PERFORMANCE ANALYSIS OF FSK MODULATION WITH
LIMITER-DISCRIMINATOR-INTEGRATOR DETECTION OVER
HOYT FADING CHANNELS
Nazih Hajri and Neji Youssef
Ecole Sup´erieure des Communications de Tunis, 2083 EL Ghazala, Ariana, Tunisia
Keywords:
BEP performance, Hoyt multipath fading, FSK modulation, LDI receivers, FM clicks, ISI, M2M channels.
Abstract:
The focus of this paper is on the performance analysis of frequency shift keying (FSK) modulation with limiter-
discriminator-integrator (LDI) detection over frequency-flat Hoyt (Nakagami-q) fading channels. Specifically,
a closed-form expression is derived for the probability density function (PDF) of the phase difference of Hoyt
faded FSK signals disturbed by additive white Gaussian noise (AWGN). This newly derived PDF is verified
to reduce to known results corresponding to the Rayleigh fading channel as a special case of the Hoyt model.
The validity of the expression is further demonstrated by simulation for the case of a Hoyt mobile-to-mobile
(M2M) fading channel. The analytical PDF of the phase difference is then applied to determine the bit-error
probability (BEP) of the LDI receiver taking into consideration the Doppler effects, the click noise as well as
the inter-symbol interference (ISI) caused by the intermediate frequency (IF) pre-detection filter. Numerical
examples, assuming a Hoyt M2M channel, are given to illustrate the analysis and examine the effects of the
FM system parameters and the fading characteristics on the BEP performance.
1 INTRODUCTION
The Hoyt statistical distribution is a general short-
term fading model which includes the one-sided
Gaussian and the Rayleigh models as special cases
(Nakagami, 1960). This multipath propagation model
was originally introduced in (Nakagami, 1960) for the
study of ionospheric scintillation. Recently, it has
been shown in (Youssef et al, 2005) that the model is
useful to accurately represent real world mobile satel-
lite channels in heavy shadowing environments, and
allows to describe more severe conditions of fading
than does the classical Rayleigh distribution. Given
the importance of the Hoyt model in statistical mod-
eling of short-term fading, it is of interest to study
and analyze its impact on the performance of wire-
less communication systems. To the best of authors
knowledge, there are only few studies that have been
reported on this topic so far. For instance, infor-
mation outage probability of orthogonal space-time
block codes over Hoyt fading channels has been in-
vestigated in (Ropokis et al, 2007). The BEP of
narrow-band digital FSK modulation with LDI detec-
tion scheme has been studied in (Hajri and Youssef,
2007). There, the methodology employed relies on
the results reported in (Tjhung et al, 1990), and con-
sists on the determination, separately, of the PDF
of the phase difference introduced by the receiver
noise and that introduced by the Hoyt fading channel.
Assuming the statistical independence between the
two random phase processes, then the resultant phase
PDF, needed for the evaluation of the BEP expres-
sion, is calculated by a convolution operation which
demands tedious numerical integrations. In addition,
the results of (Hajri and Youssef, 2007) are valid only
for small values of signal-to-noise ratio.
In this paper, and to avoid the drawback of the
method mentioned above, we approach the problem
from a different point of view. We derive a closed-
form expression for the PDF of the overall phase dif-
ference of a Hoyt faded FSK signal corrupted by addi-
tive noise. Thereafter, we apply this PDF for the cal-
culation of the desired BEP of FSK systems drawing
upon the classical work on error performance analy-
sis of LDI receivers reported in (Pawula, 1981; Ng
et al, 1994). Numerical examples are given for vari-
ous values of the FM system parameters and channel
characteristics to study their impact on the BEP per-
formance.
The remainder of the paper is structured as fol-
lows. Section II contains a review of known results on
the error performance analysis of LDI based digital
177
Hajri N. and Youssef N. (2008).
PERFORMANCE ANALYSIS OF FSK MODULATION WITH LIMITER-DISCRIMINATOR-INTEGRATOR DETECTION OVER HOYT FADING CHAN-
NELS.
In Proceedings of the International Conference on Wireless Information Networks and Systems, pages 177-181
DOI: 10.5220/0002025201770181
Copyright
c
 SciTePress

