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.net framework QR Code JIS X 0510 Moment Generating Functions in Diversity Analysis in Software Display barcode 3/9 in Software Moment Generating Functions in Diversity Analysis




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7.4 Moment Generating Functions in Diversity Analysis generate, create none none for none projectsvb.net qr code generation In this section none for none we use the MGFs introduced in Section 6.3.3 to greatly simplify the analysis of average error probability under diversity.

The use of MGFs in diversity analysis arises from the dif culty in computing the pdf p ( ) of the combiner SNR . Speci cally, although the average probability of error and outage probability associated with diversity combining are given by the simple formulas (7.2) and (7.

3), these formulas require integration over the distribution p ( ). This distribution is often not in closed-form for an arbitrary number of diversity branches with different fading distributions on each branch, regardless of the combining technique that is used. The pdf for p ( ) is often in the form of an in nite-range integral, in which case the expressions for (7.

2) and (7.3) become double integrals that can be dif cult to evaluate numerically. Even when p ( ) is in closed form, the corresponding integrals (7.

2) and (7.3) may not lead to closed-form solutions and may be dif cult to evaluate numerically. A large body of work over many decades has addressed approximations and numerical techniques to compute the integrals associated with average probability of symbol error for different modulations, fading distributions, and combining techniques (see [11] and the references therein).

Expressing the average error probability in terms of the MGF for instead of its pdf often eliminates these integration dif culties. Speci cally, when the diversity fading paths that are independent but not necessarily identically distributed, the average error probability based on the MGF of is typically in closed-form or consists of a single nite-range integral that can be easily computed numerically. The simplest application of MGFs in diversity analysis is for coherent modulation with MRC, so this is treated rst.

We then discuss the use of MGFs in the analysis of average error probability under EGC and SC.. Android 7.4.1 Diversity Analysis for MRC The simplicity none none of using MGFs in the analysis of MRC stems from the fact that, as derived in Section 7.2.4, the combiner SNR is the sum of the i s, the branch SNRS:.

=. i .. (7.40). As in the analy none for none sis of average error probability without diversity (Section 6.3.3), let us again assume that the probability of error in AWGN for the modulation of interest can be expressed either as an exponential function of s , as in (6.

67), or as a nite range integral of such a function, as in (6.68). We rst consider the case where P s is in the form of (6.

67). Then the average probability of symbol error under MRC is Ps = c1 exp[ c2 ]p ( )d . (7.

41). We assume that the branch SNRs are independent, so that their joint pdf becomes a product of the individual pdfs: p 1 ,...

, M ( 1 , . . .

, M ) = p 1 ( 1 ) . . .

p M ( M ). Using this factorization and substituting = 1 + . .

. + M in (7.41) yields.

0 M fold P s = c1 exp[ c2 ( 1 + . . . + M )]p 1 ( 1 ) . . . p M ( M )d 1 . . . d M . (7.42). Now using the product forms exp[ ( 1 +. . .+ M )] = in (7.42) yields 0 M fold M i=1 exp[ i none for none ] and p 1 ( 1 ) . . .

p M ( M ). M i=1 p i ( i ). P s = c1 exp[ c2 i ]p i ( i )d i . (7.43). Finally, switch ing the order of integration and multiplication in (7.43) yields our desired nal form. M M P s = c1 i=1 0. exp[ c2 i ]p i ( i )d i = c1 M i ( c2 ).. (7.44). Thus, the avera none none ge probability of symbol error is just the product of MGFs associated with the SNR on each branch. Similary, when Ps is in the form of (6.68), we get.

B A 0 M fold Ps = c1 exp[ c2 (x) ]dxp ( )d = A i=1 exp[ c2 (x) i ] p i ( i )d i . (7.45).

Again switching the order of integration and multiplication yields our desired nal form B M 0 B M P s = c1 A i=1 exp[ c2 (x) i ]p i ( i )d i = c1 A i=1 M i ( c2 (x))dx. (7.46). Thus, the avera none for none ge probability of symbol error is just a single nite-range integral of the product of MGFs associated with the SNR on each branch. The simplicity of (7.44) and (7.

46) are quite remarkable, given that these expressions apply for any number of diversity branches and any type of fading distribution on each branch, as long as the branch SNRs are independent. We now apply these general results to speci c modulations and fading distributions. Let us rst consider DPSK, where Pb ( b ) = .

5e b in AWGN is in the form of (6.67) with c1 = 1/2 and c2 = 1. Thus, from (7.

44), the average probability of bit error in DPSK under M-fold MRC diversity is Pb = 1 2. M i ( 1),. (7.47). where M i (s) i s the MGF of the fading distribution for the ith diversity branch, given by (6.63), (6.64), and (6.

65) for, respectively, Rayleigh, Ricean, and Nakagami fading. Note that this reduces to the probability of average bit error without diversity given by (6.60) for M = 1.

Example 7.5: Compute the average probability of bit error for DPSK modulation under three-branch MRC assuming i.i.

d. Rayleigh fading in each branch with 1 = 15 dB and 2 = 3 = 5 dB. Compare with the case of no diversity with = 15 dB.

Solution: From (6.63), M i (s) = (1 s i ) 1 Using this MGF in (7.47) with s = 1 yields Pb = With no diversity we have Pb = 1 = 1.

53 10 2 . 2(1 + 101.5 ) 1 1 2 1 + 101.

5 1 1 + 105. = 8.85 10 4 ..

This indicates that additional diversity branches can signi cantly reduce average BER, even when the SNR on this branches is somewhat low. 206. Example 7.6: Co none none mpute the average probability of bit error for DPSK modulation under three-branch MRC assuming Nakagami fading in the rst branch with m = 2 and 1 = 15 dB, Ricean fading in the second branch with K = 3 and 2 = 5 dB, and Nakagami fading in the third branch with m = 4 and 3 = 5 dB. Compare with the results of the prior example.

Solution: From (6.64) and (6.65), for Nakagami fading M i (s) = (1 s i /m) m and for Riciean fading M s (s) = 1+K Ks s .

exp 1 + K s s 1 + K s s. Using these MGF s in (7.47) with s = 1 yields Pb = 1 2 1 1 + 101.5 /2.

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