T where p($i, 92 , 63 , 94) is the joint probability density function (jpdf) of the differential phase errors 6\\, 92, 93 and 94 (see Fig. 4.3). The rest of the error probabilities in the set (P\"} can •/ / / Chapter 4. Performance of -x/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 102 be found by a similar fashion. For the determination of the occurrence probabilities of having (L + j) specific errors, the pdf of the set of the differential phase errors, i.e., #i,#2,03, and 94 is required and will be derived next. RJ I / / \\ 7 Ry^ \\ Ro X0 Xl *2 x/ In-phase x Figure 4.3 Phasor illustration of the relationship between the phasors and the corresponding differential phase errors. Because of the independency of the in-phase (xo ,x i ,x 2 ,X3,x 4 ) and the quadrature-phase (2/0)2/1? 2/2, 2/3,2/4) components, their jpdf of can be expressed as /(xo, X ! , x 2 , x 3 , x4,2/0,2/1,2/2,2/3,2/4) = /(zo, z i , x 2 , x 3 , x 4 ) • 5(2/0,2/1,2/2,2/3,2/4)- (4.18) Following [57], the pdf / ( x o , X ! , x 2 , X 3 , x 4 ) and 5(2/0,2/1,2/2,2/3,2/4) which are jointly Gaussian can be expressed as / ( x o , x i , x 2 , x 3 , x 4 ) = (27r)5det(K) 2 • exp - ( l /2)(x - s ) K _ 1 ( x - s) 5(2/o,2/i,2/2,2/3,2/4) = [(2TT)5 det ( K ) H - exp [ - ( l / 2 ) y K - 1 y T ] (4.19) Chapter 4. Performance oftr/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 103 where x = {xo,xi,X2,xz,x4), y = (2/0,2/1,2/2,2/3,2/4 det(-) denotes the determinant and (-)T denotes the transpose of a vector. K - 1 is the inverse of the covariance matrix K which is given as K = (D + N) • 1 h k2 k3 &4 h 1 fci k2 k3 ki 1 ki k2 k3 k2 ki 1 ki . k4 h k2 ki 1 . with k1 = [D/(D + N)]pfT) k2 = [D/(D + N)]p(2T) h = [D/(D + JV)H3T) k4 = [D/(D + N)]p(4T) (4.21) where p(-) was previously defined in (4.8). The determinant of K is given by (D + Nfc0 (4.22) where CQ is given in the Appendix A. Referring to Fig. 4.3, the in-phase, x, and the quadrature-phase, y, components can be expressed in polar form by using the following substitutions x0 = Ro cos ( o2) 2/i ----- Rr sin ( — o2) x2 = R2 cos 2/2 = i?2 sin (j> Chapter 4. Performance of itl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 104 x3 = R3 cos(4> + 93) y3 = fl3 sin (<£+ 03) X4 = R4 COS (

+ 94). (4.23) Substituting (4.19), (4.20) and (4.23) into (4.18) and by using the software Maple® for some algebraic simplification, f(Ro, Ri,R2,R3,R4,0i, 92,03,04,) R0R1R2R3R4 exp (2TT ) 5 ' ( /J> + N)5c0 [{D + N)c0 S f -1 exp j (j) .j. jV)co^^°' ^ ' • exp j ( I ? ^ ) e o E ( i Z o , RUR2,R3,R4, &i, 02,03,04, <£) j (4.24) where the constant A is given in the Appendix A and the variables B(RQ, RI, R2,R3,R4,0\\,92,03,94), E(RQ,Ri, R2, R3, R4,9i,92 , 93 , 94, ) are defined as follows B(R0, R\\, R2, R3, R4,9i,92 , 93 , 94) = c7 (Rl + R\\) + c8 (R2 + Rl) +CeRl+ c17R0R3 cos (01 + 6a) + CIORQRI COS (9\\ - 92) + C11R0R4 cos (0i + 94) -f C\\2R\\R2 cos92 + ci3R\\R4 cos (92 + 94) + c-i4RiR3 cos (02 +63) + c15R2R3 cos63 + ci&R3R4 cos (93 - 94) + c\\7R2R4 cos 04 + ci&R0R2 cos 9i (4.25) and E(Ro,R-y,R2, R3, R4,9i,92 , 93 , 94, - 9i) + c2R4 cos ( + 04) + c3R3 cos (<£ + 93) + c4i?i cos ((/> - 92) + c5R2 cos (j> (4.26) Chapter 4. Performance of Tr/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 105 where the mathematical expressions for the constant set {cj\\j — 1,2, • • •, 18} are given in Appendix A. Expanding (4.26), E(-Ro, Rl,R2i #3, #4,01, #2, 03, 04,) = (Ci Ro COS 01 + C2.R4 COS 04 + C3R3 COS 63 + C4R1 COS 92 + C5R2) COS + (c\\Ro sin#i + C2-R4 sin94 + C3R3 sin 03 + C4.R1 sin 92) sin0. (4.27) With the relationship [71] Pi cosw - P2 sinw = \\JPi + P22 cos (4.27) can be rewritten as (4.28) E(#O,#1,#2,#3,#4,01,02,03,04,<£) = v/Z(i^ o, #1,^2, #3, #4, ^l, 02,^3,04,^) COS ( + A) (4.29) where A is a dummy variable and Z(Ro, Ri, i?2, -R3, #4 , 0i, 02,03> 04, <£) is given by Z(i?o, R11R21R31 RAI 0I , 02,03,04, 4>) — (c-iRo cos #i + C2-R4 cos $4 + C3R3 cos#3 + C4JRI cos 62 + C5R2)2 + (ci Ro sin 0! + C2R4 sin 94 + c3R3 sin 03 + c4i?i sin 92 )2. Finally, by using (4.25) and (4.29), (4.24) can be rewritten as (4.30) f(Ro,Rl,R2,R3,R4 , 01, 02 , 03 , 04, ) S RoR\\ R2R3R4 5 / r . • Arx5 e XP (2zf(D + A/)5 Co \\ / 2 5 (D + N)c0 exp J - 1 \\ ( B + JV)co B(Ro,R\\ , i?2, - R 3 , #4 , 01 , 02 , 03 , 04) exp (D + N)co y/Z(Ro, Rr,R2,R3, R4,0i, 02 , 03 , 04) cos (0 + A) 1. (4.31) By integrating from 0 to 27r, along with the use of the relationship that for any A [72] 2ir I0(x) = J exp [x cos ( + A)]#. (4.32) 0 Chapter 4. Performance of ir/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 106 where IQ is the modified Bessel function of the first kind and zeroth order, (4.31) can be written as f(Ro,Ri,R2,Rs, #4, 9\\ , 02 , 03 , 04) RQ RI i?2 R3 R4 = v 4 / n , *n5 e X P (27r)4(D + Nfco L (P + iV)c 0 exp { -1 (D + N)c0 B(RQ, RI, i?2, R3, -#4, #1, #2, #3, 04) °^ J (P +\"/V')Co ^(-Ro, -Rl, i?2, #3, #4, #1, 02, ^3, #4) J • (4.33) Since it would be very tedious and time consuming to do the integration of the (4.33) from 0 to 00 through 5 dimensions, the .Ro, -Ri, -R2, #3 , -R4 would be replaced by using the following change of variables R0 = Rcos(fi/2) Ri = R sin (/t/2) sin (A/2) sin (t/2) cos (77/2) i?2 = .R sin (/t/2) sin (A/2) cos (t/2) -R3 = .ft sin (/t/2) sin (A/2) sin (t/2) sin (n/2) i?4 = # sin (/t/2) cos (A/2). (4.34) and the following transformation of variables dRQdRidR2dRzdR4 = \\ J\\ dRdp, dt dX drj (4.35) where J is the Jacobian matrix and is given by R4 \\J\\ — -yr sin (/t/2) sin (t/2)(l - cos/t)(l - cos A). (4.36) By substituting (4.34) - (4.36) in (4.33), followed by some algebraic manipulations along with the use of the software Maple®, (4.33) can be given as f(R, / i , A, t, 77,01,02,03,04) .R9[sin p, sin (A/2)]2 sin 77 sin A sin t (1 - cos t)(l - cos /t)2 211(-D + iV)5co(27r)4 •exp [-aR2] -I0[bR) exp (D + N)c0' (4.37) Chapter 4, Performance of itl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 107 with F(p,X,t,r1,91,92,93,94) a = 2(D + N)co b = ]j(D + N)c0 A' V'9l' °2' *3' ^ ( 4 ' 3 8 ) where F(p,X,t,r],0i,92 , 93 , 94) and G(/t,A,t,77,0X,i92,03,04) are given as F(p,X,i,ri,91,92 , 93 , 94) = c9a + cio/3 + C117 + c i 8 r + c 1 2 £ + c 1 3x + c14ip + c16p + C i 7 c r + (c6 - c8)C + ( c 8 - c 7 ) t 7 + 2c7 (4.39) and G(p,\\,i.,ri,e1,o2,o3,e4) = C o ( D + N ) • {°- 5( c 25 - CIK + (cl ~ - O- 5! 