The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

BIT ERROR PROBABILITY OF NARROW-BAND DIGITAL FM WITHLIMITER-DISCRIMINATOR-INTEGRATOR DETECTION IN HOYT MOBILE

RADIO FADING CHANNELS

Nazih Hajri and Neji YoussefEcole Superieure des Communications de Tunis, 2083 EL Ghazala, Ariana, Tunisia

ABSTRACT

Performance analysis of narrow-band digital frequency shiftkeying (FSK) modulation with limiter-discriminator-integrator(LDI) detection over Hoyt mobile fading channels is presented.Closed form expressions are derived for the probability densityfunction (PDF) of the phase angle between two Hoyt fadingvectors, for the phase noise due to additive Gaussian noise, aswell as for the average number of FM clicks occurring at theoutput of a digital FM receiver. Then, based on these investi-gated quantities, a formula for the bit error probability (BER)of FSK transmission over Hoyt channels is deduced. Since theRayleigh fading model is a special case of the Hoyt model, thederived quantities are verified to reduce to known results cor-responding to Rayleigh fading. A numerical example is givento illustrate the analysis and study the influence of the fadingseverity as well as the FM system parameters on the error per-formance.

I. INTRODUCTION

The investigation of error performance of FSK modulation withLDI detection has been widely studied for the case of Rayleighand Rice fading channels [1–4]. In [1, 2], closed form expres-sions for the bit error rate have been reported for a transmis-sion over Rayleigh fading and the validity of the results hasbeen verified by using experimental data. For the Rice fad-ing model, the error analysis of M-FSK transmission has beenstudied in [3, 4].

Recently, and besides these two frequently used channels,the Hoyt fading model [5] is being considered in several stud-ies related to mobile radio channel modeling and performanceanalysis of wireless transmission. For example, in [6] themodel was found to describe accurately mobile satellite chan-nels for heavy shadowing environments. Performance analysisfor digital transmission over Hoyt fading, using linear modu-lations, is reported in [7]. For FSK modulation with the clas-sical limiter-discriminator detection, a closed form expressionfor the error rate is derived in [8], where a rectangular shapefor the predetection filter was used and no integrate and dumpfilter was assumed (sampling discriminator).

This paper presents the derivation of the BER of a LDI baseddigital FM receiver for a transmission over Hoyt fading chan-nels. In the derivation, both the Doppler effect as well as theintersymbol interference (ISI), caused by the bandwidth limi-tation of the intermediate frequency (IF) filter, are taken intoaccount. The Jakes model is assumed for the Doppler powerspectral density, while the IF filter is selected to be of a Gaus-sian shape. The methodology, we employ, is based on the re-

Binary Source

)(tb FM

Modulation + IF Filter

( )H f )(0 te

)(te

Discriminator Circuit

Limiter

)(t•

ψ

Integrate and Dump « T sec »

)( 0tψ∆

« 0 » Threshold )(tb

∧

)(tS r ( )S t

( )w t

Figure 1: Digital FM System Model.

sults reported in [1, 9]. A numerical example is given to illus-trate the obtained theoretical BER for various values of the FMsystem parameters and the fading channel characteristics.

This paper is organized as follows. Section II is devoted tothe description of the FM system model. In Section III, wederive the PDF of the phase difference between two Hoyt vec-tors, the PDF of the phase noise due to additive noise, and theaverage number of FM clicks. The desired error rate formulais reported in Section IV. Section V contains the illustrations.Finally, the conclusion is drawn in Section VI.

II. SYSTEM MODEL

A block diagram corresponding to the digital FM system, withLDI detection, is shown in Fig. 1. The received signal, Sr(t),after a transmission over the Hoyt fading channel, can be ex-pressed as

Sr(t) =R(t) cos (θ(t) + ϑ(t))

=õ2

1(t) + µ22(t) cos

(θ(t) + tan−1

(µ2(t)µ1(t)

))(1)

where µ1(t) and µ2(t) are uncorrelated zero-mean Gaussianprocesses with variances σ2

1 and σ22 , respectively. Also, ϑ(t) =

tan−1 (µ2(t)/µ1(t)) is the Hoyt channel phase, while R(t) =√µ2

1(t) + µ22(t) is a Hoyt process, the PDF of which is given

by [6]

p(R) =R

σ1σ2exp(−R

2

4

(1σ2

1

+1σ2

2

))

