Simplified error bound analysis for M-DPSK in fading channels with diversity reception

Y.C. Chow J.P. McGeehan A.R. Nix

Indexing terms: Mobile radio, Error bound analysis, DPSK, Diversity reception

Abstract: An extremely tight upper bound symbol error probability is developed for M-DPSK over a Rician channel, incorporating the effects of AWGN, fading bandwidth and fixed Doppler shift (for the line-of-sight component). In addition, the error probability has also been determined in the above channels with the use of post-detection diversity combining. The bounds are shown to be in excellent agreement with the exact solutions produced by previous authors using numerical computation techniques for both the Rician and Rayleigh fading channels. For the case of AWGN channel, the bound is reduced to that published elsewhere in the literature. The technique is also shown to be capable of formulating the exact fading bit error probability for both binary and quaternary DPSK.

1 Introduction

Over the past 30 years or so, mobile radio in all its forms has grown at a rate in excess of 10% each year, a situ- ation which will continue for the foreseeable future given the continuing demand for channel allocations and new data-based services. This expansion has maintained the need for low-cost spectrally eficient modulation formats that support services such as digital speech, facsimile and data transmision in the fields of cellular, mobile satellite and private mobile radio.

Many years of research have already been applied to the experimental and theoretical performance of linear modulation. This work has addressed the support of both analogue speech and digital based services. Recently, these techniques have become increasingly accepted as the standard for many types of mobile service.

For example, in the UK the Department of Trade and Industry Radiocommunications Agency [ 11 has recently published MTP1376 for private mobile services, permit- ting the use of 5 kHz channelled systems within Band 111; in the USA the FCC has allocated the 220-222MHz band to 5 kHz narrowband services nationwide. It is also noteworthy that many of the linear modulation tech-

0 IEE, 1994 Paper 14121 (E8), first received first received 8th October 1993 and in revised form 15th April 1994 The authors are with the Centre for Communications Research, Faculty of Engineering, Queen’s Building, University of Bristol, University Walk, Bristol BS8 lTR, United Kingdom

IEE Proc.-Commun., Vol. 141, No. 5 , October 1994

niques are now being used, or considered for use in the cellular and mobile satellite sectors, be they narrowband or wideband in nature. The actual implementation of mobile networks using linear modulation coupled with the significant interest in M-ary level systems means the system design engineers are now looking for simple plan- ning tools which will enable them to predict error per- formance in a practical channel be this Rician or Rayleigh in nature. In the case of urban mobile commu- nications, the channel is usually modelled by the Ray- leigh distribution [2]. However, in the area of mobile satellite and aeronautical satellite communications, the channels are modelled by a Rician distribution C3-71. In addition, the vast majority of microcellular and indoor communication channels may also be represented by Rician distributions [SI.

Differential detection techniques are normally chosen for mobile systems because of their robustness in a typical fading channel. These techniques also avoid the need for carrier recovery circuits in the receiver and therefore reduce to some degree the complexity of the overall design. Currently, one of the most popular exam- ples is quadrature differential phase shift keying (Q- DPSK), which has already been accepted as the digital cellular standard for the USA and Japan.

The analytical error performance for M-DPSK on a Rician fading channel has already been published in several papers [3-71. These papers have produced exact symbol error probabilities based on a complicated numerical integration process that is computationally demanding and time-consuming to evaluate. Recently, Tjhung et al. [9] have produced a new bit error probabil- ity expression for Q-DPSK in a Rician fading channel. However, the expression only considered the case of zero fading bandwidth. The above papers also failed to take into account the theoretical benefits of post-detection diversity combining, a technique which significantly improves performance in a fading environment. For the case of the Rayleigh environment the exact solution for the error probability of M-DPSK with Lth order diver- sity has been given in References 10 and 11. Later work has also highlighted the random FM noise effect of the mobile channel. For the simple additive white Gaussian

Y.C. Chow is grateful for the CVCP Overseas Research Student Award. The authors are grateful to their many colleagues in the Centre for Com- muncations Research for their valuable comments and suggestions in relation to this work and the preparation of this paper.

341

Noise (AWGN) channel, the upper bound symbol error probability for M-DPSK has already been well docu- mented in several references [12-141.

The aim of this contribution is to obtain a simple closed-form upper bound for the symbol error probabil- ity of M-DPSK in a Rician channel. In addition, the effect of fading bandwidth, fixed line-of-sight (LOS) Doppler shift and postdetection diversity combining have all been integrated into one generic equation. These results have been extensively compared wherever possible with those that exist in previously published literature. The accuracy of the bounds has been investigated and the advantages and disadvantages of the modelling tech- niques discussed. Since our analysis is general in nature, the results can be applied without modification to Rician, Rayleigh and AWGN channels.