FM receivers. In Section III, we address the deriva-
tion of the PDF of the phase difference for a Hoyt
faded FSK signal contaminated by receiver noise. In
Section IV, the newly derived PDF is applied for the
determination of the desired BEP. Illustrations of nu-
merical examples for the case of a Hoyt M2M channel
are provided in Section V, and the conclusion is drawn
in Section VI.
2 PRELIMINARIES
The link between the transmitter and the receiver is
modeled by a narrow-band Hoyt multipath fading
channel (Nakagami, 1960). The complex low-pass
equivalent Hoyt faded FSK signal, present at the input
of the IF pre-detection filter, can be expressed as
S
r
(t) = (µ
1
(t) + jµ
2
(t))exp[ jθ(t)] + w(t) (1)
where µ
1
(t) + jµ
2
(t) is a zero-mean complex Gaus-
sian process used to model the Hoyt fading gain (Nak-
agami, 1960). The variances of µ
1
(t) and µ
2
(t) will
be denoted by σ
2
1
and σ
2
2
, respectively. Also, w(t) is
a zero-mean complex AWGN, and θ(t) stands for the
data phase after FM modulation which is given by
θ(t) =
πm
T
t
Z
−∞
b(τ)dτ. (2)
In (2), b(t) is the binary data sequence of bit rate 1

T,
and m is the FSK modulation index. Concerning the
IF band-pass filter, it is considered to be of a Gaussian
shape with an equivalent low-pass transfer function
given by
H( f ) = exp

−π f
2

2B
2

(3)
where B is the equivalent noise bandwidth. Now, for
the determination of the signal resulting at the output
of the IF filter, we follow (Tjhung et al, 1990; Ng
et al, 1994) by assuming that the Hoyt fading process
changes at a rate that is much slower than the data rate
1/T. This so called “quasi-static” analysis implies
that the channel gain µ
1
(t) + jµ
2
(t) is not affected by
its passage over the IF filter. In this case, the output
of the pre-detection filter can be written as
S
0
(t) =(µ
1
(t) + jµ
2
(t))a(t)exp[ jφ(t)]
+ n
1
(t) + jn
2
(t) (4)
where a(t) and φ(t) are the IF filtered carrier ampli-
tude and information phase, respectively, while n
1
(t)
and n
2
(t) stand for the quadrature components of the
IF filtered complex AWGN w(t). The processes n
1
(t)
and n
2
(t) have the same variance σ
2
n
and a commun
autocorrelationfunction (ACF) Γ
n
(τ). The output sig-
nal of the LDI detector is the phase difference, over
the bit time interval [t − T,t], of the IF filtered FSK
signal. This phase difference is given by (Pawula,
1981)
∆ψ = ∆φ+ ∆Ω+ 2πN(t− T,t) (5)
where ∆φ = φ(t) − φ(t − T) corresponds to the data
phase difference, ∆Ω = Ω(t) − Ω(t − T) is the phase
difference introduced by both the fading channel and
the additive noise, and N(t − T,t) stands for the num-
ber of FM clicks occurring in the time interval [t −
T,t]. From this and based on (Pawula, 1981), the
probability of making an error, when a “+1” sym-
bol is sent, is obtained by computing the quantity
Prob(∆ψ ≤ 0), where Prob(·) stands for probability,
according to
Prob(∆ψ ≤ 0) = Prob(∆Ω > ∆φ) + N (6)
where N is the average number of positive clicks oc-
curring in the time interval [t − T,t]. This quantity
was shown in (Hajri and Youssef, 2007) to be given
by
N =
1
2πqγ
t
Z
t−T
˙
φ(τ)
r

a
2
(τ) +
1
q
2
γ

a
2
(τ) +
1
γ

dτ (7)
where γ = σ
2
1
/σ
2
n
and q is the Hoyt fading parame-
ter defined as q = σ
2
/σ
1
(Nakagami, 1960). In (7)
also,
˙
φ(τ) defines differentiation of φ(τ) with respect
to τ. For the determination of Prob(∆ψ ≤ 0) accord-
ing to (6), we need also to determine the probability
Prob(∆Ω > ∆φ). This quantity can be obtained from
the knowledge of the PDF p(∆Ω) of the phase dif-
ference ∆Ω. The derivation of an expression for this
PDF will be the subject of the next section.
3 DERIVATION OF THE PDF
p(∆Ω)
To start with the derivation of the PDF of the phase
difference ∆Ω, over a bit duration T, we consider the
FSK complex baseband signals z
1
and z
2
at the two
time instants (t − T) and t, respectively, according to
z
1
= (a
1
µ
11
+ ζ
1
) + j(a
1
µ
12
+ ξ
1
) (8)
and
z
2
= (a
2
µ
21
cos(∆φ) − a
2
µ
22
sin(∆φ) + ζ
2
)
+ j(a
2
µ
21
sin(∆φ) + a
2
µ
22
cos(∆φ) + ξ
2
) (9)
where we have assumed a coordinate system that ro-
tates with an angle φ
1
, i.e., the modulated phasor
WINSYS 2008 - International Conference on Wireless Information Networks and Systems
178