× I0

(R2

4

(1σ2

2

− 1σ2

1

)), R ≥ 0 (2)

where I0(·) is the modified Bessel function of the first kind[10]. Furthermore, θ(t) is the data phase after FM modulation,given by

θ(t) =πh

T

t∫−∞

b(τ)dτ (3)

1-4244-1144-0/07/$25.00 c©2007 IEEE

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

where b(t) is the binary data sequence of bit rate 1/T and h =2∆fdT , in which ∆fd is the peak frequency deviation. The IFband-pass filter is considered to be of a Gaussian shape with anequivalent low-pass transfer function given by

H(f) = exp[−πf2

/2B2]

(4)

where B stands for the equivalent noise bandwidth. Now, byassuming a slow fading, the IF filter output signal, e0(t), canbe written as [1]

e0(t) = R0a(t) cos (φ(t) + ϑ(t)) + n(t) (5)

where a(t) and φ(t) are the filtered carrier amplitude and signalphase, respectively, n(t) is the additive Gaussian noise of aver-age power σ2

n = N0B, and in which N0 is the correspondingone sided power spectral density. Also, R0 is a Hoyt randomvariable, for which the variance of µi(t) (i = 1, 2) is reducedto σ2

i0 due to the IF filtering. When the Jakes model is assumedfor the Doppler power spectral density [11], σ2

i0(i = 1, 2) isfound to be given by [1]

σ2i0 =

σ2i

πKi, (6)

where

Ki =[(exp (−ci) + 1)

π

2+

2πci

exp (−ci/2)

×I0

(ci2

)− cosh

(ci2

)], (7)

in which ci = π(fmaxi)2/B2, and where fmaxi denotes

the maximum Doppler frequency corresponding to the pro-cess µi(t) (i = 1, 2). Here, it is worth mentioning that, inthe analysis, we assume different Doppler frequencies, fmax1

and fmax2, for the processes µ1(t) and µ2(t), respectively. Al-though, this assumption lacks a clear physical basis, it allows toincrease the flexibility of the Hoyt fading model and enables abetter fitting of measurement data [6]. Now, the integrate-and-dump filter with an integration time T gives, at a sampling timet0, the following expression for the overall phase difference

∆ψ = ∆φ+ ∆η + ∆ϑ+ 2πN(t0 − T, t0) (8)

where ∆φ is the data phase component, ∆η is the continuousphase noise due to additive Gaussian noise, ∆ϑ is the phasedifference introduced by the Hoyt fading channel, and 2πNis the click noise component generated in the time interval[t0 − T, t0]. Therefore, to evaluate the performance of the LDIdetection when digital FM signals are transmitted over Hoytfading channels, we need to calculate the PDF of the phase dif-ference ∆ψ. To obtain an expression for this PDF, the PDF’s of∆η, ∆ϑ and 2πN are required. The derivation of these quanti-ties is the subject of the next section.

III. DERIVATION OF THE PROBABILITIES DENSITIES

FUNCTIONS OF ∆ϑ, ∆η, AND N

By making the assumption that the random variables ∆η, ∆ϑand N are statistically independent, then the bit-error probabil-ity, Pe, can be expressed according to [1, 9]

Pe = P (Ω > ∆φ) +N (9)

where N is the average number of clicks, Ω = ∆η + ∆ϑ, andP (Ω > ∆φ) is the probability that Ω exceeds some angle ∆φ.This probability can be written as [1]

P (Ω > ∆φ) =

π∫∆φ−π

d∆ϑ

π∫∆φ−∆ϑ

p(∆ϑ)p(∆η)d∆η (10)

where p(∆ϑ) and p(∆η) are the PDF’s of ∆ϑ and ∆η, respec-tively.

A. Derivation of p(∆ϑ)The starting point for the determination of p(∆ϑ) is the jointPDF of the Gaussian processes µ11 = µ1(t), µ21 = µ2(t),µ12 = µ1(t + τ), and µ22 = µ2(t + τ), considered at theoutput of the IF filter. This joint PDF can easily be shown to beobtained according to [12]

p(µ11, µ21, µ12, µ22) = p(µ11)p(µ12/µ11) ·p(µ21)p(µ22/µ21)(11)

where p(µ12/µ11), and p(µ22/µ21) denote the conditionnelPDF’s of the processes µ12 and µ22 given µ11 and µ21, re-spectively. Using [12, eq. (2)], allows us to get the followingexpression for p(µ11, µ21, µ12, µ22)

p(µ11, µ21, µ12, µ22) =A1 exp[−A2

σ2

20κµ211 + σ2

10λµ221

+ σ220κµ

212 + σ2

10λµ222 − 2 (12)

