Differential Diversity Reception of MDPSK over Independent
Rayleigh Channels with Nonidentical Branch Statistics and
Asymmetric Fading Spectrum
Hua Fu and Pooi Yuen Kam
ECE Department, National University of Singapore
Singapore 117576, Email: {elefh, elekampy}@nus.edu.sg
Abstract— This paper is concerned with optimum diversity re-
ceiver structure and its performance analysis of differential phase
shift keying (DPSK) with differential detection over nonselective,
independent, nonidentically distributed, Rayleigh fading chan-
nels. The fading process in each branch is assumed to have an
arbitrary Doppler spectrum with arbitrary Doppler bandwidth,
but to have distinct, asymmetric fading power spectral density
characteristic. Using 8-DPSK as an example, the average bit error
probability (BEP) of the optimum diversity receiver is obtained
by calculating the BEP for each of the three individual bits.
The BEP results derived are given in exact, explicit, closed-form
expressions which show clearly the behavior of the performance
as a function of various system parameters.
I. INTRODUCTION
The receiver structure and bit error probability (BEP) per-
formance of differential phase shift keying (DPSK) with differ-
ential detection over nonselective, independent and identically
distributed (i.i.d.), Rayleigh fading channels with combining
diversity reception have been well known in the literature
[1]−[4]. However, reaserch shows that in some practical
systems, the independent, non-identically distributed (i.n.i.d.)
channel model is more accurate [5], [6]. In i.n.i.d. channel,
the fading processes and possibly the additive, white Gaussian
noise (AWGN) on the diversity branches have non-uniform
power profiles which are distinct from one another. The effect
of the nonidentical diversity branch statistics on the receiver
structure is studied in [7]. Recently, based on the maximum a
posteriori probability (MAP) criterion, an explicit structure of
the optimum combining differential receiver and a complete
set of closed-form BEP expressions and their Chernoff upper
bounds, for 2-, 4- and 8-DPSK, both with optimum combining
reception and suboptimum combining reception, are derived
in [8]−[10]. The purpose of this paper is to provide a further
extension. The results derived in this paper, together with those
in [8]−[10], form a benchmark counterpart to the classic ones
for the i.i.d. channel given in [1]−[4].
In a Rayleigh channel, the fading gain is usually modeled
as a zero-mean, stationary, complex, Gaussian random pro-
cess. The most widely accepted model [1]−[10] is that the
spectrum of the fading process over each diversity branch is
symmetric around the carrier so that the quadrature processes
are independent of each other. This assumption is valid for
various fading spectra. For example, see [11] and its refer-
ences. However, in some fading environments such as the land
mobile channel with Jakes model [12], the Doppler spectrum
becomes asymmetric when the multipath signals are absorbed
by obstacles or the propagation environment is characterised
by directional non-isotropic scattering [13]−[15]. Thus, it is
of great practical importance to take account of the effect of
the asymmetric fading spectrum on the receiver structure and
the performance analysis of differentially detected DPSK over
i.n.i.d. channels, the topic of this paper.
The paper is orgainzed as follows. In Section II, the signal
model is introduced and different optimum diversity receivers
are derived for different Rayleigh fading scenarios (see eqs.
(17)−(20) below). In Section III, we use 8-DPSK as an
example to study the BEP performance. Here, the average BEP
of the optimum diversity receiver is obtained by calculating the
BEP for each of the three individual bits. The results are given
in exact, explicit, closed-form expressions which show clearly
the behavior of the performance as a function of signal-to-
noise ratio (SNR), fading correlation coefficient, and diversity
order. Section IV presents numerical examples. Throughout
this paper, overhead ∼ denotes a complex quantity, superscript
∗ will denote its conjugate, E is the ensemble average operator,
δ represents the Kronecker delta, and [·]T denotes transposition
of the vector and matrix.
II. SIGNAL MODEL AND RECEIVER STRUCTURE
With space diversity reception over L frequency nonse-
lective, i.n.i.d., Rayleigh fading branches with AWGN, the
received signal over the ith branch, i = 1, 2, · · · , L, during
the kth symbol interval kT ≤ t < (k + 1)T is given, after
matched filtering and sampling at time t = (k + 1)T , by the
statistic r̃i(k), where
r̃i(k) = E
jφ(k)c̃i(k) + ñi(k). (1)
Here, Es is the energy per symbol, and for DPSK, φ(k) is
the data-modulated phase with Gray encoding of bits onto
the phase transition ∆φ(k) = φ(k) − φ(k − 1). The kth data
symbol is conveyed in ∆φ(k). We assume here that all symbol
points are equally likely. In (1), a rectangular data pulse shape
g(t), where g(t) = 1/
T for 0 ≤ t < T and zero elsewhere,
is assumed so that each matched filter has a rectangular low-
pass-equivalent impulse response hi(t) = g(T − t) for all i.
Thus, the filtered noise ñi(k) is given by
ñi(k) =
∫ (k+1)T
ñi(t)√
dt. (2)
Here, {ñi(t)}Li=1 is a set of i.n.i.d., lowpass, complex AWGN
processes with E [ñi(t)] = 0 and E[ñi(t)ñ
i (t− τ)] = Niδ(τ)
so that {ñi(k)}k is a sequence of zero-mean, complex Gaus-
sian variables with covariance function for each branch i
E[ñi(k)ñ
i (j)] = Ni δkj (3)
The multiplicative distortion c̃i(k) in (1) is given by
c̃i(k) =
∫ (k+1)T
c̃i(t)
dt. (4)
http://arxiv.org/abs/0704.1070v1
Here,
c̃i(t) = ai(t) + jbi(t)
is a set of i.n.i.d., lowpass,
zero-mean, stationary, complex, Gaussian random processes.
Each c̃i(t) represents the complex gain due to frequency
nonselective Rayleigh fading of the ith branch. For asymmetric
spectrum in each i, the inphase fading process ai(·) and the
quadrature phase fading process bi(·) are generally correlated.
At any time instant t, however, ai(t) and bi(t) are always
uncorrelated. With reference to Fig. 1, it is shown in [16] that
the covariance function of ai(·) and bi(·) can be obtained as
E[ai(t)bi(t)] = 0 (5a)
E[ai(t− τ)ai(t)] = E[bi(t− τ)bi(t)] = Ri(τ) (5b)
E[ai(t)bi(t− τ)] = −E[bi(t)ai(t− τ)] = Qi(τ) (5c)
Note that if the spectrum of each c̃i(t) is symmetric, the
processes ai(·) and bi(·) will be independent (i.e., we have
Qi(τ) = 0) with the same covariance function Ri(τ).
