A bstruct-Repetition, diversity, and single-error-correcting codes are examined for use with binary modulation techniques over the faded mobile channel.
14
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 38, NO. 1, FEBRUARY 1989
On the Use of Repetition Coding with Binary Digital Modulations on Mobile Channels
A bstruct-Repetition, diversity, and single-error-correcting codes are examined for use with binary modulation techniques over the faded mobile channel. Both bit-error performance and bandwidth efficiency are considered for phase-shift keyed (PSK), differential PSK (DPSK), and frequency-shiftkeyed (FSK) modulations. The error-correction capability of repetition coding is first considered and optimum repetition is found. M-tuple diversity with maximum-ratio-combining (MRC) is then compared with repetition coding both with a single-error-correctingcode and without coding.
I. INTRODUCTION IGITAL voice transmission that would assure high-level voice security over VHF/UHF radio mobile systems has recently become a subject of active study. Several methods of modulation and coding have been proposed for digital transmission [11-[6] over various fading conditions. Unfortunately, no conclusion has been reached yet regarding the optimum choice of signal shaping, modulation, and coding. While most communication channels are either bandlimited or powerlimited, spectrally efficient modulation techniques capable of combating the hostile propagation environment of mobile radio are needed. In an interesting article, Lee [7] has proposed the use of repetition codes with differential phaseshift keyed (DPSK) modulation to improve the system performance for slowly fading channels and for fast fading as well. Although such codes are known to degrade the system’s performance when used over a Gaussian channel, they can be advantageous in other environments, where error correction can benefit the performance more than an increase in power. This paper considers the use of repetition codes over a Rayleigh fading channel. PSK, DPSK, and frequency-shift keying (FSK) are analyzed for nonselective fading conditions. Since repetition coding can provide performance improvement similar to diversity, we check the possibility of using such codes alone and show that repetition codes improve performance when the coded bit energy is sufficiently high. We also show that an optimum repetition number exists that minimizes the bit-error rate. Using repetition codes for spread-spectrum anti-intercept mobile is useful since the code will spread the symbol energy over several “chips” (repetitions) and provide error correction as well. For bandlimited channels, however, high-rate codes are desired since they provide an efficient trade-off between spectrum efficiency and error performance.
D
Manuscript received April 24, 1987; revised September 20, 1988. The authors are with the College of Engineering, King Saud University, P.O.Box 800, Riyadh 11421, Saudi Arabia. IEEE Log Number 8927658.
Several design options are presented for the use of repetition codes, diversity, and single-error-correcting codes with binary modulations. It will be shown that optimizing the performance with the repetition code alone or diversity alone as the only correcting measure will lead to a large waste of spectrum or complexity of equipment, respectively. For bandlimited faded channels, the optimum design may involve the use of a combination of repetitions, diversity, and error-correcting codes. II. PERFORMANCE ANALYSIS Binary Signaling Over a Frequency-Nonselective Slowly Fading Channel We start by summarizing the known results of bit-error-rate @ER) over a Rayleigh fading channel. For PSK with coherent detection, it is assumed that a stable phase reference can be established, enabling coherent detection. In such a case, the probability of error is given by [8]
P, (PSI+!
2 [1+]
(la)
where Y b is the average signal-to-noise ratio (SNR) per bit. On fast fading channels where a stable phase reference is not feasible, DPSK or noncoherent FSK can be used. The probability of error over the fading channel for DPSK is given by
and for FSK by
P, (FSK) =-
1
.
2+Yb
From (1) it can be seen that the error rates decrease only inversely with the SNR, in contrast with nonfading channels where the decrease in error rate is exponential with the SNR. To keep the error rate at a tolerable level over fading channels with reasonable power, diversity, repetition coding, and/or error-correcting codes can be used.
Repetition Code, Diversity, and Block Codes for Fading Channels I ) Repetition Code: One of the simplest yet most effective coding techniques is to subdivide each information symbol into
OO18-9545/89/0200-0014$01.OO
0 1989 IEEE
15
ALI AND AL-KADI: USE OF REPETITION CODING WITH BINARY DIGITAL MODULATIONS
-32
10 Averoge signol-to-noise
18 10 14 Averoge s i g n o l - t o - n 0 i s e 7 ~dB
7, d 0
(a)
(a)
-16
I
10
2 Average signal-to-noise
Fig. 1.