at time (t − T) is taken as a reference. In (8) and
(9), µ
1i
= µ
i
(t − T) (i = 1,2), µ
2i
= µ
i
(t) (i = 1, 2),
a
1
= a(t − T), and a
2
= a(t). The noise compo-
nents ζ
i
and ξ
i
(i = 1,2) are defined relative to the
modulated phasor at time (t − T) and are related to
n
1i
= n
i
(t − T), and n
2i
= n
i
(t) (i = 1,2) according to
ζ
i
= n
i1
cos(φ
1
) + n
i2
sin(φ
1
),(i = 1,2), (10)
ξ
i
= n
i2
cos(φ
1
) − n
i1
sin(φ
1
),(i = 1,2). (11)
The components ζ
i
and ξ
i
(i = 1,2) can be shown
to be zero-mean independent Gaussian random vari-
ables having the variance σ
2
n
and the ACF Γ
n
(τ).
For the sake of convenience, we denote a
1
µ
11
+ ζ
1
,
a
1
µ
12
+ ξ
1
, a
2
µ
21
cos(∆φ) − a
2
µ
22
sin(∆φ) + ζ
2
, and
a
2
µ
21
sin(∆φ) + a
2
µ
22
cos(∆φ) + ξ
2
, respectively, by
x
1
, y
1
, x
2
, and y
2
. Clearly, for a given information
symbol, x
i
and y
i
(i = 1,2) are Gaussian distributed.
Since we assume a symmetrical Doppler PSD for the
Hoyt fading, uncorrelated noise samples, and that the
signal component is independent of the noise compo-
nent, it can easily be shown that the correlation and
cross-correlation quantities of the variables x
1
, y
1
, x
2
,
and y
2
, are given by
c = E (x
1
y
2
) = a
1
a
2
Γ
1
(T)sin(∆φ),
d = E (x
2
y
1
) = −a
1
a
2
Γ
2
(T)sin(∆φ),
e = E (x
2
y
2
) = a
2
2
σ
2
1

1− q
2

cos(∆φ)sin(∆φ),
g = E (x
1
x
2
) = a
1
a
2
Γ
1
(T)cos(∆φ) + Γ
n
(T),
l = E (y
1
y
2
) = a
1
a
2
Γ
2
(T)cos(∆φ) + Γ
n
(T), (12)
where E(·) is the expected value operator, and Γ
i
(T)
(i = 1,2) is the ACF of the process µ
i
(t) (i = 1,2).
Concerning the mean power of the underlying Gaus-
sian processes, it is found to be given by
σ
2
x
1
=E

x
2
1

= σ
2
n

a
2
1
γ+ 1

,
σ
2
y
1
=E

y
2
1

= σ
2
n

a
2
1
q
2
γ+ 1

,
σ
2
x
2
=E

x
2
2

= σ
2
n

a
2
2
γ

cos
2
(∆φ) + q
2
sin
2
(∆φ)

+ 1

,
σ
2
y
2
=E

y
2
2

= σ
2
n

a
2
2
γ

sin
2
(∆φ) + q
2
cos
2
(∆φ)