Letting c̃i(k) = ai(k) + jbi(k), it follows from (4) and (5)
that both {ai(k)}k and {bi(k)}k are sequences of zero-mean,
real-valued, Gaussian random variables with
E[ai(k)bi(k)] = 0 (6a)
E[ai(k − l)ai(k)] = E[bi(k − l)bi(k)] = Ci(l) (6b)
∫ (k+1)T
∫ (k+1−l)T
(k−l)T
Ri(u − v)
du dv
E[ai(k)bi(k − l)] = −E[bi(k)ai(k − l)] = Di(l) (6c)
∫ (k+1)T
∫ (k+1−l)T
(k−l)T
Qi(u− v)
du dv
Thus, the covariance matrix can be obtained as
Γi = E
ai(k)
ai(k − l)
bi(k)
bi(k − l)
ai(k) ai(k − l) bi(k) bi(k − l)
Ci(0) Ci(l) 0 Di(l)
Ci(l) Ci(0) −Di(l) 0
0 −Di(l) Ci(0) Ci(l)
Di(l) 0 Ci(l) Ci(0)
 (7)
For each i , c̃i(k) and ñi(k) are mutually independent. For
i 6= j, {c̃i(k), ñi(k)} are independent of {c̃j(k), ñj(k)}.
The diversity branches are nonidentical since the covariance
functions Ri(τ), Qi(τ) and Niδ(τ) depend on the branch
index i. For convenience of later application, the following
parameters are defined. The fading correlation coefficient at
the matched filter output over a symbol interval of T for the
ith diversity branch is defined as
ρ̃i =
E[c̃i(k)c̃
i (k − 1)]√
|c̃i(k)|2
|c̃i(k − 1)|2
Ci(1)− jDi(1)
Ci(0)
From (8), we note that ρ̃i is a complex quantity. It is a measure
of the fluctuation rate of the channel fading process. The mean
received SNR per symbol over the ith branch is defined as
|E1/2s ejφ(k)c̃i(k)|2
2EsCi(0)
We consider 2-, 4- and 8-DPSK with Gray encoding of bits
onto ∆φ(k) as shown in [4, Fig.1] for 4- and 8-DPSK, the
mean received SNR per bit γbi is given by γ
i = γi for 2-DPSK,
γbi = γi/2 for 4-DPSK, and γ
i = γi/3 for 8-DPSK.
Using the MAP criterion, the aim of the receiver is to
determine from the received signals {r̃i(k), r̃i(k − 1)}Li=1
which one of the possible values 2πm/M , m = 0, 1, · · · ,M−
1, of the phase difference ∆φ(k) has maximum probability
of occurrence. Following [9], it can be shown that MAP
detection is equivalent to maximum log-likelihood detection.
Specifically, based on {r̃i(k), r̃i(k − 1)}Li=1, we decide that
∆φ(k) = 2πn/M whenever the log-likelihood function
logΨm =
r̃i(k)
∣∣∣r̃i(k − 1),∆φ(k) =
is maximized for m = n.
To proceed with evaluating (10), we need to verify that
c̃i(k) = ai(k)+jbi(k) and c̃i(k−1) = ai(k−1)+jbi(k−1) are
jointly complex Gaussian. By being jointly complex Gaussian,
it means that if x̃ = xR + jxI and ỹ = yR + jyI are two
column complex random vector, then [xR
has a real multivatiate Gaussian probability density function
(PDF), and furthermore, if u = [xR
T ]T and v =
T ]T , then the real covariance matrix of [uT vT ]T
has a special form given in [18, Theorem 15.1] that satis-
fies Goodman’s theorem [19]. After careful examination, it
follows from (7) that c̃i(k) and c̃i(k − 1) are indeed jointly
complex Gaussian1. Thus, conditioned on c̃i(k − 1), c̃i(k) is
conditionally complex Gaussian with mean [18]
E [c̃i(k)|c̃i(k − 1)] = ρ̃i c̃i(k − 1) (11)
and variance
{∣∣c̃i(k)− E[c̃i(k)|c̃i(k − 1)]
∣∣∣c̃i(k − 1)
= 2Ci(0)− 2
C2i (1) +D
i (1)
Ci(0)
Moreover, conditioned on the vector [ai(k − 1) bi(k − 1)]T ,
the vector [ai(k) bi(k)]
T is conditionally Gaussian with
covariance matrix given by
Ci(0)− C
(1)+D2
Ci(0)
0 Ci(0)− C
(1)+D2
Ci(0)
which is a diagonal matrix. This shows that Re[c̃i(k)|c̃i(k−1)]
and Im[c̃i(k)|c̃i(k − 1)] are independent.
Applying (11), (12) and (13) to (10), we obtain
logΨm = ζ + (14)
2Es [Ci(1) + jDi(1)] e
−j 2πm
M r̃i(k)r̃
i (k − 1)
[2EsCi(0) +Ni]2 − 4E2s [C2i (1) +D2i (1)]
or, equivalently
logΨm = ζ + (15)
|ρ̃i| γi e−j∠ρ̃i
(1 + γi)2 − (|ρ̃i|γi)2
r̃i(k) r̃
i (k − 1)e−j
where ζ represents the constant term which does not affect
the decision. In (15), the quantities |ρ̃i| =
(1)+D2
∠ρ̃i = − tan−1
Di(1)
Ci(1)
represent the magnitude and phase of
the correlation coefficient ρ̃i given in (8), respectively.
1We also call them the proper complex Gaussian random variables [17].
Defining the real-valued weighting factors
|ρ̃i|γi
(1 + γi)2 − (|ρ̃i|γi)2
, (16)
it follows from (15) that the optimum combining differential
receiver will now compute, for the kth symbol, the decision
statistics {Λm(k)}M−1m=0 , and declares that ∆φ(k) =
Λn(k) = maxm {Λm(k)}, where
Λm(k) = Re
wi r̃i(k) r̃
i (k − 1) e−j∠ρ̃i
If the spectrum of the channel complex gain is symmetric,
ρ̃i is a real-valued quantity. Then, the optimum combining
differential receiver (17) will become [9]
Λ′m(k) = Re
wi r̃i(k) r̃
i (k − 1)
If the diversity branches are i.i.d., but the fading gains have
asymmetric spectrum, the optimum receiver will become
Λ′′m(k) = Re
M e−j∠ρ̃
r̃i(k) r̃
i (k − 1)
where ρ̃ = ρ̃i for i = 1, 2, · · · , L. For i.i.d. branches with
fading gains having symmetric spectrum, the optimum receiver
is the well-known product detector, given by [4]
Λ′′′m(k) = Re
r̃i(k) r̃
i (k − 1)
Comparing (20) with (17), we see that in the case of i.n.i.d.
channels with asymmetric power spectrum, the receiver first
rotates the product phasor r̃i(k)r̃
i (k − 1) between the two
received signal samples at each diversity branch by the angle
−∠ρ̃i, then scales each resulting phasor by the weight wi, and
finally sums all L rotated and scaled phasors to form a decision
variable. Clearly, in order to form the optimum detector (17),
besides the received signal samples r̃i(k) and r̃i(k − 1),
the receiver requires the a priori knowledge of the channel
statistics, including the power spectral densities of AWGN
Ni, both the magnitude and phase of the fading correlation
coefficient ρ̃i, and the mean received SNR γi. These quantities
can be pre-computed according to our knowledge of the
channel statistics at the receiver.
