ERROR PERFORMANCE OF NARROW-BAND DIGITAL FM WITHLIMITER-DISCRIMINATOR DETECTION UNDER FAST NAKAGAMI-Q

FADING AND CO-CHANNEL INTERFERENCE

Mohamed ZHAOUNIA and Neji YOUSSEF

Ecole Superieure des Communications de Tunis, Cite EI Ghazala, 2091 Ariana, Tunis, Tunisia.mzhaounia@yahoo.fr; nej i.youssef@supcom.mu.tn

ABSTRACT

In this paper, we evaluate the performance of narrowband digital FM with limiter-discriminator detectionin Nakagami-q mobile radio fading channels. We de­rive a closed form expression for the Bit Error Proba­bility (BEP) by taking into account the receiver noiseand the co-channel interference. The obtained resultis illustrated for several values of the signal to noiseratio (SNR) and the interference to noise ratio.

Keywords: Nakagami-q channel, FM modula­tion, discriminator, co-channel interference, BEP.

1. INTRODUCTION

In this paper, we consider digital FM transmissionover fast Nakagami-q mobile radio fading channels[1]. This fading model is found to exhibit a good fitto statistical data ofrealistic mobile satellite channels[2]. While the performance of wireless transmissionlinks over the well known Rayleigh and Rice modelshas extensively been studied in the technical literature[3-10], only few works on the performance of radiotransmission over Nakagami-q are available. For ex­ample, performance oflinear modulation is addressedin [3]. Recently, performance analysis of digital FMwith limiter-discriminator, over fast Nakagami-q fad­ing channels, has been studied in [11]. In the presentpaper, we extend the work reported in [11] by consid­ering the co-channel interference (CCI) in the analy­sis. A closed-form expression for the BEP is derived.Numerical examples are presented for the illustrationof the obtained theoretical results.

This paper is organized as follows. Section 2contains the description of the transmission systemmodel. The probability density function (PDF) of the

discriminator output is derived in Section 3. Section4 is devoted to the derivation of the BEP. Illustrationsofnumerical examples are presented in Section 5. Fi­nally, Section 6 concludes the paper.

2. SYSTEM MODEL AND MATHEMATICALDESCRIPTION

The limiter-discriminator based digital FM receiveris depicted in Fig 1.

Signal! ~ Interference

'[ ..~ Limiter HDisaiminator rtSampler riNoise

Fig. 1 Block diagram of a digital FM receiver withlimiter-discriminator detection.

Both the desired and the interference signals areassumed to be affected by fast Nakagami-q multipathfading. The equivalent complex baseband signal r(t)at in the input of the receiver can be expressed as

r(t) == s(t)ejcI>s(t) + i(t)ejcI>](t) + n(t), (1)

where s(t) == Xl (t)+jx2(t) and i(t) == XI(t)+ jYI(t)are mutually independent zero-mean baseband Gaus­sian processes standing for the Nakagami-q channelfading affecting the desired signal and the interfer-ence signal, respectively. The process n(t) == xn(t)+ jYn(t)stands for the additive Gaussian noise of the receiver.The variances of the processes Xl (t), X2 (t), XI(t),

2 1( .2(2 2) .2 ( 2 2) 2 .. ()A2 =="2 <I>s al +a2 + <I>I El +E2 -alPxl 0 +

a~px2(0) - EIPxI(O) - E~PYI(O) - 2a~Pn(0)), (6)

and

where

(5)

(7)

and YI (t) will be denoted by aI, a~, EI, and E§, re­spectively, while a; will represent the common vari­ance of Xn(t) and Yn (t). Also, <I>s and <I>I are the dataand interference phases, respectively, after FM mod­

ulation.The normalized autocorrelation functions ofthe Gaus­sian processes Xl(t), X2(t), XI(t), and YI(t) will bedenoted by pxl(r), px2(r), pxI(r), and pyI(r), re­spectively. As for the processes Xn(t) and Yn (t), theirnormalized autocorrelation function will be denoted

by Pn(T). From the above notations, the limiter-discriminator A,1A,2P = ! (<P~(ai + a~) + <PJ(Ei + E~)).input signal can be expressed as 2

(4)

r(t) == Zl(t) + jZ2(t) == R(t) exp(jO(t)), (2)

where

Zl(t) == Xl(t) cos(<I>s(t)) - X2(t) sin(<I>s(t)) +

x I (t) cos(<PI (t )) - YI (t) sin(<I>I (t )) + Xn(t),

Z2 (t) == Xl (t) sin(<I>s(t)) - X2 (t) cos(<I>s(t)) +XI(t) sin(<I>I(t)) - YI(t) COS(<I>I(t)) + y(n). (3)

Also in (2), R(t) is the signal envelope and O(t) thecorresponding phase process. The limiter-discriminatoroutput can then be written as

'( ) _ zl(t)i2(t) - il(t)Z2(t)o t - z?(t) + z~(t) 1

Using H', we can derive the PDF of the Gaussian

vector a == [Zl Z2 il i2] according to

P(a) = 1 e-~(a-E(a»tH'-l(a-E(a» (8)(211")2 Jf1PT

where IH'I == det(H') and (.)t is the transpose-matrixsymbol. This allows us to obtain

where the over dot stands for time derivative. Thedetermination of the BEP will be based on the proba­bility density function (PDF) of iJ(t). A closed formexpression for this statistical quantity is obtained inthe next section.