1 - c o s ^ )]°- 5( 1 - c o s M ) + c? + c2 - 0.5c2(l - cosp.) + 0.25(c4 - c§)(l + COST?)(C7 - C) + CiC3a + c1c4/3 + C1C27 + cic5r + c3c5f + c 4c 5£ + c2c5cr + c3c4ip + c2c3p + c2c4x}- (4.40) The expressions for the constant set {cj\\j = 0,1, • • •, 18} in (4.39) and (4.40) is given in the Appendix whereas the rest of the variables are denned as a = sin \\i sin (A/2) sin (t/2) sin (77/2) cos (61 + 93) /? = sin/i sin (A/2) sin (i/2) cos (77/2) cos (0i - 02) 7 = sin p, cos (A/2) cos (9\\ + 04) T = sin /i sin (A/2) cos (t/2) cos #i £ = sin t cos (77/2) cos 62 • 0.5(1 - cos /J) • 0.5(1 - cos A) X = sin A sin (t/2) cos (77/2) cos (02 + 94) • 0.5(1 - cos/a) ip = sin 77 • 0.5(1 - cos t) • 0.5(1 - cos A) • 0.5(1 - cos p) cos (02 + 03) Chapter 4. Performance of Tt/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 108 v = sin t sin (n/2) cos 03 • 0.5(1 - cos A) • 0.5(1 - cos p) p = sin Asin(7?/2)sin(t/2)cos(03 - 04) -0.5(1 - cos/i) <7 = sin A cos (i/2) cos 04 • 0.5(1 — cos a) C = 2[1 - 0.5(1 - cos i)] • 0.5(1 - cos p) • 0.5(1 - cos A) zu = 20.5(1 - cos A) • 0.5(1 - cos p). (4.41) Integrating R of (4.37) over (0, oo) and /i,A,t, r/ over (0,7r) together with the use of the following relationship oo j R9 exp (-aR2)l0{bR)dR = ^ exp (62/4a) • [l + 4(&2/4a) + 3(&2/4a)2 + |(62/4a)3 + ±{b2/4a)A (4.42) (4.37) becomes p(01,82 , 63,e4) OO I 1 IT » \"///// 0 0 0 0 0 f(R,p, A, t, 77, $i, $2i 03,04,) di? cfyi dA c?77 3cg 2(2TT)4 7T 7T 7T 7T exp 0 0 0 0 exp + 3 1 + 777 [F(p>, A, I, 7?, $i, 62 , 03, 04) G(/x,A,t,7?,g1,g2,g3,04)l2 L F ( M , A, t, 7?, 0i, 02,^3,04). G(/i, A, t, r),0i,62,03,04)' 24lF(p,\\,hn,O1,92 , 03,e4). 1 + 4 [•^V, <^> <\") *?) 01) 02) 03, 04)] [F{p,X,t,V,01,O2,e3,94)_ 2 + 77 G(p,X,t,rj,01,02 , 03 , 04) 3 L-F(M,'V)77>01)02,03,04). n > d/u dA di dr? (4.43) where T(/z, A, i, 77) - [sin p sin (A/2)]2 sin 77 sin A sin i (1 - cos i)[(l - cos u)f. By substituting (4.43) in (4.17), the occurrence probability can be determined. The rest of the error probabilities in the set {P\"} of (4.12) can be obtained in a similar manner. Finally, the Chapter 4. Performance of w/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 109 substitution of the values {P„, P'^, • • •, P }^ in (4.12) yields the output error probability of the single-error correcting NEC receiver. 4.4 Computer Simulation Model Description The computer simulation model used is an equivalent complex baseband representation of the digital communication system (see Fig. 4.1). The bit error rate (BER) results were obtained with Monte Carlo error counting techniques. A total of 131072 bits at 16 samples per symbol had been employed throughout the simulation for each point. Following [54], the confidence bands of the simulation points are summarized in Table 4.2. Table 4.2 Confidence bands on BER simulation results for 131072 bits observed. BER 90% Percentage Levels I O - 1 ±0.1 x 10\"1 IO\" 2 ±0.5 x IO\" 2 i o - 3 ±1.0 x I O \" 3 10\"4 ±3.7 x 10~4 The three parameters which are of interest are: C/N, K and BryP product. The carrier power C refers to the total received signal power, i.e., the summation of the direct path power S and the reflected path components power D, following which the C/N is defined as: C N S + D N ' (4.44) K is defined as the ratio of the direct path signal power divided by the ratio of the reflected path components power D, i.e., *4 (4.45) The power calculation for the C/N and K (= S/D) is defined in (3.42). Chapter 4. Performance of itl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 110 The transmitter and the receiver filters which are ideal square-root-of-raised-cosine filters have a roll-off factor of 0.2, with an x/sin(x) amplitude equalizer added to the transmitter filter so that the overall filtering strategy satisfies the first Nyquist criterion. The Rayleigh fading process was simulated by passing white Gaussian noise processes through two identical shaping filters with Gaussian spectrum (see Fig. 4.2 and equation 4.7 and equation 4.4). 4.5 Performance Evaluation Results Although the output error rate equation for the single-error correcting NEC receiver has been derived, it is too time consuming and thus expensive to be evaluated with the accessible computing facilities, including the CRAY supercomputer of the Ontario Center for Large Scale Computation of the University of Toronto. By running smaller programs in this CRAY computer, it was estimated that more than 100 CPU hours (@ $100/hour) was needed in order to obtain only one BER point for the single-error NEC receiver. This occurs because, as can be seen from (4.12), the final output error probability is a sum of nine different error probabilities and each of these error probabilities is a function of the eight-fold integrals (see equation 4.17 and 4.43). Because the correlations among the signals diminish under the circumstances of fast fading, the number of the multiple integrals in these formulas may