ON THE BIT ERROR PROBABILITIES OF GMSK IN THE RAYLEIGH FADING CHANNELS Sang U. Lee', Young M. Chung', and Jae M. Kim"

Dep4.of coatrol and Instru. Eq., sewl National University, &mI, Korea.

Electronics and Teleco"unicati0ns Research Institute, TaCpn, Korea.

consider the effect of cochanoel interfernce, which occurs froxu a neighboring cell in a cellular radio system, and adjacent channel interference. We employ the MSK-type receiver for a demodulation and correct carrier recovery and timinp are assumed. We also assume that each channel fades amording to a Rayleigh amplitude dishibution, that fading is slow and noaselective. The approach used in this pper is similar to the one used by Rhodes, Wilson and Svensson [7]. The application of this a m to GMSK leads to a closed form ercpresson for BER and rigomus numerical techniques have been used to solve the closed form

ABSTRACT Coherent detection of Gaussian minimum shift keying in a land mobile radio environment is analyzed in this pper. Both the effects of cocbannel and adjacent channel interference are Considered. We employ the MSK-type receiver for a demodulation and "ct carrier recovery and timings are assumed. A closed-form expression for the BER is derived. The numerical results are presented and compared with those for MSK. The simulation results indicate that the effect of cochannel interfemme is morc significant than that of adjacent. It is also observed that in the case of adjacent channel interference GMSK h slightly superior to MSK when the interference signal is strcmg. This confirms that GMSK poysses attractive power spectrum properties for a digital mobile radio.

expressions. GMSK SIGNAL AND MSK-TYPE RECEIVER A functional Mockdiagram of GMSC modulator is shown in Fig.1. The input data are assumed to be binary in NRZ (nonreturn to zero) form. In GMSK, the binary data are filtered by a Gaussian low pass filter befm modulation by the FM modulator with the modulation index h = l E A large class of CPM signal, including GMSK, cau be written as

INTRODUCTION In order to provide highly secure voice andor high speed data transmission, digital mobile radio communication systems are currently being studied with great interests. Bandwidthefficient digital modulation is one of the most important techniques for achieving the digital mobile radio. Thus, the demand for bandwidth efficient constant envelope modulation techniques with good error rate pcrEormance has naturally increased recently. For this purpose, continuous phase modulation(CPM) techniques has been proposed [l-31. It is a well known fact that the CPM techniques yield good spectral densities while poglessing constant envelope in time. One of the CPM techniques that has received considerable interest in recent years is Gaussian-filtered minimum shift keying(GMSK)[4]. MSK is a special case of continuous phase frequency shift keying(CPFSK) for which the modulation index h=l/2. In this case, a simple quadrature coherent detector, also known as MSK-type receiver, can be employed instead of a rather complicated maximum Likelihood estimator. The MSK-type d v e r is merely a linear quadrature matched filter, sampled at alternate times in each channel. In the literature, a number of studies have been reported focusing on the performance of coherent or noncoherent detection of MSK and raised cosine(RC) modulation techniques in additive white Gaussian nOise(AWGN) and Rayleigh fading environments[5,6]. But few results are available on GMSK with a coherent detection so far. In this pper, attempts have been made to investigate the bit error rate(BER) performance of coherent detection of GMSK on AWGN and Rayleigh fading channels. We also

s(t)= (2FJryCos(27rf$+l+(t,U,CJ)) where a=(..., an-I,a,, a.+1,...) is a sequence of data, and ai is information bit represented by independently +1 or -1 with equal probability of l/2. is the energy per bit, T is the bit duration, and fc is the carrier frequency. Also U is the bit time offset, hawever, U wwld be zero if a perfect bit timing is assumed. The phase term 4(t,u,c5) is defined as

