Iterative Decoding in Convolutionally and Turbo Coded MFSK/FH-SSMA Systems Daeyoung Park and Byeong Gi Lee Telecommunications and Signal Processing Lab. School of Electrical Engineering, Seoul National University Seoul, 151-742, Korea Tel : +82-2-880-8429, Fax : +82-2-882-4657 E-mail : fpdy,
[email protected] Abstract— This paper presents an iterative decoding method in the coded MFSK (multilevel frequency shift keying)/FH-SSMA (frequency hopping-spread spectrum multiple access) system. The kernel of the system, which is a symbol APP (a posteriori probability) calculator, accepts channel values and a priori information from the channel decoder and generates reliability values of the received symbols. The channel decoder also makes reliability outputs using the reliability values from the symbol APP calculator. Iterative decoding is performed by repeating this process. Simulation results reveal that the channel coded system with iterative decoding in the FH-SSMA system can reduce the BER significantly, thereby accommodating more users.
Channel encoder
Bit-to-symbol converter
Interleaver
Frequency hopper
(a)
Frequency dehopper
Symbol APP calculator
Symbol-to-bit converter
Channel decoder
- Deinterleaver
Interleaver
-
(b)
I. I NTRODUCTION Frequency hopping-spread spectrum multiple access systems have many attractive features in mobile communication environments. They have frequency diversity and are resistant to the near-far problem. However the system capacity is limited due to cochannel interference. In order to minimize cochannel interference from other users, Einarsson [1] proposed the optimal address assignment method which minimizes coincidence of user address assignment codes. Goodman et al. [2] investigated encoding and decoding rules of the MFSK (multilevel frequency shift keying)/FH-SSMA (frequency hoppingspread spectrum multiple access) system and analyzed the performances assuming the random user address assignment rule. Recently, Fiebig and Robertson [3] proposed soft-decision decoding in the MFSK/FH-SSMA system to provide soft inputs for channel decoder, which exhibited good performances compared to the systems using hard-decision decoding. In conventional decoding methods of concatenated systems, a decoder (or a demodulator) carried out symbol decisions without using other decoder’s information. In order to improve decoding performances, iterative decoding techniques have been applied to decoding of the concatenated systems, such as turbo coded system [4], convolutionally coded DPSK system [5], and convolutionally coded space-time code system [6]. In those cases, an encoder (or a modulator) puts constraints on transmitted symbols and the decision is aided by other soft input soft output (SISO) decoders (or SISO demodulators): Iterative decoding is performed by repeating the decision process. In this paper, we discuss how to incorporate the iterative de-
Fig. 1. Block diagram of the proposed system : (a) Transmitter, (b) receiver.
coding techniques in the coded MFSK/FH-SSMA system in order to improve the channel capacity. In this proposed system we treat a bit-to-symbol converter and a frequency hopper as an encoder and concatenate them with a channel code. II. S YSTEM M ODEL Fig. 1 shows the block diagram of the transmitter and the receiver of proposed system which employs iterative decoding scheme. We consider a time-frequency coded system where the transmitted signal from each user is a sequence of L tones, which are selected from M possible frequencies, and K users transmit FH signals using their distinct addresses. Channel coded bits are interleaved, for which a random interleaver is assumed in this paper. q (= log 2 M ) bits are grouped together and converted into M -ary symbols by the bit-to-symbol converter through the equation m
=
q 1 X
bi 2i ;
(1)
i=0
where bi denotes the ith bit and m denotes the transmitted symbol. This M -ary symbol decides which frequency is transmitted among M frequencies. The transmission time T is divided into L time slots of duration T c = T =L each. The overall bandwidth required in this system is M=T c = M L=T as the required minimum frequency separation between adjacent frequency bins for noncoherent detected orthogonal FSK is 1=T c.
7
0
6 5
0
4
1-PF
3
1-PF1 -PFE
0 PD0
0 PD
PFE
PF
PDE
E
2 1
1
0
(a)
(b)
(c)
1-PD
1
(d)
1
Fig. 2. Time-frequency plane matrices: (a) data matrix, (b) transmitted matrix T, (c) received matrix R, (d) despread matrix D.
