 ) H ( 1 ) '

0

0

. . .

0

H ( 0

0

. .

-

-

0

0

. . .

. . .

The vector y is a sufficient statistic for estimating the

transmitted bits b . The elem ents of all vectors are ordered

in accordance with their relative appearances

in

t ime. PM

is a symmetric matrix which we assume to be positive

definite and hence invertible, but most of the results we

present can be easily generalized.

All decision algorithms considered in this paper com-

prise two steps. First, a linear estimate b = Ty of the

transmitted bits b is obtained from the matched-filter out-

put y. The constraint that b

E

{

, 1

} is dropped in this

process and arbitrary real-valued es tima tes are allowed .

Second, the components of the estimate

b

are compared

to a threshold of zer o, and th e Cnal bit estim ates are taken

to be the compo nents of sgn [

b] .

111. T H E M M S E DETECTOR

We first study the linear minimum-mean-squared-error

(MM SE) detector. Formally, the problem can be stated as

follows: first a mapping T:R,"" --t R M K uch that the per-

formanc e criterion

E {

( b b )

'

b 8 ) is minimized,

where the estimate b is given by b = Ty. The mapping

that achieves this minimum is

T = ( P M + N O / 2 Z ) - ' .

( P M

+

N , / W b

=

y ,

( 3 . 1

In practice, we could estimate b by solving the linear

equation

( 3 . 2 )

a process that can be carried out efficiently using LU de-

composition, forward elimination, and back substitution

[

1 I] by the system shown in Fig. 2 . The inputs are the

MK matched-filter outpu ts y, the outputs are the estimated

bits sgn [ b ] , nd the L , and U , are matrices that arise dur-

ing the process of block LU decomposition

[

12, p .

11 1 .

The basic operations of m atrix-vector multiplication and

vector addition can be simply implemented using a tapped-

delay-line archi tectur e. To detect MK transmitted bits re-

quires 3 M K2 multiplications. Therefore, the complexity

of this detector is 3 K mu ltiplicat ions/b it, which is linear

in

K

and is independent of

t he

transmission length M.

In the absence of noise, the MMSE estimate becomes

b = PGly. At the other extreme, i . e . , when

No

>> w, or

all j , T reduces to the identity mapping and the MMSE

detector reduces to the well-known conventional receiver.

A . Performance of

the

MMSE Detector

We now con sider the performance of the MM SE detec-

tor. The performance criteria we are the bit-error prob-

ability and the asymptotic efficiency as defined by Verdu

We show in Appendix A that the probability that the ith

141.

bit of the kth user is detected in error is given by

c

,MMSE(i) = 2

- M K +

I

h l , l ( V ( / l I {

- 1 . 1 )

( 3 . 3 )

where

a

is the [ i 1 ) K + k]th diagonal entry of ( P

i s t he [ ( i

-

l ) K + k]th row of

(

P

+

iVo /2Z) - ' f lM , and

+ NO/2Z)-I PM (P I z . I + N 0 / 2 Z ) - I 3 [g l ,

g,,

. * * 9

gMK

1

dv.

u / 2 1

Q ( x ) =

I sm

Each term in the sum in ( 3 . 3 ) is the error probability for

the ith bit of the kth user, conditioned on a particular

combination of the remaining bits. Each such combina-

tion is equally likely, thus accounting for the factor

and the sum is taken over all 2 M K I such com-

binations.

Verdu [4] defines the asymptotic efficiency of the k t h

user, whose bit error probability is

P k ,

to be

qk

= sup 0

2 - M K

+

I

{

l im o + O k ( u ) / Q [ ( m ; L ) " L / u ] < + C O ] .

( 3 . 4 )

To determine t he asymptotic efficiency qpMSEf the

MMSE detector, we note that the dominant term in (3 .3 )

as N o -+ 0 is

Q

-MK I


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I EEE J O U R N A L O N S E L E C T E D A R E A S IN C O M M U N IC A T IO N S .

VOL

X. NO 4. MAY

I990

It follows that

Fig.

2 .

MMSE and

W L S

detector architecture

( 3 . 5 )

V p M S E = max2

Equations (3.3 ) and (3.5) characterize the perform ance of

the MMSE detec tor . We see tha t y p M S F s, in general,

dependent on the interfering signal power. This depen-

dence is a consequence of the fact that the estimate b

=

Ty defined by (3.2) is biased.

