Comparison of Conventional and rampDQPSK in Rayleigh Fading and Gaussian Noise

Chun Sum Ng Tjeng Thiang Tjhung Fumiyuki Adachi and Kin Mun Lye

Department of Electrical Engineering National University of Singapore Singapore NTT Mobile Communications Network Inc Japan

ABSTRACT Using a recently derived closed-form expression for

the distribution of the phase angle between two Ray- leigh vectors perturbed by Gaussian noise and inter- symbol interference we compute and compare the bit error rates (BER) of narrowband T~-DQPSK and conventional DQPSK systems We also provide physi- cal explanations on the effects of Doppler frequency spread and IF filter bandwidth on the BER which are not available in previous publications

Introduction

The quaternary differentially encoded and detected phaseshift-keyed (DQPSK) system has been widely applied in line modem terrestrial microwave radio as well as satellite communications The rationale behind such a choice is the high bit rate-bandwidth ratio of the QPSK format and the robustness of the differential detection scheme in the presence of carrier phase uncei- taint y

Conventional or classical DQPSK (0 f ~ 2 T phase changes) is the usual signalling scheme Symme- tric or T~-DQPSK (~4 amp 3 ~ 4 phase changes) had been suggested some time ago as a viable alternative to conventional DQPSK [l] Recently the symmetric DQPSK noted for its more efficient spectral shape has been chosen in the North American and Japanese digital cellular radio standards [2][3]

The bit error rate (BER) performance of the con- ventional DQPSK and T~-DQPSK in Rayleigh fading and Gaussian noise has been presented in [4]-[6] In an earlier study for the channel with only Gaussian noise Pawula [7] showed that the BER of conventional DPSK was different from the BER of the symmetric DPSK when the noise was correlated This finding was then surprising to many researchers and Pawula traced the difference in BER to the nonzero correlation between the noise samples

In this paper we will present two new results First we extend Pawularsquos discussion for DPSK in the additive white Gaussian noise (AWGN) channel to DQPSK in the channel with AWGN and Rayleigh fading Next we provide physical explanations on the

effects of Doppler frequency spread and IF filter band- width on the BER of both the conventional and ~ 4 - DQPSK This explanation which have not appeared in the many previous publications [8]-[ll] is particular- ly useful to researchers who wish to gain physical insight into the performance degrading mechanisms in fading channels

System and Signal Model

The DQPSK system under study is depicted in The lowpass equivalent form of the DQPSK Fig1

signal arriving at the receiver is given by

represents the message phase angle in which T is the symbol period p(t) is a unit pulse in (OT) and en is

the differentially encoded message phase The enrsquos are

formed using a Gray code such that a dibit in (nT (n+l)T) taking one element in 00011110 corres- ponds to a ( 0 n -0 n-1 ) mod 2~ taking a respective

phase angle in ~ 4 3714 -3~ 4 -~ 4 for the ~ 1 4 - DQPSK and in (0 T2 T 3 ~ 2 for the conventional

w(t) A TGN

Fig1 The DQPSK System

In ( l ) the carrier amplitude R(t) and phase b(t)

0-7803-0917-093$0300 0 1993 IEEE 141 1

are independent random processes where R(t) is Ray-

leigh distributed with expectation ER(t) = 2a2 and

b(t) is uniformly distributed in d The x (t) and

ys( t) are thus independent Gaussian lowpass processes

for which we assume to have a power spectral density

S 1

(PSd) of

(3) where f

shift and as = Exs(t) = Eys(t)) = ER (t)2

denotes the maximum Doppler frequency

2 2 2 2 D

An AWGN w(t) with one-sided psd No WHz is

added to Ss(t) at the receiver front end to simulate the

receiver amplification noise The IF filter is taken to have a Gaussian shaped passband with a lowpass equivalent transfer function

H(f) = ezp[-r2B2] (4)

where B is the IF equivalent noise bandwidth This IF filter removes out-of-band noise while introducing IS1 to the signal An approximate analysis of the IS1 effects is given in the following section

Quasi-tatic Analysis Model

Following Ng etal[ll] we assume that the fading speed is much slower than the symbol rate 1T so that R(t) and b(t) or x (t) and y (t) l remain relatively

constant over a few symbol durations For then the lowpass equivalent output signal of the IF fdter can be obtained in a quasi-static manner as

9

in which the asterisk denotes convolution and h(t) is the inverse Fourier transform of H(f) in (4)

It is to be noted that in (5) the fading process xs(t) + jys(t) defined in ( l ) is kept unchanged in its

passage through the IF filter This assumption holds

only if the value of f T is very small which is true in

most practical situations For example consider a 900 MHz carrier frequency and a 100 kmh travelling speed This results in fD= 833 Hz Further assuming

a bit rate of 50 kbps we have a f T value of around

0003 The analysis in this paper is based on the above approximation and is sometimes referred to as a quasi- static analysis

D

D

Pdff(A) and cdf F(A of the DifFaential Phaae

Let A$ = $(t2)-amptl) be the differential phase

due only to the message while A$ = A6+Aq where A b = 6(t2)-6(tl) and A = q(t2)-q(tl) are random

differential phases arising from the fading process (random FM noise) and AWGN respectively In order to find the BER for the DQPSK signal under Rayleigh fading we need to find the probability distribution of A$ The pdf f(A$) of A$ has been derived in [I11 with the result

2 2 1- T +A

(7)

U = T cos(A$+A$) + X sin(A$+A) (8)

The parameters r and A are the normalized auto- and cross-correlation coefficients of the combined faded signal and AWGN and are derived in [ll] as

where

where 1 2 2

C1 = 2- [a (tdj + a (t0-T)]

1412

in which E is the average signal energy per bit J ( - )

is the zeroth order Bessel function of the first kind and a(t) denotes the carrier amplitude defined in (6)

The corresponding cdf F(A$) of A$ defined by

b 0

A$ F(A$) = I f(4 dx (14)

- -T was also derived in [ll] as given in (15)

Characteristics of f( A $)

It is interesting to analyse how the shape of the pdf f(A$) given by (7) varies as a function of f T BT

SNR and the signal phase difference A$ Firstly from the right hand side of (9) it is to be noted that the

second term in the numerator (lp)exp(-TB T ) is due to the correlation between the filtered AWGN samples at t and t This term diminishes as BT gets

larger or equivalently as the filtered AWGN samples become less correlated Neglecting this term we have r = z $ ~ cos A $ and X = z $ ~ - sin A 4 where z =

1 [($- Ci)((C+ l ~ ) ~ - Ci)) Substituting these

expressions of r and into (8) and (7) it is trivial to show that f(A$) is independent of A$ for the case of no noise correlation Furthermore since ( l p ) - exp[-rB T ] also tends to zero as p tends to infinity by the same argument f(A$) will be independent of A 4 at very large SNR regions

Next we consider the general case of correlated filtered AWGN We show in Figs2(a)-2(d) curves of f(A$) for various values of fDT BT SNR and the

transmitted signal phase difference AO for the case of -T~-DQPSK Similar curves for the case of conventio- nal DQPSK are shown in Figs3(a)-3(d) From these figures it can be seen that the pdf curves are all depen- dent on the signal phase difference AO or on A$ as a result of IF band-limitation It is to be noted that in the case of no noise correlation the curves in each figure will merge into a single curve Now compare