be reduced. Later computer simulation results show that under fast fading environment, the receiver filter plays a significant role in rejecting the fading components. For accurate evaluation of the output error probabilities, the effect of the receiver filter can no longer be ignored from the previous derivation of (4.43). The reduction of the number of multiple integrals under the assumption of independency in fast fading is therefore accompanied by an increase in the complexity of the formula due to consideration of the receiver filter. As seen from (4.16), the in-phase components of the phasors in Fig. 4.3 will be approx-imately equal to y/2S for high ^ -factor so that the differential phase error between phasor Ri Chapter 4. Performance of trl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 111 and Rj can be approximated by [57] Oi = pm(V-) - lm{Vj)]/y/2S (4.46) where Im(-) refers to the imaginary part of its argument. The occurrence error probability will become a function of four integrals instead of the former eight integrals. Regardless of the the approximation by the four integrals, the output error probability is a total of nine occurrence probabilities (see equation 4.12) and would lead to inaccurate results due to accumulation error and approximation inaccuracies. It was thus decided to employ the computer simulation approach for the evaluation of the performance of the 7r/4-shift DQPSK signals employing the proposed NEC receivers. In the rest of this section, various BER performance results of 7r/4-shift DQPSK systems operated in a frequency nonselective fading environment and which employ single- and double-error correcting NEC receivers are presented. In the evaluation of the overall performance, the following three parameters are of great importance: the C/N, the ^-factor, and the B^T product. The obtained performance evaluation results are given in terms of bit error rate (BER) as shown in Figs. 4.4 - 4.6. In these figures, the values of the Jf-factor considered are 1 dB, 10 dB, and 15 dB whereas the BDT takes values of 0.01, 0.05, 0.13, and 1.26. Fig. 4.7 presents the performance of the proposed NEC receivers in an extremely fast fading environment, for example, an aeronautical channel. Finally, Fig. 4.8 illustrates a comparison of the performance of the conventional differentially detected 7r/4-shift DQPSK systems operated in the presence of fading and CCI. Chapter 4. Performance of z/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 112 -4 10 1— 1— 1— 1— 1— 1— 1—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I I 1 6 11 16 21 26 C/N [dB] Figure 4.4 BER performance of differential detected 7r/4-shift DQPSK systems employing NEC receivers operated in a Rician fading environment at K = 1 dB with various values of BDT. Chapter 4. Performance ofit/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 113 10 1 1 1 1 1 1 1 1 1 1 1 1 L 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C/N [dB] Figure 4.5 BER performance of differential detected 7r/4-shift DQPSK systems employing NEC receivers operated in a Rician fading environment at K = 10 dB with various values of BDT. Chapter 4. Performance of it!4-shift DQPSK Systems with NEC in a Multipath Fading Environment 1 Figure 4.6 BER performance of differential detected 7r/4-shift DQPSK systems employing NEC receivers operated in a Rician fading environment at K = 15 dB with various values of BDT. Chapter 4. Performance of w/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 115 1 6 11 16 21 26 31 C/N [dB] Figure 4.7 BER performance of differential detected 7r/4-shift DQPSK systems employing NEC receivers in an aeronautical channel with BDT of 6.29. Chapter 4. Performance of tr/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 116 -1 10 -3 8 9 10 11 12 13 14 C/N (dB) Figure 4.8 Comparison of the error rate performance of the conventional differentially detected 7r/4-shift DQPSK systems in the presence of CCI and fading with equal power. Chapter 4. Performance of itl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 117 4.6 Analysis and Discussion of Performance Evaluation Results As it can be seen from Figs. 4.4 - 4.7, in general the performance improvements obtained by the employment of the NEC receivers are not as high as the ones reported in Chapter 3 where CCI was the source of interference. In fact, for AT-factor with relatively low values (for example, 1 dB) and not very fast fading (for example, Br)T=0.0l, 0.05, and 0.13) the performance of the NEC systems is worse as compared to the conventional differential detected 7r/4-shift DQPSK systems, i.e., without NEC. For higher values of BryT, significant improvements have been obtained. For example, for BDT = 1.26, K = 1 dB, and at BER = 3 x I O - 3 , a performance gain of approximately 4 dB is obtained. Based upon the performance results presented in Figs. 4.4 - 4.8, the most important findings, including some heuristic interpretation and explanation of these results, will be summarized. i. By considering first the BER performance of conventional differentially detected 7r/4-shift DQPSK systems operated in a fading environment, it is clear that they perform better than in an \"equivalent\" CCI environment. We refer to \"equivalent\" interference, if we have the same signal-to-interference ratio, i.e., K—C/l. The main reason for that is the fact that CCI corrupts the 7r/4-shift DQPSK signals independently from symbol to symbol and thus the performance of differential detection is not satisfactory. On the other hand, for the fading channel where, depending upon the value of BpT, there is a strong correlation between adjacent symbols, differential detection will at least partially cancel its effects on the overall performance. In fact, the higher this correlation is (i.e., low BDT), the better the differential detection performs. It should be mentioned that, as it was pointed out by Mason [56], for very high values of BDT (for example, >1.0), the receiver filter #R ( / ) attenuates the fading component, thus causing a reduction in the error probability. This phenomena can also be seen from our results for JE?