the effects of time delay spread

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IEEE JOUR NAL ON SELECTED AREAS IN CO MMUNICATIONS, VOL. SAC-5, NO. 5 , JUNE

1987 879

The Effects of Time Delay Spread on Portable Radio

Communications Channels with Digital Modulation

Abstract-Frequency-selective fading caused by multipatb time de-

lay spread degrades digital commu nication channels by cau sing inter-

symbol interference, thus resulting in an irreducible

BER

and impos-

ing a upper limit on the data symbol rate. In this paper, a frequency-

selective, slowly fading channel is studiedy computer simulation. The

unfiltered BPSK, QPSK, OQPS K, and MSK modu lations are consid-

ered first to illustrate the physical insights and the error mechanisms.

Two classes of modu lation with spectral-shaping filtering are studied

next to assess the tradeoff between spectral occupancy and the perfor-

mance under the influence f time delay spread. The simulation is very

flexible so that different channel parameters can be studied and opti-

mized either ndividually or collectively. The rreducible BER aver-

aged over fading samples with a given delay rofile is used to compare

different modulation/detection methods, while the cumu lative distri-

bution of short-term BER s employed to show allowable data symbol

rates for given values f delay spread. It is found that both MSK and

QPSK withaised-cosineyquistulsereuitableor

TDM /TDMA digital portable commun ications channel.

I. INTRODUCTION

T

E emergence of a demand for personal portable

communications, along with a need for integrated

voice and data in our mobile society, has made universal

digital portable communications

[l]

highly desirable. A

candidate for providing a flexible, high-capacity digital

service is the TDM / TDMA (time-division multiplexing/

time-division multiple-access) channel allocation scheme

[l],

[2].

TDM architecture has several advantages over

the more conventional FDM (frequency-division multi-

plexing) such as simpler radio hardware and variable-rate

voice communications

[3];

however, the signaling rate re-

quired for such a system is higher. As a result, the influ-

ence of frequency-selective fading caused by delay spread

becomes a crucial issue

[4]-[6].

A flexible computer simulation is presented in this pa-

per to assess the effects of delay spread and to optimize

system parameters such as modulation/detection scheme

and data symbol rate. Adaptive equalization of the fading

channel is not assumed for this study.

Section I1 illustrates the channel model, and Section 111

describes the simulation. Numerical results for some un-

filtered modulations are shown in Section IV. In Section

V modulations employing spectral-shaping filters, as

would be used in a practical system are studied. Section

VI summarizes the results of the paper.

Manu script received January 14, 1987; revised February 17, 1987.

The author is with Bell Comm unications Research Incorporated, Red

IEEE

Log Number 8714459.

Bank, NJ 07701.

JUSTIN C-I CHUANG

11. MODEL

A multipath radio propagation channel can be described

mathematically by its impulse response

h

(

t ) .

A baseband

model of a multipath radio channel is shown in Fig.

1 .

A

quasi-static channel is considered in this study, that is,

the channel is assumed to be time-invariant over many

symbol periods. This is a reasonable assumption fora

portable communications system with around 1 GHz car-

rier frequency and several hundred kbits/s signaling rate,

since the maximum Doppler frequency is only on the or-

der of lop5 imes the bit rate.

Let t )be the baseband representation of the modu-

lated waveform. The total received waveform is then

z ( t ) = r ( t ) + n ( t ) ( 1 )

r ( ’ t )= u t ) *

h t ) . (2)

where n t ) s the additive Gaussian noise and

Intersymbol interference (ISI) caused by the delay spread

of h

( t ) results in the frequency-selective fading. With in-

creasing signal-to-noise ratio (SNR), an irreducible

“floor” of bit error rate (BER) is approached because IS1

increases in proportion to the signal level.

In this paper, the short-term, small-scale (“micro-

scopic”) signal variations due to multipath fading are

studied; therefore, the “macroscopic” effects such as

shadowing and distance-related attenuation are normal-

ized as unity. When an overall system design is consid-

ered, proper scaling factors should be incorporated in a

power budget to account for these long-term, large-scale

variations [ 1

The impulse response of the channel can be expressed

as

[71

h t ) = C

Arnej m6(t T m ) .