D

2 2

1 2lsquo

2 2

FigP(a) and FigZ(d) where f T and E N are fixed

at 002 and 5 dB respectively It can be seen that as BT is brought down from 10 to 06 the curves for A 8 = 025x are seperated further apart from the curves for AO= f 0 7 5 ~ The pdf curves for AB= k025-T are peakier than the pdf curves for AB= A075r Further pdf curves with positive ABS are skewed to the left while those with negative A h are skewed to the right The skewness is also observed to be more pronounced for a lower BT or a higher noise correlation Next we compare FigS(a) and Fig2(c) for the same set of BT and f T values As can be seen

as the SNR gets larger the pdf curves becomes peakier Lastly by comparing Fig2(b) and FigZ(c) for the same set of BT and SNR values the pdf curves for the larger f T value are spread broader than those for the

smaller f T value Similar observations are obtained

for the pdf curves in FigsS(a)-S(d) for the case of conventional DQPSK However for conventional DQPSK f(A$) is symmetric or unskewed when Ad equals zero or f ~

D b o

D

D

D

Conditional BER of DQPSK In this section we will employ (15) to evaluate the

BER of DQPSK signals The transmitted phase A 8 is distorted to A$ due to the IS1 produced by the narrowband IF filter With the ideal differential phase detector output represented by A$+A$ and for a Gray coded message the conditional BER can be calculated from

P(E( Ad) = P r o b [ - r I X lt - 3 ~ 4 ]

+ 05 Prob[-3-~4 5 X lt 4 4 1 + 05 Prob[-~4 X lt 3 ~ 4 ]

where x = (A$+AampAfl mod 2 ~ For an IF band- width not unreasonably narrow so that A $-A 8 lies in the range of f r 4 (16) reduces to

rsin(A$+A$)-Xcos(A$+A$) rsin A$ - X c o s A $ -a 1 + s i n - ~ ( p ) ] +

ji- ( r c o s A + Xsin A)rsquo

-1 - [ E 2 - sin (rcos A + Xsin

1413

ldquo I ldquo

Fig2 DQPSK System

(a)

PDF of A$ in 7ramp

BT = 1 f T = 002 D E N = 5 dB b o

BT = 1 f T = 00002

EbN = 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D E N = 5 dB

(b) D

0

(c)

0

(d)

b o

Fig3 ventional DQPSK System

(a)

PDF of A$ in con-

BT = 1 f T = 002 D E N = 5 dB b o

(b) BT = 1 f T = 00002 D EbNo= 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D EbN = 5 dB

(c)

0

(d)

0

~

At9 =251~ A B =-25 TT ~ t 9 = 2 5 ~ A8=-25 7~ At9=751~ A0=-757~ At9=757r A 0 =-75 IT

-7r 0 A+ n

(b) At9 =-25 n At9=-757~ 1

-7T 0 A 9 n

f(W

I I -n 0 A 9 7~ -IT 0

At9 =O A d =--5 n

At9 =5n

- AO=O - At9=-5n

At9 =O At9 =-5 IT

-7r I

(4 1 1

-7T 0 A+ T -7r 0

1414

I

P(EJ A8) = fDT where J0(27rfT) --t 0 The ensuing BER are now

governed only by the f TI and for non-zero f T F(AB-A$-~T~) - F(A~-A$+T~) 1 D D values the BER tends to some constant or floor values as the SNR tends to infinity

(17) where the cdf F( - ) is defined in (15) IS1 Analysis Technique

We now have closed-form formulae (17) and (15) for computing BER for narrowband DQPSK systems in Rayleigh fading channels Before we proceed further we will first discuss two limiting cases where the

A final source of randomness which we have to include in the computation of the average bit error probability Pb or BER arises from the different

effects of either the AWGN or the random phase noise are dominant

sequences of the message phase angle 8(t) From (6) we observe that 8(t) is distorted to $(t) and the unity normalized carrier amplitude is distorted to a(t) by IS1 due to the band-limitation imposed by the IF filter The Case of Quasidationary Fading

To account for the effect of IS1 through a(t) and $(t) we apply the IS1 analysis technique presented in [4] and [ll] for conventional DQPSK and T~-DQPSK respectively to compute a(t) and $(t)

In this case the random phase noise due to the fading mechanism varies so slowly that its effects on the BER is negligible This means Ab 0 and A$ AV that is the differential phase noise comes only from the AWGN This slowly fading case also implies that f T + 0 so that from (12) T= J (2n f T)

approaches unity and

BER Computation Results and Discussions

We consider the bandwidth parameter BT to be sufficiently large so that the distortion due to IS1 is restricted to I A 8-A 4 I lt ~ 4 It was computational- ly found that I AampA$I lt ~ 4 if BT 2 06 We also consider the fading induced parameter f T to be D restricted to a useful range from zero to 002 Within this range of f T values 0 -J (2rfDT) is noted to

decrease monotonically from unity as f T increases

D O D

- =(E T) 1 lt- c C O S A$ + ( l p ) e 1 T =

IFl+ 1 P I 2 - lt (18) D T- 0

D (19) from zero

s in A$ A =

We have computed BER for both r4-DQPSK and conventional DQPSK systems for various values of BT andf T

[ [ C l + U P - cp The pdf f(A$) and the resulting BER are then D controlled by the SNR p the IS1 through the parame- ters C and C and the signal phase difference A$

In Fig4 we show BER curves for a low BT value of 06 for which IS1 effects are large and for various values of f T As can be seen from these curves the 1 2

D The Case of an Intde SNR

(Error Rate Floor)

In this case the differential phase noise comes only from the fading process since the contribution from the AWGN tends to zero By letting p -I m in (9) and (10) we have r = cos A$ X = OTsin A$ and by

substituting these values of r and into (8) we get p = Tcos A$ which is independent of the signal

phase difference A$ The pdf f(A$) is in turn inde- pendent of A 4 and therefore is symmetric or unskewed For this infinite p case it can also be shown

T that f(A$) approaches an impules function as 0

T

BER at first fall off with increasing E N indicating a b o behaviour expected of an AWGN channel Then a certain value of E N is reached beyond which the

BER attains a constant floor value indicating the influence of only the random FM phase noise As expected the BER floor for a higher f T value is

higher than that for a lower f T value indicating the

rapid spreading of the random phase noise as f T gets

larger This can be explained from the fact as noted earlier in Section 52 that the pdf f(A$) is spread broader with an increase in the f T value Because the

b o

D D

D

D approaches unity and f(A$) tends to be a uniform distribution as 0 tends to zero The former occurs

when f T + 0 while the latter occurs at larger values of

error probability terms in (16) are computed within fixed ranges of the phase difference the broader the f( A$) the larger is the error probability