III. PERFORMANCE ANALYSIS
In this section, we will derive exact, explicit and closed-
form BEP expressions for differentially detected DPSK for
the optimum receiver (17). Due to space limitation, we only
consider 8-DPSK in this paper. The signal constellation, bit
mapping and the decision region Rm for 8-DPSK is shown in
Fig. 2. In [4] and [9], the average BEP is computed using
the binary reflected Gray code (BRGC) approach through
Hamming weight spectrum [20]. It is shown in [21] that the
BRGC approach with Hamming weight is less accurate for
M ≥ 16. In this paper, we adopt a new approach, namely,
the average BEP is obtained by calculating the BEP for each
of the three individual bits in 8-DPSK. This approach has
the advantage of showing explicitly the BEP performance
differently for the three different transmitted information bits.
Therefore, using the bit which has lower BEP to convey more
important information can improve communication reliability.
From Fig. 2, we see that each signal point is represented
by a 3-bit symbol (j1, j2, j3). We use Pj1 , Pj2 and Pj3 to
denote the corresponding individual BEP. Since the three bits
are equally likely, the average BEP is given by
(Pj1 + Pj2 + Pj3) (21)
We begin with computing Pj1 . Without loss of generality, it
is assumed that j1 = 0. The case where j1 = 1 gives an
identical result. From Fig. 2, we see that the bit j1 = 0 is
associated with the symbols 000 (∆φ(k) = 0), 001 (∆φ(k) =
π/4), 011 (∆φ(k) = π/2), and 010 (∆φ(k) = 3π/4). Thus,
conditioning on j1 = 0, the BEP Pj1 will be given by
Pj1 =
Pj1 (e|∆φ(k) = 0) + Pj1(e|∆φ(k) = π/4) (22)
+Pj1(e|∆φ(k) = π/2) + Pj1 (e|∆φ(k) = 3π/4)
Here, Pj1(e|∆φ(k) = mπ/4),m = 0, 1, 2, 3, is the probabil-
ity that conditioning on ∆φ(k) = mπ/4, the decision j1 = 1
is made. With reference to Fig. 2, this is equivalent to the
probability that conditioning on ∆φ(k) = mπ/4, the phasor∑L
i=1 wi r̃i(k) r̃
i (k − 1) e−j∠ρ̃i lies outside the half-plane
region R0+R1+R2+R3 (i.e., in the region R4+R5+R6+R7).
The BEP Pj1(e|∆φ(k) = mπ/4) is thus obtained as
e|∆φ(k) = mπ/4
8 (23)
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
∣∣∣∣∣
∆φ(k) =
To evaluate (23), first, it follows from (11) and (12) that con-
ditioning on ∆φ(k) = mπ/4 and on r̃i(k− 1) ej∠ρ̃i ej3π/8 =
α̃i, for i = 1, 2, · · · , L, the quantity r̃i(k) is condition-
ally Gaussian with mean α̃i
ρ̃iγi
e−j∠ρ̃ie−j3π/8ejmπ/4 =
|ρ̃i|γi
e−j3π/8ejmπ/4, where ρ̃i = |ρ̃i|ej∠ρ̃i has been used,
and variance (1+γi)
2−(|ρ̃i|γi)2
Ni. Then, in (23) the quantity
Re[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)] is conditionally
Gaussian with mean cos
mπ/4− 3π/8
i=1 wi
|ρ̃i|γi
|α̃i|2
and variance 1
i=1 w
(1+γi)
2−(|ρ̃i|γi)2
Ni |α̃i|2. Finally,
following the derivation procedure detailed in [9], the BEP
in (23) can be obtained as
e|∆φ(k) = mπ
= (24)
+ 1/λi
where the quantity Gi is given by
j=1,j 6=i
λi − λj
, and λi =
(|ρ̃i|γi)2
(1 + γi)2 − (|ρ̃i|γi)2
Putting (24) into (22) leads to the BEP Pj1. An interesting
observation from (24) is that the BEP does not depend on the
phase, ∠ρ̃i, of the fading correlation coefficient ρ̃i. Intuitively,
this is because the optimum receiver (17) can provide “phase
compensation” for each diversity branch before combining
using the channel statistic knowledge e−j∠ρ̃i . As such, we
expect that the receivers (18) and (20) are suboptimum over
the channel with asymmetric fading spectrum.
Next, we compute Pj2 in (21). The procedure for obtaining
the conditional BEP for j2 = 0 is parallel to that followed in
the case for j1 = 0. From Fig. 2, the bit j2 = 0 is associated
with the symbols 001 (∆φ(k) = π/4), 000 (∆φ(k) = 0),
100 (∆φ(k) = 7π/4), and 101 (∆φ(k) = 3π/2). Hence,
conditioning on j2 = 0, the BEP Pj2 is given by
Pj2 =
Pj2(e|∆φ(k) = π/4) + Pj2(e|∆φ(k) = 0) (26)
+Pj2(e|∆φ(k) = 7π/4) + Pj2 (e|∆φ(k) = 3π/2)
where Pj2(e|∆φ(k) = nπ/4), n = 0, 1, 6, 7, is the conditional
probability that the phasor
i=1 wir̃i(k)r̃
i (k− 1)e−j∠ρ̃i lies
in the half-plane region R2 +R3 +R4 +R5, i.e.,
e|∆φ(k) = nπ/4
8 (27)
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
∣∣∣∣∣
∆φ(k) =
which has solution
Pj2 (e|∆φ(k) = nπ/4)= (28)
+ 1/λi
Putting (28) into (26) leads to the BEP Pj2.
Finally, we compute Pj3 in (21). From Fig. 2, the bit j3 = 0
is associated with the symbols 100 (∆φ(k) = 7π/4), 000
(∆φ(k) = 0), 010 (∆φ(k) = 3π/4), and 110 (∆φ(k) = π).