2 System modelling

2.1 General characteristics The overall transmission system with diversity reception is modelled as shown in Fig. la. For mathematical conve- nience, an equivalent lowpass signal representation is used. For M-DPSK transmission, the equivalent lowpass signal u(t) can be written as

u(t) = A&*., nT, < t < (n + 1)T, (1) where 6, = 2n(i - l)/M, with i = 1, 2, ..., M. M is the number of different transmit phases, and A defines the amplitude of the signal and is a constant for phase shift keying modulation. A rectangular pulse shape of dura- tion T, is assumed, where T, represents the symbol period of the system. The signal generated is transmitted over L independent frequency nonselective Rician fading chan- nels. In addition, the delay between the diffuse signal component relative to the direct signal component is taken to be much shorter than T.. The bandwidth of the receiver's filter is assumed to be wide enough to neglect

pFlzj diversity receiver I

from other branches

b

Fig. 1 lowpass representation 4 Overall transmission system b Channel and receiver at kth diversity branch

342

Block diagram for M-DPSK transmission system in equivalent

the filtering distortion for both the direct and diffuse components of the received signal. This contribution also assumes that the fading variations are slow enough so that the fading process remains constant over a one- symbol interval, T,, but fluctuates from one symbol inter- val to the next (i.e. the effect of non-zero fading bandwidth is incorporated).

2.2 The fading channel For the general case of Rician fading, the received signal consists of three components: a single specular com- ponent, a diffuse or Rayleigh component and the channel noise. The channel model for the kth diversity branch is shown in Fig. lb. At the nth time interval, the equivalent lowpass received signal in the kth diversity channel rk(t) can be written as

r,(t) = u,u(t)e-jz"fdkr + q,(t)u(t) + z,(t),

n K < t < ( n + l ) T , , k = l , 2 ,..., L (2)

where U, and f,, represent, for the direct component in the kth channel, the attenuation factor and fixed LOS Doppler shift, respectively. For the kth diversity branch, q,(t) represents the time-variant multipath component, which is modelled as a zero mean complex Gaussian fading process and zk(t) denotes the zero mean complex additive white Gaussian noise. The autocorrelation func- tions for gk(t) and zk(t) are stated below.

Rg(T) = E[gk(t)g:(t - pg(T)Rg(o)>

k = 1, 2, ..., L (3) Rz(T) E[z,(t)z:(t - T)] = 2No 6 ( T ) ,

k = 1, 2, ..., L (4) where E ( . ) denotes the ensemble average and * denotes the complex conjugate value. Rg(0) and pg( t ) are the power and the autocorrelation coeficients, respectively, of the diffuse process, and N o is the value of the noise spectral density. The various types of autocorrelation coefficients appropriate to this field of research are listed in Reference 4. For the aeronautical satellite channel, the following form is used [3,4] :

and for the land or satellite mobile channel [2, 51 = J O ( 2 n B D 5) (6)

Here Jo(.) is the zero-order Bessel function of the first kind and ED is the fading bandwidth.

A parameter to determine the nature of the channel at the kth branch is the ratio of the direct component power P,, to the diffuse power Pdk . This ratio POJP,, is referred to as the K,-factor. Psk and P,, can be represented as u:EJT, and R&O)EJT,, where E, = A2T$2 is the trans- mitted signal energy. The K,-factor can be written as

(7)

Thus, when K, = 0 the fading in the kth channel is reduced to a Rayleigh distribution, when K, = 00 the channel is AWGN, and when 0 c K, < CO the fading in the channel is described by the Rician distribution. Another important parameter in our general analysis is the average received total signal energy-to-noise power spectral density ratio per channel, which can be specified as follows:

Yrk = Yak + Y d k , k = 1, 2, ... , L (8) I E E Proc.-Commun., Vol. 141, No. 5, October I994

where ysk = a:EdNo is the average received direct signal energy-to-noise power spectral density ratio per channel, and ydk = R,(0)E$No is the average received diffuse signal energy-to-noise power spectral density ratio per channel. As shown in Reference 9, ya can also be written in two different forms:

y r k = y S k 1 + - , k = l , 2 ,..., L ( :,> (9)

2.3 Receiver processing The receiver block diagram for the kth order diversity branch is shown in Fig. lb. As described in Reference 4, it is assumed that the receiver contains automatic frequency control (AFC) to compensate for the fixed LOS Doppler shift (equivalent to multiplying the incoming signal rk(t) by &'''st'). Further, since the noise zk(t) is white, it will not be affected by frequency shift in the receiver. The resultant signal can be written as follows:

x k ( f ) = ak u(t) + g,(t)U(f)82n'st' + z k ( t ) >

n T , < t < ( n + l ) T , , k = l , 2 ,..., L (11)

The signal xk(t) is passed through a matched filter and sampled at time t = (n + 1)T. For a rectangular pulse, the matched filter is reduced to an integrator. The output of the filter is therefore