To compare the performance of the MMSE detec tor

with that of the conventional receiver, we consider the

special case for which No


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S U B O P T I M U M D E T E C T O K S F O K M U L T I U S k K C O M M U N I C A T I O N S 687

it is reasonable to expect this type of behavior for any

detection scheme that operates

on

only a subset of the

matched-filter outputs.

I V. L I N E A R Q U A L I Z E R

In this section, we formulate the MM SE detection prob-

lem under a constraint that the detector delay be finite,

say approximately JK T seconds with J


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As in the case of the MMSE detector, the estimate of

(5.2 ) can be efficiently determined using block

LU

de-

composition followed by forward and backward substi-

tution [12]. The system shown in Fig. 2can also be used

to implement the WLS detector, and t he computational

complexity of the WLS detector is 3 K multiplica-

tions /bit .

In a manner similar to that described in Appendix A, it

can be shown that the probability that the ith bit of the

kth user is in error is

a great simplification over the expressions for PpMSE)

and P k E ( i ) .Furthermore, PyLs i ) s independent of the

interfering signal power [8].

The asymptotic efficiency of the WLS detector is

which is also independent of the interfering signal power

and is often substantially supe rior to the efficiency o f the

conventional receiver. This result is identical to that ob-

tained fo r the MM SE detector when No


7/8

fective way of examining the performance of the DFE is

through simulation. We describe the results of several

simulations in the next section.

VII . S I M U L A T I O N S

To provide a basis for com paring the various detectors

discussed in this paper, we conducted an extensive series

of Monte-Carlo simulations. The schemes we evaluated

were finite-delay versions of the MM SE and W LS detec-

tors, each with sub sequence length

I

= 9K and an overlap

of K bits between adjacent subsequences; a linear equal-

izer with J = 4; t he DF E; the conventional receiver; and

the ML sequence detector. In all cases, the modulation

was equiprobable BPSK with rectangular bit and chip

waveforms. In cases for which the ML receiver was too

complex to simulate, we used the performance of the sin-

gle-user system to provide a bound.

Among the examples studied were two asynchronous

DSISSMA configurations with K = 2 and 18 and spread-

ing codes of lengths

S

and 12 7, respectively. T he cross-

correlation coefficients in the two-user system were taken

to be H 1 , ? (0 ) 0.3 and HI,?( ) = 0.1, and the code

sequences for K =

18

were taken from [ 131. We used

relative time delays of 0.05 i

1 )

T , i =

1 ,

2 , . . . 18,

for the eighteen-user system, and we set all its phase lags

to zero.

Fig . 6 shows plots of the average bit-error probabilities

of several different receivers as functions of the signal-to-

noise ratio for K =

18.

In this example, the powers of all

users are equal. Fig. 7 shows plots of the bit-error prob-

ability

of

the first-user versus the power of the interfering

user

in

the two-user asynchro nous system to illustrate the

near-far capabilities of the various detectors. These re-

sults are typical of the many configurations we simulated.

From these plots , and from the many other simulations

we conducted, we m ake the following observations. First,

all the suboptimum detectors proposed in this paper per-

form much better than the conventional receiver and

nearly as well as the ML receiver under

a

wide variety of

conditions. S econd, these detectors are quite robust

in

the

face of widely different signal pow ers. T he linear equal-

izer and the DFE perform somewhat better than the sub-

optimum finite-delay versions of the MM SE and W LS de-

tectors. A s predicted, the DFE is more robust than the

linear equalizer in the face of widely different signal power

levels.

To compare these results to those obtained previously

using a sequential decoding algorithm [SI we plot in Fig.

8

the bit-error probability achieved in the eighteen-user

configuration using the linear equalizer, the D FE , and the

sequential decoding algorithm. We see that these three

detectors exhibit about the same performance. Th e advan-

tage of the detectors studied in this paper, as compared

with the sequential decoding algorithm, is their s impler

structure. Their s im plicity, combined with their excellent

performance as demonstrated by these examples, makes

them very attracti ve practical algorithm s.

VIII. C O N C L U S I O N S

The primary contribution of this paper is the develop-

ment and evaluation of a family of suboptimum DS/SSMA

C on ve nud R ece iver

opunum

ingle-Usa Detector

SuboptimumWLS Detector

Suboplimum MMSE

etector

Decision Fe ed kk Equalizer

10

0 1 2 3 4 5 6 7 8 9

Fig.