Thus, conditioning on j3 = 0, the BEP Pj3 is given by
Pj3 =
Pj3 (e|∆φ(k) = 7π/4) + Pj3(e|∆φ(k) = 0) (29)
+Pj3(e|∆φ(k) = 3π/4) + Pj3 (e|∆φ(k) = π)
where Pj3(e|∆φ(k) = lπ/4), l = 0, 3, 4, 7, is the conditional
probability that the phasor
i=1 wir̃i(k)r̃
i (k− 1)e−j∠ρ̃i lies
in the region R1 + R2 + R5 + R6. This is equivalent to the
conditional probability that after rotating by −π/8, the product
of the inphase and quadrature-phase components of the phasor∑L
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i is greater than zero, i.e.,
e|∆φ(k) = l π/4
= (30)
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
∣∣∣∣∣
∆φ(k) =
From the argument for deriving (24), we note that condi-
tioning on ∆φ(k) = lπ/4 and on r̃i(k − 1) ej∠ρ̃i ejπ/8 =
β̃i, for i = 1, 2, · · · , L, the inphase component in (30),
Re[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)] is conditionally
Gaussian with mean cos
lπ/4− π/8
i=1 wi
|ρ̃i|γi
|β̃i|2
and variance 1
i=1 w
(1+γi)
2−(|ρ̃i|γi)2
Ni |β̃i|2. Similarly,
the component Im[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)] in
(30) is also a conditionally Gaussian random variable,
with mean sin
lπ/4− π/8
i=1 wi
|ρ̃i|γi
|β̃i|2 and vari-
ance 1
i=1 w
(1+γi)
2−(|ρ̃i|γi)2
Ni |β̃i|2. Moreover, it fol-
lows from (13) and the properties of the complex Gaussian ran-
dom variables [18] that the conditional inphase and quadrature-
phase components Re[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)]
and Im[e−j
i=1 wir̃i(k)r̃
i (k−1)e−j∠ρ̃i)] in (30) are also
independent. Therefore, conditioning on ∆φ(k) = lπ/4 and
on r̃i(k − 1) ej∠ρ̃i ejπ/8 = β̃i, and denoting the inphase and
quadrature-phase components as
X ∼ N
lπ/4− π/8
u, η2
Y ∼ N
lπ/4− π/8
u, η2
where u and η2 are given, respectively, by
|ρ̃i|γi
1 + γi
|β̃i|2
(1 + γi)
2 − (|ρ̃i|γi)2
1 + γi
Ni |β̃i|2 (32)
the conditional BEP Pj3
∣∣∆φ(k) = lπ
, β̃i
is given by
∣∣∆φ(k) =
, β̃i
X Y > 0
∣∣∆φ(k) =
, β̃i
This is probability that the product of two independent real-
valued Gaussian random variables with non-zero, nonidentical
means and identical variances is greater than zero. This is a
special case of the results given in [2, Appendix B] concerning
the probability that a general quadratic form in complex-valued
Gaussian random variables is less than zero. Using [2, (B-21)
of Appendix B], (33) can be evaluated as
∣∣∆φ(k) =
, β̃i
= 1− (34)
g[1− sin (lπ/2− π/4)],
g[1 + sin (lπ/2− π/4)]
I0 [g| cos (lπ/2− π/4)|] exp(−g)
where, Q1(a, b) is first-order Marcum’s Q-function and Ik(x)
is the kth-order modified Bessel function of the first kind. In
(34), the quantity g =
i=1 w
i |β̃i|2 has PDF given by [9]
p(g) =
w′i Ni (1 + γi)
w′iNi (1 + γi)
where w′i =
(|ρ̃i|γi)2
(1+γi)[(1+γi)2−(|ρ̃i|γi)2] . Averaging the condi-
tional probability (34) over g using the PDF (35) gives the
BEP Pj3
∣∣∆φ(k) = lπ/4
in (30), i.e.,
∣∣∆φ(k) =
∣∣∆φ(k) =
, β̃i
p(g)dg
Substituting (34) and (35) into (36), we obtain, after manipu-
lation and simplification,
∣∣∆φ(k) =
A2i − cos2 ( lπ2 −
1− | cos (
−cos2 ( lπ
− 1/2√
A2i − cos2 ( lπ2 −
where Ai is given by
1 + γi
|ρ̃i|γi
Putting (37) into (29) leads to the BEP Pj3. Substituting (22),
(26) and (29) in (21), we obtain the average BEP P .
IV. NUMERICAL EXAMPLE
Fig. 3 plots the BEP performance for the three individual
bits in (22), (26) and (29) and the average BEP in (21) of
8-DPSK against the total average received SNR per bit. The
order of diversity is set to L = 2. The abscissa represents the
total mean SNR per bit which is given by γb =
i=1 γ
i=1 γi. The average received bit energy distribution among
the two branches is set to γb1 : γ
2 = 30% : 70%. It is
assumed that the fading correlation coefficient (the normalized
covariance function) model follows [14, eq.(10)], given by
E[c̃i(t)c̃
i (t− τ)]
|c̃i(t)|2
κ2 − 4π2f2dτ2 + j4πκfdτ
I0(κ)
where fd is the Doppler frequency, and κ is a parameter that
controls the width of the angle of arrival of scatter components
[14, eq.(1)]. Note that if κ = 0, (38) results in the correlation
coefficient for the Jakes two-dimensional isotropic scattering
model, i.e., E[c̃i(t)c̃
i (t− τ)]/E
|c̃i(t)|2
= I0(j2πfmτ) =
J0(2πfmτ), where J0(·) is the zeroth-order Bessel function.
We assume that the normalized Doppler spread fdT = 0.03
and 0.05 for diversity branches 1 and 2, respectively, and the
parameter κ is set to 3. Thus, we have ρ̃1 = 0.9871+ j0.1519
and ρ̃2 = 0.9642 + j0.2511. It is seen from Fig. 3 that the
third bit j3 has the lowest BEP, whereas, the BEP Pj1 for the
first bit j1 is equal to the BEP Pj2 for the second bit j2.
REFERENCES
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[3] M. K. Simon and M. S. Alouini,, Digital communication over fading
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Rayleigh fading channel with diversity reception,” IEEE Trans. Com-
mun., vol.39, pp.220-224, February 1991.
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combining of diversity channels with unequal SNR in Rayleigh fading,”
Proc. 49th IEEE VTC, pp.215-220, May 16-20, 1999.
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[7] F. Adachi, “Postdetection optimal diversity combiner for DPSK differen-
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identical Rayleigh Fading Channels,” Proc. 40th IEEE ICC, pp.2367-
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[9] H. Fu and P. Y. Kam, “MDPSK diversity receiver over Rayleigh fading
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Proc. 63rd IEEE VTC, pp.1660-1664, May 7-10, 2006.
[10] H. Fu and P. Y. Kam, “Performance of Optimum and Suboptimum
Combining Diversity Reception for Binary and Quadrature DPSK over
Independent, Nonidentical Rayleigh Fading Channels,” to appear in the
IEEE Trans. Commun., May 2007.
[11] L. J. Mason “Error probability evaluation for systems employing differ-
ential detection in a Rician fast fading environment and Gaussian noise,”
IEEE Trans. Commun., vol.35, pp.39-46, January 1987.
[12] W.C. Jakes, Microwave Mobile Communications, NJ: IEEE Press, 1974.
[13] M. Patzold, Y.C. Li and F. Laue “A study of a land mobile satel-
lite channel model with asymmetrical Doppler power spectrum and
lognormally distributed line-of-sight component,” IEEE Trans. Veh.
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[14] A. Abdi, J.A. Barger and M. Kaveh “A parametric model for the
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[16] W. B. Davenport and W. L. Root, An introduction to the theory of
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[17] F. D. Neeser and J.L. Massey “Proper complex random processes with
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pp.1293-1302, July 1993.