U , = r x k ( t ) dt = akAgenT, + ~ g k ( t ) d z Z f d k r dt

+ rz&) dt

and similarly, for x,(t - T,) the output of filter is

K , r x k ( t - T,) dt

= a,A@-lT , + ra&@--lgk(t - T , ) & Z " f d d I - T d dt

+ [z& - T,) dt (13)

As indicated in Fig. lb, the demodulator at each branch forms the product between the two complex Gaussian random variables of eqns. 12 and 13, so that

Z k = U k K : (14) For Lth-order diversity, the combiner sums all the demodulator outputs and forms a combined vector, which can be expressed as

L L

z = cz ,= C U k K : (15) k = l k = l

If the transmission system is ideal (no fading and AWGN), the phase of 2, or Z is simply the phase of the actual transmitted symbol. Therefore, Z can be used as a decision variable for the diversity receiver.

3 Mathematical analysis

In this Section a mathematical description of the system performance is developed that enables the error probabil- ity of the system to be predicted in Rician, Rayleigh and AWGN channels.

IEE Proc.-Commun., Vol. 141, No. 5, October 1994

3.1 Upper bound For M-DPSK signalling, the transmitted phase at time nT, can be written as below.

On = On-1 + A i (16) where Ai is given by 2z(i - 1)/M (for i = 1, 2, . . . , M) and represents the absolute phase of the data symbol si. Eqn. 16 shows that the information is encoded into the phase differences between 0, and On- l . At the receiver, a correct decision is made only if the difference between the two successive received phases 8, and 8,- is such that

where 8, and on-, are the noisy versions of the corre- sponding transmitted phases. The decison rule expr. 17 can be represented in graphical form as shown in Fig. 2a. An incorrect decision is made if the phase of the decision vector 2 (or 8" - falls inside the area R e . If inter- symbol interference (ISI) free transmission is assumed together with rotational symmetry of the signal constella- tion, then the symbol error probability of M-DPSK PS(M) is simply the probability of the vector 2 falling inside the area R e . The region Re can be represented by

0 b

C d Fig. 2 symbol is transmitted LI Incorrect decison is made when received vector falls into region R, b Reson R. can be hounded by the sum of the two half-plane regions c RegionR, d RegionR,

Decision region for M-DPSK signal when ith transmitted

the union of the two half-plane regions R I and R , shown in Fig. 26. The shaded region R, represents the overlap area between R, and R,. Benedetto et al. [12] and Ziemer and@eterson [13] had shown that a simple upper bound of P,(M) can he written as

(18) As M becomes larger, it can immediately be seen that the shaded region R, will be reduced in area, and hence the bound should become increasingly tight. In References 12-14, eqn. 18 was used to evaluate the symbol error bound for M-DPSK in an AWGN channel.

3.2 Formulation of the upper bound To evaluate the upper bound shown in expr. 18, the objective is to calculate the probability of Z falling in the regions R , and R,. Figs. 2c and 2d show the half-plane

343

PAM) < Pr (2 E R I } + Pr {Z E R,}

regions R, (m = 1, 2). Following the approach developed by Edbauer [16], a new decision variable d, is intro- duced :

d , = ZgYm (19) where Y, is the angle of rotation of the constellation mapping that transforms the half-plane area into the left- hand plane (LHP). Now the probability of Z lying in the region R, is equal to the probability of d,,, lying in the LHP, hence

Pr {Z E R,} = Pr {d, + dz < 0) (20)

Using eqn. 15, d , can be described as L L

d , = ZeJ'Ym = Zk&Ym = lJkK:ei'y" k = l k = 1

and introducing two new complex Gaussian random variables:

X , = U k , & = K,e-jYm (22)

Hence eqn. 20 can be rewritten as follows.

Pr {Z E R,} = Pr 1 ( X , Y: + X:&) < 0 { k r l

The right-hand side of eqn. 23 is a special case of the derivation given by Proakis [17] concerned with the probability that a general quadratic form in complex Gaussian random variables is less than zero. Proakis has shown that to solve eqn. 23, only the mean and the second central moment of X , and & must be determined. Using eqns. 12, 13 and 22, the mean of X , and & are given below :

X , = E[Xk] = a, A@'"T,,

E = E [ & ] = akAe'em-lT,e-JYm (24)

and the second central moments of X, and & are

m, = E[(X, - Xk)(Xk - I,)*] = A2R,(0)T: + 2N0 T ,

my, = E[(& - - E)*] = A2Rg(0)T,2 + 2N0 T, (26)

(27)

The final result of the general form is detailed in Refer- ence 17, and for the special case of eqn. 23, the result for L > 1 is given below:

m,, = E[(x, - I&& - Fk)*] = A2Rq(TJT3&(& + 2=/dkTs+vm)