It is also noted in Fig4 that conventional DQPSK

T

D

1415

are independent random processes where R(t) is Ray-

leigh distributed with expectation ER(t) = 2a2 and

b(t) is uniformly distributed in d The x (t) and

ys( t) are thus independent Gaussian lowpass processes

for which we assume to have a power spectral density

S 1

(PSd) of

(3) where f

shift and as = Exs(t) = Eys(t)) = ER (t)2

denotes the maximum Doppler frequency

2 2 2 2 D

An AWGN w(t) with one-sided psd No WHz is

added to Ss(t) at the receiver front end to simulate the

receiver amplification noise The IF filter is taken to have a Gaussian shaped passband with a lowpass equivalent transfer function

H(f) = ezp[-r2B2] (4)

where B is the IF equivalent noise bandwidth This IF filter removes out-of-band noise while introducing IS1 to the signal An approximate analysis of the IS1 effects is given in the following section

Quasi-tatic Analysis Model

Following Ng etal[ll] we assume that the fading speed is much slower than the symbol rate 1T so that R(t) and b(t) or x (t) and y (t) l remain relatively

constant over a few symbol durations For then the lowpass equivalent output signal of the IF fdter can be obtained in a quasi-static manner as

9

in which the asterisk denotes convolution and h(t) is the inverse Fourier transform of H(f) in (4)

It is to be noted that in (5) the fading process xs(t) + jys(t) defined in ( l ) is kept unchanged in its

passage through the IF filter This assumption holds

only if the value of f T is very small which is true in

most practical situations For example consider a 900 MHz carrier frequency and a 100 kmh travelling speed This results in fD= 833 Hz Further assuming

a bit rate of 50 kbps we have a f T value of around

0003 The analysis in this paper is based on the above approximation and is sometimes referred to as a quasi- static analysis

D

D

Pdff(A) and cdf F(A of the DifFaential Phaae

Let A$ = $(t2)-amptl) be the differential phase

due only to the message while A$ = A6+Aq where A b = 6(t2)-6(tl) and A = q(t2)-q(tl) are random

differential phases arising from the fading process (random FM noise) and AWGN respectively In order to find the BER for the DQPSK signal under Rayleigh fading we need to find the probability distribution of A$ The pdf f(A$) of A$ has been derived in [I11 with the result

2 2 1- T +A

(7)

U = T cos(A$+A$) + X sin(A$+A) (8)

The parameters r and A are the normalized auto- and cross-correlation coefficients of the combined faded signal and AWGN and are derived in [ll] as

where

where 1 2 2

C1 = 2- [a (tdj + a (t0-T)]

1412

in which E is the average signal energy per bit J ( - )

is the zeroth order Bessel function of the first kind and a(t) denotes the carrier amplitude defined in (6)

The corresponding cdf F(A$) of A$ defined by

b 0

A$ F(A$) = I f(4 dx (14)

- -T was also derived in [ll] as given in (15)

Characteristics of f( A $)

It is interesting to analyse how the shape of the pdf f(A$) given by (7) varies as a function of f T BT

SNR and the signal phase difference A$ Firstly from the right hand side of (9) it is to be noted that the

second term in the numerator (lp)exp(-TB T ) is due to the correlation between the filtered AWGN samples at t and t This term diminishes as BT gets

larger or equivalently as the filtered AWGN samples become less correlated Neglecting this term we have r = z $ ~ cos A $ and X = z $ ~ - sin A 4 where z =

1 [($- Ci)((C+ l ~ ) ~ - Ci)) Substituting these

expressions of r and into (8) and (7) it is trivial to show that f(A$) is independent of A$ for the case of no noise correlation Furthermore since ( l p ) - exp[-rB T ] also tends to zero as p tends to infinity by the same argument f(A$) will be independent of A 4 at very large SNR regions

Next we consider the general case of correlated filtered AWGN We show in Figs2(a)-2(d) curves of f(A$) for various values of fDT BT SNR and the

transmitted signal phase difference AO for the case of -T~-DQPSK Similar curves for the case of conventio- nal DQPSK are shown in Figs3(a)-3(d) From these figures it can be seen that the pdf curves are all depen- dent on the signal phase difference AO or on A$ as a result of IF band-limitation It is to be noted that in the case of no noise correlation the curves in each figure will merge into a single curve Now compare

D

2 2

1 2lsquo

2 2

FigP(a) and FigZ(d) where f T and E N are fixed

at 002 and 5 dB respectively It can be seen that as BT is brought down from 10 to 06 the curves for A 8 = 025x are seperated further apart from the curves for AO= f 0 7 5 ~ The pdf curves for AB= k025-T are peakier than the pdf curves for AB= A075r Further pdf curves with positive ABS are skewed to the left while those with negative A h are skewed to the right The skewness is also observed to be more pronounced for a lower BT or a higher noise correlation Next we compare FigS(a) and Fig2(c) for the same set of BT and f T values As can be seen

as the SNR gets larger the pdf curves becomes peakier Lastly by comparing Fig2(b) and FigZ(c) for the same set of BT and SNR values the pdf curves for the larger f T value are spread broader than those for the

smaller f T value Similar observations are obtained

for the pdf curves in FigsS(a)-S(d) for the case of conventional DQPSK However for conventional DQPSK f(A$) is symmetric or unskewed when Ad equals zero or f ~

D b o

D

D

D

Conditional BER of DQPSK In this section we will employ (15) to evaluate the

BER of DQPSK signals The transmitted phase A 8 is distorted to A$ due to the IS1 produced by the narrowband IF filter With the ideal differential phase detector output represented by A$+A$ and for a Gray coded message the conditional BER can be calculated from

P(E( Ad) = P r o b [ - r I X lt - 3 ~ 4 ]

+ 05 Prob[-3-~4 5 X lt 4 4 1 + 05 Prob[-~4 X lt 3 ~ 4 ]

where x = (A$+AampAfl mod 2 ~ For an IF band- width not unreasonably narrow so that A $-A 8 lies in the range of f r 4 (16) reduces to

rsin(A$+A$)-Xcos(A$+A$) rsin A$ - X c o s A $ -a 1 + s i n - ~ ( p ) ] +

ji- ( r c o s A + Xsin A)rsquo

-1 - [ E 2 - sin (rcos A + Xsin

1413

ldquo I ldquo

Fig2 DQPSK System

(a)

PDF of A$ in 7ramp

BT = 1 f T = 002 D E N = 5 dB b o

BT = 1 f T = 00002

EbN = 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D E N = 5 dB

(b) D

0

(c)

0

(d)

b o

Fig3 ventional DQPSK System

(a)

PDF of A$ in con-

BT = 1 f T = 002 D E N = 5 dB b o

(b) BT = 1 f T = 00002 D EbNo= 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D EbN = 5 dB

(c)

0

(d)

0

~

At9 =251~ A B =-25 TT ~ t 9 = 2 5 ~ A8=-25 7~ At9=751~ A0=-757~ At9=757r A 0 =-75 IT