Each user has a distinct address occupying L chips. We take randomly selected addressing in this paper, as its performance is comparable to that of Einarsson’s optimal addressing when M; L, and K are large-valued. Let ak = (ak;1 ; ak;2 ; ; ak;L ), ak;i 2 GF (M ) denote the address of user k , and we assume that the k th user transmits symbol m 2 GF (M ), where GF (M ) indicates a finite field of M elements. Then, the transmitted FH pattern is represented by x = ak m 1; (2) where 1 (1; 1; ; 1). If M is a prime number, the rule of addition is ordinary addition with modulo M , but otherwise the rule is different. The FH patterns are transmitted by the frequency synthesizer which generates actual frequency sequences corresponding to the components of x. The channel is modeled as a frequency selective Rayleigh fading channel. All chips are assumed to be faded independently and aligned in time. The received signal is noncoherently detected by M filter banks matched to the M frequencies. The output values of the filter banks are hard-limited and despread according to the user’s address code. The resulting outputs are M L binary values. Time-frequency planes (i.e., an M L matrix) are used to describe the transmitted and the received signals [2]. The transmitted signal, the received signal after hard-limiting, and the despread signal are listed in the transmitted matrix T, the receiver matrix R, and the despread matrix D, respectively. M rows represent the corresponding frequencies, and L columns represent the chips. Fig. 2 shows an example of the original data, the transmitted, the received, and the despread matrices. In this example, the user transmits a symbol, ‘1’. If the user’s address code is (1, 6, 0, 3) and is an addition with modulo 8, then the transmitted pattern is (2, 7, 1, 4). This FH pattern is shown in Fig. 2 (b). If the output of the detector exceeds the decision threshold, the received matrix takes the shape shown in Fig. 2 (c): There are three users (; 2; 4) who share this FH channel. A false alarm (FA) occurs (which is indicated by ), if a chip is not actually transmitted but noise power exceeds the decision
PF1
(a)
1- PD0 - PDE
1
(b)
Fig. 3. Discrete memoryless channel model of FH system: (a) 2-level hard decision, (b) 3-level hard decision.
threshold. If the power drops below the threshold due to severe fading, no entry is put in the corresponding slot (which is indicated by ). A user-specific despreading yields the matrix whose M rows correspond to the M possible transmitted symbols of the desired user. For the decision, a majority voting is used, and a random decision is made in the case of a tie. Fig. 2 (d) illustrates the despread matrix for the case when rows 1 and 3 have three entries. III. R ELIABILITY INFORMATION We modify the soft decision decoding method [3] to incorporate the iterative decoding. In addition to the 2-level hard decision, we consider the 3-level hard decision as well, in order to avoid the ambiguous bit problem in the 2-level hard decision. A. 2-Level Hard Decision The reliability information on each received bit can be derived from the 2-level hard decision [3]. Let n m ; m = 0; 1; ; M 1, denote the number of entries in the mth row of D, and let Ntot be the total number of entries in D. For a normalized receiver threshold Æ , each transmitted chip 2 is deleted with the probability P D = 1 e (Æ =2(1+Ec =N0 )) , where N0 is the one-sided noise-power spectral density. The combination of two or more chips of interferers always yields an entry. An entry may be yielded due to noise even if no chip 2 was transmitted, and this probability is P F = e (Æ =2) . Fig. 3 (a) shows the transition probabilities of the discrete memoryless non-symmetric channel model of 2 inputs and 2 outputs. The probability P I that an entry occurs which does not correspond to a chip of the desired user is PI
= (P
P1 ) + P1 (1
PD ) + (1
P )PF ;
(3)
where P and P1 are the probabilities that a particular element in D corresponds to at least 1 chip and to exactly 1 chip, re-