( 3 )

A general form of h t ) an be expressed as a continuous-

time function. Both

A ,

and 4rn re slowly varying random

quantities that introduce a small Doppler shift for the

communications channel.A commonly accepted model

[8], [9] suggests that

A m

be a random variable with a Ray-

leigh distribution and 4 be a random variable uniformly

distributed from 0 to

27r;

therefore,

h

( t ) s a zero-mean

complex Gaussian random variable. Physically, at a spe-

cific time delay, the received signal approaches a complex

Gaussian distribution according

to

the central limit theo-

m

0733-8716/87/0600-0879 01OO

O

1987 IEEE



880

IEEE JOURNAL ON

SELECT ED AREAS IN COMMUNICATIONS, VOL. SAC-5, NO.5.,JUNE

1987

Digital

Source

a k = ’

Bits

Encoderodulator

Wave‘orm

Transmitter

I

Receiver

Demodulator

Decoder

n(t)

Fig.

1 .

A baseband model for a portable radio communications channel.

rem because it is a combination of signals from a large

number of unresolved paths.

For t # t ’ ; it is reasonable to assume that h ( t ) and

h ( ’ ) are uncorrelated since they are composedof signals

from ndependent sets of paths with different Doppler

spectra (‘ ‘uncorrelated scattering” [9] . Consequently,

( h * ( t ) ( t ’ ) )

=

p ( t ) ( t ‘ ) (4)

where

(

) denotes the ensemble average, and

P O ) = ( l h ( t , l ’ > . 5 )

The function p ( t ) s a measurable profile [101 called the

“power delay profile.” Some measured results for p ( t )

in the portable radio environment are documented by De-

vasirvatham [ l], [121.

A measure of the width of p ( t ) s theroot mean-square

(rms) delay spread

T

defined as the square root of the sec-

ond

central moment

[9].

That

is

1

/*

( t D f p ( t ) t

T = [‘

j P ( t ) d t

]

( 6 )

where the average delay D , i.e., the centroid of p ( t ) , s

p ( t ) , d t

O ) d l *

D = 7)

The error rates for transmission through a channel with a

small delay spread are most strongly dependent on the

normalized rms delay spread [4],

[5],

[13]. Normalized

delay spread is defined as

d = T

where

T is

the symbol period. In this study,

d 0.2 is

considered.

111.

SIMULATION

Fig. 2 is the flowchart for coherent detection of the

modulations considered except GMSK. The simulations

rMP

NUMBER

Processes

Gausslan

A hin, hqn n=1,2

,...,

Impulse

Carrier

Recovery

1 (t)e-j

Receiver

Filterlng

fm(t)

INPUT

BITS

kO(k)<

0

I

Svmbol

t Error

BERER

Compu-istri-

tationsution

Fig. 2. The flowchart of a computer simulation

of

a frequency-selective

fading channel.

for GMSK and differential detection involve minor mod-

ifications to what is shown.

A . Sampling p ( t )

The powerdelay profilep ( ) s sampled at = tn where

n = 1 , 2 ,

,

N. The spacing between ime instants is

chosen according to the accuracy needed.

In addition to a measured profile [111 as’shown in Fig.

3 ,

three idealized profiles are considered:

a) Gaussian profile

b) One-sided exponential profile

1

exp { - t / 7 } for t 2 o

for t < 0 .

P ( t ) =

r;

(9b)

c) Equal amplitude two-ray profile

p ( t )

=

; [ a c t

T +

6 ( t + T ] . ( 9 4

Both

the Gaussian and he two-ray profiles are represented



CHUANG: EFFECTSFIME DELAYPREAD

ON

PORTABLE RADIO COMMUNICATIONS

881

V Y / L

Z .4 .6

B 1

1.2 1.4

1 E

Time

In

I ic rowwonds

Fig.

3 .

A

power delay profile obtained from measurements done in an

of-

fice building; the

rms

delay spread

is

approximately

250

ns.

by noncausal functions because the average delays are re-

moved for mathematical convenience.