3. PDF OF THE LIMITER-DISCRIMINATOROUTPUT SIGNAL

In order to determine the PDF of iJ (t), we first con­sider the covariance matrix H among the Gaussian

random variables Xl, X2, Xl, X2, XI, YI, XI, YI, Xn,Yn, x'n and y"n. Calculating all the coefficients of thismatrix we find that H is diagonal where the diagonal

1 . b 2 2 2·· (0) 2·· (0)va ues are gIven y: aI, a2' -al Pxl , -a2 Px2 ,2 2 2·· (0) 2·' (0) 2 2 2·· (0) dEl' E2' -El PxI , -E2 PyI , an' an' -an Pn an-a~ Pn(O).

By making transformation of the Cartesian coordi­nates to polar coordinates, we can express the jointPDF of the variables R, R, 0 and iJ as

.. R 2

P(R, R, 0, 0) = (211")2~IA~(1_p2)

-1 R2 R2+ R2iJ2 2pR2iJ. exp( 2(1 _ p2) (A,T + A,~ - A,1A,2))' (to)

The desired PDF of iJ(t) is obtained from (10) ac­cording to

P(8) =100

i: i11"11" P(R, il, 0, 8) dO dR dR.

(11)Performing the integration in (11), we obtain the fol­lowing closed-form expression for P(O)

From the above, we can deduce the following covari­

ance matrix of the Gaussian variables Zl, Z2, Zl and

Z2

H'==

Ai 0 0 AlA2Po Ai -AlA2P 0o -AlA2P A~ 0

AlA2P 0 0 A~

4. DERIVATION OF THE BEP

In order to obtain the desired BEP Pe, we assumethat mark and space transmissions in the desired andnon-desired signals are given by M s, Ss, Mi and Si

respectively. Then, the average BEP is expressed asfollows

5. NUMERICAL EXAMPLE

(17)1 A"

Pe == 2(1 - BII)'

We assume the Jakes' model [7] for the Doppler powerspectral density (PSD) of the fading channel and theideal rectangular PSD for the receiver noise. In thiscase, we have the following quantities for 'rJi, 'rJn and7i[11 ]'rJi == (V21ffmaxi)2 ,i == 1,277n = (~)2

7i == (V21f fmaxi)2 ,i == 1,2where fmax i is the maximum Doppler frequency, andB denotes the bandwidth of the pre-detection filter.In the following, fmax 1 will be denoted as fmax.

(13)

1Pe == 2(1 - Pspip(Ms, Mi) +Ps(l - pi)p(Ms, Si) +

(1 - Ps)PiP(Ss, Mi) + (1 - Ps)(1 - pi)p(Ss, Si)) (l<Qhere A" = m'Yh' + v + 2)-1/2, and

where p is determined from (7). B" == (,(m2+2(fmaxT)21t;aG)+v(m2+2(fmaxT)2

1ik2Jl ) + 2B23T2)1/2. Also, m denotes the modula-

For simplicity, we consider the special case where tio~Lindex defined according to I~ I = lI"m. For thePs = Pi = 1/2. By noticing that p(Ms,Si). = -p(Mi, Ss) quasi-stationary Nakagami-q fadin~, i.e., T

and.p(Ms, Mi ). == -p(Si, Ss), we can write the fol- fmaxT -t 0, (17) simplifies tolOWing expression for Pe

Also in (13), Ps and Pi are the probabilities of marktransmission in s(t) and i(t), respectively. The quan­tity p(0/Ms , Mi ) denotes the conditional PDF of 0under a mark transmission in s(t) and in i(t), whilep( iJ j M s , Si) represents the conditional probability of

ounder a mark transmission in s(t) and a space trans- A d· 1 th BEp·. . ccor ing y, e is nowmission in i(t). Concemingp(O / 5s, M i ) andp(O / 5s, Si),they are defined similarly. Now, by evaluating (13)we obtain

Pe == PsPi I1 + Ps(l - Pi)I2 +

(1 - Ps)PiI3 + (1 - Ps)(l - pi)I4'o· .

where II == f_oop(OjMs, M i ) dO,o· . 00· •

12 == f-oo p(0/ Ms, Si) dB, 13 == fa p(Bj 5s, Mi) dO,

and 14 == fooo p(0j 5s, Si) dO.