£>T=1.26. Chapter 4. Performance of ir/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 118 i i . For small values of BDT (for example, 0.01), the gains acquired by the NEC receivers increase with increasing K. However, when K is small (for example, 1 dB) the improvements in the BER performance are small. Moreover, the improvement of the double-error correcting NEC receiver is negligibly small as compared to that of single-error correcting NEC receiver. In order to understand the above behavior, it should first be recalled that the error correction capability of the NEC receivers comes from the parity symbols, i.e., the outputs of the higher-order detectors. In other words, the correctness of the parity symbols determines the error correction performance of the NEC receivers. Thus, it would be interesting to examine the performance of each of the differential detectors, i.e., first-, second-, and third-order differential detectors. These error rate results have been obtained by means of computer simulation and are summarized in Figs. 4.9-4.12. For comparison purposes, the same values of BDT and the .fif-factor which have previously been used, have also been selected here. It can be seen from Fig. 4.9, that for K—\\ dB the performance of the third-order differential detectors is the worst, whereas, the second-order differential is inferior to that of the conventional differential detector, i.e., first-order differential detector. As a result, the performance of the NEC is not satisfactory for this small K and small BDT, as illustrated in Fig. 4.4. On the other hand, as the values of the /sf-factor increases, the performance of all three differential detectors becomes better, therefore, the performance of the NEC becomes better for these higher K. The above observations clearly explain most of their performance results for the relatively small BDT, such as 0.01, reported in Figs. 4.4-4.6. iii. For very high BDT (1.26 as illustrated in Figs. 4.4-4.6 or 6.29 as illustrated in Fig. 4.7) the gains offered by the NEC receivers increase with decreasing K. Furthermore, it appears here that the double-error correcting NEC receiver provides a noticeable improvement as compared to the single-error correcting NEC receiver. Clearly these observations are opposite to the results and comments discussed in the previous paragraph. In order to explain this behavior, Chapter 4. Performance of n/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 119 the correlation characteristics in fading has to be taken into account. As pointed out earlier, for very high BDT, the receiver filter rejects the fading components and thus decreases the correlation between signals. As a result, the error probability of the overall systems is relatively less and thus the higher order differential detectors perform more effectively. This can be seen from Fig. 4.12 that the error probabilities of the first-, second- and the third-order differential detectors are approximately the same. This explains the increased improvement acquired by the double-error correcting NEC systems. Although decreasing the values of the AT-factor would increase the overall error probability, the fact the signals become somewhat uncorrelated, allows the NEC receivers to effectively improve the overall performance. Therefore, the effectiveness of the NEC receivers becomes more noticeable at smaller values of K, despite the overall higher probability of error. Finally, it is worth to mention that the highest gain obtained is approximately 6 dB for a mobile-satellite channel with BDT=6.29, as shown in Fig. 4.7. This suggests that the NEC technique will perform well in application with very fast fading, such as in the aeronautical channel, iv. At moderate BDT values, for example, at BDT = 0.05 or 0.13, the performance of the NEC receivers is similar to the situation when BDT is very small, i.e., increasing K enhances the gain provided by the NEC receivers. However, it is worth notice that when K = 1 dB, the performance of the NEC receivers is worse than the situation when the BDT is very small. Moreover, increasing the BDT (from 0.05 to 0.13 in our case here) deteriorates the performance of the NEC receivers at this small K value. This can be explained by the poor performance of the higher order detectors as illustrated in Figs. 4.10-4.11. In addition, increasing the BDT decreases the correlation among the signals, and lead to a higher error probability in conventional system. As a result, the overall system performance deteriorated. Chapter 4. Performance of ir/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 120 1 6 11 16 21 26 C/N [dB] Figure 4.9 Output symbol error rate for differential detectors of different orders at BDT = 0.01. Chapter 4, Performance of Ttl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 121 C/N [dB] Figure 4.10 Output symbol error rate for differential detectors of different orders at BDT = 0.05. Chapter 4. Performance ofn/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 122 C/N [dB] Figure 