The channel impulse response at

t,,

denoted by

h ( t , )

= hi,

+

jhqn , s a complex zero-mean Gaussian random

process with variance

p ( t , ) ;

therefore,

hi,

and

h,,

are un-

correlated Gaussian processes each with variance

P ( t , ) / 2 .

B. Generating Gaussian Random Processes and h ( t )

From (4), the covariance matrix for the 2N-dimension

Gaussian random “vector”

{ hi

,

hql

,

*

,

hiN hqN

can be expressed as

M =

P(tl)

2

P(t l )

2

0

0

A subroutine is called to generate the 2 -deviate Gauss-

ian random vectors with a covariance matrix

M .

A sample

of the channel impulse response can then be constructed

by the following formula:

N

h ( t ) = C (hi, + j h , , ) ( t , ) . (11)

n = l

C . Convolving the Modulated Waveform with h ( t )

All the modulations treated here except GMSK can be

simulated as quadrature amplitude modulation (QAM)

with a signaling waveform g ( ) of symbol period T.

GMSK is simulated as a form of frequency modulation

(FM).

1 UnJiltered BPSK , QP SK, O QPSK : Signaling with a

rectangular pulse,

Note that 1) BPSK is simulated by one input bit sequence;

2) QPSK and OQPSK are simulated by two independent

input bit sequences; and

3)

a time offset of T/2 offset is

introduced between the two sequences for OQPSK.

2) MSK: Signaling with a sinusoidal pulse,

g ( t > = s m 0

a t

={;F -T/2 /2

.

(12b)

otherwise.

Note that MSK is simulated in the same way as OQPSK

except with a different signaling waveform.

3) QPSK with raised-cosine Nyquist signaling pulse

7]

(RC-QPSK):

Signaling with a pulse with the following

form.

g ( t )

=

g c ( t )

( 1 2 4

where

g ,

( t ) can be realized by a filter that has the square

root of a raised-cosine spectrum with a roll-off factor CY.

This signaling waveform and its matched filter form an

ISI-free Nyquist pulse in the absence of delay spread.

4) GMSK: Signaling with a pulse with the following

form.

s ( t )= g p ( t ) g g ( t ) (12d)

where

g p

t ) s a rectangular pulse expressed by (12a) and

g g

t )

s the impulse response of a Gaussian filter whose

width is controlled by the BT, ( 3 dB bandwidth normal-

ized by bit rate) product

[

141.

The linear combination of binary pulses is used to fre-

quency-modulate the carrier, with the total phase change

for each pulse being a 1 2 .

With the exception of GMSK, the received waveform’

is a linear combination of g

( t )

*

h ( t ) .

D. C arrier Recovery for Coherent Detection

For a portable communications system using around 1

GHz for the carrier and several hundred Kbits/s for the

signaling rate, the following assumptions are reasonable.

a) The bandwidth of the carrier recovery circuit is as-

sumed to be much higher than the channel fading rate;

therefore, the carrier phase can be tracked as if h ( t ) s

time-invariant.

b) The bandwidth of the carrier recovery loop is much

lower than the symbol rate; herefore, for an idealized

carrier recovery circuit that removes the modulation, the

carrier phase is extracted from the average of h ( t ) over

several symbol periods.

Phase jitter on the recovered carrier caused by Gaussian

noise is neglected as we focus on the effects of delay

spread; herefore, he low SNR performance should be



882 IEEEOURNALNELECTED AREASN COMMUNICATIONS,

VOL.

SAC-5,O. 5 , J U N E

1987

viewed as the idealized case, which is reasonable because where

E b / N o

s average signal-to-noise ratio per bit at the

of b), above.

input of the receiver and “erfc” is the complementary

For

d 0.2,

he average of h

( t )

over several symbols

error function.

can be approximated by the average over the whole time

b) If O k ) d k

<

0, errors occur if AWGN causes no

axis; as a result, the recovered carrier phase is then errors; therefore,

h ( t ) dt = phasef H ( f ) l f = o (13) 1

P , ( k )

=

1 erfc

2

where H ( f is the Fourier transform of h

( t ) .