J

-m

In (2), At) is a shaping pulse function. In the ca5e of GMSK, the desirable duration of the g(t) is infinite however, for practical reasons, it is neeesuy to truncate the g(t) aver a small integer number of bit period. In addition, g(t) should be nonnalized in such a way that /-:g(t)dt=le. Gmsequently, the change in the phase due to one bit IS restricted to the values of f7rD. The shape of g(t) determines the smoothness of the transmitted information canying phase. The sbaping puke for GMSK is

249 CH2622-9/88/0000-0249 $1.00 0 1988 IEEE

n(t) is AWGN with two-sided spedral density N&? W f i . Then, in the Rayleigh fading AWGN channel case, the received signal is

where a t ) is defined as

Ji

In (3a) is 3dB-bandwidth of the premodulation filter. In G k L T=0.25 is widely used. Fig.2 depicts the phase tree of G with o=(a-4,a-3 ,...,a,,a,). As is expected, all phase changes are very " 0 t h . It is a well known fact that a general binary partial response CPM signals with modulation index h = v 2 can be detected by a simple MSK-type receiver. The MSK-type receiver based on the parallel MSK receive#] is shown in Fig.3. Thc received signal is first multiplied by the in-phase and quadram components. Here wc assume a pexfect carrier recovery. Outputs of the multiplier are applied to two zonal filters to s u p the high frequency components without degrading basebend signal. Th: I,Q baseband signals are then applied to baseband matched filter, each with impulse =P= h(O=

s.wMm} (0

,Or t< 2T ,otherwise

assume that the channel fading is sufficiently slow 90 that the ra" pmceses &(t) and ei(t) can be described by random variable & and Oi, respectivelyIl0]. When this signal is processed with the receiver in Fig.3, it is sufficient to consider oaly one of the a m , say the in-phase channel, in the receiver. Then it is easy to show that the sampler output in the I channel at the idcai sempliag time b is

X - ( ~ ~ m r l p ( ~ + ( " b T ) m W ~ ~ ~ ( B , A f , v , B )(74 +N

(4)

Where

P o = m h ( t ) * W 4 ( t , o . $ N I,-(7b) recove~s the data sequence by alternately sampling at 2T intervals. Finally, a A@,Af ,v,8)= lE{h(t)*COS(2mAft++(t ,v,@ +8-el+@,)} I ,=,(7~) differential decoder recoven the infomation bit a.. It is noted that the filter of (4) is not optimum for 'GMSK. From (7a), we obsme that the first term is due to the Themetically, for a given criterion, it is possible to find the desired signal, the second term is a am~bution from optimum filter for GMSK. For example, recently El-Tanany interferer and the Wise component N is a zero mean and Mahwnrd[9] an optimum filter mp0n.w for Gaussian random variable with variance CPM baaed 011 s the m e a ~ ~ ~ ~ ~ ~ - i n t e ~ t d intexference subject to the a " ' t that filter wise varw= o2 = No h(t),dt (8) bandwidth is held constant. But our'study has focused on using the filter of (4) for G M X , and we shall show that Then, it can be shown that the conditional error probability the performance degradation, in terms of BER, is negligible given the amplitude of the desired and interfering signal is compared with the minimum BER in the AWGN enviroament[4]. P.(g,,$,Af,V,W

Then, in each channel, the receiver

w.

J

BIT ERROR RATE PROBABILITIES

=PrC(2EbT)*R,p(g,)+

The bit error rate of CPM with the MSK-type receiver in the c " e l and adjacent channel interference on the AWGN channel has previously analyzed in [q. Also in [A,the BER in the Rayleigh fading channel was obtained by averaging the BER in the AWGN chanoel. In this paper we analyze the BER performance of GMSK in the Rayleigh fading channel specifically. The approach used in this paper is similar to the one followed to derive the BER of CPM in [q.However, parts of this paper are repeated here for the sake of completeness. Assuming that undesired interference and AWGN m added to the desired signal at the receiver is given by

< N)