Pr { Z E R,J

= Q(a, b) - Io(ab) exp

Io(ab) exp [ - ( a 2 + b2)/2] + (1 + u 2 / u 1 ) 2 L -

2L - 1 u2 ' exp [ - ( a 2 + b2)/2] L - 1

k = O ' ( k )(&) + (1 +u2/u1)2L-1

and for L = 1

Pr {Z E R,,,} a2 + b2

= Q(a, b) - ~ u2'u1 Io(ab) exp (- 7) (29) 1 + u2/u1

where

Q(a, b) = e-(a2+bz)/2 " = O f ( i r I n ( a b ) , if a < b (30)

Q(a. b) = 1 - e--(gz+bZ)'Z " = l f (:)"(ab), if a > b (31)

is the Marcum Q-function [18], and I " ( . ) is the n-order modified Bessel function of the first kind. For a and b

(32)

(33)

with

(37)

where Re { .} and Im { .} are the real and imaginary parts of a complex variable. Eqn. 18 can now be evalu- ated by using eqns. 28 and 29 with parameters from eqns. 32-37.

3.2.1 Rician fading channel (0 < K < m): For this type of channel, the received signal is already written in eqn. 11 and assumptions are made that the attenuation and the fixed LOS Doppler shift of the direct path are equal for all diversity branches (i.e. a, = a, hk =fd for k = 1, . . . , L). The average received total signal energy-to- noise power spectral density ratio per channel in eqn. 8 can therefore be reduced to

Ya=Yt=Y.+Yd, k = l , & . . . , L (38) To calculate Pr {Z E R,} shown in eqns. 28 and 29, the parameters in eqns. 32-37 have to be determined. By sub- stituting eqns. 24-27 into eqns. 32-37, the following parameters can be derived.

(40) 4V1 V$[(Ky,/(K + l))& V, + i2]

b = [ (Vl + V2I2 1 where

344 I E E Proc.-Comun., Vol. 141, No. 5, October 1994

Without loss of generality, suppose that s, is the trans- mitted data symbol as indicated in Fig. 3; thus

(48) = A, =e. - On-l = o

Fig. 3 Corresponding regions R , and R , when s, is the transmitted symbol The diagcam also shows the two rotation angles for R , and R, required to cover the left-hand plane region

Fig. 3 shows the half-plane regions R , and R , , with the corresponding rotation angles for R , and R , given by

(49)

Consequently, the upper bound for the M-DPSK symbol error probability can be written as

Ps(M) < Pr { Z E R , } + Pr { Z E R 2 } (50) where Pr {Z E R , } and Pr { Z E R, } can be evaluated by substituting eqns. 39-41 into eqn. 28 or 29 with the corresponding angles as stated in eqn. 49. In general, when the fixed LOS Doppler shiftf, is present, the values of Pr { Z E R , } and Pr { Z E R, } are not equal. This is because the Doppler shift introduces a fixed phase shift in the received signal, so the probabilities that the decision variable Z will fall into regions R, or R , are not equal. However, when the fixed LOS Dopper shift,f, = 0, it can be shown that eqns. 39-47 are independent of Y , and Y2. Hence, for this particular case, the upper bound is reduced to

(5 1) Once the upper bound symbol error probability of M-DPSK in Rician fading channel has been formulated, the error bound for two special cases of the fading chan- nels, namely the Rayleigh and AWGN channels, can easily be obtained.

32.2 Rayleigh fading channel ( K = 0) : In this case the received signal at any of the k diversity branches do not have a specular component (i.e. ak = 0) and hence the fixed LOS Doppler shiftf, is reduced to zero. Therefore eqn. 2 can be expressed as rk(t) = gk(t)u(t) + z,(t), with

PAM) < 2 Pr { Z E R , }

I E E Proc.-Commun., Vol. 141, No. 5, October 1994

k = 1, 2, . . . , L. The average received total signal energy- to-noise power spectral density ratio per channel in eqns. 8 or 10 is reduced to

(52) As a result, the receiver's automatic frequency control circuit can now be omitted for convenience. By inserting K = 0 andf, = 0 into eqns. 44-47 produces the following expressions

ytk = yr = y d , k = 1, 2, ._., L

j 1 = L { ) I d c l - &7(K)1 + (53)

s, = 0 (54) m X X m Y Y = (7, + (55)

&xr = ydpAT,)&("'+ym' (56) and the parameters a and b in eqns. 39 and 40 are reduced to zero. Eqns. 28 and 29 are simplified to

L - 1 2L-1 k

Pr { Z E R , } = (1 + v ~ ~ v , ) ~ ~ - ' zo ( k )($ (57)

Eqn. 58 is a special form of eqn. 57; as a result eqn. 57 can be used as more general for any order of diversity L.