-7r 0 A+ n

(b) At9 =-25 n At9=-757~ 1

-7T 0 A 9 n

f(W

I I -n 0 A 9 7~ -IT 0

At9 =O A d =--5 n

At9 =5n

- AO=O - At9=-5n

At9 =O At9 =-5 IT

-7r I

(4 1 1

-7T 0 A+ T -7r 0

1414

I

P(EJ A8) = fDT where J0(27rfT) --t 0 The ensuing BER are now

governed only by the f TI and for non-zero f T F(AB-A$-~T~) - F(A~-A$+T~) 1 D D values the BER tends to some constant or floor values as the SNR tends to infinity

(17) where the cdf F( - ) is defined in (15) IS1 Analysis Technique

We now have closed-form formulae (17) and (15) for computing BER for narrowband DQPSK systems in Rayleigh fading channels Before we proceed further we will first discuss two limiting cases where the

A final source of randomness which we have to include in the computation of the average bit error probability Pb or BER arises from the different

effects of either the AWGN or the random phase noise are dominant

sequences of the message phase angle 8(t) From (6) we observe that 8(t) is distorted to $(t) and the unity normalized carrier amplitude is distorted to a(t) by IS1 due to the band-limitation imposed by the IF filter The Case of Quasidationary Fading

To account for the effect of IS1 through a(t) and $(t) we apply the IS1 analysis technique presented in [4] and [ll] for conventional DQPSK and T~-DQPSK respectively to compute a(t) and $(t)

In this case the random phase noise due to the fading mechanism varies so slowly that its effects on the BER is negligible This means Ab 0 and A$ AV that is the differential phase noise comes only from the AWGN This slowly fading case also implies that f T + 0 so that from (12) T= J (2n f T)

approaches unity and

BER Computation Results and Discussions

We consider the bandwidth parameter BT to be sufficiently large so that the distortion due to IS1 is restricted to I A 8-A 4 I lt ~ 4 It was computational- ly found that I AampA$I lt ~ 4 if BT 2 06 We also consider the fading induced parameter f T to be D restricted to a useful range from zero to 002 Within this range of f T values 0 -J (2rfDT) is noted to

decrease monotonically from unity as f T increases

D O D

- =(E T) 1 lt- c C O S A$ + ( l p ) e 1 T =

IFl+ 1 P I 2 - lt (18) D T- 0

D (19) from zero

s in A$ A =

We have computed BER for both r4-DQPSK and conventional DQPSK systems for various values of BT andf T

[ [ C l + U P - cp The pdf f(A$) and the resulting BER are then D controlled by the SNR p the IS1 through the parame- ters C and C and the signal phase difference A$

In Fig4 we show BER curves for a low BT value of 06 for which IS1 effects are large and for various values of f T As can be seen from these curves the 1 2

D The Case of an Intde SNR

(Error Rate Floor)

In this case the differential phase noise comes only from the fading process since the contribution from the AWGN tends to zero By letting p -I m in (9) and (10) we have r = cos A$ X = OTsin A$ and by

substituting these values of r and into (8) we get p = Tcos A$ which is independent of the signal

phase difference A$ The pdf f(A$) is in turn inde- pendent of A 4 and therefore is symmetric or unskewed For this infinite p case it can also be shown

T that f(A$) approaches an impules function as 0

T

BER at first fall off with increasing E N indicating a b o behaviour expected of an AWGN channel Then a certain value of E N is reached beyond which the

BER attains a constant floor value indicating the influence of only the random FM phase noise As expected the BER floor for a higher f T value is

higher than that for a lower f T value indicating the

rapid spreading of the random phase noise as f T gets

larger This can be explained from the fact as noted earlier in Section 52 that the pdf f(A$) is spread broader with an increase in the f T value Because the

b o

D D

D

D approaches unity and f(A$) tends to be a uniform distribution as 0 tends to zero The former occurs

when f T + 0 while the latter occurs at larger values of

error probability terms in (16) are computed within fixed ranges of the phase difference the broader the f( A$) the larger is the error probability

It is also noted in Fig4 that conventional DQPSK

T

D

1415

in which E is the average signal energy per bit J ( - )

is the zeroth order Bessel function of the first kind and a(t) denotes the carrier amplitude defined in (6)

The corresponding cdf F(A$) of A$ defined by

b 0

A$ F(A$) = I f(4 dx (14)

- -T was also derived in [ll] as given in (15)

Characteristics of f( A $)

It is interesting to analyse how the shape of the pdf f(A$) given by (7) varies as a function of f T BT

SNR and the signal phase difference A$ Firstly from the right hand side of (9) it is to be noted that the

second term in the numerator (lp)exp(-TB T ) is due to the correlation between the filtered AWGN samples at t and t This term diminishes as BT gets

larger or equivalently as the filtered AWGN samples become less correlated Neglecting this term we have r = z $ ~ cos A $ and X = z $ ~ - sin A 4 where z =

1 [($- Ci)((C+ l ~ ) ~ - Ci)) Substituting these

expressions of r and into (8) and (7) it is trivial to show that f(A$) is independent of A$ for the case of no noise correlation Furthermore since ( l p ) - exp[-rB T ] also tends to zero as p tends to infinity by the same argument f(A$) will be independent of A 4 at very large SNR regions

Next we consider the general case of correlated filtered AWGN We show in Figs2(a)-2(d) curves of f(A$) for various values of fDT BT SNR and the

transmitted signal phase difference AO for the case of -T~-DQPSK Similar curves for the case of conventio- nal DQPSK are shown in Figs3(a)-3(d) From these figures it can be seen that the pdf curves are all depen- dent on the signal phase difference AO or on A$ as a result of IF band-limitation It is to be noted that in the case of no noise correlation the curves in each figure will merge into a single curve Now compare

D

2 2

1 2lsquo

2 2

FigP(a) and FigZ(d) where f T and E N are fixed

at 002 and 5 dB respectively It can be seen that as BT is brought down from 10 to 06 the curves for A 8 = 025x are seperated further apart from the curves for AO= f 0 7 5 ~ The pdf curves for AB= k025-T are peakier than the pdf curves for AB= A075r Further pdf curves with positive ABS are skewed to the left while those with negative A h are skewed to the right The skewness is also observed to be more pronounced for a lower BT or a higher noise correlation Next we compare FigS(a) and Fig2(c) for the same set of BT and f T values As can be seen

as the SNR gets larger the pdf curves becomes peakier Lastly by comparing Fig2(b) and FigZ(c) for the same set of BT and SNR values the pdf curves for the larger f T value are spread broader than those for the

smaller f T value Similar observations are obtained

for the pdf curves in FigsS(a)-S(d) for the case of conventional DQPSK However for conventional DQPSK f(A$) is symmetric or unskewed when Ad equals zero or f ~

D b o

D

D

D

Conditional BER of DQPSK In this section we will employ (15) to evaluate the

BER of DQPSK signals The transmitted phase A 8 is distorted to A$ due to the IS1 produced by the narrowband IF filter With the ideal differential phase detector output represented by A$+A$ and for a Gray coded message the conditional BER can be calculated from

P(E( Ad) = P r o b [ - r I X lt - 3 ~ 4 ]