The recovered carrier can be represented by For differential detection of BPSK, sampled values of

cible BER. To compute the short-term BER when AWGN

(14) is present, the BER formula, ( 1 2 )

PEbINo,

s not exact

because differential detection is not a inear operation.

Carrier recovery is then treated

h

( t , However, we can get a reasonable approximation forhe

h ( t ) - j in theimulation. conditions assumed in this paper by multiplying the sig-

nal-to-noise ratio by I

k ) O k

1) I and then using

E. Receiveriltering the above BER formula. -The. easons ares follows:

For all the modulations considered except GMSK, the

a) This fOrmula is exact for Eb

/ N o

-

O

i.e. when

optimal receiver filter in the absence of delay spread is AWGN is not present; therefore, the irreducible BER is

the matched filter with impulse response

g*

( - t ) ; this is

curate and the short-term BER at high SNR is a good

the receiver filter assumed in the simulation. For GMSK,

approximation.

we simulate the type of receiver described by Murota and

b) This fOrmula is exact if k ) = I k 1 I be-

Hirada [14]; this is a parallel implementation of the MSK

cause the noise Processes are effectively changed by the

receiver, which is a suboptimal receiver for GMSK in the

Same ratio for the two bits. It Will be Seen from the next

absence of delay spread [151. section that IS1 is much smaller than the signal component

unless the signal is in a deep fade as aresult of multipath

F. Timing Rec ove rynd Sampling cancellation forhe range .0.02 .2 simulated;

therefore,

I

O(

k ) = I

O k 1

)

I in most cases. When

( k ) k )

is

very

1 r m( ) r z ( t T ) are computed to determine the irredu-

H (O )

= r ‘

A squaring timing loop

[

161 is used in the simulation to the signal is

in

a deep fade, I

recover

the

timing The timing

td caused by a Small, the BER is very large, the channel is unusable, and

multipath channel was analyzed in an earlier paper [17]. the exact value

of

the BER is of less interest.

The timing jitter caused by Gaussian noise is neglected;

Since

the of the delay-spread

effects

on

the

tion is a lower bound.

cible BER) performance, the low SNR region where the

therefore, the BER computed under the low SNR condi- two detection methods is based on the high SNR (irredu-

For small

d ,

the recovered timing tracks the centroid of approximation is less accurate does not effect the conclu-

I h

( t ) 2

Twocases Of timing are considered:

If

sions reached in the paper. The approximate BER calcu-

timing recovery loop,

p ( t )

is

the fading rate is much higher than

timing; whereas the short-term Ih

( t )

is used when the large.

fading rate is much lower than the bandwidth of the tim-

For

the

ing recovery loop. It will be seen later that the BER per-

is

computed.

fonnances are about the same forboth

the bandwidth of the lation for differential detection

should

be

used

with tau-

t’ generate detection tion, especially when SNR is low and the value of

d

is

of

GMSK, only the irreducible BER

cases f

d

is small.

For

all

the modulations treated except GMSK, ( k ) s

Once td

is

computed, the waveform

rm t ) a weighted sum of the samples off,

( t )

(coherent detec-

o ( k ) ,

tion) or

f m ( t )f t T )

(differential detection) where

s sampled at t = kT

+

td for the

kth

bit; the

is used for symbol detection.

f , ( t )

=

g ( t )

*

h ( t )

*

g * (

- t ) .

For example, O ( k ) can

G .

Symbol

Detection and BER Computation

be expressed in the following form for coherent QPSK,

The quantity k ) s normalized in such a way that it

has a 0 dB rms value in theabsence of delay spread.The o ( k ) = Re [

( d n l + j d n 2 ) f m [ ( k

n ) T +

t d ] ]

effect of AWGN is then included in the following ways

for the coherent detection: 16 )

a) If

O k ) d k 0,

errors occur if AWGN causes er-

.rors; the resulting short-term BER, P,

) ,

is

where .Re denotes the real part and

d n l , d n 2

= 1. It is

clear that

dklfm

td

) is the desired signal term in the kth

interval, while other terms cause intersymbol interfer-

ence. Interference between

I

and Q rails in general exists

m

n =

- m

P, )

=

erfc

i . . x i ) .