(2EJ)*WC)*R&&,Af,v,W

rp(RI&Jl,r,Wd'dr,

(9)

where g, represents a particular data sequence giving a positive p. Enur will be ocaured if we detect a negative X when a . is sent. In (9), rp(RIw(n r r ) is a joint probability density &on of raadom -abIL1g, R, and But N and .&, % are statistically independent. n u s can be rewritten

3.

as

Pe(aJ3i,Af,v,W

fRl(r,)%} fRZ(rz)dr* where a is the desired information sequence, @ is the interferer's information sequence, Af is the frequency separation between the interference signal and desired signal. Also v is the random variable representing bit time offset and 8 is the random phase offset. But it can be assumed that v and 8 are uniformly distributed random variables on [ O , q and [0,2m], respedively. c/I is the ratio of the desired signal power to the interfering signal power at the receiver.

250

From (lo), we need to know the distribution of random variables Y , and q. Since the random phase Y = 8 - 8 l+C92 is a sum of three uniformly distributed independent random variable, the probability density function is close to a Gaussian. However, we assume that Y is uniformly distributed on [0,2?r] for the sake of Convenience. On the other hand, Hansea[ll] showed that the short-term Rayleigh fading model can be described by the Rayleigh probability density function. fR(r)= r/% exp(-?/(W) (0

,r2 0 ,otherwise

(11)

where r is the amplitude and % is the mean power. Then we can express the density function of ra” variables R q such that the mean &ved power of the desired signal IS rl= IT and the mean received power of the interferer is r,=& WC), respectively. The form of fR(r) is %(r)= 2r e M - 3 ) (0

,r2 0

,otherwise

(12)

Thus, the error probability in the I channel is obtained by averaging (10) over the random vector (s,b,v,V).

performance degradations. For example, the maximum average eye opening value for GMSK is 1.931, as compared with the corresponding value 2 for MSK. In GMSK, the BER performance bound in the high SNR condition is approximately represented as [4]

where d

is the minimum value of the signal distance d between%” and ”1” in the Hilbert spnce. While the BER prfbound given by (15) is obtahed only when the Ideal manm~m.IikdWoddetection is employed however, it givesanapprmmate solution for the ideal BER performance of GMSK with coherent dete&d4]. Tbe BER behavior of GMSK in the AWGN chsneel is shown in FigS. For the sake of comj”, we ako include the minimum BER of

degradation as compared with the minimum BER. Thus it would appear that our results are very doee to the optimal and the simple MSK-type receiver are capable of providing adequate performance. Fig.6 shaws the BER performance of GMSK in the Rayleigh fading channel. comparing with GMSK experiences about 0.4dB degradation at P -10 ~lso it an be seen that the K / N must ~ exceed 2 1 1 ~ in order to hold Pb<10-3. As is expected, MSK is superior to GMSK. We also observe that the numerical result is very dose to the result in [4]. Next we Consider the cochannel and adjacent channel interference case. In the case of cochannel interference (Af=O), Fig.7 and Fig.8 present the Pb performance for GMSK in the AWGN and Rayleigh fading environment, respedively. In the AWGN channel, it is seen that the effect of cochannel interference for UI>UMB is almost negligible. But GMSK is more sensitive to the COchaMel interference than MSK. For example, when UI varies from 2wB to lWB, MSK degrades about 2 . 1 d ~ ,while GMSK suffers about 3.m at ~ , = 1 0 - ~ . In the Rayleigh fading channel, the degradations are more signifhnt. Even for CII==2wB,wbicb gives little effect in the AWGN channel, we can see much more degradation. It may be seen that GMSK and MSK both require a UI of more than 3OdB to attain even P In addition, we an obeer~e irreducible error &ties exist even at high cocbannel case. In geaeral, MSK enjoys a slight GMSK in the cocbannel case. F i y we amsider the adjacent interference(Af#O). Fig.9 and Fig.10 are the results. Fig.9 presents the P behavior for Af=1.5 in the AWGN channel. observe that $th GMSK and MSK, given interference level, are quantitatively much less degrading than in the cochannel case. For example, when UIMdB, the degradation is almaft negligible. But it is noted that when the interference signal is strung, GMSK is slightly better than MSK in the BER p e r f o ” c . MSK has an irreducible error floor, but GMSK maintains respedable performance far Af= 1.5 and Cn=-2odB. But, for UI=-lOdB, both GMSK and MSK @de almost same performance and MSK enjoys a slight advantage when c/I=WB. Tbe Pb performance in the Rayleigb fading chanael case is Shawn in Fig.10 with Af= 1.5. We see similarly shaped performance c w e for the AWGN channel. We can s i x that GMSK is slightly Superior to MSK when the CIIddB. Since the bandwidth for GMSK is narrower than that a€ UM, GMSK is expected to be less semitive to the i n t e r f e r a ~ ~This . explains why GMSK outperforms MSK when Af-1.5 a d interference is strong. For such case, p e f f o “ a k primarily sensitive to the level of spectral s,-i and GMSK exhibits substantially lower level than MSK.