By using eqns. 41-43 in conjunction with eqns. 55 and 56, v2/v1 can be simplified to

where

p COS (Ai + Y,,,) [I - p z sin2(Ai + Y,,,)]'/' P, =

(59)

From eqns. 57 and 59, the probability of Z falling into region R , can be written as

Pr { Z E R,}

As before, by assuming that s, is the transmitted data symbol, the upper bound for the symbol error probability of M-DPSK transmitted over a frequency nonselective Rayleigh fading channel can then be evaluated by using eqn. 50. The parameter p, is reduced to

p sin (KIM) [I - p2 c ~ s ~ ( n / M ) ] ' / ~

p = p =

This implies that Pr { Z E R , } and Pr { Z E R,} are equal. The symbol error probability can be shown to be

32.3 A WGN channel ( K = 03): For the sake of com- pleteness and to underpin the theoretical analysis pre- sented in this paper, the classical case of the additive white Gaussian noise channel is analysed and compared with the body of work of other researchers. For this type of channel, the received signal contains only the specular component and channel noise. Both fixed LOS Doppler shift f , and the fade component g(t) are reduced to zero.

345

For mathematical convenience, single-channel transmis- sion is assumed in this analysis, i.e. L = 1, and the channel attenuation is assumed to be unity (U = 1). Con- sequently, the received signal in eqn. 2 can be written as r(t) = u(t) + z(t), and the average received total signal energy-to-noise power spectral density ratio per channel in eqns. 8 or 9 is reduced as shown below.

Ytk = Yr = Y s , k = 1 (65)

Under these conditions, eqns. 42-47 become

v = VI = v2 = 1

i1 = 1

i2 = y , COS (Ai + Y,,,) (68)

r?lxxmrr = 1 (69)

r?lxr = 0 (70)

Hence eqn. 29, which shows the probability of Z falling into the region R,, is reduced to

a2 + b2 Pr {Z E R,} = Q(a, b) - flo(ab) exp (- 7) (71)

The parameters a and b can be obtained by substituting eqns. 66-68 into eqns. 39 and 40

(72)

By assuming that s1 is the transmitted data symbol, the upper bound for the symbol error probability P J M ) can be evaluated by using expr. 50, and the result is shown below.

PsW) < 1 + Q(a, b) - Q(b, a) (73)

with a and b given by

{ ; } = J [ y s ( l +sin$)] (74)

The bound shown in expr. 73 is identical to that devised by Benedetto et al. [12] in which a different method has been applied to determine the bound. This agreement for the special case of AWGN channel provides further proof of the general close form analysis presented in this paper.

3.3 Exact bit error probability for 6-DPSK and 0-DPSK

3.3.1 B-DPSK (M =2): For B-DPSK the two possible transmitted phase values are zero and n. Therefore, to calculate the exact error probability of the modulation in AWGN, only the probability of the received signal that falls into a half-plane region needs to be evaluated [lo, 121. Applying the same idea, the symbol (or bit) error probability of B-DPSK can be written as below (see the Appendix).

Pb(2) = Pr {Z E R,}, with Yl = 0 (75)

Therefore, the exact bit error probability for B-DPSK over various types of channel can be calculated by using eqns. 28 and 29 together with eqns. 39-41.

For a Rician fading channel (0 < K < CO), if it is assumed that f, = 0, the closed-form equation for the

346

exact bit error probability can be simplified as

2L - 1 L - 1 1

k = O

where

L > 1 (76)

(77)

When pg(7J = 0 (i.e. zero fading bandwidth), eqn. 76 is equal to the solution obtained by Jones [19]. Mason [4] gave the expression for B-DPSK over a Rician fast- fading channel but did not consider diversity (however, his equation did allow fading bandwidth in the order of the bit rate to be investigated). The expression is equiva- lent to eqn. 77 when the fading bandwidth is small com- pared with the bit rate. For a nonzero fixed LOS Doppler shift, the bit error probability can be calculated using eqn. 28 or eqn. 29 by inserting parameters of eqns. 39-41 into eqn. 75.

For a Rayleigh fading channel (K = 0), y t = y s and eqn. 76 is reduced to the following form.

k = O

L > 1 (79)

where p is given by eqn. 61. Although written in a some- what different form, eqn. 79 is equivalent to the solution given by Kam [ll]. Also, the equation is equivalent to the one given by Proakis [lo] when zero fading band- width is assumed.

For an AWGN channel (K = CO), y t = y. and eqns. 76 and 77 are reduced to

pb(2) = )e-Ys, L = 1 (81)

These are well known results and have already been given by Proakis [17].