(I sa ) (“cross-rail nterference’’).



CHUANG:EFFECT S OF TIME DELAYSPREADON PORTABLE RADIOCOMMUNICATIONS

883

0.10

10-5

0.01 0.02 0.04 0.06 , 0.08

0.10

d

Fig. 4. Comparison

of

results

of

the simulationwith those obtainedby

analysis and experiment.

IV. RESULTS

OR

THE UNFILTERED MODULATIONS

In this section, results for the unfiltered modulations are

presented. Major parameters in the simulation are

1)

type

of modulation: BPSK, QPSK, OQPSK, and MSK; 2) type

of detection: coherent and differential;

3)

delay profile and

its normalized

rms

delay spread; 4) ratio of the bandwidth

B

of the timing recovery loop to the fading rate

F ,

and

5

the signal-to-noise ratio: Eb N o .

Simulations using different profiles indicate that the

BER performance is not sensitive to the shape of the delay

profile, for the range of d simulated. Examples involving

different profiles are shown in the following; however, the

results apply regardless of the profile used.

A . Verijication of the Simula tion with Results in the

Existing Literature

Bello and Nelin [4] calculated the irreducible BER av-

eraged over themultipath fading samples for differentially

detected BPSK in a channel with a Gaussian delay profile.

Y shida

et al. [

181 performed microscopic BER measure-

ments for coherent BPSK with a two-ray profile. Fig. 4

indicates that the results of the present simulations com-

pare well with the available analytical and experimental

results.

B .

The

Error

Mechanisms and the Efects

of

the Timing

Recovery Circuit

In Figs.

5

and

6 ,

two examples are shown to illustrate

the error mechanisms for small d ; a rectangular signaling

pulse with a matched receiver filter is considered. Both

figures indicate how three consecutive symbols combine

and interfere with one another to form the output

0

k ) at

the kth detection timing. The channel impulse response in

each example has two equal-amplitude “rays” which are

in-phase for Fig. 5 and out-of-phase for Fig. 6.

O k)=dkfm O)+dk.lfm T)+dk+lfm -T)= l-d)dk+0.5d dk+l+dk-l)

O(k)dk>O NO IRREDUCIBLE ERROR

Fig. 5 . Illustration

of

the

error

mechanism by using a two-ray model (equal-

amplitude and in-phase rays).

\

)LO.5d T

\

-1

O

1

T

+-I

k -4

2d

2dd

I

0.5d I

t

T

\

O(k)=0.5d(dk+l-dk-1)

P,(O(k)dk <

1 = 0.5

Fig. 6 . Illustration of the error mechanism by using a two-ray model (equal-

amplitude and out-of-phase rays).

The “.irreducible” errors (i.e., those made at very high

SNR)

cannot occur unless IS1 outweighs the signal com-

ponent at the sampling instant; therefore, there are three

error mechanisms: 1) A faded signal component caused



IEEE JOURNAL ON

SELECTED AREAS

IN

COMMUNICATIONS,

VOL.

SAC-5, NO. 5 , J U N E 1987

-40

-35

71J

-25 -20 -15 -10 -5 0 5

10

Signal

Level, dB

Fig. 7 . Amplitude distributions of the detected waveform w hen irreducible

symbol errors occur for coherent BPSK in a channel with a two-ray pro-

file. T he Rayleigh distribution results for the entire set

of h t ) .

by multipath cancellation; 2) IS1 caused by nonzero d;

and

3)

shift of sampling timing as a result of delay spread.

It

is

clear from

both figures

that,

for

small d,

1) IS1

causes

a very small perturbation at the tail end; and

2)

a small

shift of sampling timing has a negligible effect or t h e ir-

reducible BER. Therefore, the major error mechanism is

signal fading in this case. For example, the signal com-

ponent is much stronger than IS1 in Fig. 5 ; as a result,

irreducible errors will not occur. On the contrary, the sig-

nal component for Fig.