7, dB
(b) Performance of repetition code with binary modulation over fading channels. (a) PSK. @) FSK.
equal energy subsymbols that are then transmitted over independent channel states. This is referred to as “time” diversity transmission or a repetition code. Time diversity techniques can significantly reduce the effect of fading on the P, performance. In particular, optimum diversity restores the exponential relationship between P, and &/No (energy per bit-to-noise power spectral density). If a repetition code with K repetitions is used over a channel with error probability P,, the resulting bit-error rate (same as word error rate) Per is given by
Per=
6
(7)
PL(l-P,)k-’.
i = ( k + 1)/2
(7)
The channel probability of error P, is given by ( l ) , and is the ith coefficient of the binomial expansion = k ! / ( k i ) ! (i ) ! . It will be assumed in this analysis that a majority vote decoding is used, and hence only odd values of K are chosen. Per is depicted in Fig. 1 with K = 1 , 3 , * * - , 1 1 . 2) M-Branch Diversity with Maximal Ratio Combiner (MRC): Assume that M diversity channels are carrying the same signal, and each is slowly fading with Rayleighdistributed envelope statistics. We further assume the channels to be mutually statistically independent. If an optimum maximum-ratio combiner (MRC) is used, it can be verified that the error rate with M-diversity channels Pd is
Fig. 2.
I
I
10
14 Averoge signal-to-noise
I
I
18 r b dB
(b) Performance of optimal MRC diversity with binary modulation over fading channels. (a) PSK. (b) FSK.
given by [8]
M-l+i
(3)
i=O
with P, given in ( 1 ) . The above closed-form expression is valid for PSK, DPSK, and FSK with the proper P,. Fig. 2 depicts Pd for M = 5 , 10, 30. 3) Error-Correcting Codes: We assume the fading to be fast enough so that two adjacent bits are independent. Interleavers and deinterleavers may be used at the expense of added complexity. For an error-correcting code which can correct up to t errors in a world of N bits, the word error rate is given by
-
Pa=
a ,
5 (y)
Pi(1-PY.
(4)
i=f+l
The BER P in (4) is given by (1)-(3) depending on whether no diversity or repetition is used, a repetition code is applied, or diversity is utilized, respectively. When repetition coding is used over a channel with the M branch diversity described, P, in (2) is replaced by Pd of (3). Similarly, code concatenation may be employed with repetition coding as the inner code and block code with t-error correction as the outer code. If a channel with M-diversity
16
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 38, NO. 1, FEBRUARY 1989
branch is used. P in (4) is given by Perin (2) with P, replaced by ped. 111. RESULTS
The Effect of Repetition Code Figs. l(a) and (b) show the error rate performance of PSK and FSK, respectively, with repetition K = 1, 3, * , 11. For PSK, the code always improves performance; however, such improvement becomes faster as the SNR Y b increases above 10 dB. Similarly, for DPSK, the improvement starts at Yb > 10 dB with about 3 dB less performance compared with PSK. For FSK, however, repetition code improvement starts at Y b > 13 dB. At smaller values of Y b , coding will degrade the performance. Again, FSK is 3 dB inferior to DPSK. For binary modulation techniques over faded channel, it can be seen that as the repetition increases, performance improvement approaches the exponential relation; hence repetition code can be used as an effective measure to combat multipath fading.
-
8
3
z - Eb/No
4
(DPSK)
32
LO
K
(a)
"1
The Effect of Diversity Repetition coding with majority voting decoding differs essentially from the optimal diversity in that a hard decision on each symbol of the K-symbol codeword is made at the output of the K receivers. The decoded information bit is then in favor of the majority of the K symbols codeword. Fig. 2 depicts the improvement in performance offered by the use of diversity over multipath fading channel with PSK and FSK. Clearly, the improvement is essentially the same as in the case of repetition coding. Careful evaluation, however, reveals that diversity of order n outperforms n-repetitions code. This is an expected result, since repetition code employs hard decision decoding in which some of the information is lost. Fig. 3 depicts the improvement gained by repetition code and by n-branch MRC diversity for FSK. Clearly, 1) diversity outperforms repetition since P, of requires an SNR of 20 dB at an optimum repetition of 20, while optimum diversity of order 12 requires an SNR of 16.5 dB at the same error rate. Similarly, Perof requires an SNR of -22 dB at an optimum repetition of -40, while the same error rate can be reached with optimum diversity on the order of 20 at an SNR of 18.5 dB. Such a result is due to the soft decision advantage of about 3 dB over hard decision decoding. Fig. 4 shows the optimum order of diversity as a function of SNR, from which it can be concluded that the optimum diversity and average SNR &/No are related by
16 24 Repetition
,
8
,
,
I6
~
,
32
,
ol
,
-31
10
0 Diversity Order M
Fig. 3.