× (σ220κγτ1µ11µ12 + σ2

10λγτ2µ21µ22

)]where γτi

(i = 1, 2) is the normalized autocorrelation functionof the process µi(t), and the quantities λ, κ, A1, and A2 aregiven by

λ = 1 − γ2τ1, κ = 1 − γ2

τ2,

A1 = 1/(

4π2σ210σ

220

√λκ), A2 = 1

/(2σ2

10σ220λκ

).

(13)Now, by making use of the variable transformation given by

µ11 = R1 cos θ1µ21 = R1 sin θ1

,

µ12 = R2 cos θ2µ22 = R2 sin θ2

, (14)

we obtain the following expression for the joint PDF,p(R1, R2, θ1, θ2), of the variables R1, R2, θ1, and θ2

p(R1, R2, θ1, θ2) =A1R1R2 exp[−A2

R2

1

(σ2

20κ cos2 θ1+ σ2

10λ sin2 θ1)

+ R22

(σ2

20κ cos2 θ2 + σ210λ sin2 θ2

)− 2R1R2

(σ2

20κγτ1 cos θ1 cos θ2+ σ2

10λγτ2 sin θ1 sin θ2)]

(15)

where 0 ≤ R1, R2 <∞, and −π ≤ θ1, θ2 < π. Therefore, thejoint PDF, p(θ1, θ2), of the phases θ1 and θ2 can be computedaccording to

p(θ1, θ2) =

∞∫0

∞∫0

p(R1, R2, θ1, θ2)dR1dR2. (16)

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

The integrals involved in (16) can be solved analytically bymaking use of the multiple integrals evaluation approach re-ported in [13]. Based on that approach, we start by consideringthe integral I given by

I =

∞∫0

∞∫0

exp[−A2σ

210ER

21 −A2σ

210FR

22

+2A2σ210DR1R2

]dR1dR2 (17)

where the quantities, E, F , and D are expressed by

E = β0κ cos2 θ1 + λ sin2 θ1,F = β0κ cos2 θ2 + λ sin2 θ2,D = β0κγτ1 cos θ1 cos θ2 + λγτ2 sin θ1 sin θ2,

(18)

and in which,

β0 =σ2

20

σ210

= β · K2

K1. (19)

In (19), β stands for the fading parameter and is given byβ = σ2

2

/σ2

1 . To solve the integrals in (17), we first let

x1 = R1

(A2σ

210E)1/2

, and x2 = R2

(A2σ

210F)1/2

, and thenwe make the following linear change of variables

x1 = y1 + g(1 − g2

)−1/2y2

x2 =(1 − g2

)−1/2y2

(20)

where g = D(EF )1/2 . The application of these transformations

results in the following expression for the integral I

I =

(1 − g2

)−1/2

A2σ210 (EF )1/2

∞∫0

dy2

∞∫−g(1−g2)−1/2y2

exp[−y2

1 − y22

]dy1.

(21)Now, by making use of the transformation of the cartesian coor-dinates (y1, y2) to polar coordinates, and performing the neces-sary algebraic manipulations, the integral I is found to be givenby

I =1

2A2σ210 (EF )1/2

1

(1 − g2)1/2cot−1

(− g

(1 − g2)1/2

).

(22)Finally, differentiating (22), with respect to the variable g, re-sults in the following expression for p(θ1, θ2)

p(θ1, θ2) =β (λκ)3/2

4π2

1(EF −D2)

×(

1 +D

(EF −D2)1/2

×[π

2+ tan−1

(D

(EF −D2)1/2

)]). (23)

Next, by letting θ1 = θ, and θ2 = θ + ∆ϑ, the desired PDFof the phase difference ∆ϑ, p(∆ϑ), can be obtained from (23)according to

p(∆ϑ) =

π∫−π

p(θ, θ + ∆ϑ)dθ. (24)

0 0

0.5

1

1.5

2

2.5

∆ ϑ

p(

∆

ϑ)

γτ1

=0.95

γτ1

=0.6

γτ1

=0

β0=0.2

−π −π/2 π/2 π

Figure 2: The PDF p(∆ϑ) for various values of γτ1 .