[18] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation
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[19] A. Papoulis and S.U. Pillai, Probability, Random Variables and Stochas-
tic Processes, 4th Ed., MA: McGraw-Hill, 2002.
[20] P. J. Lee, “Computation of the bit error rate of coherent M-ary PSK
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Fig. 1. Illustration of complex channel fading process.
Fig. 2. 8-DPSK constellation and decision region.
10 15 20 25 30 35 40
SNR (γ
), dB
Fig. 3. BEP comparison of the three individual bits and the average of all
bits for 8-DPSK.
	INTRODUCTION
	SIGNAL MODEL AND RECEIVER STRUCTURE
	PERFORMANCE ANALYSIS
	NUMERICAL EXAMPLE
	References

  This paper is concerned with optimum diversity receiver structure and its
performance analysis of differential phase shift keying (DPSK) with
differential detection over nonselective, independent, nonidentically
distributed, Rayleigh fading channels. The fading process in each branch is
assumed to have an arbitrary Doppler spectrum with arbitrary Doppler bandwidth,
but to have distinct, asymmetric fading power spectral density characteristic.
Using 8-DPSK as an example, the average bit error probability (BEP) of the
optimum diversity receiver is obtained by calculating the BEP for each of the
three individual bits. The BEP results derived are given in exact, explicit,
closed-form expressions which show clearly the behavior of the performance as a
function of various system parameters.

Differential Diversity Reception of MDPSK over Independent
Rayleigh Channels with Nonidentical Branch Statistics and
Asymmetric Fading Spectrum
Hua Fu and Pooi Yuen Kam
ECE Department, National University of Singapore
Singapore 117576, Email: {elefh, elekampy}@nus.edu.sg
Abstract— This paper is concerned with optimum diversity re-
ceiver structure and its performance analysis of differential phase
shift keying (DPSK) with differential detection over nonselective,
independent, nonidentically distributed, Rayleigh fading chan-
nels. The fading process in each branch is assumed to have an
arbitrary Doppler spectrum with arbitrary Doppler bandwidth,
but to have distinct, asymmetric fading power spectral density
characteristic. Using 8-DPSK as an example, the average bit error
probability (BEP) of the optimum diversity receiver is obtained
by calculating the BEP for each of the three individual bits.
The BEP results derived are given in exact, explicit, closed-form
expressions which show clearly the behavior of the performance
as a function of various system parameters.
I. INTRODUCTION
The receiver structure and bit error probability (BEP) per-
formance of differential phase shift keying (DPSK) with differ-
ential detection over nonselective, independent and identically
distributed (i.i.d.), Rayleigh fading channels with combining
diversity reception have been well known in the literature
[1]−[4]. However, reaserch shows that in some practical
systems, the independent, non-identically distributed (i.n.i.d.)
channel model is more accurate [5], [6]. In i.n.i.d. channel,
the fading processes and possibly the additive, white Gaussian
noise (AWGN) on the diversity branches have non-uniform
power profiles which are distinct from one another. The effect
of the nonidentical diversity branch statistics on the receiver
structure is studied in [7]. Recently, based on the maximum a
posteriori probability (MAP) criterion, an explicit structure of
the optimum combining differential receiver and a complete
set of closed-form BEP expressions and their Chernoff upper
bounds, for 2-, 4- and 8-DPSK, both with optimum combining
reception and suboptimum combining reception, are derived
in [8]−[10]. The purpose of this paper is to provide a further
extension. The results derived in this paper, together with those
in [8]−[10], form a benchmark counterpart to the classic ones
for the i.i.d. channel given in [1]−[4].
In a Rayleigh channel, the fading gain is usually modeled
as a zero-mean, stationary, complex, Gaussian random pro-
cess. The most widely accepted model [1]−[10] is that the
spectrum of the fading process over each diversity branch is
symmetric around the carrier so that the quadrature processes
are independent of each other. This assumption is valid for
various fading spectra. For example, see [11] and its refer-
ences. However, in some fading environments such as the land
mobile channel with Jakes model [12], the Doppler spectrum
becomes asymmetric when the multipath signals are absorbed
by obstacles or the propagation environment is characterised
by directional non-isotropic scattering [13]−[15]. Thus, it is
of great practical importance to take account of the effect of
the asymmetric fading spectrum on the receiver structure and
the performance analysis of differentially detected DPSK over
i.n.i.d. channels, the topic of this paper.
The paper is orgainzed as follows. In Section II, the signal
model is introduced and different optimum diversity receivers
are derived for different Rayleigh fading scenarios (see eqs.
(17)−(20) below). In Section III, we use 8-DPSK as an
example to study the BEP performance. Here, the average BEP
of the optimum diversity receiver is obtained by calculating the
BEP for each of the three individual bits. The results are given
in exact, explicit, closed-form expressions which show clearly
the behavior of the performance as a function of signal-to-
noise ratio (SNR), fading correlation coefficient, and diversity
order. Section IV presents numerical examples. Throughout
this paper, overhead ∼ denotes a complex quantity, superscript
∗ will denote its conjugate, E is the ensemble average operator,
δ represents the Kronecker delta, and [·]T denotes transposition
of the vector and matrix.
II. SIGNAL MODEL AND RECEIVER STRUCTURE
With space diversity reception over L frequency nonse-
lective, i.n.i.d., Rayleigh fading branches with AWGN, the
received signal over the ith branch, i = 1, 2, · · · , L, during
the kth symbol interval kT ≤ t < (k + 1)T is given, after
matched filtering and sampling at time t = (k + 1)T , by the
statistic r̃i(k), where
r̃i(k) = E
jφ(k)c̃i(k) + ñi(k). (1)
Here, Es is the energy per symbol, and for DPSK, φ(k) is
the data-modulated phase with Gray encoding of bits onto
the phase transition ∆φ(k) = φ(k) − φ(k − 1). The kth data
symbol is conveyed in ∆φ(k). We assume here that all symbol
points are equally likely. In (1), a rectangular data pulse shape
g(t), where g(t) = 1/
T for 0 ≤ t < T and zero elsewhere,
is assumed so that each matched filter has a rectangular low-
pass-equivalent impulse response hi(t) = g(T − t) for all i.
Thus, the filtered noise ñi(k) is given by
ñi(k) =
∫ (k+1)T
ñi(t)√
dt. (2)
Here, {ñi(t)}Li=1 is a set of i.n.i.d., lowpass, complex AWGN
processes with E [ñi(t)] = 0 and E[ñi(t)ñ
i (t− τ)] = Niδ(τ)
so that {ñi(k)}k is a sequence of zero-mean, complex Gaus-
sian variables with covariance function for each branch i
E[ñi(k)ñ
i (j)] = Ni δkj (3)
The multiplicative distortion c̃i(k) in (1) is given by
c̃i(k) =
∫ (k+1)T
c̃i(t)
dt. (4)
http://arxiv.org/abs/0704.1070v1
Here,
c̃i(t) = ai(t) + jbi(t)
is a set of i.n.i.d., lowpass,
zero-mean, stationary, complex, Gaussian random processes.