+ 05 Prob[-3-~4 5 X lt 4 4 1 + 05 Prob[-~4 X lt 3 ~ 4 ]

where x = (A$+AampAfl mod 2 ~ For an IF band- width not unreasonably narrow so that A $-A 8 lies in the range of f r 4 (16) reduces to

rsin(A$+A$)-Xcos(A$+A$) rsin A$ - X c o s A $ -a 1 + s i n - ~ ( p ) ] +

ji- ( r c o s A + Xsin A)rsquo

-1 - [ E 2 - sin (rcos A + Xsin

1413

ldquo I ldquo

Fig2 DQPSK System

(a)

PDF of A$ in 7ramp

BT = 1 f T = 002 D E N = 5 dB b o

BT = 1 f T = 00002

EbN = 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D E N = 5 dB

(b) D

0

(c)

0

(d)

b o

Fig3 ventional DQPSK System

(a)

PDF of A$ in con-

BT = 1 f T = 002 D E N = 5 dB b o

(b) BT = 1 f T = 00002 D EbNo= 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D EbN = 5 dB

(c)

0

(d)

0

~

At9 =251~ A B =-25 TT ~ t 9 = 2 5 ~ A8=-25 7~ At9=751~ A0=-757~ At9=757r A 0 =-75 IT

-7r 0 A+ n

(b) At9 =-25 n At9=-757~ 1

-7T 0 A 9 n

f(W

I I -n 0 A 9 7~ -IT 0

At9 =O A d =--5 n

At9 =5n

- AO=O - At9=-5n

At9 =O At9 =-5 IT

-7r I

(4 1 1

-7T 0 A+ T -7r 0

1414

I

P(EJ A8) = fDT where J0(27rfT) --t 0 The ensuing BER are now

governed only by the f TI and for non-zero f T F(AB-A$-~T~) - F(A~-A$+T~) 1 D D values the BER tends to some constant or floor values as the SNR tends to infinity

(17) where the cdf F( - ) is defined in (15) IS1 Analysis Technique

We now have closed-form formulae (17) and (15) for computing BER for narrowband DQPSK systems in Rayleigh fading channels Before we proceed further we will first discuss two limiting cases where the

A final source of randomness which we have to include in the computation of the average bit error probability Pb or BER arises from the different

effects of either the AWGN or the random phase noise are dominant

sequences of the message phase angle 8(t) From (6) we observe that 8(t) is distorted to $(t) and the unity normalized carrier amplitude is distorted to a(t) by IS1 due to the band-limitation imposed by the IF filter The Case of Quasidationary Fading

To account for the effect of IS1 through a(t) and $(t) we apply the IS1 analysis technique presented in [4] and [ll] for conventional DQPSK and T~-DQPSK respectively to compute a(t) and $(t)

In this case the random phase noise due to the fading mechanism varies so slowly that its effects on the BER is negligible This means Ab 0 and A$ AV that is the differential phase noise comes only from the AWGN This slowly fading case also implies that f T + 0 so that from (12) T= J (2n f T)

approaches unity and

BER Computation Results and Discussions

We consider the bandwidth parameter BT to be sufficiently large so that the distortion due to IS1 is restricted to I A 8-A 4 I lt ~ 4 It was computational- ly found that I AampA$I lt ~ 4 if BT 2 06 We also consider the fading induced parameter f T to be D restricted to a useful range from zero to 002 Within this range of f T values 0 -J (2rfDT) is noted to

decrease monotonically from unity as f T increases

D O D

- =(E T) 1 lt- c C O S A$ + ( l p ) e 1 T =

IFl+ 1 P I 2 - lt (18) D T- 0

D (19) from zero

s in A$ A =

We have computed BER for both r4-DQPSK and conventional DQPSK systems for various values of BT andf T

[ [ C l + U P - cp The pdf f(A$) and the resulting BER are then D controlled by the SNR p the IS1 through the parame- ters C and C and the signal phase difference A$

In Fig4 we show BER curves for a low BT value of 06 for which IS1 effects are large and for various values of f T As can be seen from these curves the 1 2

D The Case of an Intde SNR

(Error Rate Floor)

In this case the differential phase noise comes only from the fading process since the contribution from the AWGN tends to zero By letting p -I m in (9) and (10) we have r = cos A$ X = OTsin A$ and by

substituting these values of r and into (8) we get p = Tcos A$ which is independent of the signal

phase difference A$ The pdf f(A$) is in turn inde- pendent of A 4 and therefore is symmetric or unskewed For this infinite p case it can also be shown

T that f(A$) approaches an impules function as 0

T

BER at first fall off with increasing E N indicating a b o behaviour expected of an AWGN channel Then a certain value of E N is reached beyond which the

BER attains a constant floor value indicating the influence of only the random FM phase noise As expected the BER floor for a higher f T value is

higher than that for a lower f T value indicating the

rapid spreading of the random phase noise as f T gets

larger This can be explained from the fact as noted earlier in Section 52 that the pdf f(A$) is spread broader with an increase in the f T value Because the

b o

D D

D

D approaches unity and f(A$) tends to be a uniform distribution as 0 tends to zero The former occurs

when f T + 0 while the latter occurs at larger values of

error probability terms in (16) are computed within fixed ranges of the phase difference the broader the f( A$) the larger is the error probability

It is also noted in Fig4 that conventional DQPSK

T

D

1415

Fig2 DQPSK System

(a)

PDF of A$ in 7ramp

BT = 1 f T = 002 D E N = 5 dB b o

BT = 1 f T = 00002

EbN = 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D E N = 5 dB

(b) D

0

(c)

0

(d)

b o

Fig3 ventional DQPSK System

(a)

PDF of A$ in con-

BT = 1 f T = 002 D E N = 5 dB b o

(b) BT = 1 f T = 00002 D EbNo= 15 dB

BT = 1 f T = 002 D EbN = 15 dB

BT = 06 f T = 002 D EbN = 5 dB

(c)

0

(d)

0

~

At9 =251~ A B =-25 TT ~ t 9 = 2 5 ~ A8=-25 7~ At9=751~ A0=-757~ At9=757r A 0 =-75 IT

-7r 0 A+ n

(b) At9 =-25 n At9=-757~ 1

-7T 0 A 9 n

f(W

I I -n 0 A 9 7~ -IT 0

At9 =O A d =--5 n

At9 =5n

- AO=O - At9=-5n

At9 =O At9 =-5 IT

-7r I

(4 1 1

-7T 0 A+ T -7r 0

1414

I

P(EJ A8) = fDT where J0(27rfT) --t 0 The ensuing BER are now

governed only by the f TI and for non-zero f T F(AB-A$-~T~) - F(A~-A$+T~) 1 D D values the BER tends to some constant or floor values as the SNR tends to infinity

(17) where the cdf F( - ) is defined in (15) IS1 Analysis Technique

We now have closed-form formulae (17) and (15) for computing BER for narrowband DQPSK systems in Rayleigh fading channels Before we proceed further we will first discuss two limiting cases where the

A final source of randomness which we have to include in the computation of the average bit error probability Pb or BER arises from the different

effects of either the AWGN or the random phase noise are dominant

sequences of the message phase angle 8(t) From (6) we observe that 8(t) is distorted to $(t) and the unity normalized carrier amplitude is distorted to a(t) by IS1 due to the band-limitation imposed by the IF filter The Case of Quasidationary Fading