Unfortunately, the integral involved in (24) can be evaluatedonly using numerical techniques. It should be mentioned thatfor β0 = 1, and fmax1 = fmax2 , i.e., the case of Rayleighfading, the calculation of the above integral using (13), (18),(23), and [10, eq. (1.624(2))] yields [1, eq. (A4)]. The behaviorof p(∆ϑ), for β0 = 0.2, fmax1 = fmax2 , and various values ofγτ1 , can be studied from Fig. 2.

B. Derivation of p(∆η)

For the derivation of p(∆η), we use the result given by [1, 9]

p(∆η/R0) =exp (−U)

2π

coshV +

12

π∫0

dα [(U sinα

+√U2 − V 2 cos ∆η

)· cosh (V cosα) (25)

× exp(√

U2 − V 2 sinα cos ∆η)]

where

U = 12 [ρ(t0) + ρ(t0 − T )] = R2

02σ2

nC1,

V = 12 [ρ(t0) − ρ(t0 − T )] = R2

02σ2

nC2.

(26)

Here, the parameters C1 and C2 are functions of the specificbit patterns to be considered in the evaluation of the BER, andtheir expressions will be given in the next section. Also, ρ(t) isthe time varying signal-to-noise ratio (SNR) defined by

ρ(t) =R2

0

2a2(t)σ2

n

=Eb

N0

1BT

R20a

2(t)(β + 1)σ2

1

(27)

where Eb = (β+1)2 σ2

1T is the average received signal energyper bit at the input of the IF filter. Then, the desired quantityfor p(∆η) can be obtained by averaging p(∆η/R0) over theHoyt random variable R0 according to

p(∆η) =

∞∫0

p(∆η/R0)p(R0)dR0. (28)

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

Using [10, eq. (6.611(1)), eq. (6.623(2)), eq. (8.406(3))],p(∆η) can be written as

p(∆η) = d

(1√

[(C1−C2)+e]2−f2+ 1√

[(C1+C2)+e]2−f2

)

+d2

π∫0

dα (C1 sinα+ C3 cos ∆η)

×[

e−(C3 sin α cos ∆η+C2 cos α−C1)

[e−(C3 sin α cos ∆η+C2 cos α−C1)]2−f23/2

+ e−(C3 sin α cos ∆η−C2 cos α−C1)

[e−(C3 sin α cos ∆η−C2 cos α−C1)]2−f23/2

](29)

where

C3 =√C2

1 − C22 , d = 1+β0

8π√

β0r, e = (1+β0)

2

4β0r,

g = 1−β20

4β0r, r = 1

/((1+β0)σ

210

2σ2n

).

(30)

The integral in (29) is difficult to handle, and it can be evaluatedonly numerically.

C. Derivation of N

Based on the assumption that the FM clicks can be describedstatistically by a Poisson distributed discrete random variable[14], the average number of clicks, N , occurring during thetime interval [t0 − T, t0], corresponding to the bit duration, T ,can be derived according to the following relation [1, 9]

N =12π

∞∫0

t0∫t0−T

φ(τ) exp(−R

20

2a2(τ)σ2

n

)p(R0)dτdR0.

(31)After some algebraic manipulations, we obtain the followingquantity for desired click rate N

N = 2d

t0∫t0−T

φ(τ)√[(a2(τ) + e)2 − g2

]dτ. (32)

Again, for the case where β = 1, and fmax1 = fmax2 , (32)simplifies to the result given in [1, eq. (27)], as expected.To summarize, the quantity given by (9) can be evaluated using(10), (24), (29), and (32).

IV. BIT ERROR RATE PROBABILITY

According to [1, 9], for BT ≥ 1, the distortion caused by theISI extends only to one bit either side of the bit under detection.Thus, only the three bits patterns of “010”, “111” and “011” in-tervened in the calculation of the average BER. For these threepatterns, the BER, Pe, can be approximated by [1]

Pe∼= Pe,1 + Pe,2 (33)

where

Pe,1 =14

[P (Ω > ∆φ |111) + P (Ω > ∆φ |010)

+ 2P (Ω > ∆φ |011)] , (34)

and

Pe,2 =14[N |111 +N

∣∣010 + 2N |011]. (35)

From Section III, we can verify that the quantities Pe,1 and Pe,2

depend on φ(t), ∆φ, a2(t), C1 and C2. The evaluation of thesequantities must be performed for the three bits patterns givenby “111”, “010”, and “011”. By considering that the samplingtime is t0 = 0, and according to [1], we deduce the followingquantities needed for the calculation of the average BER

• For “111” bit pattern

∆φ = πh, C1 = |H(h/2T )|2 , C2 = 0,N = 2πd · h√[

(|H(h/2T )|2+e)2−g2] .