4.11 Output symbol error rate for differential detectors of different orders at BDT = 0.13. Chapter 4. Performance of itl4-shift DQPSK Systems with NEC in a Multipath Fading Environment 123 -3 10 1— 1— 1— 1— I—I— I—I— I—I—I— I—I— I—I— J—I—I— I—I—I—LJ I I L 1 6 11 16 21 26 C/N [dB] Figure 4.12 Output symbol error rate for differential detectors of different orders at BDT = 1.26. Chapter 4. Performance of tr/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 124 4.7 Summary In this chapter, we have applied the NEC technique to the 7r/4-shift DQPSK system operated in a frequency nonselective fading environment. First, the theoretical error rate equation for a single-error correcting NEC receiver has been derived. Afterwards, the performance of single- and double-error correcting NEC receivers have been obtained by means of computer simulation. It was found that for very fast fading the NEC receivers resulted in significant improved performance with a gain up to 6 dB, as compared to the conventional 7r/4-shift DQPSK systems. Furthermore, it was found that in general the performance of a conventional differential detected 7r/4—shift DQPSK systems operated in the CCI is inferior to that operated in fading. Chapter 5. Conclusions and Future Research Suggestions 5.1 Conclusions This thesis has dealt with the application of the NEC technique to the 7r/4-shift DQPSK modulation format, which is the new transmission standard for the new North American and Japanese digital mobile cellular network. It was assumed that the environment in which such systems would operate is the CCI as well as the fading channel. First, the NEC technique was analyzed from the convolutional coding point of view. It was shown that it could be viewed as a feedback decoding technique. Various issues inherent of the NEC technique, such as infinitive error propagation, decoding depth and constraint length were discussed in detail. Afterwards, based on a phase transformation concept, the NEC receiver, originally suggested by Samejima et al. [33] for DQPSK signals, is modified in order to accommodate for the 7r/4-shift DQPSK scheme. For the first time, we have theoretically analyzed the performance of the NEC receivers with up to three error correction capability. Furthermore, the performance of the 7r/4-shift DQPSK systems in the presence of a mixture of AWGN and CCI was theoretically evaluated. The obtained performance evaluation results have indicated significant gains in performance as well as reductions in error floors as compared to the conventional differential detected 7r/4—shift DQPSK. These gains increased with increasing number of interferers and decreasing C/I ratio. Finally, in this thesis, the NEC technique was applied to the 7r/4-shift DQPSK systems operated in a frequency nonselective Rician fading channel. The error rate equation for the single-error correcting NEC receiver was for the first time derived. Performance evaluation results for both single- and double-error correcting NEC receivers obtained by means of computer simulation have shown that for very fast fading and decreasing AT-factor, the obtained gains increase. In general, these gains were not as significant as those obtained for the CCI. It was found that 125 Chapter 5. Performance of Tt/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 126 the reason behind this observation was the fact that the CCI corrupts the transmitted symbols independently and thus the performance of differential detection was poor. Next, the most important findings would be summarized, so as to provide a guideline for the future researchers whom would be interested in working with the NEC technique. i. Since the NEC technique makes use of the output symbols from the higher order detectors as parity symbols for the output symbols from the conventional detector (or the information symbols), the output error probability of the higher order detectors has to be at least close to that of the conventional detector in order for the NEC technique to operate effectively. Otherwise, increasing the number of higher order detectors may not increase the error correction capability as it should. These lead to two other important observations which would be described as follows. • Optimum improvement from the NEC technique can be achieved when the signals are independently corrupted. It is because if signals are independently corrupted, the error probability of any two signals should give the same probability of error despite the time separation between these two signals. If there is a strong correlation between adjacent signals, increase the number of higher order detectors may not improve the performance significantly, unless the higher order detectors can yield at least approximately the same error probability as the conventional detectors. ii. Under very low C/N condition, for example, 1 dB, virtually no improvement can be obtained from the NEC technique. It is because error may occur so frequent that the error correction capabilities of the NEC receiver have already exceeded. Furthermore, because of the feedback path in the NEC receiver, the frequent occurrence of errors does not give a chance to the system to recover. Consequently, the errors continue to propagate over time, thus resulting Chapter 5. Performance of n/4-shift DQPSK Systems with NEC in a Multipath Fading Environment 127 in possible higher probability of error than conventional system without employing NEC. It should, however, be noted that this error propagation is different from the infinite error propagation where one error propagates throughout the network even though there is no more new incoming errors. However, these situations usually occur at error probability higher than 10 - 2 which is usually of no practical interest in actual environment, iii. Since both the information and the parity symbols of the NEC receivers come from the detectors, well-designed detectors are important and will directly affect the performance of NEC receivers. Thus, rather than seeking ways to improve the decoding algorithm, it is more important to have a well-designed detectors. It is also worth notice that although more sophisticated decoding algorithm may possibly lead to a better improvement, the extra hardware circuitry involved may diminish the attractiveness of the NEC schemes in which simplicity is one of the prime concern. 