Each c̃i(t) represents the complex gain due to frequency
nonselective Rayleigh fading of the ith branch. For asymmetric
spectrum in each i, the inphase fading process ai(·) and the
quadrature phase fading process bi(·) are generally correlated.
At any time instant t, however, ai(t) and bi(t) are always
uncorrelated. With reference to Fig. 1, it is shown in [16] that
the covariance function of ai(·) and bi(·) can be obtained as
E[ai(t)bi(t)] = 0 (5a)
E[ai(t− τ)ai(t)] = E[bi(t− τ)bi(t)] = Ri(τ) (5b)
E[ai(t)bi(t− τ)] = −E[bi(t)ai(t− τ)] = Qi(τ) (5c)
Note that if the spectrum of each c̃i(t) is symmetric, the
processes ai(·) and bi(·) will be independent (i.e., we have
Qi(τ) = 0) with the same covariance function Ri(τ).
Letting c̃i(k) = ai(k) + jbi(k), it follows from (4) and (5)
that both {ai(k)}k and {bi(k)}k are sequences of zero-mean,
real-valued, Gaussian random variables with
E[ai(k)bi(k)] = 0 (6a)
E[ai(k − l)ai(k)] = E[bi(k − l)bi(k)] = Ci(l) (6b)
∫ (k+1)T
∫ (k+1−l)T
(k−l)T
Ri(u − v)
du dv
E[ai(k)bi(k − l)] = −E[bi(k)ai(k − l)] = Di(l) (6c)
∫ (k+1)T
∫ (k+1−l)T
(k−l)T
Qi(u− v)
du dv
Thus, the covariance matrix can be obtained as
Γi = E
ai(k)
ai(k − l)
bi(k)
bi(k − l)
ai(k) ai(k − l) bi(k) bi(k − l)
Ci(0) Ci(l) 0 Di(l)
Ci(l) Ci(0) −Di(l) 0
0 −Di(l) Ci(0) Ci(l)
Di(l) 0 Ci(l) Ci(0)
 (7)
For each i , c̃i(k) and ñi(k) are mutually independent. For
i 6= j, {c̃i(k), ñi(k)} are independent of {c̃j(k), ñj(k)}.
The diversity branches are nonidentical since the covariance
functions Ri(τ), Qi(τ) and Niδ(τ) depend on the branch
index i. For convenience of later application, the following
parameters are defined. The fading correlation coefficient at
the matched filter output over a symbol interval of T for the
ith diversity branch is defined as
ρ̃i =
E[c̃i(k)c̃
i (k − 1)]√
|c̃i(k)|2
|c̃i(k − 1)|2
Ci(1)− jDi(1)
Ci(0)
From (8), we note that ρ̃i is a complex quantity. It is a measure
of the fluctuation rate of the channel fading process. The mean
received SNR per symbol over the ith branch is defined as
|E1/2s ejφ(k)c̃i(k)|2
2EsCi(0)
We consider 2-, 4- and 8-DPSK with Gray encoding of bits
onto ∆φ(k) as shown in [4, Fig.1] for 4- and 8-DPSK, the
mean received SNR per bit γbi is given by γ
i = γi for 2-DPSK,
γbi = γi/2 for 4-DPSK, and γ
i = γi/3 for 8-DPSK.
Using the MAP criterion, the aim of the receiver is to
determine from the received signals {r̃i(k), r̃i(k − 1)}Li=1
which one of the possible values 2πm/M , m = 0, 1, · · · ,M−
1, of the phase difference ∆φ(k) has maximum probability
of occurrence. Following [9], it can be shown that MAP
detection is equivalent to maximum log-likelihood detection.
Specifically, based on {r̃i(k), r̃i(k − 1)}Li=1, we decide that
∆φ(k) = 2πn/M whenever the log-likelihood function
logΨm =
r̃i(k)
∣∣∣r̃i(k − 1),∆φ(k) =
is maximized for m = n.
To proceed with evaluating (10), we need to verify that
c̃i(k) = ai(k)+jbi(k) and c̃i(k−1) = ai(k−1)+jbi(k−1) are
jointly complex Gaussian. By being jointly complex Gaussian,
it means that if x̃ = xR + jxI and ỹ = yR + jyI are two
column complex random vector, then [xR
has a real multivatiate Gaussian probability density function
(PDF), and furthermore, if u = [xR
T ]T and v =
T ]T , then the real covariance matrix of [uT vT ]T
has a special form given in [18, Theorem 15.1] that satis-
fies Goodman’s theorem [19]. After careful examination, it
follows from (7) that c̃i(k) and c̃i(k − 1) are indeed jointly
complex Gaussian1. Thus, conditioned on c̃i(k − 1), c̃i(k) is
conditionally complex Gaussian with mean [18]
E [c̃i(k)|c̃i(k − 1)] = ρ̃i c̃i(k − 1) (11)
and variance
{∣∣c̃i(k)− E[c̃i(k)|c̃i(k − 1)]
∣∣∣c̃i(k − 1)
= 2Ci(0)− 2
C2i (1) +D
i (1)
Ci(0)
Moreover, conditioned on the vector [ai(k − 1) bi(k − 1)]T ,
the vector [ai(k) bi(k)]
T is conditionally Gaussian with
covariance matrix given by
Ci(0)− C
(1)+D2
Ci(0)
0 Ci(0)− C
(1)+D2
Ci(0)
which is a diagonal matrix. This shows that Re[c̃i(k)|c̃i(k−1)]
and Im[c̃i(k)|c̃i(k − 1)] are independent.
Applying (11), (12) and (13) to (10), we obtain
logΨm = ζ + (14)
2Es [Ci(1) + jDi(1)] e
−j 2πm
M r̃i(k)r̃
i (k − 1)
[2EsCi(0) +Ni]2 − 4E2s [C2i (1) +D2i (1)]
or, equivalently
logΨm = ζ + (15)
|ρ̃i| γi e−j∠ρ̃i
(1 + γi)2 − (|ρ̃i|γi)2
r̃i(k) r̃
i (k − 1)e−j
where ζ represents the constant term which does not affect
the decision. In (15), the quantities |ρ̃i| =
(1)+D2
∠ρ̃i = − tan−1
Di(1)
Ci(1)
represent the magnitude and phase of
the correlation coefficient ρ̃i given in (8), respectively.
1We also call them the proper complex Gaussian random variables [17].