1 APe ==2(1-

B), (15)

where A == q)s(,l +'2)('1+,2+V1+V2+2)-1/2, and· 2 . 2 . 2

B == ('1 (<I>s + 'rJ1) + ,2 (<I>s + 'rJ2) + VI (<I>I + 71) +

V2(<Ii I2

+ 72) + 2'rJn) 1/2. In these quantities, we have2 2,i == ~, Vi == ~, 'rJi == -pi(O) (i == 1,2), 'rJn ==

(Tn (Tn

-Pn(0), 71 == -PxI(O), and 72 == -PyI(O).

By assuming that'2 == a,l, V2 == j1V1, 'rJ2k1'rJ1, and 72 == k271, (15) becomes

1 A'Pe == 2(1 - BI)' (16)

where A' == Q;s1'(1' + v + 2)-1/2, and

B' = 'Y(~/ + 7711i:~Q) + v(~l + T11i:~lL) +217n)1/2.

We notice that in the case ofabsence ofthe co-channelinterference, i.e., v == 0, (16) reduces to the BEP ex­pression obtained in [11]. In addition, for a == J-L ==

k1 == k2 == 1, which corresponds to the Rayleighcase, (16) simplifies to the result reported in [10].

p. - !(1- m'Yh' + v + 2)-1/2) (18)e- 2 (m2(,+v)+2B23T2)1/2'

and in the absence of Cel, i.e, v -t 0, (17) reducesto

(19)

The BEP is shown in Fig. 2 for the case of MinimumShift Keying (MSK) modulation, i.e., m == 0.5 andslow Nakagami-q fading channel, i.e., fmaxT -t 0,and several values of the interference to noise ratiov. The behavior of Pe for a fast Nakagami-q fadingcan be studied from Fig. 3. It can be noted from thisfigure that Pe exhibits an error floor due to the timeselective behavior of the channel.

lcf

10.1 Y=35

25

10.2

:.-15

10 3 fmu=O

(1,=1,5

Y=OJl=I,5

10.4k}=1.5

k =1,52

105

0 10 15 20 25 30 35 40 45 50

1 endB

Fig. 2 BEP of digital FM with discriminator detec­tion under slow Hoyt fading.

Y=35

[2] N. Youssef, C. Wang, and M. Paetzold, "A Studyon the Second Order Statistics of Nakagami­Hoyt Mobile Fading Channels," IEEE Trans.Veh. Technol." vol. 54, no. 4, pp. 1259-1265, Jul.2005.

[3] M. K. Simon and M. S. Alouini, A un{fied ap­proach to the performance analysis of digitalcommunication over generalized fading chan­nels, John Wiley & Sons, New York, 2000.

[4] C. X. Wang, N. Youssef, and M. Paetzold, "Levelcrossing rate and average duration of fades ofdeterministic simulation models for Nakagami­q fading channels," WPMC'02, Honolulu, Oct.2002, pp. 272-276.

[5] P. D. Shaft, "Error rate of PCM - FM using dis­criminator detection," IEEE Trans. Space Elec­tron. Telem. SET-9, p. 13, Dec. 1963.

[6] J. Klapper, "Demodulator threshold performanceand error rates in angle - modulation digital sig­nals," RCA Review, vol. 27, p. 223.

Y=25

fmax =0.05

10 2(1.=;1,5

Y=15

[7] W. C. Jakes, "Microwave Mobile Communica­tions," Piscataway, NJ: IEEE Press, 2nd edition,1993.

Y=O

10-3 '-------'-_--'--------'--_-'-------'-_-L----L._~_______'__ __J

o 10 15 20 25 30 35 40 45 50

T en dB

Fig. 3 BEP of digital FM with discriminator detec­tion under fast Hoyt fading.

6. CONCLUSION

We have considered the performance of FSK mod­ulation under fast Nakagami-q fading with limiter­discriminator detection. Closed from expression forthe BEP is derived. In the derivation, the co-channelinterference has been taken into account. The de­rived expression has been verified to include partic­ular cases of already published results.

7. REFERENCES

[1] R. S. Hoyt, "Probability functions for the mod­ulus and angle of the normal complex variate,"Bell Syst. Tech. J, vol. 26,pp. 318-359 , Apr.1947.

[8] T. T. Tjhung, K. M. Lye, K. A. Koh, and K. B.Chang, "Error rates for narrow band digital FMwith discriminator detection in mobile radio sys­tems," in IEEE Trans. Commun., vol. 38, no. 7,pp. 999-1005 ,Jul. 1990.

[9] I. Kom, "Relation between bit and symbol errorprobabilities for FSK with LDI detection," IEEETrans. Commun., vol. 42, no. 8, pp. 2512-2514 ,Aug. 1994.

[10] K. Hirade, M. Ishizuka and F. Adach, "Errorrate performance of digital FM with discrimina­tor detection in the presence of co-channel inter­ference under fast Rayleigh fading environment,"Trans. ofthe IEICE, vol. E61, no. 9, pp 704-709,Sept. 1979.

[11] M. Zhaounia and N. Youssef, "Bit error proba­bility of narrow band digital FM with limiter dis­criminator detection in Nakagami-q mobile radiofading," IEEE ISCCSP'04,Tunisia, pp. 673-676,Mar. 2004.