5.2 Suggestions for Future Research 5.2.1. 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Appendices 135 Appendix A Definition of Constants 136 Appendix A Definition of Constants Co = 1 - 8*1*3*! + 4*2Jb3ifc3 + 6*2*? - 3*?, - 4*? + 8*1*2*3 ~ 6*?*2*4 + 4*1*3*4 +3fc4 + 2*4 - 4*3*? - 2*1*? - 2k\\ + 2*22*4 + 2*1*1 + fc4 - 4*!* 2* 3*4 - 4 * 2 * i 4 - f 2*4*i4 + 2*23*2 + 2k\\k\\ - 2*24*4 + 4*3* 2 2*1*4 + 4* 2* 2* 4 + 4*2*2*? - 4* 3 *4* 3 - 4*!*3*3 - 4* 2 * |* i + 2*1*4*? ~ 2*1*2*4 A = 16*1*3*1 + 16*2*4*i + 4*1*2*1 + 8*3*4*i2 - 2 f c 2 * ? + 7*2 - 12*1*3 +12*3*? + 12*|*! - 8*2*3 - 20*i*2 + 6*1 + 8*1 + 6*2 + 4*3 + 2*4 + 4 * i * 2 * 3 -20*2*3*! - 12*1*4*1 + 2*2*4*? - 4*3*4*1 + 8*3*2*4 ~ 8*? - 8fc3 - fc4 +4*3*? - 8*1*? + 6*1 - 6*1*? - 4*4*1 + 2*4*? - 4*2*4 + 4* 2*3 + 12*2*3 - 4 * 4 * 3 - 4*3*4 - 2*2*1 - 6*1*4 - 8*1*1 + 8*1*4 - 4*1 - k* - 4* i* 2 f c 3 *4 +2*1*2*4 + 3*2 - h\\k\\ + 2*1*4 - 2*2*2 - 5 + 4*2*1 - 4*1*4*1 + 4*1*1 + 4 * 2 * 2 * 1 - 4 * 3 * 4 * 1 - 4 * 1 * 2 (A.2) C i = - * 3 * 2 - * l * i + 1 - 2*1*3*1 + *1 - *3 - 2*? + fc2*3 - * l * 2 * 3 + * f ~ *1 - * ! - 3*1*2*4 + * ? - *i 2 *2*4 + * i - *4 + 3*1*2 - 2*2*1*1 + * 3 * 4 + 4*2*3*? — * 2 — * | * 4 + * i * 4 — 2*1*1 — * 4 * 3 * l — * 2 * 3 * 4 — 2*2*? — *1*1 + 2 * l * 4 * i + * 2 * i 2 + * 4 f c ? + * 2 * 4 + *!*? + 3*1*3 - 3*3*i3 + 3*1*? + * i * 2 * 3 * 4 + *1*4 C 2 = * 2 * 4 + * 2 * 3 — * i - * ! + * ! * ! + * 3 * 4 - * i * 3 * 4 — * 3 * ? - * l * i + fcf — 2 * 2 * l * i -3*2*4*1 + *1*4 + *1 - *3 ~ *2*3*4 + * i + 2*1*4*1 - 2* l* i + 3*i* 3 - * 3 * 4 + * 1 * 2 * 3 * 4 - *1*2*3 + * ? * ! + *?*2 - *?*2*4 - * 2 ~ *4 + 1 + k\\kA - 2 * l * 3 * i -3*3fc3 + *4*? + 3*1*? + 3*1*2 ~ 2*? + *? - * 2 + 4*2*3*? - 2* 2*i 3 - * f * l C 3 = - * ! - fcifcj*2 + * i * 2 + * 3 * 4 - *2 + * i * 4 * l - 3*?*3 + 2*1*4 + 2*2*3 + 2* i* 3 * 4 -f\"&4&^ — ~\\~ 3Ar^\"4\" 5^2^3^i &2 \"r\" 2&^ — ^4^3 \"T~ ^3 — 2&4&2 ~r~ 2^ 2^ *3 + 2*1 + * 2 * 2 - * 2 - *? + *i*4 - 3*1*1 - 4*1*2*4 - 2*3*4*? ~ 3*2*? - *2*1*1 - * 2 * 1 - * 4 * ? - 4 * l * 3 * i + *1* 3 *4 + 2*i*3 + 2*?*1 - 2*2*3*4 ~ *1*3 ~ *3 - 2 * i + 1 C4 = —*1 + * l * 4 * i — *2 — 2*2*3*4 + *1*3*4 + *1 *2 + * l *4 + *2*2 — *2*2*1 + 5*2*3*? - * 1 * 3 + *4*? + 2*2*3 - 3*2*? - 4 * 2 * 4 * l + 5*i* 2 + 3 * l * 4 * i + * 3 * 4 - * 2 * 1 * , + 2*? - 3*3*? ~ 2*3*4*? ~ 2*, + 2* 4*3*1 - 3*1*1 - * 2 * 1 - *2 + * ! - * 3 + 2*1*? + 2*1*3 - *4fc? - 2*1 - 4*1*3*1 - *? + 2*1*1 - 2*2*4 +2*1*4 + 2fc3 + 1 - * 1 * 4 Appendix A Definition of Constants 137 CS = -2*2*4 + 2/fc2Jt4 - * 4 + 2*2*3**1 + 2*4*^*3 \" *4 + 4*!*2 + 2*2*3*? - 4*?*3 -2*3**2 - 4*3*4*2 + *^ + 2*2*3 - 2*|*! - 2*!* 2*4 + 2*2*^ - 2*|* 2* 4 + 2*!*3 - 2 * i * 2 * 3 + 2*1*3*4 + 2*2*2 - 2*?*4 - 2*2 - 2*3*! - 2* 2* 3* 4 - 2*2 4- 2*!*2 +2*4*! * 2 + 2* 3 - 2*, - 2* 2* 3 - k\\ - 2*2*1 k\\ + 2* 2 * i + 2 * 2 * 4 * i + 2 * 4 * f - 2 * 2 * 3 + k\\k\\ + 2* 3* 3 + 2*2*? - 4*2fc3*i + 2*1*2*3*4 + 2*23 + 1 ce = -(* 2 - 0.5*!4 - 0.5 + 0.5*2 ~ 0-5fc3 + °-5 f c2 + *i *2*< - 0.5k]k24 - 2*!* 3* 4 + * 2 * | * 4 + *? + *?*2 - 2*1*2*3) c 7 = -(0.5*2 - 0.5k* + k\\ + * 2 * 3 * i - 0.5*2*? - 2* 2* 2 - 2*!*2*3 + * 3 * 3 - 0.5 +*2*2 - 0.5*4 + 1.5*?) c 8 = _(_0.5*2*? + * , 2 + * 1 f c 2 * 3 * 4 - kxk3kA - 0.5*2*2 + 0.5*2 + * 2 * 4 * ! 2 - 0.5 -0.5*2*? + * 2 * l * 3 - *2*? + *2 + 0.5*| - k\\kA - * i * 2 * 3 ) C 9 = - ( - * 3 * 4 * 2 + 3*2*3*1 + * i * 2 * 4 - 2*2*3*? ~ *2* i\" + * 2 * 3 * 4 ~ ^2^3 -*2* 3 + 2*i*2 + *1 - fcf^fcl - M\"4 ~ 2*1*2 + * ? * 4 + + * 3 - *33) C 1 0 = - ( * | * 3 + *2* 3 - 2*2*3* 2 + 3 t 2 * i \" ~ *4* f - *1*2*3 + * ? * 3 - * 2 * 4 * j - f * 3 * 4 * ? + 2*!*2*4 + * i + * i * 3 - * 3 * 4 ~ *1*2 ~ *1 * f ~ fc2*3 ~ 2*f) Cl l = ~ ( * 4 + *2 - * 2 + 2*j 2 * 2 * 4 - 2*2*3*1 + 2*i*2*3 - * * * 4 + * 2 * 3 ~ 2*3*1 -2*2*? + 3*2*2 + 2* 3* 3 - 2*4*? - k\\k\\ - *4) c 12 = ~ (*1 + *2*3*4 + 2*?*3* 4 - fc2*2*i - * 4 * ? - * 2 * i + 2*?*3 — fc2*4*l + * 2 * 4 * i - * 2 * l * 4 + * ! * 2 * 4 - *2*3*4 + *3*2 + *2^1 + * 2 * ? ~ *2*3 + *2^3 — *3*1 — 2*i*2 ~ * i * 2 * 3 — * i ) C 1 3 = —(* 3 — 2*2*3*? - *3*4*? + *2*3*4 + * l * f ~~ ^2*3 + 2 * i * | — * i * 2 * 4 ~ *1*4 - * f * 2 + *j 3*4 - *2* 3 - 2*i*2 + 3*2*1*1 + * i * 2 * 4 - *3 + * ? ) C14 = —(*4*? — * 2*? + 2*2*3*1 — 2*1*2*3*4 — 2*i*3 -f\" *?*4 \"T\" 2 * i* 2 *3 — *2*2 + fc4*| - * 2 * 2 + 2*2*? - 2*2*4*? + 2*3*4 + * 2 - 2* 3 - *?) Cj5 — \"\"(\"\"\"^^ '^4 \" T \" ^k^k^k^ — k^k^ — ^1^3^'4 — ^j^2^4 ^\\^2 \" T * ^1^2 ^2^3 + *1*4*2 — *2*3*? + *2*3 — *2*3*4 — *2* 3 *1 + *2*3*4 + *1*2*4 ~ *2*3 - * ! * | + 2*?*3 - 2*1*2 + *1 - * ? ) C i 6 = - ( * i + fc3*3 - * 4 * i 3 + *2* 3 - * i * 2 + 3*,3*2 + * ? * 3 ~ *1*2 + *1*3 + 2 * i * 2 * 4 \"r\"^1^3^4 — k\\k^kt\\ — 2fcj^3^i — k^kt\\ — ^2^3 *^1 — ^2^3 — Cj7 — ~(^'2 \" T * ^1^3 — ^2 \"~ ^j^2 \"~ k^k^ — ^3^1 ™\" ^2^4 — ^2^3 k^k^ ~f* k^k^ -k\\k\\ + 3*1*2*3 ~ *?*2*4 + *1*3*4 + 2*2*3*1 - *3*1 ~ *1*2*3*4 + * i ~ *? ) C 1 8 = - ( * 2 + * ? * 4 - * 3 * 3 - *f - * ? * 2 * 4 + * 3 * 4 - *2*| ~ *2*4 + 3*1*2*3 + * 1 * 3 * 4 - fct*3 - k\\k\\ - *?