To account for the effect of IS1 through a(t) and $(t) we apply the IS1 analysis technique presented in [4] and [ll] for conventional DQPSK and T~-DQPSK respectively to compute a(t) and $(t)

In this case the random phase noise due to the fading mechanism varies so slowly that its effects on the BER is negligible This means Ab 0 and A$ AV that is the differential phase noise comes only from the AWGN This slowly fading case also implies that f T + 0 so that from (12) T= J (2n f T)

approaches unity and

BER Computation Results and Discussions

We consider the bandwidth parameter BT to be sufficiently large so that the distortion due to IS1 is restricted to I A 8-A 4 I lt ~ 4 It was computational- ly found that I AampA$I lt ~ 4 if BT 2 06 We also consider the fading induced parameter f T to be D restricted to a useful range from zero to 002 Within this range of f T values 0 -J (2rfDT) is noted to

decrease monotonically from unity as f T increases

D O D

- =(E T) 1 lt- c C O S A$ + ( l p ) e 1 T =

IFl+ 1 P I 2 - lt (18) D T- 0

D (19) from zero

s in A$ A =

We have computed BER for both r4-DQPSK and conventional DQPSK systems for various values of BT andf T

[ [ C l + U P - cp The pdf f(A$) and the resulting BER are then D controlled by the SNR p the IS1 through the parame- ters C and C and the signal phase difference A$

In Fig4 we show BER curves for a low BT value of 06 for which IS1 effects are large and for various values of f T As can be seen from these curves the 1 2

D The Case of an Intde SNR

(Error Rate Floor)

In this case the differential phase noise comes only from the fading process since the contribution from the AWGN tends to zero By letting p -I m in (9) and (10) we have r = cos A$ X = OTsin A$ and by

substituting these values of r and into (8) we get p = Tcos A$ which is independent of the signal

phase difference A$ The pdf f(A$) is in turn inde- pendent of A 4 and therefore is symmetric or unskewed For this infinite p case it can also be shown

T that f(A$) approaches an impules function as 0

T

BER at first fall off with increasing E N indicating a b o behaviour expected of an AWGN channel Then a certain value of E N is reached beyond which the

BER attains a constant floor value indicating the influence of only the random FM phase noise As expected the BER floor for a higher f T value is

higher than that for a lower f T value indicating the

rapid spreading of the random phase noise as f T gets

larger This can be explained from the fact as noted earlier in Section 52 that the pdf f(A$) is spread broader with an increase in the f T value Because the

b o

D D

D

D approaches unity and f(A$) tends to be a uniform distribution as 0 tends to zero The former occurs

when f T + 0 while the latter occurs at larger values of

error probability terms in (16) are computed within fixed ranges of the phase difference the broader the f( A$) the larger is the error probability

It is also noted in Fig4 that conventional DQPSK

T

D

1415

P(EJ A8) = fDT where J0(27rfT) --t 0 The ensuing BER are now

governed only by the f TI and for non-zero f T F(AB-A$-~T~) - F(A~-A$+T~) 1 D D values the BER tends to some constant or floor values as the SNR tends to infinity

(17) where the cdf F( - ) is defined in (15) IS1 Analysis Technique

We now have closed-form formulae (17) and (15) for computing BER for narrowband DQPSK systems in Rayleigh fading channels Before we proceed further we will first discuss two limiting cases where the

A final source of randomness which we have to include in the computation of the average bit error probability Pb or BER arises from the different

effects of either the AWGN or the random phase noise are dominant

sequences of the message phase angle 8(t) From (6) we observe that 8(t) is distorted to $(t) and the unity normalized carrier amplitude is distorted to a(t) by IS1 due to the band-limitation imposed by the IF filter The Case of Quasidationary Fading

To account for the effect of IS1 through a(t) and $(t) we apply the IS1 analysis technique presented in [4] and [ll] for conventional DQPSK and T~-DQPSK respectively to compute a(t) and $(t)

In this case the random phase noise due to the fading mechanism varies so slowly that its effects on the BER is negligible This means Ab 0 and A$ AV that is the differential phase noise comes only from the AWGN This slowly fading case also implies that f T + 0 so that from (12) T= J (2n f T)

approaches unity and

BER Computation Results and Discussions

We consider the bandwidth parameter BT to be sufficiently large so that the distortion due to IS1 is restricted to I A 8-A 4 I lt ~ 4 It was computational- ly found that I AampA$I lt ~ 4 if BT 2 06 We also consider the fading induced parameter f T to be D restricted to a useful range from zero to 002 Within this range of f T values 0 -J (2rfDT) is noted to

decrease monotonically from unity as f T increases

D O D

- =(E T) 1 lt- c C O S A$ + ( l p ) e 1 T =

IFl+ 1 P I 2 - lt (18) D T- 0

D (19) from zero

s in A$ A =

We have computed BER for both r4-DQPSK and conventional DQPSK systems for various values of BT andf T

[ [ C l + U P - cp The pdf f(A$) and the resulting BER are then D controlled by the SNR p the IS1 through the parame- ters C and C and the signal phase difference A$

In Fig4 we show BER curves for a low BT value of 06 for which IS1 effects are large and for various values of f T As can be seen from these curves the 1 2

D The Case of an Intde SNR

(Error Rate Floor)

In this case the differential phase noise comes only from the fading process since the contribution from the AWGN tends to zero By letting p -I m in (9) and (10) we have r = cos A$ X = OTsin A$ and by

substituting these values of r and into (8) we get p = Tcos A$ which is independent of the signal

phase difference A$ The pdf f(A$) is in turn inde- pendent of A 4 and therefore is symmetric or unskewed For this infinite p case it can also be shown

T that f(A$) approaches an impules function as 0

T

BER at first fall off with increasing E N indicating a b o behaviour expected of an AWGN channel Then a certain value of E N is reached beyond which the

BER attains a constant floor value indicating the influence of only the random FM phase noise As expected the BER floor for a higher f T value is

higher than that for a lower f T value indicating the

rapid spreading of the random phase noise as f T gets

larger This can be explained from the fact as noted earlier in Section 52 that the pdf f(A$) is spread broader with an increase in the f T value Because the

b o

D D

D

D approaches unity and f(A$) tends to be a uniform distribution as 0 tends to zero The former occurs

when f T + 0 while the latter occurs at larger values of

error probability terms in (16) are computed within fixed ranges of the phase difference the broader the f( A$) the larger is the error probability

It is also noted in Fig4 that conventional DQPSK

T

D

1415

outperforms T~-DQPSK in the region of low E N b o

where the effects of AWGN is dominant This can be explained by comparing Fig2(d) and FigJ(d) where there are two pdPs with A 0 = 0 7 5 ~ for T~-DQPSK which are spread broader and skewed as compared with only one pdf with ne= T for conventional DQPSK which is spread broader and unskewed Broadening of f(A$) would inevitably lead to a larger BER Also because the skewing occurs in a direction that tends to steer the peak of f(A$) away from the decision region it would also lead to a larger BER Thus when the BER is averaged over the four possible Aamps in each case it is reasonable to expect that conventional DQPSK will have a lower overall BER than ~ 4 - DQPSK