+ 1

.
(13)
Finally, we introduce the quantities R
i
=
q
x
2
i
+ y
2
i
and ϕ
i
= tan
−1
(y
i
/x
i
) (i = 1,2), to denote, respec-
tively, the overall envelope and phase of the Hoyt
faded FSK signal contaminated by additive noise.
Then, this transformation of random variables results
after some lengthy algebraic manipulations (Rice,
1945) in the following expression for the joint PDF
p(ϕ
1
,ϕ
2
) of the random phases ϕ
1
and ϕ
2
, as
p(ϕ
1
,ϕ
2
) =
A
1
4A
2
2
1
(EF − D
2
)
× 1+
D
(EF − D
2
)
1/2
×
"
π
2
+ tan
−1
D
(EF − D
2
)
1/2
!#!
,
(14)
where A
1
and A
2
are expressed in the Appendix, and
the quantities D, E, and F are given by
D =cosϕ
1
{χ
13
cosϕ
2
+ χ
14
sinϕ
2
}
+ sinϕ
1
{χ
23
cosϕ
2
+ χ
24
sinϕ
2
},
E =χ
11
cos
2
ϕ
1
+ χ
22
sin
2
ϕ
1
+ 2χ
12
cosϕ
1
sinϕ
1
,
F =χ
33
cos
2
ϕ
2
+ χ
44
sin
2
ϕ
2
+ 2χ
34
cosϕ
2
sinϕ
2
.
(15)
In (15), the quantities χ
11
, χ
22
, χ
12
, χ
33
, χ
44
, χ
34
, χ
13
,
χ
14
, χ
23
, and χ
24
are expressed in terms of the statis-
tical parameters described by (12) and (13), and are
given in the Appendix. To the best of our knowledge,
(14) is new and it constitutes the basis for the deter-
mination of the PDF p(∆Ω). In fact, using (14) and
noting that ϕ
2
− ϕ
1
= ∆φ + ∆Ω, allows us to obtain
the PDF p(∆Ω) of the phase difference ∆Ω according
to
p(∆Ω) =
π
Z
−π
p(ϕ
1
, ϕ
1
+ ∆φ+ ∆Ω)dϕ
1
. (16)
Unfortunately, the finite range integral involved in
(16) is difficult to handle and it can be evaluated only
numerically. For the special case given by q = 1, i.e.,
the Rayleigh channel case, the underlying integral is
easily solved and it is found that (16) is in agreement
with the result of (Ng et al, 1994). For further verifica-
tion of the validity of (16), we have also compared it
with correspondingsimulation results obtained for the
case of a M2M Hoyt fading channel. For this channel
type, the ACF Γ
i
(τ) (i = 1, 2) is given by (Akki and
Haber, 1986)
Γ
i
(τ) = σ
2
i
J
0
(2πf
T,max
τ)J
0
(2πf
R,max
τ) (17)
where J
0
(·) is the zeroth-order Bessel function of the
first kind (Gradshteyn and Ryzhik, 1994), and f
T,max
and f
R,max
are the maximum Doppler frequenciesgen-
erated by the motion of the transmitter and the re-
ceiver, respectively. Figure 1 shows the theoretical
PDF p(∆Ω) of the phase difference ∆Ω, along with
the corresponding simulation data for two values of
the Hoyt fading parameter q. It can be observed from
this figure that there exists a reasonable mutual agree-
ment between the theoretical and the simulation re-
sults.
PERFORMANCE ANALYSIS OF FSK MODULATION WITH LIMITER-DISCRIMINATOR-INTEGRATOR
DETECTION OVER HOYT FADING CHANNELS
179

x12 x13 x14 x15 x16
v12
v13
v14
v15
v16
v17
v18
s01
s02
s06
s07
s21
s22
s23
s24
s29
s31
s32 s33
s34
s35
Figure 1: Theoretical and simulated PDF p(∆Ω) for two
values of the Hoyt fading parameter q.
4 BIT ERROR PROBABILITY
According to (Pawula, 1981; Tjhung et al, 1990), the
BEP for the FSK system under consideration is given
by
P
e
= Prob(∆ψ ≤ 0) = P
e,1
+ P
e,2
(18)
where, for the case of absence of ISI, P
e,1
=
Prob(∆Ω > ∆φ), which can be computed from (16)
according to
Prob(∆Ω > ∆φ) =
π
Z
∆φ
p(∆Ω)d∆Ω. (19)
Also in (19), P
e,2
= N, which is given directly by (7).
Now, to take into consideration the effect of the ISI
caused by the bandwidth limitation of the IF filter, we
follow (Pawula, 1981) by assuming a time-bandwidth
product BT ≥ 1. In this case, when a “+1” symbol is
sent, only the three bit patterns given by “111”, “010”
and “011” are considered in the ISI evaluation. Then,
by considering these bit patterns, the quantities P
e,1
and P
e,2
can be obtained according to the details re-
ported in (Pawula, 1981; Hajri and Youssef, 2007).
5 NUMERICAL EXAMPLES
In this section, computed numerical results for the
closed-form BEP formula given by (18), taking into
account the ISI effects, are presented for the Doppler
PSD of a M2M Hoyt fading channel. For this channel
type, the ACF is given by (17). Concerning the ad-
ditive Gaussian noise, we assume uncorrelated noise
samples, i.e., Γ
n
(T) = 0. The E
b
/N
0
parameter, ver-
sus which the BEP is plotted, is related to the param-
x12 x13 x14 x15 x16 x17 x18 x19 x20
v12
v13
v14
v15
v16
v17
s01
s02
s06
s07
s49
s50
s51
s56
Figure 2: BEP for various values of the Hoyt fading param-
eter q.
eters q and γ according to
E
b
/N
0
=