spectively, of interferers. They take the expressions P P1
= 1 (1 1=M )K 1 ; 1 K 1 = (K 1) 1 M
M
P{m|D}
(4) 2
:
(5)
The probability pfDjmg for a given m is p
fDjmg = PIN n (1 PI ) M (1 PD )n PDL n tot
(
m
1)L (Ntot
m
m
nm )
:
(6)
We obtain the a posteriori probability
f jDg = PMpfDjmgP fmg = PMPy (nm )P fmg pfDjm e gP fme g Py (nm eg e )P fm m e m e
p m
1 =0
1 =0
[(1 PD )(1 where Py (nm ) = (PD PI ) P m is the a priori probability of symbol m, L nm
f g
PI )]
nm
jmk P fmg = P (bj ); bj = 2j mod 2: j =0 Y
(7) and
q 1
(8)
Computation of (7) and (8) is done in the symbol APP(a posteriori probability) calculator in Fig. 1 (b). When performing iterative decoding, we use the extrinsic values from the channel decoder for P (b j ). Since soft-decision decoding of binary codes requires reliability information on each bit rather than that on each received symbol, we need to derive the probability P (b j = 0jD). Considering the binary-to-M -ary conversion mapping rule (1) we obtain P (bj
= 0jD) =
X
(M=2) 1
r =0
f jDg;
p m
m
j k
= r + 2rj 2j :
(9)
Finally, we consider the log-likelihood ratio
1jD) ln P (bj = 1) j (D) = ln P (b = 1 jP=(b 0j=D)0jD) P (bj = 0) = ln P (b =j 0jD) fj ; j P (bj
(10)
fj is the log-likelihood of the a priori probability of the where bit bj . The reliability information j (D) is used as the input to fj to calculate the channel decoder. Note that we already used the first term in the right hand side in (10) by using (7) and (8), fj in (10) not to reuse it in the channel decoder. so we subtract We use the extrinsic values that the channel decoder generates fj and decode the encoded bits iteratively. for Fig. 4 shows an example of extracting reliability information. In Fig. 4 (a), there are two rows that have the largest number of entries in D and cause the largest values for the corresponding pfmjDg, namely, pf1jDg = pf3jDg = 0:474.
P{m|D}
7
2.50 · 10-2 7
3.18 · 10-2
6
1.32 · 10-3 6
1.38 · 10-3
5
6.99 · 10-5 5
5.96 · 10-5
4
6.99 · 10-5 4
5.96 · 10-5
3
4.74 · 10-1 3
2
1.32 · 10-3 2
1.37 · 10-3
1
4.74 · 10-1 1
7.33 · 10-1
0
2.50 · 10-2 0
3.18 · 10-2
E
(a)
2.00 · 10-1
(b)
D
Fig. 4. Despread matrix and P fmj g corresponding to each row: (a) 2-level hard decision, (b) 3-level hard decision.
Using (9) and (10) and assuming that all the symbols are equally likely, we obtain the reliability information 0 (D) = 3:55; 1 (D) = 0:005, and 2 (D) = 3:60. So bits b0 and b2 are reliable but bit b 1 is unreliable. Notice that the rows corresponding to symbols 1(=001 (2)) and 3(=011(2)) have the largest number of entries and are associated with bits b 0 = 1 and b2 = 0, providing these bits with high reliability. But symbol 1 is associated with b1 = 0 and symbol 3 is associated with b1 = 1, so bit b1 is unreliable. B. 3-Level Hard Decision To alleviate such an unreliable bit problem, we adopt a 3level hard decision rule. When a 3-level hard decision is adopted instead of the 2-level hard decision, the above equations should be modified accordingly. It is natural to expect a better performance when ‘erasure’ region is inserted between ‘0’ and ‘1’ decision regions. Then the continuous channel reduces to a discrete memoryless channel (DMC) with 2 inputs and 3 outputs. (See Fig. 3 (b).) If the output of a matched filter is less than the first threshold Æ1 , the received symbol is decided to be ‘0’; if the output is greater than the second threshold Æ 2 (Æ2 > Æ1 ), it is decided to be ‘1’; otherwise it is decided to be erasure ‘E’. Let nm;1 ; nm;E ; m = 0; 1; ; M 1 denote the number of ‘1’ and ‘E’ in the mth row of D, respectively. And let N 1 and NE be the total number of ‘1’ and ‘E’ in D. If the transmitted chip is severely faded, the received signal may not be greater than Æ 2 , i.e., a deletion occurs with the probabilities PD0 and PDE , PD0 PDE
=1 =1
e
2 =2(1+E =N )) (Æ1 c 0
e
(Æ22 =2(1+Ec =N0 ))
;
(11) PD0 ;
(12)
where PD0 is the probability that the signal is less than Æ 1 and PDE is the probability that the signal is greater than Æ 1 but less than Æ2 .