6

is in a deep fade and 0 k ) de-

pends only one two adjacent symbols; a BER of about 0.5

is expected because the output bit is uncorrelated with the

corresponding input bit.

It is indicated by,the simulation that the occurrence of

irreducible symbol errors is very bursty. For example, for

a channel with a Gaussian profile and d =

0.05,

a simu-

lation for differential detection of BPSK yields a

1.5

X

average irreducible BER. 1n.this simulation,only 10

out of

2000

samples of h ( t ) result in irreducible symbol

errors; in these

10,

the BER is very high.

To determine whether the burst of errors is a result of

envelope fading, cumulative distributions of the ampli-

tude of O

k )

when the irreducible symbol error occurs

are computed. The distributions shown in Fig. 7 corre-

spond to two-ray channels with d =

0.05,

0.1 and 0.2 for

coherent detection of BPSK, along with a Rayleigh dis-

tribution, which is the amplitude distribution of the entire

1

I

I I

I l l 1

NRZ, Two-Ray Profile

U/D

=

1 OO

Delay

d =

2T

10-1

O T

10-5

I I I

I 1 I I I

0.01

0.02

0.04 0 06 0.08 0.10

d

Fig. 8. The irreducible

BER

performance for BPSK with two kinds of de-

tection and two B / F ratios. (two-ray profile)

sample space of

h (

t ) with or without symbol errors). The

rms signal level for the Rayleigh-faded signal is 0 dB. All

cumulative distributions are computed as probabilities in

the

entire sample space. It is indicated that

O k s

al-

ways in a fade when irreducible symbol errors occur; for

example, the depth of fading is at east 15dB with respect.

to the

rms

signal level for d .1.

To determine the influence of timing error on the irre-

ducible BER, more simulations were performed. In the

case of a two-ray propagation environment, a comparison

of the average irreducible BER for a system using a very

fast squaring timing recovery circuit (i.e., loop band-

width

B

is much higher than the fading rate F ) with one

using a very slow timing circuit (i.e., B << F ) or BPSK

is shown in Fig. 8 . It shows that if delay spread is not

severe, a very fast timing loop improves the irreducible

BER performance only slightly. This suggests that timing

error is not the major mechanism for the bursty irreduci-

ble symbol errors. Simulations using other profiles indi-

cate the same result. In the following, only results for B

<<

Fa re shown.

In summary: 1) For small delay spread, envelope fad-

ing is the most important mechanism causing error bursts;

and

2

for severe delay spread, extrapolation

o f

Fig. 8

suggests that timing error could be a significant factor if

the timing recovery circuit is not fast enough.

The most significant implication o f this result is that

diversity selection can be effective in this case of small



CHUANG: EFFECTS OF TIMEELAYPREAD ON PORTABLE RADIO COMMUNICATIONS

885

+

BPSK

Coherent Detection

OQPSK

* Q p SK M o d u la s

]

X

MSK

10-4I

I

I I 1 1 1 1 I I I I I l l

10-2

d

Fig. 9. The irreducible BER performance for different modu lations with

coherent detection for a channel with a Gaussian profile. Th e parameter

d is the rms delay spread normalized by symbol period .

delay spread because an irreducible symbol error rarely

occurs fora diversity branch that has a high received

power.

C. Comparison

of

Modulation and Dete ction Methods

Fig. 8 indicates that coherent detection performs better

than differential detection; we shall focus the discussion

on coherent detection.

Fig.

9

shows the average irreducible BER as functions

of

d

for different unfiltered modulation methods with co-

herent detection; the multipath channel is simulated by

using a Gaussian delay profile. This figure indicates that

the delay spread performance of various unfiltered mod-

ulations when normalized to the same symbol period is

ranked in the following order: 1) BPSK, 2 ) QPSK, 3)

OQPSK, 4) MSK. The performance of BPSK is the best

because cross-rail interference does not exist. Both

OQPSK and MSK have a

T / 2

iming offset between two

bit sequences, hence the cross-rail IS1 is more severe;

therefore, their performances are inferior to that of QPSK.