On the other hand in the region of very large EbNo where the random FM noise is the only source

of phase disturbance we observe that T~-DQPSK gives a better BER performance This can be explain- ed from the fact that as the SNR gets larger the depen- dence of f( A $) on A 0 diminishes due to the diminish- ing effect of the noise correlation term ( l p )

exp(-TB T ) in (9) and eventually all f(A$)s for different A d merge into a single pdf Therefore the difference in the BER of these two DQPSK systems does not come from the dependence of f(A$) on A8 However it is known that T~-DQPSK has a more confined power spectrum than conventional DQPSK because of less spectral splatter during phase transi- tions between symbols Consequently under the same IF banddimitation a T~-DQPSK signal suffers less phase distortion than a conventional DQPSK signal This leads to a smaller value of I AampA$I and as a result a lower BER for T~-DQPSK In the case of low SNR this band occupancy advantage of ~ 4 - DQPSK over conventional DQPSK is reduced because of the dominant effect of the AWGN

Fig5 shows the same set of BER curves as those in Fig4 except that the BT value has been raised to 1 0 For this larger BT value the two sets of BER curves for ~ 4 - and conventional DQPSK appear merged The coming together of the BER values of these two DQPSK systems shows the decreasing amount of noise correlation as well as symbol phase distortion as ET gets larger The BER values of these two DQPSK systems are actually different In order to show this slight difference we have plotted a ratio

2 2

8 (b T~-DQP S K)b conv DQ P S K 1 (20)

in Fig6 In Fig6 it can be seen that conventional DQPSK outperform T~-DQPSK in the lower SNR region where P bgt 1 while the opposite behaviour

occurs in the higher SNR region where P blt 1 This

F

A - fDT=OO100 B - f~T=00050 C - fDT=00025 D - fDT=OOOlO E - fnT=OO004 F - f~T=OOOOO

(dB)

Fig4 BER for various values off T with BT=06 D

1 D - fDT=00010 F - E - fDT=Oo004 - F - fDT=00000

lo- i o i o $0 80 100

EbNo (dB)

Figd BER for various values off T with B T d 0 D

BER behaviour as a function of SNR is consistent with that observed in the case of BT = 06

Fig7 compares the BER performance of ~ 4 - and conventional DQPSK systems at two f T values and D for different BT values The behaviours of these BER curves follow the explanation given in the preceeding two paragraphs Fig8 shows BER curves for the T~-DQPSK system for various values of BT with f T D

141 6

PE

09999 0 20 40 60 80 100

EbN (dB) Fig6 rampDQPSK to conventional DQPSK BER

ratio for various values of f T with BTA0 D

- BT=l O n4 DQPSK

lo- 20 I 40 I 60 I 80 I 100 EbN (dB1

Fig8 BER of rllrDQPSK for various values of BT with fDT=00025

irreducible error is seen to decrease as BT increases These observations are consistent with that observed in [4] for conventional DQPSK

Figs9 and 10 compare BER floors between ~ 4 - and conventional DQPSK with the former as a func- tion of BT for various f T values and the latter a

function of f T for various BT values These two

figures are plotted with very fine increments of BT and f T respectively to reflect the rate at which the BER D varies with respect to these two system parameters

D

D

Conclusion

In this paper using a recently derived closed-form formulae for the pdf and cdf of the differential phase of a phase modulated carrier in Rayleigh fading channels

C - BT-068 we evaluate and compare the BER of T~-DQPSK and D - BT-100 conventional DQPSK systems Using the dependence

of the shape of the pdf of the message phase difference when the noise samples are correlated we explain why the BER of conventional DQPSK and T~-DQPSK are different We also provide physical explanations not available in previous publications on the effects of Doppler frequency spread IF filter bandwidth and SNR on the BER The difference in BER between ~ 4 - DQPSK and conventional DQPSK under very narrow IF band-limitation and low SNR conditions is attribut- ed to the dependence of the differential phase pdf on

EbN (dB)

D Fig7 BER for various values of BT with f T=OOl

and 0004

set at 00025 Here it is observed that prior to the appearence of the BER floor BT = 10 gives the best BER performance On the other hand the amount of

1417

the signal phase difference Under the same narrow IF band-limitation but at high SNR the T~-DQPSK outperforms the conventional DQPSK because the former has less spectral splatter during phase transi- tions between symbol An IF bandwidth of about one times the symbol rate yields minimum error probability in the presence of various Doppler frequency shifts

0050 -

0025 IO

DQPSK n4 DQPSK

055 065 075 085 10 -

BT Fig9 BER floor values as a function of BI for

various values off T D

10 -1k I I I

r ERROR RATE FLOOR BT

10

10

10

10

060 062 - 068 100 1

-

-

DQPSK ------ - n4 DQPSK

10 - 0000 0004 0008 001 2

f DT Fig10 BER floor values as a function of f T for

various values of BT D

PI REFERENCES

W-M-Hubbard The effect of intersymbol inter- ference on error rate in binary differentially coherent phase shift keyed systems Bell System Tech J Vo146 pp1149-1172 JUl-Aug 1967

DMHoover An instrument for testing North American digital cellular radios Hewwlett- Packard J Vo142 No2 pp65-72 Apr1991

N-Nakajima and KKinoshita A system design for TDMA mobile radios in h o c IEEE Veh Tech conf Orlando Florida May 6-9 1990

CSNg Francois PSChin TTTjhung and KMLye Closed-form error probability formula for narrowband DQPSK in slow Rayleigh fading and Gaussian noise IEICE Trans on Commun VolE75-B pp401-412 May 1992

CLLiu and K-Feher Bit error performance of T~-DQPSK in a frequency-selective fast Rayleigh fading channel IEEE Trans on Veh Tech VolVT-40 pp558-568 Aug1991

F-Adachi and KOhno BER performance of QDPSK with postdetection diversity reception in mobile radio channels IEEE Trans on Veh Tech VolVT-40 pp237-249 Feb1991

RFPawula Offset DPSK and a comparison of conventional and symmetric DPSK with noise correlation and power imbalance IEEE Trans on Commun VolCOM-32 No3 pp 233-240 Mar1984

LJMason Error probability evaluation for systems employing differential detection in a Rician fast fading environment and Gaussian noise IEEE Trans on Commun volCOM- 35 pp39-46 1987

IKom Offset DPSK with differential phase detector in satellite mobile channel with narrow- band receiver filters IEEE Trans on Veh Tech VolVT-38 pp193-203 Nov1989

IKorn M-ary FSK with limiter discriminator integrator detection and DPSK with differential phase detection in Rician fading channel Int J of Satellite Commun ~018 pp363-368 1990

CSNg TTTjhung FAdachi and KMLye On the error rates of differentially detected narrowband T~-DQPSK in Rayleigh fading and Gaussian noise To appear in the IEEE Trans on Veh Tech