(b) Optimum repetition and diversity for FSK. (a) Repetition code. (b) Diversity.
I
I
13 -
G
Mop1
X
24
16
32
40
G
Diversity Order M
(a)
(5)
where Eb/Noand Lo, are the total SNR (per bit) and optimum diversity order, respectively. The relation in (5) holds for DPSK. For FSK, however, a similar relation exists where 2 Loptz-Eb/No (FSK). 3
(6)
The error performance can be further improved by combin-
(b) Fig. 4. Optimum diversity for DPSK and FSK. (a) DPSK. (b) FSK.
17
ALI AND AL-KADI: USE OF REPETITION CODING WITH BINARY DIGITAL MODULATIONS
TABLE I PERFORMANCE OF FSK WITH REPETITION CODE AND DIVERSITY Diversity Only SNR (dB) M=2 M=4 Mop,
No Diversity No Repetition SNR (dB)
K=3
K=7
KO*
10-4
40
27
22
20
22
19
16.5
10-6
60
32
26
22
23
25
18.5
Error Rate
Repetition Only SNR (dB)
Both Diversity and Repetition SNR (dB)
M=2, K=3 20 dB M=2, K=7 20 dB M=4, K=3 20 dB
ing both diversity and bit repetition. Table I gives the required &/No in dB for error rate of and
Combined Diversity and Repetition Code When no repetition or diversity is used, a high signal power Eb/No = 40 dB is needed to combat fading. For P, = and decreasing the error probability to requires Eb/Noof 60 dB. When the optimum repetition is used, 21 repetitions with Eb/No 20 dB are needed at P, of However, a simple 3-bit repetition reduces the &/No to 27 dB for an error rate of The use of diversity alone may become the only option to improve performance when bandlimited channels are used, and hence no bandwidth expansion (as required by repetition coding) is permitted. A simple twofold diversity can at &/No of 22 dB as sevenprovide the same Pd of repetition code. The optimum diversity (twelvefold) requires only 16.5 dB at A compromise between bandwidth expansion required by repetition coding and equipment complexity of high order diversity is shown in the last column of Table I where the use of simple two-order diversity and 3-bit repetitions can provide the required performance at a realistic Eb/No. Fig. 5 shows the error performance for FSK as a function of diversity at 3bit and 7-bit repetitions. Table I outlines some possible strategies in system design using time diversity (repetition code) and space diversity.
e .' ;10 -
-
10 10
2
10 14 Average signol-to-noise
18
6
7,
dB
(a)
Combined Repetition and Block Codes and Diversity Code concatenation can be used to advantage in fading and interference environments. We assume repetition code as the inner code with a single-error-correcting code as the outer code. The code rate is assumed to be 0.5 in the present analysis, with codeword lengths of 10, 20, 30, and 40 bits. Table I1 summarizes the three cases: using two layers of coding, using a single-error-correcting code (high rate code) with diversity, and finally, using coding, repetition, and diversity. Numerical results, not shown in Table 11, indicate that codeword error increases by about 0.5 dB when the word length is increased from 10 to 20, etc. When used with a repetition code, the block code offers a large improvement of more than 10 dB at K = 3, decreasing to less than 6 dB at K = 7. Higher gains are even attained at an error rate of Similar results are shown for the case of coding and diversity; however, fewer gains are obtained. The use of coding, diversity and repetition may provide an
1
7, dB (b) Combined effect of diversity and repetition code for FSK. (a) Three repetitions. (b) Seven repetitions. Average signal-to-noax
Fig. 5 .
acceptable performance at a moderate complexity (twofold diversity M = 2), low bandwidth expansion required by three repetitions (K = 3). A modest signal-to-noise of 17 dB can provide an error rate of while 19 dB of SNR can reduce as shown in Table 11. the error rate to IV. CONCLUSION Repetition codes, diversity, and single error-correcting codes have been examined for use with binary modulation techniques over the faded mobile channel. It has been shown that a minimum of 10 dB SNR is required for an improved performance with repetition code or diversity. The repetition code is about 3 dB inferior compared to optimal diversity.