6 .

Bit-error probability for conventional receiver, optimum single-user

detector, suboptimum M M S E detector. suboptimum

W L S

detector. lin-

ear equalizer, and decision feedback equalizer with

K

= 18 and

L

=

127.

/.

/.

- - - - onvenuonal Receiver

Subopumum

L

Receiver

WLS

Detector

Suboplmum MMSE

Dewtor

_ _ _ _

eclsion

FeedbaJ: qualizer

Llnear F4uallzer

--__

689

IE ~ f

r l

S U B O P T IM U M D F T F C T O R Y FOR M U L T IU Y F R C O M M U N I C A T I O N \

I

~

_ _ ~~

~ _ _ _ _ _ ~ ~

~~

~~ ~

t

1

I

3

6 9 12 15

10..

Fig.

7 .

Bit-error probability versus interfering user's power. for conven-

tional receiver,

M L

receiver. suboptimum

M M S E

detector. suboptimum

W L S detector. linear equalize r, and decision feedback equa lizer with

K

=

2 .

L = 5 . and w , / N , ) = 5 dB.

lo-

10

~o~oglo(wlmolo)

Fig.

8.

Bit-error probability for linear equalizer. decision feedback equal-

izer. and sequential decoding algorithm with K

=

18 and L

=

127.

detec tors whose computational complexity is only linear

in the number of users and yet which perform much better

than the conventional receiver and nearly as well as the

ML sequence detector in many practical circumstances.

The simple structures and excellent performances of these


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8.

N O

4.

MAY 1990

detectors make them very attractive altern atives to the

conventional receiver and the

M L

detector.

The results described in this paper were obtained under

the assumption that all system parameters (signal phase.

power, and delay) are known.

In

practice, these parame-

ters must be estimated and the receiver structure contin-

ually modified to reflect the updated estimates. The ul t i -

mate evaluation and comparison of the detectors studied

here cannot be carried out without considering the effects

of noisy estimates

on

their performances. We are cur-

rently investigating this issue.

A P P E N D I X

In this Appendix, we consider the bit-error probability

of the MM SE detector. T he probability that the k t h user's

ith bit is detected in erro; is given by

pFMSE

Pr { [sgn ( T Y ) l , l - , ) , + , 1 I b , ( i ) = 1 )

b , i )

=

1 )

quences.

/EI,.E Trtrm

Coimrtu. . vol. COM-35. pp. 87-98. Jan.

1987. .

131

S .

Verdu. Miniinurn prohability of error for asynchronous Gaussian

multiple-access channels. / F E E

Trtii i. . /fifitrrti.

Tlioor?, vol. IT-32.

pp. 85-96, J a n . 1986.

141 -.

Optimum inultiuser asymptotic eflicicncy. l E E E

Trctm.

Cornmuti..

vol. COM-34, pp.

890-897.

Sept.

1986.

IS] Z. Xie, C. K . Rushforth, and R . T. Short, Multi-user signal detec-

tion using sequential decoding.

l E E E

Trutis.

Cotntnun. , vol. 37,

1989.

161

R . T. Short, Multiple user receiver structures. Ph.D. dissertation.

Dcp. Elcc. En g . .

U n i v .

Utah. Dec.

1988.

171 M.

K .

Varanasi and

B .

Aazhang. An iterative detec tor

l o r

asyn-

chronous spread-spectrum multiple-access syateins, in Proc..

181

R .

Lupas and S . Vcrdu, Linear multiuser detectors for synchronous

code-division multip le-ac cess channels.

/ E E E

Trtrris /r fitriti. T l i ~

o r y ,

vol. 35, pp. 123-136, Jan. 1989.

191 R . Lupas and S . Verdu, Optimum near-far resistance of linear de-

tectors for code-division multiple-access chan nels, in Proc.

I n f .

Symp.

Inform.

Theorv,

June

19-24, 1988,

Kobe. Japan. p.

14.

1101 -. Near-far re\istance of linear detectors for code-division mul-

tiple-access channels.

/ E E E

Truris.

Conirtiuu.,

vol. COM-37,

1989.