p, .

s,

where M, is the number of distinct data sequence for and is determined by the time support of the filter. In t h ~ s paper, we choose 5T for g(t), thus -32. For A we must consider all Iv$ relevant data In our case, %=4M1, since % IS the number of all possible combinations of sequences & wth positive or negative A and there is time offset T. Since the sampler output in the Q channel is shifted by T in time, the m probability in the Q channel is same as (13). Finally, due to differential decoding, the actual bit error probability Pb is related to Pe,the probability of error, by the relation when deciding on symbol aa or

3:;;.

Pb= 2Pe(1-Pe)

(14)

In fact, if we perform the integration in (13), we obtain the same results in [7]. The identical results are expected since we assume that the distribution of the random phase Y is a uniform over [0,2?r]. Otherwise, the e m probability in the Rayleigh fading channel can not be obtained simply by averaging the error probability in the AWGN channel. NUMERICAL RESULTS AND DISCUSSIONS The BER of GMSK with the MSK-type receiver was evaluated numerically using (13) and (14). In order to evaluate Pb, it is necasary to perform the numerical integration m V v and q.In computing the Q function, we use 1 5 t h - o r d ~ k u s d a nquadrature rule and trapezoidal rule is used for other integrations. As stated earlier, &) is extended over infinite time interval hawever, we only &der g(t) over I t k 2 . n and zero outside. In this case, the integration value over this region is 0.4997, which is very dose to the ideal value 0.5. For the purpose of demonstrating the effect of ISI(intersymb0l intderence) on the phase of GMSK, Fig.4(a) gives the eye diagram at the output of the postdetection filter. Fig.4@) also shows the eye diagram of MSK for the comparison purpose. It is observed that the ideal sampling time. 6 for GMSK is 0 . s . Also,compared with MSK, GMSK is expected to suffer some

251

CONCLUSIONS

In this paper we described the BER performance of GMSK with &rent detection in AWGN and Rayleigh fading environiaentS. Bath the effecrs of cochaanel and adjacent channel interference were a"d. We emplayled the MSK-type d v e r for a demodulation and correct camer aawmed. A dored-fonn expaession recoyerY and is derived for the BER. Tbe numericd nrulg are m t e d and compand with MSK. It was found that the degradation due to using the MSK-type receiver is about 0.at BERlo-? nus the simpie --type raceivsr is came of providing adcqFte perfWe obamed that the effect of L"d interference ls more signifkant than that of adjacent. For example, in the AWGN -1, UI==2OdB is required to &Wide roughly rKKdumEl performance, while UI-OdB is enough for no-adjacent channel performance. In the slow Rayleigh fading environment, UIm 3OdB gives no degradation in the cocbannel performance and c/I=lodB is required to provide no-adjacent performance. It is also oberved that in the case of adjacent channel interference GMSK is slightly superior to MSK when the interfexence signal is strong. This observation coafirms that GMSK potmxws anractivc power spechum pmperties for a digital mobile radio.