3.3.2 0-DPSK (M = 4 ) : For Q-DPSK the four possible transmitted phase values are zero, 4 2 , n and 3x12. When Gray coding is assumed in the constellation mapping, Edbauer [16] had shown that the bit error probability of Q-DPSK can be stated as (see the Appendix)

Pb(4) = Pr { Z E R , } , with Yl = n/4 (82)

However, the equation is only suitable when the fixed LOS Doppler shiftf, is equal to zero. This is because the formulation of eqn. 82 is based on the received signal falling into incorrect regions, the probabilities of which are no longer equal once Doppler shifts are assumed.

For a Rician fading channel (0 < K < CO), by assuming that the first diversity (L = 1) is employed, the

IEE Proc.-Commun., Vol. 141, No. 5, October 1994

closed-form expression for the exact bit error probability can be written as

where

(86) with D = [2(F + 1)' - (FJL,(T,))']'/~ and F = yJ(K + 1). For higher-order diversity, the exact bit error probability can be calculated directly by using eqn. 28 with param- eters defined in eqns. 39-41 substituted into eqn. 82.

For a Rayleigh fading channel (K = 0), yt = yd and eqn. 83 is reduced to

1 1 - p Pb(4) = - - ~ 1 + c - 2 (87)

where p = p/[2 - p23'I2 and p is given by the general result of eqn. 61.

For Lth-order diversity, the exact bit error probability can be obtained directly from eqn. 62, resulting in the expression

Although written in a different form, eqn. 88 is equivalent to the solution given by Kam [ll]. Also, the equation is equivalent to the one given by Proakis [lo] when zero fading bandwidth is assumed.

For an AWGN channel (K = CO), yr = ys and eqn. 83 is reduced to

pb(4) = Q(a, b) - i lo(ab) exp ( - - " l b2 ) (89)

where a and b are given by

The above result is again in agreement with that given in Reference 17.

4 Results and discussion

4.1 Exact bit error probability for B- and Q-DPSK in fading and noise

For B-DPSK, the error performance over the Rician fading channel with postdetection diversity is &own in Fig. 4. The parameters of the channel depicted here are typical of those in a mobile satellite channel [ lS]. For the case with first-order diversity and zero fixed LOS Doppler shift, it is equivalent to the result given by Qu [20, Fig. 21. The graph shows the effect of the fixed LOS Doppler shift, which is seen to degrade the performance of the system. It can thus be deduced that the direction of

IEE Prof.-Commun., Vol. 141, No. 5, October I994

the receiver unit relative to the transmitter is an impor- tant parameter in the error performance consideration.

10-8- 0 5 IO 15 20 25 30

average direct EblNo per channel, dB

Fig. 4 Exact bit error probability for B-DPSK in a Rician fading channel with single- and two-branch diversity (post-detection combining, mobile fading model) K = S d B , B , T , = O . I ____ f , T , = o ~-~~ f, 7 = 0.025 . . . . . . . f, T, = 0.05

For Q-DPSK the bit error performance over the Rician fading channel is shown in Fig. 5. For first-order diversity and zero fixed LOS Doppler shift, eqn. 83 can be applied. In this calculation, 50 terms of summation were used to evaluate the Marcum Q-function. The results were found to be equivalent to the result given by

' r

lo-7L, ; Ib ' 1; i o ;5 3b average total EbINa per channel,dB

Fig. 5 channel (order of diversity, L = I , mobile fading model) f d T = o

~ B,T ,=O

Exact bit error probability for Q-DPSK in a Rician fading

B, 7 = 0.02 _ _ _ _

341

Tjhung et al. [9] for the slow Rician fading channel of mobile communication. However, the expression shown in Reference 9 involved a numerical integration process and is only valid when the fading bandwidth is equal to zero. The results presented here show that the effect of the non-zero fading bandwidth increases when the channel experiences Rayleigh fading. This is because, for the Rician fading channel, the received signal contains a direct path as well as a diffuse component.

The advantage of postdetection diversity, as calculated by using eqn. 82, is shown in Fig. 6.. The results show that diversity reception increases the communication reli- ability. However, the relative effectiveness of the diversity decreases as the value of the K-factor is increased. A similar effect is observed by Lindsey [21] for binary signalling.

For a Rayleigh fading channel, both exact and upper bound symbol error probabilities for 4-, 8- and 16-DPSK

’I

average direct EslNo per channel, dB

Fig. 7 Comparison between exact and upper bound symbol error prob- ability for M-DPSK in a Rician fading channel (order of diversity, L = I , aeronautical fading model) K = 9 dB,f,T, = 0, B. T = 0

i , \ , 0 5 IO 15 20 25 30

overage total EblNo per channel, dB

10-7L

Fig. 6 Exact bit error probability for Q-DPSK in a Rician fading channel with single and two-branch diversity (postdetection combining, m b i l e f d i n g model) Jr T = 0, E , T = 0.02