In Fig. 9, the normalization factor for parameter d is

the symbol period T , during which two bits are transmit-

ted over the channel for QPSK, OQPSK, and MSK, while

only one bit is sent for BPSK. A fairer comparison of

performance for the same information capacity should be

based on

d’ =

r

/ Tb

where

r

is the rms delay spread and

Tb is the bit period. Fig. 1 0 is the same set of functions

as those in Fig. 9 plotted against

d ’

. When this normali-

10-1

I I

I I

I

I l l

I

I I

I I

I l l

Coherent Detection

I

PSK 1

OQPSK

* QPSKodulation

X MSK

d’ = rms delay spread

bitper iod

Fig. 10. Thesame set of curves as in Fig. 9, plottedagainstrmsdelay

spread normalized by bit period.

zation is applied, it is clear that 4-level modulations

(QPSK, OQPSK, and MSK) are more resistant to delay

spread than BPSK for constant information throughput.

Higher level modulations were also considered. For ex-

ample, Fig.

11

indicates that the performance of 8-PSK

as SNR approaches infinity is not superior to that of QPSK

even though it derives 3 bits per symbol. Since higher

level modulations are less efficient than 4-level modula-

tions at low SNR, we shall concentrate on 4-level modu-

lations in the next section.

It is also interesting to note that all the curves shown in

Figs. 9,

10,

and

11

are nearly parallel to a straight line

o f

slope

2 ;

that is, an order of magnitude increase in delay

spread results in about two orders of magnitude increase

in the irreducible BER within the range of

d

simulated.

By showing that the group delay is a Student’s

t

distri-

bution with two degrees of freedom, Andersen

et al. [13]

have proved that the irreducible BER caused by fre-

quency-selective fading is proportional to

d 2

when d is

small and that the proportionality constant depends on the

method of modulation and detection. The results of the

simulation are consistent with this earlier result.

D . Cumulative Distribution of BER

Because symbol errors are very bursty, it is important

to predict the cumulative distribution functions (cdf) of

BER, namely, the probability that short-term BER per-

formance is worse than a certain value, say,

lop3.

Fig.



10.4

I I I I

I I l l

I I I I 1 1 1 1

10-2

10-1 1 0

d ’= ms delay spread

bit eriod

Fig . 11. The irreducible BER performance

for

QPSK and 8-PSK . The pa-

rameter

d ‘

is the rms delay spread normalized by bit period.

12 is aypical cdf plot which includes a set of distribution

curves for BPSK with two detections (coherent and dif-

ferential) simulated by using a Gaussian profile with d

=

0.08.

In

the

example

set

of

curves,

the

average

& , / N o

(sometimes called the ‘‘local mean”) is varied from 0 dB

to 40 dB in 5 dB steps so that a change of signal level due

to shadow fading and other large-scale variations can be

accounted for.

For each cdf plot, the abscissa indicates a better BER

performance in the left-hand part of the figure while the

ordinate indicates a more reliable coverage in the lower

part of the figure; therefore, the curve appearing in the

left-most and lowest position represents the “best” com-

bination. Both axes are expressed in a log scale. It is easy

to see from Fig. 12 that, as expected, 1) coherent detec-

tion

is

better than differential detection and 2) as SNR

increases, the performance gets better; however, dimin-

ishing returns are observed as the “irreducible BER” is

approached.

V. RESULTSFOR MODULATIONS

ITH

SPECTRAL-

SHAPING FILTERS

Two classes of spectrally efficient modulation methods

that have been considered for practical applications in

mobile and satellite communications are considered: 1)

Gaussian-filtered minimum shift keying (GMSK)

[

141 and

2) QPSK with a raised-cosine Nyquist pulse (RC-QPSK)

1191.

102

C: Coherent Detection

D Differential D

1o- I l I 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 “ 1 1 1 1 1 1 1 l l f i l l l l l

I

‘ l u L L J

10.5

10.4 10-3 10-2

10-1

100

BER

Fig.