18
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 38, NO. I , FEBRUARY 1989 TABLE I1 PERFORMANCE OF FSK WITH REPETITION CODE, BLOCK CODE, AND DIVERSITY Error Rate
10-4 10-6
Coding and Repetition SNR (dB)
Coding and Diversity SNR (dB)
K=3
K=5
K=7
M=2
M=3
M=4
21 28
19 24
18 22
21 24
18.5 22.5
17 21
An optimized design of moderate complexity and bandwidth expansion can be reached with twofold diversity and three repetitions as an inner layer of coding concatenated with an outer single-error-correcting code. It is particularly shown that 1) optimum diversity or repetition code can effect an exponential exchange of signal-to-noise and performance, 2) the use of combined diversity and repetition code reduces equipment complexity and bandwidth expansion, and 3) concatenated codes with a repetition code as an inner code will provide further improvement in combatting fading degradations.
Coding, Diversity and Repetition SNR (dB) M=2,K=3 M=2,K=5
17 19.5
16 18.5
Adel A. Ali (M’82) was born in Alexandria, Egypt. He received the degree B.Sc. from Alexandria University, and the M.Sc. and Ph.D. degrees from the University of Manitoba, Canada, in 1967, 1973, and 1976, respectively, all in electrical engineering. From 1976 to 1978 he was a Transmission and Special Service Engineer with Manitoba Telephone System, Winnipeg, MB, Canada. Since 1978 he has been with the Electrical Engineering Department, King Saud University, Riyadh, Saudi Arabia, where he is now a Professor. He also served as a consultant to the Minis!t? of FTT, Saudi Arabia. His research interests include communication theory microwave propagation, and mobile and military communications.
REFERENCES M. A. Clark, “Digital modes for land mobile radio,” Proc. Inst. Elec. Eng., vol. 132, pt. F, no. 5, pp. 348-362, Aug. 1985. K. Murota, “Spectrum efficiency of GMSK landmobile radio,” IEEE Trans. Veh. Technol., vol. VT-34, no. 2, pp. 69-75, May 1985. D. R. Hpmmels and F. W. Ratcliffe, “Calculation of error probability for MSK and OQPSK systems operating in a fading multipath environment,” IEEE Trans. Veh. Technol., vol. VT-30, pp. 112120, 1981. [41 T. Aulin and C. E. Sundberg, “Detection performance of bandlimited continuous phase modulation,” in Proc. IEEE Global Commun. C o n f , Miami, FL, 1982, vol. 3, pp. 1119-1125. H. W. Arnold and W. F. Bodtmann, “The performance of FSK in frequency selective Rayleigh fading,” IEEE Trans. Commun., vol. COM-31, pp. 568-572, 1983. L. Milstein, S. Dvidovici, and D. Schilling, “Coding and modulation techniques for frequency-hopped spread spectrum communication over a pulse-burst jammed Rayleigh fading channel,” IEEE J. Sel. Areas Commun.,vol. SAC-3, no. 5, pp. 644-652, Sept. 1985. W. C. Lee, “The advantages of using repetition coding in mobile radio communication,” in Proc. 36th IEEE Vehicular Technology Conf., Dallas, TX, May 1986, pp. 157-161. J . G. Proakis, Digital Communications. New York: McGraw-Hill, 1983, ch. 7.
Ibrahim A. AI-kadi (S’83-M’84) was born in Onaizah, Saudi Arabia, in December 1953. He received the B.S.E.E. degree from Riyadh University (now King Saud University), Riyadh, Saudi Arabia, the M.S.E.E. degree from the University of Michigan, Ann Arbor, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1978, 1980, and 1984, respectivelv. He worked for a year (1978/1978) as a Graduate Assistant at Riyadh University. During 1981-1984 he worked as a Research Assistant at the Communication Satellite Planning Center of Stanford University. Since 1984 he has been with the Electrical Engineering Department at the College of Engineering of King Saud University, Riyadh, Saudi Arabia, where he is now an Associate Professor. He teaches graduate and undergraduate courses in electromagnetics and communications and performs research and consultation works. His areas of interest include communication systems, digital communications, spectrum management, radio wave propagation, remote sensing, power electronics, solar cells, technology transfer, and satellite surveying.
