SIMULATION OF ERROR TRAPPING DECODERS ON A FADING CHANNEL

Transcription

1 Proc. IEEE Vehicular Technoloy Conference, Atlanta, GA, 28 Ar.-1 May 1996, vol. 2, SIMULATION OF ERROR TRAPPING DECODERS ON A FADING CHANNEL G. Sharma, A. Dholakia, and A. Hassan ECE Det., North Carolina State University, Raleih, NC IBM Cor., 3039 Cornwallis Rd., Research Trianle Park, NC Ericsson Inc., 1 Trianle Drive, Research Trianle Park, NC ABSTRACT In this aer, we evaluate the erformance of error train decodin when used to rotect data over a moile radio channel. The channel is reresented y the Gilert-Elliott(GE) model, with its arameters related to the hysical quantities determinin the fadin statistics. Simulations are used to evaluate codeword and it error roailities for three different error correctin codes for a numer of scenarios. Usin simlyin assumtions, analytical exressions are derived for the codeword error roailities and these are shown to e in excellent areement with simulation results. 1. INTRODUCTION The received sinal enveloe in a moile diital cellular system is known to dislay Rayleih statistics. This Rayleih fadin is characterized in the diital domain y a channel havin urst errors. The Gilert-Elliott model [3, 4] rovides a useful discrete model for such a channel where the arameters of the model can e readily related to the statistics of the fade [1, 6]. The simlicity of the model makes it attractive for analyzin the erformance of error control codes throuh simulations or throuh exact/aroximate analysis. Analytical exressions for error roailities were first derived in [3, 4] and more recently in [8, 9] for a simlified versions of the GE model. However, all the analytical exressions were aimed at determinin codeword error erformance for random error correctin codes and it error statistics and urst error correctin codes were not considered. In this aer, we consider the erformance of three inary codes: the (23,12) Golay Code and a (14,6) cyclic urst error correctin code (BECC) with urst correctin caaility (BCC) of 4, and a (23,13) shortened cyclic code with BCC of 5. The erformance of the three codes on a Gilert-Elliott channel are comared throuh simulations for BPSK and DPSK modulation schemes. We also resent a simli- This research was erformed while A. Dho lakia was at NCSU. fied analysis for a restricted rane of model arameters and comare the theoretical results with simulations. 2. CHANNEL MODEL The Gilert-Elliott(GE) model [3, 4] was chosen to model the fadin channel in the simulations. In the model, the channel is a inary symmetric channel (BSC) with memory determined y a two state Markov chain. This model is shown schematically in Fi. 1. A channel has two states, a ood (G) state and a ad (B) state with transition roailities α and β as shown in the fiure. In either state the channel is reresented as a inary symmetric channel (BSC) with the roaility of it error iven y intheoodstateand in the ad state. 1 β G α β B 1 α Fiure 1: The Gilert-Elliott Channel Model. Assumin that the channel fades slowly with resect to a it interval the arameters of the model can e related to various hysical quantities. To otain such a relation, note that Rayleih fadin results in an exonentially distriuted multilicative distortion of the sinal. As a result, the roaility density function of the sinal-to-noise ratio (SNR),λ, is iven y [1, 6] f(λ) = 1 λ 0 ex( λ λ 0 ) λ>0 (1)