[ I J . G. Proakis,

D i g i f t i /

Co, r i r t r io i i c . t r r i t~ r i .

New York: McGraw-Hill.

1983.

[ 121

G. H. Goluh and

C. F.

Van Loan.

Mtirri i

Compuftrf iot i . \ . Baltimore.

MD: The Johns Hopkins University Press, 1983.

[

131

F . Gebhardt and C. L. Weber. Aperiodic correlation properties of

pseudonoise codes. Tech. Rep., USCEE-430. Dcp. Elec. Eng..

U n iv .

Southern California. Oct.,

1972.

GLOBECOM'88.

Hollywood. FL. N O V . 1988, pp. 556-560.

Zhenhua Xie wa\ born

in

Zhejidng, China, on

July

15,

1958 He received the B

S

degree

in

electrical engineering from Zhejiang Univer\ity,

China.

in

1982, and the M S dnd Ph D degrees

in

electrical engineering from the Univer\ity ot

Utah.

i n

1985 and 1989, respectively

He

\

currently d po\t-doc fellow in Electrical

Engineering at the University of Utah

Hi5 re-

search interest\ include digital signal proces\ing,

vision. and multiu\er communication\

where

(

.

I

symbolizes the

t h e k t h

element of the vector

inside the parentheses. Since the noise component [ (

PM

+ N 0/ 2Z )- 'n l c l - i s Ga us si an w ith z e ro me an a nd

of

E {

Pw

+ N 0 / 2 N '

n n T ( f i w

+

N , ) / 2 C }

variance equal to the [ i - 1

)

K

+

k]th diagonal element

= O w + / 2 Z ) - '

P,( @V +

N I ) / ~ Z ) - ' , ( A . 2 )

P y M S E ( i )an be expressed as a sum of Q-functions,

where

Specifically, if we denote the

[

i 1 )

K

+

k]th diagonal

entry of the matrix in (A .2 ) a s

N U / 2 a

nd the [ i 1 ) K

+

k]th row of the matrix

( P M + N 0 / 2 Z

) - ' P w a s

[g I , g , ,

* *

,,I, then p,MMsE(i an be written as

c Q

Y M S E ( j ) = 2-MK+I

( A . 3 )

. l , l (V (

/ ) ) E ( 1 . 1

)

of Electrica l Engineerii

research interests lie

in

cessing.

Robert

T.

Short

was born in Boulder, CO. on

May 8, 1954. He received the B.S. degree

in

elec-

trical engineering from New Mexico State Uni-

versity

in

December of

1975.

He worked for Motorola Government Electron-

ics Division, Scottsdale. AZ, (1976-1981). Hew-

lett-Packard Corporation, Fort Collins, CO (1981-

1984). and Unisys Corporation, Salt Lake City,

UT (1984-1989). before receiving the Ph.D. de-

gree in electrical engineering in December of

1988. He is now on the faculty in the Department

ng at the University of Utah in Salt Lake City. His

the areas of communication theory and signal pro-

which can, in princip le, be calculated if ( p,,,

+

N 0 / 2 Z ) - '

is given.

REF ERENCES

I ] M. B. Pursley, D. V. Sarwate. and W. E . Stark. Error probability

for direct-sequence spread-spectrum multiple-access coinmunica-

tions-Part

I :

Upper and lower bounds. / F E E

Trtrris.

Cortirtlrtrr. . vol.

121 J . S . Lehnert and M. B. Pursley. Error probability tor binary direc t-

sequence spread-spectrum communication with random signature se-

COM-30. pp. 975-984. May 1982.

Craig K. R ushforth was born

in

Ogden. UT. He

received the B.S.. M.S., and Ph.D. degrees in

electrical engineering from Stanford University,

Stanford. CA. in

1958, 1960.

and

1962.

respec-

tively.

He has served

on

the facultieb of Utah State

University and Montana State University and has

held research positions at Stanford Research In -

stitute. the Institute for Defense Analyses. and

ESL. Inc. He has been with the Department

o f

Electrical Engineering at the University of Utah

since 1974. He has served as Department Chairman from 1982 to 1985,

and from

1988

to the present. His research interests lie in communication

theory and signal processing. with current emphasis on multiple-access

communications. coded modulation. VLSI implementation

o f

algorithms.

and signal restoration.

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A Family of Suboptimum Detectors for Coherent MUD

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