~ L = l L = 2 ~~~~

4.2 Upper bound symbol error probability for

Both exact and upper bound solutions for 4-, 8- and 16-DPSK in Rician fading channels are shown in Fig. 7. The parameters of the channel are typical values of the L-band aeronautical satellite channel [3]. The exact solu- tion is calculated by using Reference 20, eqn. 18. The bound is obtained by using expr. 51 and eqn. 29 with parameters of eqns. 39-41. The plot shows that the bound is increasingly tight when M is large. This justifies the comments made at the end of Section 3.1 concerning the upper bound analysis. The effect of the nonzero fading bandwidth and the fixed LOS Doppler shift on the error performance is shown in Figs. 8 and 9. The results clearly show the improvement of the error performance when postdetection diversity is employed. The bound for Q-DPSK in Fig. 8 is in very good agreement when com- pared with the exact solutions given Reference 3, Fig. 1. The simulated results for Q-DPSK with postdetection diversity combining is also given in Reference 3., Fig. 7. Again the bound is in good agreement with the simulated data.

M-DPSK

348

I I

4 8 12 16 20 2L

overage direct EslNa per channel, dB

Fig. 8 fading channel (order of diversity, L = I , aeronauticalfading model)

~ f , T , = O

Upper bound symbol error probability for M-DPSK on Rician

K = 9 dB, B,T, = 0.1

- - fd T, = 0.025

schemes are shown in Fig. 10. The expression for the exact solution is given by Proakis [17, eqn. 7.6.11, which can be modified to take into account the effect of nonzero fading bandwidth. The upper bounds are calculated by using eqn. 64. Improvement of the bound is also shown when M is large. Fig. 11 shows the improvement of the bound when postdetection diversity is employed for

IEE Proc.-Commun., Vol. 141, No. S, October 1994

Q-DPSK. The reason for this is that by using diversity, the probability of deep fades occurring is considerably reduced, and the received vector Z is less likely to fall into the shaded region R , . Therefore, the probability that 2 lies in R , (or Pr {Z E R 3 } ) becomes low, resulting in a very tight bound.

L e c

I

10-71 8 I2 16 20 2L average direct EslNo per channel, dB

Upper bound symbol error probability for M-DPSK in a Rician Fig. 9 fading chnnnel (order of diversity, L = 2, neronnutical fading model)

~ & T = O K = 9 dB, B,T, = 0.1

fa T = 0.025 _ _ _ _

0 5 IO 15 20 25 30 35 40 average diffuse EblNo per channel, dB

Fig. 10 Comparison between exnct and upper bound symbol error probability for M-DPSK in a Rayleighfading channel (order of diuersity, L = I , mobile fading model) K = -mdB,B.T,=O

5 Conclusions

In this paper the upper bound symbol error probability of M-DPSK over a Rician fading channel with postdetec-

IEE P r o c . - C m u n . , Vol. 141, No. 9, October 1994

tion diversity combining has been formulated. The bound is able to investigate the effect of AWGN, fading band- width and fixed LOS Dopper shifts introduced by the

overage diffuse EblNo per channel, dB Fig. 11 Comparison between exnct and upper bound symbol error probability for Q-DPSK in a Rayleigh fading chnnnel (mobile fading model) K = - m dB. B, T, = 0.02

channel. Unlike the expression for the exact solution, the upper bound obtained in this paper avoids any numerical integration process, yet the comprehensive set of results obtained shows that the bound is very tight when com- pared with the exact results.

New expressions to calculate the exact bit error prob- ability for binary and quaternary DPSK over a Rician fading channel with post-detection diversity are also evaluated. For B-DPSK, the closed-formed expression is capable of determining the effect of the AWGN, fading bandwidth and fixed LOS Doppler shift. For Q-DPSK, the expression is only applicable when zero fixed LOS Doppler shift is considered.

Since the expressions for the upper bound and the exact solutions of the Rician fading channel are a func- tion of the K-factor of the channel, the expressions can therefore be easily applied to Rayleigh and AWGN chan- nels by merely changing the factor.

The results presented in this paper show that the channel parameters such as the K-factor, fading band- width and fixed LOS Dopper shift are important when designing a reliable communications system. In general, post-detection diversity is shown to be an effective tech- nique to combat the effects of the nonzero fading band- width. The results also show that the effectiveness of diversity is increased when the channel fading becomes more Rayleigh-distributed in nature.