2 where λ 0 is the averae SNR. The channel is said to e in the ood(g) state while the SNR is aove a threshold λ T and once the SNR falls elow λ T the channel oes into the ad(b) state. Usin the level crossin rate and the SNR density function, the transition roailities can e found in terms of hysical quantities as follows [1, 6]: α = f d T 2 π Γ (2) ex(γ) 1 β = f d T 2 π Γ (3) where T is the symol interval, Γ = λt λ 0 and f d is the maximum Doler seed iven y f d = vf c c, where v is the vehicle seed, f c is the carrier frequency, and c is the seed of liht ( m/s). The symol interval is usually secified in terms of the symol rate R s = 1 T. The it error rates in each state are determined for a iven modulation scheme y aroriately averain the for an AWGN exressed as a function of SNR. Thus if E(λ) is the for an Additive White Gaussian Noise (AWGN) channel with SNR of λ for the iven modulation scheme then, λ = T E(λ)f(λ) dλ (4) λ T f(λ) dλ = λt 0 E(λ)f(λ) dλ λt 0 f(λ) dλ (5) We shall concentrate on two modulation schemes BPSK and DPSK. Simlified exressions for and for these modulation schemes are rovided in [6]. 3. SIMULATION METHODOLOGY Three linear lock codes were chosen for erformance evaluation 1. The first is the cyclic Golay (23, 12) code with a minimum distance of 7 which has the enerator olynomial, 1 (x) =x 11 + x 10 + x 6 + x 5 + x 4 + x The other two codes chosen were urst error correctin codes with redundancy and lock lenth similar to the Golay code. The second code is a cyclic (14, 6) urst error correctin code from [2,. 152] caale of correctin all error ursts havin urst lenth 4 or less. The enerator olynomial for this code is 2 (x) =x 8 +x 6 +x The third code chosen for study 1 Note that for the study of linear lock codes an iid it-stream enerator is unnecessary for simulations. was a (23,13) code otained y shortenin a (341,331) cyclic urst error correctin code with urst correctin caaility of 5, havin the enerator olynomial 3 (x) =x 10 + x 8 + x 7 + x 5 + x 4 + x Systematic encodin was used for all codes. Decodin for the Golay code was erformed usin the Meitt decoder otained y modifyin an error train decoder [2]. Since the Golay code is a erfect code this corresonds to comlete decodin and hence there were no uncorrectale error events detected. For the urst error correctin codes error train decoders were used. For the (14, 6) cyclic urst error correctin code (BECC) the error train decoder corrects all ursts of cyclic lenth 4 or less. Since the shortened (23,13) code is not cyclic, it is caale of correctin only non-cyclic ursts with urst lenth 5 or less. For oth the urst error correctin codes, the error train decoder erforms incomlete decodin and there are error events detected that cannot e corrected. In these cases, the codeword was declared to e in error and the information its were assed on for comutation of it error statistics without any correction. 4. SIMPLIFIED ANALYSIS For tyical vehicle velocities and symol rates for American Diital Cellular and GSM, it can e seen that the channel arameters α and β in (2) and (3) are extremely small in relation to the inverse of tyical codeword lock lenths. As a result, for these scenarios the transitions etween states are extremely infrequent, and one may assume that for the duration of a codeword the channel remains in the state in which the codeword ean. With this aroximation, the roaility of codeword error on the GE channel is readily otained as P w = π G P c ( )+π B P c ( ) (6) where π G = α/(α + β) andπ B = β/(α + β) arethe steady state roailities of ein in the ood and ad state, resectively, and P c () reresents the roaility of codeword error on a BSC with symol error roaility. For an (n,k) random error correctin code caale of correctin t random errors, P c () P REC c (n, t, ) =1 t i=0 ( ) n (1 ) n i i. (7) i The corresondin exression for a BECC caale of correctin all cyclic ursts with urst lenth (BL) l is otained from, P c () P CBEC c (n, l, )

3 = 1 Pro(No error) Pro(BL <land i errors). Since on a BSC all error atterns with i errors are equiroale Pro(BL <l i errors) = n( (l 1)) (i 1) ( n, (8) i) from which it follows that, P CBEC c (n, l, ) = 1 (1 ) n ( ) (l 1) n (1 ) n i i. In a similar fashion, it can e shown that for a noncyclic BEC with BCC of l, P c () Pc BEC (n, l, ) = 1 (1 ) n ( ) (l 1) (n +1 l) (1 ) n i i l i ( ) (l 1 r) (1 ) n i i. (9) r=1 5. SIMULATION RESULTS Usin the channel model, Monte-Carlo simulations were erformed. The threshold Γ was set to 0.1 sothat an SNR 10 db elow the averae SNR reresents the transition to the ad state. In order to reresent a wide rane of moile communication environments, the roduct f d T was considered as an indeendent arameter and simulations were erformed for f d T =, 0.01, 0.05, and 0.1. These values encomass the whole amut of moile communication environments ranin from cellular telehony to PCS. Two modulation schemes, BPSK and DPSK, were considered in the simulations. For each comination of these arameters simulations were erformed for an averae SNR varyin from 0 db to 26dB in stes of 2 db and codeword and it error rates were estimated. For extremely small values of α and β channel transitions may ecome extremely rare events, hence the simulations determined the codeword error and it error roailities for codewords einnin in a iven state. These were then weihted y the steady state roaility of the corresondin state and added to otain the overall codeword and it error roailities. Error Proaility CER DPSK CER BPSK DPSK BPSK Fiure 2: CER and estimates for the Golay code with BPSK and DPSK. We first consider the difference in erformance of the BPSK and DPSK modulation schemes. Fiure 2 contains a lot of the averae SNR vs. estimated it error rate () and codeword error rate (CER) for the Golay code for f d T =0.01, for BPSK and DPSK modulation schemes. From the rahs one can see that over the moderate and hih SNR reion BPSK is 3 db etter than DPSK. This result is in areement with the uncoded it error roailities for BPSK and DPSK on a fadin channel [7,. 718]. The results for other codes and arameter values have similar characteristics. Hence, in the susequent discussion results are resented only for BPSK modulation. One can also see from the lots that the codeword and it error rates show identical trends and hence we shall use only the estimates in most of the followin discussion. Next we consider the imact of the arameter f d T on each of the codes. Fiure 3 contains lots of the estimated it error rate () vs. averae SNR for the Golay code with BPSK modulation, for f d T =, 0.01, 0.05, and 0.1 and Fi. 4 and 5 contain the corresondin lots for the (14,6) cyclic urst error correctin code and the (23,13) urst error correctin code, resectively. From the fiures, it is clear that an increase in f d T imroves the erformance of the Golay code sinificantly whereas it leads to only marinal imrovement in the erformance of the two urst error correctin codes. These results aree with the intuition that the errors tend to e more random as the channel arameters α and β increase. Since the Golay