1418

PE

09999 0 20 40 60 80 100

EbN (dB) Fig6 rampDQPSK to conventional DQPSK BER

ratio for various values of f T with BTA0 D

- BT=l O n4 DQPSK

lo- 20 I 40 I 60 I 80 I 100 EbN (dB1

Fig8 BER of rllrDQPSK for various values of BT with fDT=00025

irreducible error is seen to decrease as BT increases These observations are consistent with that observed in [4] for conventional DQPSK

Figs9 and 10 compare BER floors between ~ 4 - and conventional DQPSK with the former as a func- tion of BT for various f T values and the latter a

function of f T for various BT values These two

figures are plotted with very fine increments of BT and f T respectively to reflect the rate at which the BER D varies with respect to these two system parameters

D

D

Conclusion

In this paper using a recently derived closed-form formulae for the pdf and cdf of the differential phase of a phase modulated carrier in Rayleigh fading channels

C - BT-068 we evaluate and compare the BER of T~-DQPSK and D - BT-100 conventional DQPSK systems Using the dependence

of the shape of the pdf of the message phase difference when the noise samples are correlated we explain why the BER of conventional DQPSK and T~-DQPSK are different We also provide physical explanations not available in previous publications on the effects of Doppler frequency spread IF filter bandwidth and SNR on the BER The difference in BER between ~ 4 - DQPSK and conventional DQPSK under very narrow IF band-limitation and low SNR conditions is attribut- ed to the dependence of the differential phase pdf on

EbN (dB)

D Fig7 BER for various values of BT with f T=OOl

and 0004

set at 00025 Here it is observed that prior to the appearence of the BER floor BT = 10 gives the best BER performance On the other hand the amount of

1417

the signal phase difference Under the same narrow IF band-limitation but at high SNR the T~-DQPSK outperforms the conventional DQPSK because the former has less spectral splatter during phase transi- tions between symbol An IF bandwidth of about one times the symbol rate yields minimum error probability in the presence of various Doppler frequency shifts

0050 -

0025 IO

DQPSK n4 DQPSK

055 065 075 085 10 -

BT Fig9 BER floor values as a function of BI for

various values off T D

10 -1k I I I

r ERROR RATE FLOOR BT

10

10

10

10

060 062 - 068 100 1

-

-

DQPSK ------ - n4 DQPSK

10 - 0000 0004 0008 001 2

f DT Fig10 BER floor values as a function of f T for

various values of BT D

PI REFERENCES

W-M-Hubbard The effect of intersymbol inter- ference on error rate in binary differentially coherent phase shift keyed systems Bell System Tech J Vo146 pp1149-1172 JUl-Aug 1967

DMHoover An instrument for testing North American digital cellular radios Hewwlett- Packard J Vo142 No2 pp65-72 Apr1991

N-Nakajima and KKinoshita A system design for TDMA mobile radios in h o c IEEE Veh Tech conf Orlando Florida May 6-9 1990

CSNg Francois PSChin TTTjhung and KMLye Closed-form error probability formula for narrowband DQPSK in slow Rayleigh fading and Gaussian noise IEICE Trans on Commun VolE75-B pp401-412 May 1992

CLLiu and K-Feher Bit error performance of T~-DQPSK in a frequency-selective fast Rayleigh fading channel IEEE Trans on Veh Tech VolVT-40 pp558-568 Aug1991

F-Adachi and KOhno BER performance of QDPSK with postdetection diversity reception in mobile radio channels IEEE Trans on Veh Tech VolVT-40 pp237-249 Feb1991

RFPawula Offset DPSK and a comparison of conventional and symmetric DPSK with noise correlation and power imbalance IEEE Trans on Commun VolCOM-32 No3 pp 233-240 Mar1984

LJMason Error probability evaluation for systems employing differential detection in a Rician fast fading environment and Gaussian noise IEEE Trans on Commun volCOM- 35 pp39-46 1987

IKom Offset DPSK with differential phase detector in satellite mobile channel with narrow- band receiver filters IEEE Trans on Veh Tech VolVT-38 pp193-203 Nov1989

IKorn M-ary FSK with limiter discriminator integrator detection and DPSK with differential phase detection in Rician fading channel Int J of Satellite Commun ~018 pp363-368 1990

CSNg TTTjhung FAdachi and KMLye On the error rates of differentially detected narrowband T~-DQPSK in Rayleigh fading and Gaussian noise To appear in the IEEE Trans on Veh Tech

1418

the signal phase difference Under the same narrow IF band-limitation but at high SNR the T~-DQPSK outperforms the conventional DQPSK because the former has less spectral splatter during phase transi- tions between symbol An IF bandwidth of about one times the symbol rate yields minimum error probability in the presence of various Doppler frequency shifts

0050 -

0025 IO

DQPSK n4 DQPSK

055 065 075 085 10 -

BT Fig9 BER floor values as a function of BI for

various values off T D

10 -1k I I I

r ERROR RATE FLOOR BT

10

10

10

10

060 062 - 068 100 1

-

-

DQPSK ------ - n4 DQPSK

10 - 0000 0004 0008 001 2

f DT Fig10 BER floor values as a function of f T for

various values of BT D

PI REFERENCES

W-M-Hubbard The effect of intersymbol inter- ference on error rate in binary differentially coherent phase shift keyed systems Bell System Tech J Vo146 pp1149-1172 JUl-Aug 1967

DMHoover An instrument for testing North American digital cellular radios Hewwlett- Packard J Vo142 No2 pp65-72 Apr1991

N-Nakajima and KKinoshita A system design for TDMA mobile radios in h o c IEEE Veh Tech conf Orlando Florida May 6-9 1990

CSNg Francois PSChin TTTjhung and KMLye Closed-form error probability formula for narrowband DQPSK in slow Rayleigh fading and Gaussian noise IEICE Trans on Commun VolE75-B pp401-412 May 1992

CLLiu and K-Feher Bit error performance of T~-DQPSK in a frequency-selective fast Rayleigh fading channel IEEE Trans on Veh Tech VolVT-40 pp558-568 Aug1991

F-Adachi and KOhno BER performance of QDPSK with postdetection diversity reception in mobile radio channels IEEE Trans on Veh Tech VolVT-40 pp237-249 Feb1991

RFPawula Offset DPSK and a comparison of conventional and symmetric DPSK with noise correlation and power imbalance IEEE Trans on Commun VolCOM-32 No3 pp 233-240 Mar1984

LJMason Error probability evaluation for systems employing differential detection in a Rician fast fading environment and Gaussian noise IEEE Trans on Commun volCOM- 35 pp39-46 1987

IKom Offset DPSK with differential phase detector in satellite mobile channel with narrow- band receiver filters IEEE Trans on Veh Tech VolVT-38 pp193-203 Nov1989

IKorn M-ary FSK with limiter discriminator integrator detection and DPSK with differential phase detection in Rician fading channel Int J of Satellite Commun ~018 pp363-368 1990

CSNg TTTjhung FAdachi and KMLye On the error rates of differentially detected narrowband T~-DQPSK in Rayleigh fading and Gaussian noise To appear in the IEEE Trans on Veh Tech

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