*2 + 2*2*1*3 - * i * 2 * 3 * 4 + * | * 1 ~ *2*2 +fc 4-*f) (A.3) Appendix B Program Listings Appendix B Program Listings necl.s ) a = b = c = s i = s i _ l = output = e i _ l = pass = 0; /* necl.s *l j******************************************************************************* return (0); end necl() main_oode ******************************************************************************** i f (avail(0) & avail(1)) Programmer : Dominic Wong { Date: Sep 1990 (vl.O) b = a; Feb 1991 (v2.0) i t i n(0); a = dl ( 0 ) ; Mar 1991 (v3.0) i t i n ( l ) ; c = d2(0); i f ( ( s i = (a + b - c) % 4) < 0) s i += 4; This star models the SINGLE ERROR CORRECTION c c t . I t has two inputs and 1 e i _ l = ( ( s i = s i _ l ) SS ( s i != 2)) ? s i : 0; output. Among the two inputs, one comes from the IT period delay, while the i f ((output = ( b - e i 1 ) * 4) < 0) output += 4; other one comes from 2T period delay. The output i s matched to the o r i g i n a l i f ( ( s i _ l = (si - e i _ l ) % 4) < 0) s i _ l += 4; values to maintain consistency with the transmitter end, followed by transform-ing these numbers to b i t stream. /* convert the decoded output to o r i g i n a l values to maintain consistency */ */ switch (output) { case 0: input buffers output = 2; in t d l ; break; i n t d2; case 1: end output = 0; break; output_buffer case 2: in t r x b i t ; output = 1; end break; case 3: states output = 3; in t a; break; i n t b; default: i n t c; output = 0; in t s i ; break; i n t s i _ l ; ) /* switch */ i n t e i _ l ; i n t output; i f (pass = 1) i n t pass; ( end M i n t * ) i t out(0) = output/2; /* MSB goes f i r s t . */ M i n t *)it_out(0) = output%2; i i n i t i a l i z ation_code 1 pass = 1; /* Fix input and output c e l l sizes */ ) /* i f */ s e t _ c e l l s i z e _ i n (0, s i z e o f ( i n t ) ) ; s e t _ c e l l s i z e _ i n (1, s i z e o f ( i n t ) ) ; return (0); set_cellsize_out(0, s i z e o f ( i n t ) ) ; end /* main */ i f (NARR.narrative > MEDIUMJJARRATION) ( nec2.s f p r i n t f (NARR . f i l e . *********************************************** *********\\ n\\ n* ^ . /* nec2.s */ f p r i n t f (NARR.file, \"no parameters for star necl.s\\n\\n\") ; /******************************************************************************* f p r i n t f (NARR. f i l e , \"no of input buffers : 2 (int)\\n\"),-fprintf(NARR.file,\"no of output buffers : 1 (int)\\n\"|; fprintf(NARR.file. nec2 () ********************************************************\\n\\nt<* . *********************** ******************** ********** Programmer : Dominic Hong Date : Sep 1990 (vl.O) Feb 1991 (v2.0) Mar 1991 (v3.0) This s t a r models the DOUBLE ERROR CORRECTION cct. I t has three inputs and 1 output. Among the three inputs, one comes from the IT period delay, while the others comes from 2T period delay and 3T period delay. The output i s matched to the o r i g i n a l values t o maintain consistency with the transmitter end, followed by transforming these numbers to b i t stream. */ input_buffers i n t d l ; i n t d2; i n t d3; end output_bu f fer i n t r x b i t ; end states i n t s l i ; i n t s l i _ l ; i n t s l i _ 2 ; i n t s 2 i ; i n t s 2 i _ l ; i n t a2i_2 i n t ei_2; i n t output; i n t x; i n t found; i n t as; i n t bs; i n t cs; i n t ds; i n t es; i n t pass; i n t SH; end declarations s t a t i c i n t a [34]; s t a t i c i n t b[34]; s t a t i c i n t ct34]; s t a t i c i n t d[34]; s t a t i c i n t e[341; s t a t i c i n t f(34); i n t n; i n t m; i n t t ; end i n i t i a l i z a t i o n code /* Fix input and output c e l l sizes */ s e t _ c e l l s i z e _ i n (0, s i z e o f ( i n t ) ) ; s e t _ c e l l s i z e _ i n (1, s i z e o f ( i n t ) ) ; s e t _ c e l l s i z e _ i n (2, s i z e o f ( i n t ) ) ; set_cellsize_out(0, s i z e o f ( i n t ) ) ; i f (NARR.narrative > MEDIUM_NARRATION) { fprintf(NARR.file, * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * f r * * * \\ n \\ n n ^ . fprintf(NARR.file,\"no parameters for star necl.s\\n\\n\"); fprintf(NARR.file,\"no of input buffers : 3 (int)\\n\"); fprintf(NARR.file,\"no of output buffers : 1 (int)\\n\"); fprintf(NARR.file, *******************************************************\\n\\n\"); ) /* I n i t i z l i z e the syndrome patterns. */ t = 0; for (n=l; n<=3; n+=2) ( /* single error */ a[t) = 0; b[t) = c[t) = d[t) = e[tj = f [ t ] = n; t++; /* double error */ for (m=l; m<=3; m+=2) ( a[t) = m; btt] = n; c[t] = n; d[t) = (m + n)%4; e[t) = n; f[t ) = n; t++; ) for (m=l; m<=3; m+=2) ( a[t] = m; b[t] = (m + n)%4; c(t] = n; d[t) = (m + n)%4; e[t) = (m + n)%4; f[ t ) = n; t++; for ( (m=l; nK=3; m+=2) a[t) = m; b[t) = n; c[t) = n; d[t] = n; e[t] = n; f[t ) = n; t++; for (m=l; nK=3; m+=2) ) a[t] = 0; i f (b[t] = (n ctt ] = n; d[t] = n; e[t] = n; f [t] = n; t++; > m)%4 < 0) b[t] += 4; for (m=l; m<=3; m+=2) ( a[tj = 0; b[t] = n; i f (c[t] = (n d[t] - n; e[t] = n; f [ t ] = n; t++; ) for (m=l; nK=3; m+=2) ( a[t] = 0; b[t] = n; c[t] = n; i f (d[t) = (n e[t) = n; f [t l = n; t++; > m)%4 < 0) o[t] += 4; m)%4 < 0) d[t] += 4; for (m=l; nK=3; m+=2) ( a[t] = 0; b(t] = n; c[t] = n; d[t] = n; i f (e[t] = (n f tt] = n; t++; ) m)%4 < 0) e[t] += 4; for (m=l; m<=3; m+=2) { a[t] = 0; b[t] = n; ctt ] = n; d[t] = n; e[t] = n; i f (f[t ] = (n - m)%4 < 0) t++; ) f ft) a l i = s l i _ l = s l i _ 2 = s2i = s 2 i _ l = s2i_2 = ei_2 = output as = bs = cs = ds = es = pass = SW = 0; return (0); end found = 0 main code i f (avail(0) 5 avail(1) & avail(2)) ( i t _ i n ( 0 ) ; i t _ i n ( l ) ; i t _ i n ( 2 ) ; cs = bs,\" bs = as; as = d l ( 0 ) ; ds = d2 (0); es = d3(0); i f ( ( s l i = (as + bs - ds)%4) < 0) s l i += 4; i f ((s2i = (as + bs + ca - es)%4) < 0) s2i += 4; /* Error detection */ x = 0; /* I n i t . to the 1st element of the patterns */ found = 0; /* Zero means not found. */ while ((x < 34) SS (found == 0)) ( i f ( •include int z8[3],z9[3],y7[3],y9[3],t6[3] ,t9[3] ; int 3l i , s l i_l ,s l i_2,s2i ,s2i_l ,s2i_2; int error=0,testing; ~ int pattern00=0,pattern01=0,pattern02=0 ; void patternl(), pattern2(), pattern30, pattern*(), patterns(), pattern6(), pattern7(), pattern8(), pattern9(), patternlOO, patternl lO, patternl2(), patternl3(), patternl4(), patternl5(), patternl6(), patternl7(), patternl8(), patternl9(), pattern200 ; int check(int *); int mod(int); void checkPattern(); void checkPattern () { switch (error-testing) ( case 0: pattern00++; break; case 1: pattern01++; break; case 2: pattern02++; break; default: printf(\"error\\n\"); ) I int check (int *corr) ( i f ( SS (s l i_ l=l) SS (sli_2=l) SS (s2i=2) SS (s2i 1=1) SS (s2i_2=l)) ( *corr = 1; return 0; ) i f ((sli=3) SS (sli_l=3) SS (sli_2=3) SS (s2i=2) SS (s2i_l=3) SS (s2i_2=3)) ( *corr = 3; return 0; ) i f ((sli==l) SS (sli_l=0) ss (sli_2==3) SS (s2i=0) SS (s2i_l=0) SS (s2i_2==3)) ( •corr = 3; return 0; ( i f ( U» LB LB LB UJ LB Uf LB LB UJ Ml LB LB LB LB LB LB UJ US m rH ~ . rH — co —. ro ~ H — rH —. m — cn —. rH — 11 7 II II II II II 7 II 7 II II 11 rH It rn II CM II CN A II CN II CN II CN CN II CN II CN II CN II CN II A II CM, CN, CM, N , |CN, ICN, CM |CN CM, |CN, -H | rt | •rt | •rt | -rt | •rt | •rt | \"rt ) - r t •rt | rH -H rH -H rH -H rH -H rH -H rH -H rH -H rH -H rH -H rH -H a CN 0} CN n CN 03 CN «Q CN a CN 03 CM 01 CM 01 CN 01 CN — 01 ta —' 03 — 3^ — 01 — 01 — ' _TO LB U» LB LB US u> UJ LB LB UJ u» UJ UJ UJ LB LB LB LB LB LB LB LB UJ LB UJ LB LB UJ US UJ LB UJ LB UJ LB LB LB UJ LB LB cn — H —- rH — co CO —. rH -~ rH -- CO — co .—. rH —-[1 ro 1! 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CD O H H H N CM ( N H N H H H N N II O & II II m A, ii c M U u a O 4J US — —• o 1 n _ M U U 3 O 4J U 3 O 4-> U 0) 31 2 E C *• U Q, 3 — s s a — +J o tp V H a -— v to •4-1 *H •H 0) \\ -H — .8 = •P 4J •rt T) -rt * O > • >. 41 <3\\ 4J sli=0; ( s2i=0; for (k=0;k<=l;k++) check(Scorr) ; ( i f (corr != 0) error++; corr=0; checkPattern () ; ) te3ting=error; ) sli=0; ) s l i l = z 8 [ i ] ; ) sli _ 2 = z 8 [ i ] ; s2i=z8[i); s2i l=z8[i]; /* z8,y9,t6 are i n error. */ s2i_2=mod(z8[i]-t6[j]); void pattern3() check(Scorr) ; ( i f (corr != z8[i)) error++; i n t i , j,k; i n t corr=0; s l i _ 2 = mod(sli_l - c o r r ) ; s l i _ l = s l i ; f o r (i=0;i<=l;i++) s2i_2 = mod(s2i_l - c o r r ) ; ( s2i 1 = mod(s2i - c o r r ) ; for (j=0;j<=l; sli=0; { s2i=mod<-t9[k]); for s l i 2=mod(-y7[i) ); } s2i=0; s2i 1=0; s2i_2=mod(-t6[j]) ; /* z9,t6,t9 are i n error. */ check(Scorr); vo i d patternl5() i i f (corr != 0) error++; i i n t i , j , k ; s l i _ 2 = mod(sli_l - co r r ) ; i n t corr=0; s l i _ l = s l i ; s2i_2 = mod(s2i_l - co r r ) ; for (i=0;i<=l;i++) s2i 1 = rood(s2i - co r r ) ; ( sli=0; for (j=0;j<=l; s2i=mod(-t9[k]); ( check(Scorr); for (k=0;k<=l;k++) i f (corr != 0) error++; { checkPattern () ; corr=0; ) testing=error; ) s l i = z 9 [ i ] ; } ) s l i l = z 9 [i]; sli_2=0; s2i=z9[i]; /* z8,z9,t6 are i n error. */ s2i l = z 9 [ i ] ; void patternl7() s2i_2=mod(-t6[j]); ( check(Scorr) ; i n t i , j,k; i f (corr != 0) error++; i n t corr=0; s l i _ 2 = mod(sli_l - corr); for (i=0;i<=l;i++) s l i _ l = s l i ; ( s2i_2 = mod(s2i_l - corr); for (j=0;j<=l;j++) ( s2i 1 = mod(s2i - corr); sli=0; for (k=0;k<=l;k++) s2i=mod(z9[i]-t9[k)); { check(Scorr); corr=0; i f (corr != z9[i)) error++; testing=error; checkPattern(); ) s l i = z 9 [ j ] ; ) s l i I=(z9[j]+z8[i])%4; ) sl i _ 2 = z 8 ( i ] ; ) s 2 i = s l i _ l ; s2i l=s2i; s2i_2=mod(z8[i]-t6[k]); /* y7,t6,t9 are i n error. */ check(Scorr) ; void patternl6() { i f (corr != z8[i]) error++; i n t i , j , k ; s l i _ 2 = mod(sli_l - co r r ) ; i n t corr=0; s l i _ l = s l i ; s2i_2 = rood(s2i_l - co r r ) ; for (i=0;i<=l;i++) s2i 1 = mod(s2i - co r r ) ; ( sli=0; for (j=0;j<=l;j++) s2i=z9tj]; ( check(Scorr) ; for (k=0;k<=l,-k++) i f (corr != z9[jl) error++; ( checkPattern () ; corr=0; ) testing=error; ) sli=0; ( ) s l i _ l = 0 ; i f (corr != 0) error++; / * z8,z9,y9 are i n error. */ void patternl8() s l i _ 2 = mod(sli _ l - corr); ( s l i _ l = s l i ; i n t i , j,k; s2i_2 = mod(s2i_l - corr); i n t corr=0; s2i 1 = mod(s2i - corr); sli=0; for (i=0;i<=l;i++) s2i=mod(z9[i)-t9[k]|; ( check(Scorr); for (j=0;j<=l;j++) i f (corr != z9[i]) error++; < checkPattern() ; for (k=0;k<=l;k++) ) i corr=0; ) 1 testing=error; ) sli=mod(z9[j)-y9[k]); s l i I=(z9[j]+z8[i])*4; / * z8,y9,t9 are i n error. */ sli_2= z 8[ij; void pattern20() s 2 i = s l i _ l ; ( s2i l = s l i 1; int i , j,k; s2i_2=z8[i); i n t corr=0; check(Scorr) ; i f (corr != z8[i]) error++; for (i=0;i<=l;i++) s l i _ 2 = mod(sli_l - cor r ) ; { for (j=0;j<=l;j++) ( s l i _ l = s l i ; s2i_2 = mod(s2i_l - corr); for (k=0;k<=l;k++) s2i 1 = mod(s2i - corr); ( sli=0; corr=0; s2i=z9tj); testing=error; check(Scorr); i f (corr != z9[jj) error++; sli=mod ( -y9[j]); checkPattern () ; s l i l=z8[i); ) sli_2= z 8[i]; ) s2i=z8[i]; ) s2i_l= z 8[i]; ) s2i_2=z8[i) ; / * z9,y9,t9 are i n error. */ check(Scorr) ; void patternl9() { i f (corr != z8[il) error++; int i , j,k; s l i _ 2 = mod(sli_l - corr); int corr=0; s l i _ l = s l i ; 32i_2 = mod(s2i_l - corr); for (i=0;i<=l;i++) s2i 1 = mod(s2i - corr); ( sli=0; for (j=0;j<=l;j++) s2i=mod(-t9[k]); ( check(Scorr) ; for