4 Fiure 3: for the Golay code with BPSK. code is caale of correctin random errors it enefits sinificantly from the increase in randomness whereas the urst error correctin codes make only marinal ains. We now comare the erformance of the different codes. Fiure 6 contains a lot of the vs. averae SNR for the three codes for BPSK modulation and f d T =0.01. From the lots one can see that for these arameter values the erformance of the three codes is similar for values of the averae SNR elow 12 db. Aove 12 db the erformance of the two urst error correctin codes is similar while that of the Golay code is sinificantly etter. For the channel model that we are considerin, at hih SNR s the roaility of error in oth the ood state and the ad state decreases. As a consequence, the errors at hih SNR s are less correlated. This works once aain to the advantae of the Golay code caale of correctin random errors, whereas the urst error correctin codes only make marinal ains due to the decreased numer of symol errors. Since ursts of lenth 4 and 5 will rarely have more than 3 symol errors the Golay code erforms etter or as well as the urst error correctin codes in most scenarios. In this reard, it is worth notin that a urst error correctin code will ive otimum erformance over a channel roducin ursts close to its urst correctin caaility, whereas a random error correctin code is more roust to channel arameter variations. Finally, we investiate the validity and accuracy of the simlified exressions for codeword error rates derived in section 4. Fiure 7 contains lots of comar- Fiure 4: for the (14,6) BECC code with BPSK. Fiure 5: for the (23,13) BECC code with BPSK.

5 000 Modulation BPSK (23,13) BECC (14, 6) BECC (23,12) Golay Pro. Codeword Error (23,13) BECC (14, 6) BECC (23,12) Golay Theoretical Curves Fiure 6: for the three codes with BPSK and f d T =0.01. Fiure 7: Comarison of theoretical CER estimates with simulations. isons of simulations and the simlified theoretical exressions for the three codes with BPSK modulation and for f d T = For this case, α and β and hence the assumtion that the channel rarely chanes state in the middle of a codeword is valid and therefore the theoretical results aree extremely well with the simulations. However, for hiher values of f d T =0.01 the assumtion is no loner valid and hence the simulation results deviate sinificantly from the simlified theoretical exressions. 6. CONCLUSIONS In this aer, we simulated the erformance of three linear lock codes on a fadin channel modeled y the Gilert-Elliott model and considered the imact of several arameters on the erformance of the codes. The relative erformance for different modulation schemes was seen to e in areement with their uncoded it error rates. For the arameters considered the random error correctin Golay code erformed etter than the urst error correctin codes. For some channel arameter values of our interest, analytical exressions for codeword error roaility were derived and found to e in excellent areement with the simulations. 7. REFERENCES [1] L. Ahlin, Codin Methods for the moile radio channel, Nordic Seminar on Diital Land Moile Communication, Fe. 1985, Esoo, Finland. [2] R.E. Blahut, Theory and Practice of Error Control Codes, Addison-Wesley, Menlo Park, California, [3] E.N. Gilert, Caacity of a urst noise channel, Bell Syst. Tech. J., vol. 39, , Set [4] E.O. Elliott, Estimates of error rates for codes on urst-noise channels, Bell Syst. Tech. J., vol. 42, , Set [5] S. Lin and D.J. Cosetllo,Jr., Error Control Codin: Fundamentals and Alications, Prentice Hall, NJ, [6] R. Krishnamurthi, An Analytical Study of Block Codes in a Portale Diital Cellular System, Ph.D. Thesis, SMU, [7] J.G. Proakis, Diital Communications, 2nd ed., McGraw Hill, [8] B. Won and C. Leun, On comutin undetected error roailities on the Gilert channel, IEEE Trans. Comm., vol. 43, no. 11, , Nov [9] J.R. Yee and E.J. Weldon, Jr., Evaluation of the erformance of error correctin codes on a Gilert channel, IEEE Trans. Comm., vol. 43, no. 8, , Au