Even if no chip is transmitted, noise may cause false alarm with the probabilities PF1
Æ22 =2
=e
;
PFE
=e
Æ12 =2
PF1 ;
(13)
where PF1 is the probability that the received noise is greater than Æ2 and PFE is the probability that the received noise is greater than Æ 1 but less than Æ2 . The probability P I1 that ‘1’ occurs which does not correspond to a chip of the desired user is
= (P
PI1
P1 ) + P1 (1
PDE ) + (1
PD0
P )PF1 (14)
and the probability P IE that ‘E’ occurs which does not correspond to a chip of desired user is PIE
= P1 PD + (1
P )PFE ;
E
(15)
where P and P1 are as defined in (4) and (5). The probability pfDjmg is calculated as in the case of the 2-level hard decision:
fDjmg = PIN1 1 n 1 PIN n (1 PI1 PI ) M L (1 PD0 PD )n 1 PDn m;
p
E
m;E
E
(
(N1
1)
E
E
m;E
m;
E
nm;1 ) (NE nm;E ) L nm;1 nm;E
PD0
;
(17)
1
with
= PIN1 1
L
N
PIEE
L
(1
PIE )(M
PI1
1)L N1
NE
PD0L ;
(18)
Py (nm;1 ; nm;E )
= (PD0 PI1 )L n 1 (PD0 PI )L n [(1 PD0 PD )(1 PI1 PI )]n [PD (1 PI1 PI )]n : m;
m;E
E
E
E
E
E
m;1
m;E
(19)
Based on this, we obtain the a posteriori probability
f jDg = PMpfDjmgP fmg pfDjm e gP fme g m e Py (nm; ; nm;E )P fmg = PM : Py (nm; eg e ; nm;E e )P fm m e
p m
1 =0
1
1 =0
The reliability information, j (D) is transferred to the channel decoder after being deinterleaved. The channel decoder also generates the reliability values of the bits using the channel coder constraints. The optimal MAP decoder, i.e., BCJR decoder [7] can be used for the soft input soft output (SISO) decoder. The reliability information that the channel decoder generates is fedback to the symbol APP calculator as shown in Fig. 1 (b): It is first interleaved, and then transferred to the symbol APP calculator. The calculated output is delivered to the symbol-to-bit converter. Its deinterleaved output is used as a priori probabilities to regenerate new reliability information in equations (7) and (20). We may iterate this decoding process, thereby increasing the reliability of a posteriori probability of the bits in the channel decoder. IV. S IMULATION R ESULTS We have conducted simulations with the numbers M = 32, = 6, Eb =N0 = 25dB to examine the performance of the proposed iterative decoding scheme of the FH-SSMA system. We generated binary data randomly and encoded them using the channel encoder. We used the data frame consisting of 192 bits and the generator polynomials of the convolutional codes and turbo codes (35,23) in octal. We used 4 tail bits to terminate the code and appended 3 stuffing bits to the encoded (192+4) 2 bits to make the resulting bits per frame divisible by log 2 M . Then the size of random interleaver is 395. As it is hard to derive the optimal decision thresholds, we selected them by trial-and-error. We used BCJR decoder for channel decoder to generate reliability information from channel coder constraints. Figs. 5, 6, and 7 plot the resulting BER performances for the proposed system. In the figures, the numbers 1 to 6 denote the number of iterations of the proposed iterative decoding system. Fig. 5 plots the performance of the turbo coded system. Ref in Fig. 5 indicates the conventional turbo coded system with no iterative decoding and the turbo decoding is carried out with five iterations [3]. We can find that the iterative decoding brings forth performance improvements. Figs. 6 and 7 plot the performance of the convolutionally coded system with the 2- and the 3-level hard decisions, respectively, where 1 iteration implies a conventional decoding which can be used as a performance reference. The figures show that the iterative decoding yields significant performance gain but the gain saturates after about 6 iterations. We find that at the BER of 10 4 , the conventional system with no it-
L
fDjmg = Px Py (nm; ; nm;E )
Px
C. Iterative Decoding
(16)
which can be easily decomposed into two parts, p
Symbol 1 can be considered as the most likely transmitted symbol, so ambiguity problem does not occur. Using (9) and (10) and assuming all the symbols equally likely, we obtain the reliability information 0 (D) = 3:33; 1 (D) = 1:18, and 2 (D) = 3:37. So all the bit b0, b1 , and b2 are reliable and the ambiguous bit problem in the previous 2-level hard decision is resolved.