12.

BER distr ibutions

for

BPSK in a channel with a Gaussian profile

d

=

0 . 0 8 .

A .

General

Similar to the unfiltered modulations in Section IV, we

find that: 1 ) Timing jitter caused by small delay spread is

not crucial; 2) coherent detection is he more desired

choice;

3

the

BER

performance

is

relatively

insensitive

to the shape of the delay profiles, but this sensitivity in-

creases with

d ;

and 4) the irreducible BER increases about

two orders of magnitude as

d

increases from 0.02 to 0.2.

A measured power delay profile with about 250 ns rms

delay spread as shown in Fig. 3 was used to simulate a

real-world portable communications channel. Only re-

sults for coherent detection will be shown.

B. RC-QPSK

Fig.13 shows the average irreducible BER perfor-

mance of RC-QPSK as a function

of

roll-off factor

a ,

along with that for unfiltered QPSK, for different values

of

mis

delay spread.

As

a

increases, the irreducible BER

for a given value of d decreases monotonically due to de-

creasing ISI;however, he spectral occupancy also in-

creases. It is interesting to note that RC-QPSK with a 2

0.75 is more resistant to delay spread than the unfiltered

NRZ-QPSK.

C. GMSK

Fig. 14 shows the BER performance as a function of

the BTb product of the GMSK premodulation filter, along

with that for unfiltered MSK modulation. It is found that

the best BER performance is achieved by choosing BTb =



Coherent

a

=

0.25

RC-QPSK

EdNo

=

30

dB

-e d’ =

0.40

o d’

=

0.20

+

d‘=0.10

A

d ‘=0 .04

10-1 1

J

1 0 - 5 10-4 10-3 10-2

10-1

1

BER

Fig. 15. D etermining the maximum al lowable s ignal ing rate by using a cdf

plot . The parameter d’ s the rms delay spread normalized by bit period.

higher signaling rate is possible with even better perfor-

mance if diversity is implemented.

VI.CONCLUSION

A flexible simulation for evaluating the BER perfor-

mance of a frequency-selective, slowly fading digital ra-

dio channel has been described. Results from the simu-

lation for normalized delay spread in the range of 0.02

d 0.2 are consistent with the following performance

characteristics:

a) Only a small fraction of. the multipath channel im-

pulse responses encountered will exhibit “irreducible” bit

errors; however, once channel conditions which cause er-

rors occur, the resulting short-term BER is very high.

b) The major error mechanism is envelope fading; the

degradation due to timing error caused by a small delay

spread is not significant.

c) Since the irreducible errors occur only when the sig-

nal is in a deep fade, diversity will be an effective way to

lower the irreducible BER or to permit higher transmis-

sion rates for a given delay spread.

d) The BER performance is more sensitive to the rms

value of the delay spread than to the shape of the delay

profile. By measuring the rms delay spread, both symbol

rate and the modulation/detection scheme can be chosen

using the simulation. A power-of-two dependence on the

rms delay spread for the irreducible BER is found.

e) The BER averaged over fading samples of a given

delay profile provides a comparison among different mod-

ulation/detection schemes; cumulative distributions can

be used to determine the allowable signaling rate.

f ) Coherent detection is more resistant to delay spread

than is differential detection.

g

4-level modulation yields greater information rates

for a given delay spread than does 2-level modulation.

h) GMSK with BT,

=

0.25 is near optimum for resis-

tance to delay spread.

i)RC-QPSK performance degrades monotonically as

the roll-off factor is reduced from 1 to 0.

j) It is possible to transmit a few hundred kbits/s using

a TDM /TDMA architecture in a typical portable, radio

environment without diversity or equalization.

ACKNOWLEDGMENTS

The author would like to thank D. C. Cox,

P.

T. Por-

ter, and H . W . Arnold, for theirdiscussions and guidance

during the course of this work. Special thanks are due.to

D. Devasirvatham for providing a delay profile measured

by him to be used in the simulation. Finally, the author

is grateful to theanonymous reviewers for their comments

that improve the quality of this paper.

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