Fn rodulator

modulation index4.5

Fig.1. Hockdiagram for GMSK modulator.

REFERENCES T.Aulin and C-E.W.Sundberg, "Continuous phase modulation-part I: Full respoase signaling," IEEE Tram. Cornm~n.,vd.COM-29, ~p.1%-209, Mar. 1981. T.Aulin, N.Rydbeck and GE.W.Sundberg, "Continuous phase modulation-Part II: Partial respoase signaling," IEEE Trans. Co"., vol.COM-29, pp.210-225, Mar. 1981. GE.Sundberg, "Continuous phase modulation," IEEE Cornm~n.Soc. Magazine, vol.24, pp.25-38, Apr. 1986. K.Murota and K.Hirade, "GMSK modulation for digital mobile radio telephony," IEEE Trans. C o n " . , vd.COM29, pp.1044-1050, July 1981. A.&and GE.Sunbeg, "Optimum MSK-type receivcxs for CPM on G a d a n and Rayleigh fading channels," IEE Proc., 4.131, Pt. F, pp.480-490,Aug. 1984. S.M.Elnoubi, "Analysis of GMSK with two-bit differential detection in land mobile radio channels," IEEE Tram. C o n " , vol.COM-35, pp.237-240, Feb. 1987. R.B.Rhodcs,Jr., S.G.Wilsoa and A.Svensson, "MSK-type reception of continuous phase modulation: &channel and adjacent channel interference," IEEE Trans. Commwr., vol.COM-35, pp.185-193, Feb. 1987. Spaspathy, "Minimum shift keying: A apoctrally efficient modulation," IEEE Commwr. Soc. Magazine, vol.19, pp.1422, July 1979. M.S.El-Taasny and S.A.Mahmoud, "Mean-square error Optimization of quadrature receivers for CPM with modulation index ll2," IEEE J . Select. Areas Commun., vol.SAG5, pp.8969M, June 1987. [lo] J.G.Praakis, Digitnl C d h , NIW York: McoraWHill, 1983. [ll] F.Hamen and F.I.Mew, "Mobile fading-Rayleigh and lognormal superimpad," IEEE Trans. Veh. Technd., vd.VT-26, ~p.332-335, NOV. 1977.

Fig.2. Wase tree for GMSK si@.

xos 12-rrfctl

2nTtto input

r It1

decision logic

-2SIN12lrfctj

2htllTtto

Fig.3. Blockdiagram for MSK-type receiver.

252

loo

1

0 -8

10 --

.

-1

, . . . . 4d

0

+

6119( (optiur receiver1

cl A

6119( (experilentall

1

ancr)

l o - - 11 0

2

I

4

I

10

12

14

11

w 6%) FigS. Pb oomparison with an optimum receiver.

loo

10-1

-la! -1

. . . . . . . . . . . . . . . . . . . . . . . . -am

0

"

W

1

in

0 0

A

6119( lexperirental]

Fig.4. Eye pattern. (a)G=

a

4

I

12

11

a0

24

a,

w (e) Fig.6. Pb under Rayleigb fading with no interference.

253

loo

,

10

0

10-2

10-~ P

10-6

lo-*

i

\ \'

10 -

1

0

2

4

q

I

I

10

j

12

14

10-1

2

0

11

4

I

6

B/lb

B P (9)

10

14

12

16

(a

Fig.9. P under AWGN with adjacent channel interference (At= 1S).

Fig.7. Pb under AWGN with cochannel interference.

loo

10-1

;lo-2

I

X 6MSK,C/I=2OdB

0

I

12

11

*(4

1D

H

a

Fig.10. P under Rayleigh fading with adjacent channel inter!erence(Af=l.S).

fig.8. Pb under Rayleigh fading with cochannel interference.

254

I

4

On the bit error probabilities of GMSK in the Rayleigh ... - IEEE Xplore

detector, also known as MSK-type receiver, can be employed instead of a rather complicated ... literature, a number of studies have been reported focusing.

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