Defining the real-valued weighting factors
|ρ̃i|γi
(1 + γi)2 − (|ρ̃i|γi)2
, (16)
it follows from (15) that the optimum combining differential
receiver will now compute, for the kth symbol, the decision
statistics {Λm(k)}M−1m=0 , and declares that ∆φ(k) =
Λn(k) = maxm {Λm(k)}, where
Λm(k) = Re
wi r̃i(k) r̃
i (k − 1) e−j∠ρ̃i
If the spectrum of the channel complex gain is symmetric,
ρ̃i is a real-valued quantity. Then, the optimum combining
differential receiver (17) will become [9]
Λ′m(k) = Re
wi r̃i(k) r̃
i (k − 1)
If the diversity branches are i.i.d., but the fading gains have
asymmetric spectrum, the optimum receiver will become
Λ′′m(k) = Re
M e−j∠ρ̃
r̃i(k) r̃
i (k − 1)
where ρ̃ = ρ̃i for i = 1, 2, · · · , L. For i.i.d. branches with
fading gains having symmetric spectrum, the optimum receiver
is the well-known product detector, given by [4]
Λ′′′m(k) = Re
r̃i(k) r̃
i (k − 1)
Comparing (20) with (17), we see that in the case of i.n.i.d.
channels with asymmetric power spectrum, the receiver first
rotates the product phasor r̃i(k)r̃
i (k − 1) between the two
received signal samples at each diversity branch by the angle
−∠ρ̃i, then scales each resulting phasor by the weight wi, and
finally sums all L rotated and scaled phasors to form a decision
variable. Clearly, in order to form the optimum detector (17),
besides the received signal samples r̃i(k) and r̃i(k − 1),
the receiver requires the a priori knowledge of the channel
statistics, including the power spectral densities of AWGN
Ni, both the magnitude and phase of the fading correlation
coefficient ρ̃i, and the mean received SNR γi. These quantities
can be pre-computed according to our knowledge of the
channel statistics at the receiver.
III. PERFORMANCE ANALYSIS
In this section, we will derive exact, explicit and closed-
form BEP expressions for differentially detected DPSK for
the optimum receiver (17). Due to space limitation, we only
consider 8-DPSK in this paper. The signal constellation, bit
mapping and the decision region Rm for 8-DPSK is shown in
Fig. 2. In [4] and [9], the average BEP is computed using
the binary reflected Gray code (BRGC) approach through
Hamming weight spectrum [20]. It is shown in [21] that the
BRGC approach with Hamming weight is less accurate for
M ≥ 16. In this paper, we adopt a new approach, namely,
the average BEP is obtained by calculating the BEP for each
of the three individual bits in 8-DPSK. This approach has
the advantage of showing explicitly the BEP performance
differently for the three different transmitted information bits.
Therefore, using the bit which has lower BEP to convey more
important information can improve communication reliability.
From Fig. 2, we see that each signal point is represented
by a 3-bit symbol (j1, j2, j3). We use Pj1 , Pj2 and Pj3 to
denote the corresponding individual BEP. Since the three bits
are equally likely, the average BEP is given by
(Pj1 + Pj2 + Pj3) (21)
We begin with computing Pj1 . Without loss of generality, it
is assumed that j1 = 0. The case where j1 = 1 gives an
identical result. From Fig. 2, we see that the bit j1 = 0 is
associated with the symbols 000 (∆φ(k) = 0), 001 (∆φ(k) =
π/4), 011 (∆φ(k) = π/2), and 010 (∆φ(k) = 3π/4). Thus,
conditioning on j1 = 0, the BEP Pj1 will be given by
Pj1 =
Pj1 (e|∆φ(k) = 0) + Pj1(e|∆φ(k) = π/4) (22)
+Pj1(e|∆φ(k) = π/2) + Pj1 (e|∆φ(k) = 3π/4)
Here, Pj1(e|∆φ(k) = mπ/4),m = 0, 1, 2, 3, is the probabil-
ity that conditioning on ∆φ(k) = mπ/4, the decision j1 = 1
is made. With reference to Fig. 2, this is equivalent to the
probability that conditioning on ∆φ(k) = mπ/4, the phasor∑L
i=1 wi r̃i(k) r̃
i (k − 1) e−j∠ρ̃i lies outside the half-plane
region R0+R1+R2+R3 (i.e., in the region R4+R5+R6+R7).
The BEP Pj1(e|∆φ(k) = mπ/4) is thus obtained as
e|∆φ(k) = mπ/4
8 (23)
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
∣∣∣∣∣
∆φ(k) =
To evaluate (23), first, it follows from (11) and (12) that con-
ditioning on ∆φ(k) = mπ/4 and on r̃i(k− 1) ej∠ρ̃i ej3π/8 =
α̃i, for i = 1, 2, · · · , L, the quantity r̃i(k) is condition-
ally Gaussian with mean α̃i
ρ̃iγi
e−j∠ρ̃ie−j3π/8ejmπ/4 =
|ρ̃i|γi
e−j3π/8ejmπ/4, where ρ̃i = |ρ̃i|ej∠ρ̃i has been used,
and variance (1+γi)
2−(|ρ̃i|γi)2
Ni. Then, in (23) the quantity
Re[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)] is conditionally
Gaussian with mean cos
mπ/4− 3π/8
i=1 wi
|ρ̃i|γi
|α̃i|2
and variance 1
i=1 w
(1+γi)
2−(|ρ̃i|γi)2
Ni |α̃i|2. Finally,
following the derivation procedure detailed in [9], the BEP
in (23) can be obtained as
e|∆φ(k) = mπ
= (24)
+ 1/λi
where the quantity Gi is given by
j=1,j 6=i
λi − λj
, and λi =
(|ρ̃i|γi)2
(1 + γi)2 − (|ρ̃i|γi)2
Putting (24) into (22) leads to the BEP Pj1. An interesting
observation from (24) is that the BEP does not depend on the
phase, ∠ρ̃i, of the fading correlation coefficient ρ̃i. Intuitively,
this is because the optimum receiver (17) can provide “phase
compensation” for each diversity branch before combining
using the channel statistic knowledge e−j∠ρ̃i . As such, we
expect that the receivers (18) and (20) are suboptimum over
the channel with asymmetric fading spectrum.
Next, we compute Pj2 in (21). The procedure for obtaining
the conditional BEP for j2 = 0 is parallel to that followed in
the case for j1 = 0. From Fig. 2, the bit j2 = 0 is associated
with the symbols 001 (∆φ(k) = π/4), 000 (∆φ(k) = 0),
100 (∆φ(k) = 7π/4), and 101 (∆φ(k) = 3π/2). Hence,
conditioning on j2 = 0, the BEP Pj2 is given by
Pj2 =
Pj2(e|∆φ(k) = π/4) + Pj2(e|∆φ(k) = 0) (26)
+Pj2(e|∆φ(k) = 7π/4) + Pj2 (e|∆φ(k) = 3π/2)
where Pj2(e|∆φ(k) = nπ/4), n = 0, 1, 6, 7, is the conditional
probability that the phasor
i=1 wir̃i(k)r̃
i (k− 1)e−j∠ρ̃i lies
in the half-plane region R2 +R3 +R4 +R5, i.e.,
e|∆φ(k) = nπ/4
8 (27)
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
∣∣∣∣∣
∆φ(k) =
which has solution
Pj2 (e|∆φ(k) = nπ/4)= (28)
+ 1/λi
Putting (28) into (26) leads to the BEP Pj2.