(20)
1
After carrying out M -ary-to-binary conversion using equation (9), we can calculate the log-likelihood ratio j (D) as in equation (10). The reliability information j (D) is used as the input to the channel decoder. Fig. 4 (b) shows an example of extracting reliability information in the 3-level hard decision case. The additionally detected symbol in symbol 3 is detected as an erasure in the 3-level decision and P fmjDg’s are changed according to (20).
10
-1
10
10
1
-1
-2
1 10
-2
10
2
10
-3
-3
BER
BER
Ref
3 4 5
10
2
-4
3 10
-5
4 5
10
-4
21
10 22
23 Users
24
Fig. 5. BER performance of a turbo coded system with 2-level hard decision. (Numbers 1-5 denote the number of iterations and Ref denotes the performance reference. M = 32; L = 6; Eb =N0 = 25 dB.) 10
-6
16
25
17
18
19
20
21
22
23
24
25
Users
Fig. 7. BER performance of a convolutionally coded system with 3-level hard decision. (Numbers 1-5 denote the number of iterations. M = 32; L = 6; Eb =N0 = 25 dB.)
-1
APP calculator generates the a posteriori probabilities using the fedback reliability information, which implies that the de10 coder fully utilizes the information of the received symbols. 1 Through simulations we have confirmed that the iteration processing yields significant performance improvements over the 10 2 conventional system: In terms of the number of accommodated 3 users, the iterative decoding system has increased the num4 5 10 ber by about 50 %, from 16 to 25. We have observed that 6 the convolutionally coded system outperforms the turbo coded system in the FH-SSMA environment, which contrasts to the 10 tendency in the AWGN channel at the low SNR. In addition, the 3-level hard decision did not produce much performance 10 gain over the 2-level hard decision. Consequently, the convo16 17 18 19 20 21 22 23 24 25 lutionally coded system with the 2-level hard decision turned Users out best-performing among all iterative decoding systems we Fig. 6. BER performance of a convolutionally coded system with 2-level hard considered in the simulation. decision. (Numbers 1-6 denote the number of iterations. M = 32; L = -2
BER
-3
-4
-5
-6
6;
Eb =N0
= 25 dB.)
R EFERENCES erative decoding accommodates 16 users, which increases to 25 for the proposed system with iterative decoding. We obtain a larger gain in the convolutionally coded system than in the turbo coded system, when we apply the proposed iterative decoding rule. The systems with the 3-level hard decision exhibit only a slightly better performance gain than the 2-level hard decision case. This indicates that the 2-level hard decision may be sufficient to extract the reliability values from the symbols. Therefore we may conclude that the convolutionally coded system with the 2-level hard decision performs best among the three systems. V. C ONCLUSION In this paper we have presented an iterative decoding technique in the FH-SSMA system. In the system, the symbol
[1] [2] [3]
[4] [5] [6] [7]
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