6 References

1 DTI Radio communications Agency: ‘Opening 5 kHz channel will meet increasing private mobile radio demand‘. DTI Press Notice, 17 August 1993

2 CLARKE, R.H.: ‘A statistical theory of mobile-radio reception’, Bell Syst. Tech., 1968.47, pp, 957-1000

3 MIYAGAKI, Y., MORINAGA, N., and NAMEKAWA, T.: ‘Error rate performance of M-ary DPSK systems in satellite/aircraft com- munications’. Proc. IEEE Int. Cod. Communications, 1979, pp. 34.6.1-34.6.6

349

4 MASON, L.J.: ‘Error probability evaluation for systems employing differential detection in a Rician fast fading environment and Gauss- ian noise’, I E E E Trans. Commun., 1987.35, pp. 39-46

5 MASON, L.J.: ‘An error probability formula for M-ary DPSK in a Rician fast fading and Gaussian noise’, I E E E Trans. Commun., 1987, 35, pp. 916-918

6 KORN, I.: ‘Offset DPSK with differential phase detector in satellite mobile channel with narrow-band receiver filter’, I E E E Trans. Vehicul. Technol., 1989,38, pp. 193-203

7 KORN, I.: ‘M-ary frequency shift keying with integrator detection and DPSK with differential phase detection in a Rician fading channel‘, Inc. J. Satellite Commun., 1990,8, pp. 363-368

8 PARSONS, J.D.: ‘The mobile radio propagation channel’ (Pentech . . - Press, London, 1992)

9 TJHUNG, T.T., LOO, C., and SECORD, N.P.: ‘BER performance of DQPSK in slow Rician fading’, Electron. Lett., 1992, 28, pp. 176?-1765 ~ . -_ - , --

10 PROAKIS, J.G.: ‘probabilities of error for adaptive reception of M-phase signals’, I E E E Trans. Commun., 1968,16, pp. 71-81

11 KAM, P.Y.: ‘Bit error probabilities of MDPSK over the non- selective Rayleigh fading channel with diversity reception’, I E E E Trans. Commun., 1991,39, pp. 220-224

12 BENEDETTO, S., BIGLIERI, E., and CASTELLANI, V.: ‘Digital transmission theory’ (Prentice-Hall, Englewood Cliffs, New Jersey, 1987)

13 ZIEMER, R.E., and PETERSON, R.L.: ‘Introduction to digital communication’ (Macmillan Publishing Company, New York, 1992)

14 PRABHU, V.K.: ‘Error rate bounds for differential PSK, I E E E Trans. Commun., 1982,30, pp. 2547-2550

15 KORN, I.: ‘M-ary frequency shift keying with limiterdiscriminator- integrator detector in satellite mobile channel with narrow-band receiver filter’, I E E E Trans. Commun., 1990.38, pp. 1771-1778

16 EDBAUER, F.: ‘Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection’, I E E E Trans. Commun., 1992.40, pp. 457-460

17 PROAKIS, J.G.: ‘Digital communications’ (McGraw-Hill, New York, 1989,2nd edn.)

18 VAN TREES, H.L.: ‘Detection, estimation and modulation theory: Part I’ (John Wiley, New York, 1968), p. 395

19 JONES, J.J.: ‘Multichannel FSK and DPSK reception with three- component multipath’, I E E E Trans. Commun. Technol., 1968, 16, pp. 808-821

20 QU, S.: ‘Double differential MPSK on the fast Rician fading channel’, I E E E Trans. Vehicul. Technol., 1992,41, pp. 278-295

21 LINDSEY, W.C.: ‘Error probabilities for Rician fading multi- channel reception of binary and N-ary signals’, I E E E Trans. Inform. Theory, 1964,10, pp. 339-350

7 Appendix: Formulation of bit error probabilities of B-DPSK and Q-DPSK

The signal constellation for B-DPSK is shown in Fig. 12a. Assuming that s1 is the transmitted signal, the bit

error probability of B-DPSK is equal to the received vector Z that falls in region RI, that is

Pb(2) = Pr { Z E R , } (91)

Since the region R, already lies on the LHP, the corre- sponding rotation angle Y is equal to zero.

The signal constellation for Q-DPSK is shown in Fig. 12b. Gray coding is used in the mapping. Without loss of

0 b

Fig. 12 The constellation mapping D B-DPSK. If s, is transmitted, an incorrect decision is made if the received vector falls into the half plane region R , b Q-DPSK. I f s , is transmitted, an incorrect decision is made if the received vector falls into one of the regions r 2 , r, or 1,

generality, by assuming s1 is the transmitted symbol, the probability of the received vector Z falling into regions ri (i = 2, 3, 4) is Pr {Z E ri) (i = 2, 3, 4). By assuming that Pr {Z E r 2 } equals Pr {Z E r4}, and since double bit errors occur at Pr {Z E r3 } , Edbauer [16] has shown that the bit error probability is given by

(92) P,(4) = Pr { Z E r 2 } + Pr { Z E r,}

By letting the half-plane region R, equal the combined regions r2 and r , , eqn. 92 can be rewritten as

Pb(4) = Pr {Z E R,} (93) The corresponding rotation angle Y to transform R , into the left-hand-plane region is n/4.

350 I E E Proc.-Commun., Vol. 141, No. 5 , October 1994