1+ q
2

BT
2
γ (20)
where E
b
stands for the average received signal en-
ergy per bit at the input of the IF filter. Figure 2 shows
the effect of the Hoyt fading parameter q on the BEP,
when BT = 1.0, f
T,max
= f
R,max
= 40 Hz, T = 10
−4
s, and the modulation index m = 0.7. As expected,
these results indicate that the BEP P
e
improves as q
increases. The best performanceis obtained for q = 1,
i.e., the case of Rayleigh fading channel.
6 CONCLUSIONS
In this paper, the BEP performance for LDI receivers
of narrow-band digital FSK modulation has been an-
alyzed considering Hoyt mobile radio fading chan-
nels. Specifically, a closed-form expression for the
PDF of the phase difference, over a symbol period,
between Hoyt faded FSK signals perturbed by addi-
tive Gaussian noise, has been derived. The validity of
the derived PDF has been demonstrated based on its
comparison against corresponding simulation results
obtained for the case of Hoyt M2M fading channels.
This newly derived PDF is then applied for the deter-
mination of the desired BEP performance. Numerical
results of the BEP have been presented for several val-
ues of the FM system parameters and the Hoyt M2M
fading channel characteristics.
WINSYS 2008 - International Conference on Wireless Information Networks and Systems
180

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APPENDIX
In this Appendix, we give the statistical parameters
A
1
, A
2
, χ
11
, χ
22
, χ
12
, χ
33
, χ
44
, χ
34
, χ
13
, χ
14
, χ
23
, and
χ
24
, used in the description of (14) and (15). These
quantities are expressed as
A
1
=
1
4π
2
σ
x
1
σ
y
1
σ
x
2
σ
y
2
K
1/2
, A
2
=
1
2K
, with
K =1−
"
g
2
σ
2
x
1
σ
2
x
2
+
l
2
σ
2
y
1
σ
2
y
2
+
c
2
σ
2
x
1
σ
2
y
2
+
d
2
σ
2
y
1
σ
2
x
2
+
e
2
σ
2
x
2
σ
2
y
2
−
(gl − cd)
2
σ
2
x
1
σ
2
y
1
σ
2
x
2
σ
2
y
2
− 2
e
σ
2
x
2
σ
2
y
2
×
ld
σ
2
y
1
+
gc
σ
2
x
1
!#
.
χ
11
=
1
σ
2
x
1
"
1−
l
2
σ
2
y
1
σ
2
y
2
+
d
2
σ
2
y
1
σ
2
x
2
+
e
2
σ
2
x
2
σ
2
y
2
− 2
lde
σ
2
y
1
σ
2
x
2
σ
2
y
2
!#
,
χ
22
=
1
σ
2
y
1
"
1−
g
2
σ
2
x
1
σ
2
x
2
+
c
2
σ
2
x
1
σ
2
y
2
+
e
2
σ
2
x
2
σ
2
y
2
− 2
gce
σ
2
x
1
σ
2
x
2
σ
2
y
2
!#
,
χ
12
=
1
σ
2
x
1
σ
2
y
1
"
lc
σ
2
y
2
+
gd
σ
2
x
2
−
1
σ
2
x
2
σ
2
y
2
(cde+ gle)
#
,
χ
33
=
1
σ
2
x
2
"
1−
c
2
σ
2
x
1
σ
2
y
2
+
l
2
σ
2
y
1
σ
2
y
2
!#
,
χ
44
=
1
σ
2
y
2
"
1−
g
2
σ
2
x
1
σ
2
x
2
+
d
2
σ
2
x
2
σ
2
y
1
!#
,
χ
34
=
1
σ
2
x
2
σ
2
y
2
"
ld
σ
2
y
1
+
cg
σ
2
x
1
− e
#
,
χ
13
=
1
σ
2
x
1
σ
2
x
2
"
g−
ce
σ
2
y
2
+
1
σ
2
y
1
σ
2
y
2

lcd − gl
2

#
,
χ
14
=
1
σ
2
x
1
σ
2
y
2
"
c+ −
ge
σ
2
x
2
+
1
σ
2
x
2
σ
2
y
1

gld − cd
2

#
,
χ
23
=
1
σ
2
y
1
σ
2
x
2
"
d +
1
σ
2
x
1
σ
2
y
2

cgl − dc
2

−
le
σ
2
y
2
#
,
χ
24
=
1
σ
2
y
1
σ
2
y
2
"
l +
1
σ
2
x
1
σ
2
x
2

gcd − lg
2

−
de
σ
2
x
2
#
. (A-1)
PERFORMANCE ANALYSIS OF FSK MODULATION WITH LIMITER-DISCRIMINATOR-INTEGRATOR
DETECTION OVER HOYT FADING CHANNELS
181