• For “010” bit pattern

∆φ = 2 tan−1 m1−n ,

m = 2 |H(1/2T )| h2

1−h2 cot(πh/2),n = 2 |H(1/T )| h2

4−h2 ,

C1 =[

sin πh2

πh2

]2 [(1 − n)2 +m2

], C2 = 0,

φ(t) = πT

[(n cos 2πt

T −1)m sin πtT −2mn cos πt

T sin 2πtT

(1−n cos 2πtT )2

+m2 cos2 πtT

],

a2(t) =[

sin πh2

πh2

]2·[(

1 − n cos 2πtT

)2 +m2 cos2 πtT

].

• For “011” bit pattern

∆φ = 12 [∆φ| 111 + ∆φ| 010] ,

N = 12

[N∣∣ 111 + N

∣∣ 010].

V. NUMERICAL RESULTS AND DISCUSSION

In this section, we present computed numerical results for N ,and the average BER as a function of Eb/N0 which is relatedto the parameter r, given in (30), according to

Eb/N0 =π (1 + β)K1 (1 + β0)

BT

r. (36)

Fig. 3 shows the average number of FM clicks, N , as a func-tion of Eb/N0, for fmax1 = fmax2 , the bit pattern “010”, andseveral values of the maximum Doppler frequency fmax1 . Asexpected, N decreases as Eb/N0 increases, while it increaseswith increasing values of fmax1 . For the computation of theBER, we need to evaluate γ

τi(i = 1, 2), at τ = T , i.e.,

γTi

=1σ2

i0

fmaxi∫−fmaxi

σ2i |H(f)|2

π√f2maxi

− f2exp (j2πfT ) df. (37)

Using (6), (7), and [1, eq. (A8)], (37) can be computed numer-ically according to

γT i

=2Ki

π/2∫0

exp(−pi sin2 χ

)cos (qi sinχ)

dχ

, (38)

where pi = π (fmaxiT )2/

(BT )2

, and qi = 2πfmaxiT .

In Fig. 4, the average BER is plotted versus Eb/N0 for

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

8 10 12 14 16 18 20 22 24

10−5

10−4

10−3

10−2

Eb/N

0(dB)

Average number of FM clicks

fmax

1

=20 Hz

fmax

1

=20 Hz

fmax

1

=60 Hz

fmax

1

=110 Hz

β=0.2

β=1h=0.7T=2⋅10−3

Figure 3: The average number of FM clicks N .

6 8 10 12 14 16 18

10−2

10−1

100

Eb/N

0 (dB)

Pe

β=0.2β=0.5β=1.0

h=0.7BT=1.0

fmax

1

T=4⋅10−3

Figure 4: The influence of β on the BER.

fmax1 = fmax2 , h = 0.7, BT = 1, fmax1T = 0.004, and sev-eral values of the fading parameter β. As can be seen, the errorperformance improves if β increases. The best performance isobtained when β = 1, i.e., the case of Rayleigh fading model.Fig. 5 represents the average BER, Pe, as a function of Eb/N0

for fmax1 = fmax2 , β = 0.2, h = 0.7, fmax1T = 0.004 andBT = 1, 2, and 3. It can be noted from this figure that theperformance improves with decreasing the values of BT .

VI. CONCLUSION

In this paper, capitalizing on the known theory of digital FMtransmission over Rayleigh fading channels, performance anal-ysis for narrow band FM systems with LDI detection has beenconsidered in the case of Hoyt fading models. The PDF of thephase difference between two Hoyt vectors, and that inducedby the additive noise, as well as the average number of FMclicks have been first derived. Based on these quantities, anexpression for the corresponding BER has been deduced. Nu-merical results have been given, and by which the effects of thefading severity as well as the system parameters on the averageBER have been examined.

10 15 20 25 30 35 40 45 50 5510

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

Pe

BT=1.0

BT=2.0

BT=3.0

β=0.2h=0.7

fmax

1

T=4⋅10−3

Figure 5: BER for various values of BT .

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