Finally, we compute Pj3 in (21). From Fig. 2, the bit j3 = 0
is associated with the symbols 100 (∆φ(k) = 7π/4), 000
(∆φ(k) = 0), 010 (∆φ(k) = 3π/4), and 110 (∆φ(k) = π).
Thus, conditioning on j3 = 0, the BEP Pj3 is given by
Pj3 =
Pj3 (e|∆φ(k) = 7π/4) + Pj3(e|∆φ(k) = 0) (29)
+Pj3(e|∆φ(k) = 3π/4) + Pj3 (e|∆φ(k) = π)
where Pj3(e|∆φ(k) = lπ/4), l = 0, 3, 4, 7, is the conditional
probability that the phasor
i=1 wir̃i(k)r̃
i (k− 1)e−j∠ρ̃i lies
in the region R1 + R2 + R5 + R6. This is equivalent to the
conditional probability that after rotating by −π/8, the product
of the inphase and quadrature-phase components of the phasor∑L
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i is greater than zero, i.e.,
e|∆φ(k) = l π/4
= (30)
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i
∣∣∣∣∣
∆φ(k) =
From the argument for deriving (24), we note that condi-
tioning on ∆φ(k) = lπ/4 and on r̃i(k − 1) ej∠ρ̃i ejπ/8 =
β̃i, for i = 1, 2, · · · , L, the inphase component in (30),
Re[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)] is conditionally
Gaussian with mean cos
lπ/4− π/8
i=1 wi
|ρ̃i|γi
|β̃i|2
and variance 1
i=1 w
(1+γi)
2−(|ρ̃i|γi)2
Ni |β̃i|2. Similarly,
the component Im[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)] in
(30) is also a conditionally Gaussian random variable,
with mean sin
lπ/4− π/8
i=1 wi
|ρ̃i|γi
|β̃i|2 and vari-
ance 1
i=1 w
(1+γi)
2−(|ρ̃i|γi)2
Ni |β̃i|2. Moreover, it fol-
lows from (13) and the properties of the complex Gaussian ran-
dom variables [18] that the conditional inphase and quadrature-
phase components Re[e−j
i=1 wir̃i(k)r̃
i (k − 1)e−j∠ρ̃i)]
and Im[e−j
i=1 wir̃i(k)r̃
i (k−1)e−j∠ρ̃i)] in (30) are also
independent. Therefore, conditioning on ∆φ(k) = lπ/4 and
on r̃i(k − 1) ej∠ρ̃i ejπ/8 = β̃i, and denoting the inphase and
quadrature-phase components as
X ∼ N
lπ/4− π/8
u, η2
Y ∼ N
lπ/4− π/8
u, η2
where u and η2 are given, respectively, by
|ρ̃i|γi
1 + γi
|β̃i|2
(1 + γi)
2 − (|ρ̃i|γi)2
1 + γi
Ni |β̃i|2 (32)
the conditional BEP Pj3
∣∣∆φ(k) = lπ
, β̃i
is given by
∣∣∆φ(k) =
, β̃i
X Y > 0
∣∣∆φ(k) =
, β̃i
This is probability that the product of two independent real-
valued Gaussian random variables with non-zero, nonidentical
means and identical variances is greater than zero. This is a
special case of the results given in [2, Appendix B] concerning
the probability that a general quadratic form in complex-valued
Gaussian random variables is less than zero. Using [2, (B-21)
of Appendix B], (33) can be evaluated as
∣∣∆φ(k) =
, β̃i
= 1− (34)
g[1− sin (lπ/2− π/4)],
g[1 + sin (lπ/2− π/4)]
I0 [g| cos (lπ/2− π/4)|] exp(−g)
where, Q1(a, b) is first-order Marcum’s Q-function and Ik(x)
is the kth-order modified Bessel function of the first kind. In
(34), the quantity g =
i=1 w
i |β̃i|2 has PDF given by [9]
p(g) =
w′i Ni (1 + γi)
w′iNi (1 + γi)
where w′i =
(|ρ̃i|γi)2
(1+γi)[(1+γi)2−(|ρ̃i|γi)2] . Averaging the condi-
tional probability (34) over g using the PDF (35) gives the
BEP Pj3
∣∣∆φ(k) = lπ/4
in (30), i.e.,
∣∣∆φ(k) =
∣∣∆φ(k) =
, β̃i
p(g)dg
Substituting (34) and (35) into (36), we obtain, after manipu-
lation and simplification,
∣∣∆φ(k) =
A2i − cos2 ( lπ2 −
1− | cos (
−cos2 ( lπ
− 1/2√
A2i − cos2 ( lπ2 −
where Ai is given by
1 + γi
|ρ̃i|γi
Putting (37) into (29) leads to the BEP Pj3. Substituting (22),
(26) and (29) in (21), we obtain the average BEP P .
IV. NUMERICAL EXAMPLE
Fig. 3 plots the BEP performance for the three individual
bits in (22), (26) and (29) and the average BEP in (21) of
8-DPSK against the total average received SNR per bit. The
order of diversity is set to L = 2. The abscissa represents the
total mean SNR per bit which is given by γb =
i=1 γ
i=1 γi. The average received bit energy distribution among
the two branches is set to γb1 : γ
2 = 30% : 70%. It is
assumed that the fading correlation coefficient (the normalized
covariance function) model follows [14, eq.(10)], given by
E[c̃i(t)c̃
i (t− τ)]
|c̃i(t)|2
κ2 − 4π2f2dτ2 + j4πκfdτ
I0(κ)
where fd is the Doppler frequency, and κ is a parameter that
controls the width of the angle of arrival of scatter components
[14, eq.(1)]. Note that if κ = 0, (38) results in the correlation
coefficient for the Jakes two-dimensional isotropic scattering
model, i.e., E[c̃i(t)c̃
i (t− τ)]/E
|c̃i(t)|2
= I0(j2πfmτ) =
J0(2πfmτ), where J0(·) is the zeroth-order Bessel function.
We assume that the normalized Doppler spread fdT = 0.03
and 0.05 for diversity branches 1 and 2, respectively, and the
parameter κ is set to 3. Thus, we have ρ̃1 = 0.9871+ j0.1519
and ρ̃2 = 0.9642 + j0.2511. It is seen from Fig. 3 that the
third bit j3 has the lowest BEP, whereas, the BEP Pj1 for the
first bit j1 is equal to the BEP Pj2 for the second bit j2.
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Fig. 1. Illustration of complex channel fading process.
Fig. 2. 8-DPSK constellation and decision region.
10 15 20 25 30 35 40
SNR (γ
), dB
Fig. 3. BEP comparison of the three individual bits and the average of all
bits for 8-DPSK.
	INTRODUCTION
	SIGNAL MODEL AND RECEIVER STRUCTURE
	PERFORMANCE ANALYSIS
	NUMERICAL EXAMPLE
	References
