2.4 SLOW FADING UNIFORM CHANNELS
2.4.2 Broadband Jamming—Diversity and Coding
Consider diversity of order mwhere each MFSK signal of duration Tsec- onds is transmitted as mMFSK chip tones each of duration TcT/msec- onds. Each chip is independently hopped over the spread-spectrum frequency band. Also assume that each chip tone has independent Rayleigh fading of the same statistics. This is a reasonable assumption for FH/MFSK signals that hop over a wide frequency band Wsswhere
(2.211) where fcis the “coherence bandwidth” of the fading channel [24].
If we denote
(2.212) as the cosine and sine components of the received signal at the l-th frequency during the k-th chip interval and
(2.213) as the sequence of such components for the l-th tone, then the conditional probability of the entire set of mMchip cosine and sine components given the l-th tone was transmitted as
(2.214) where
(2.215) (2.216) and
(2.217) p1m1Yj2 q
m k1
p11yjk2. p0m1Yj2 q
m k1
p01yjk2. F1Y1, Y2,p, YM2 q
M
j1p0m1Yj2,
p1Y1, Y2,p, YM0l is sent2F1Y1, Y2,p, YM2p1m1Yl2 p0m1Yl2 Yl 1yl1, yl2,p, ylm2
ylk 1ylkc, ylks2 Wss W ¢fc
Eb>NJ
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The ML decision rule chooses lthat maximizes
(2.218) or equivalently maximizes
(2.219) where
(2.220) is the energy of the l-th tone in the k-th chip interval. is the average energy per MFSK chip.
E elkylkc2 ylks2
el a
m k1
elk p1m1Yl2
p0m1Yl2 a NJ
ENJbmexpe E>NJ ENJ a
m
k11ylkc2 ylks2 2 f
Slow Fading Uniform Channels 605
Figure 2.107. Rayleigh fading BFSK—diversity.
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Note that here the ML decision rule is based on the sum of the mchip energies for each of the Mtones. The energy metric is thus optimum for the Rayleigh fading channel with equivalent white Gaussian noise or broadband noise jamming. This was not the case with coding over a non-fading white Gaussian noise channel. There, however, the energy metric was used since it is convenient.
For mdiversity, the bit error probability is given by [24],
(2.221) a
l1m12 k0
bkl1m1k2!
1m12! a 1 1E>NJ2
1l31 1E>NJ2 4bk Pb
1 2M M1 a
M1 l1
aM1
l b112l1 51l31 1E>NJ2 4 6m .
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Figure 2.108. Rayleigh fading 4FSK—diversity.
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where ,M2K, and bklsatisfies
(2.222) For the binary case where M2, the bit error probability is simply
(2.223) where
(2.224) Figure 2.107 shows this bit error probability versus for various values of diversity. Note that, for each value of Eb>NJ, there is an optimum
Eb>NJ
d m
2m1Eb>NJ2 . Pbdm a
m1
k0am1k
k b 11d2k
a a
m1 k0
xk
k!bl a
l1m12 k0
bklxk. E 1K>m2Eb
Slow Fading Uniform Channels 607
Figure 2.109. Rayleigh fading 8FSK—diversity.
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diversity. Excessive diversity results in non-coherent combining losses that begin to cancel the beneficial effects of having independent observations.
Next, a Chernoff bound will be derived for the bit error probability when M 2. Assuming the ML metric, the optimized Chernoff bound is the Bhattacharyya bound given by
(2.225)
)
But F(Y1,Y2) is the joint Gaussian density function where all random vari- ables are i.i.d with zero mean and variance NJ/2. Thus
(2.226) Figure 2.107 shows this bound (dotted lines) together with the exact bit error probability.
For M 2 the union bound can be combined with the above Chernoff bound. The symbol error probability is union bounded by
(2.227) where l 1 was assumed to be the transmitted symbol and is the pairwise error bound, which is the same as the M2 bit error proba- bility. Thus, using the Chernoff bound
(2.228) Pr51S26 1
2 c4m1m1E>NJ2 2 12m 1E>NJ2 22 dm
Pr51Slˆ6 1M12Pr51S26
a
M k2
Pr51Sk6 a
M k2
Pr5ˆl k0l16 PsPr5ˆl10l16 Pb 1
2c4m1m1Eb>NJ2 2 12m 1Eb>NJ2 22 dm expe E>NJ
ENJ ay1kc2 y1ks2 y2kc2 y2ks2
2 b fdY1dY2. 1
2F1Y1, Y22kqm1a NJ ENJb. 1
2F1Y1, Y22Bpp1m0m11YY1122pp1m0m11YY2222 dY1dY2 Pb 1
22p1Y1, Y20l12p1Y1, Y20l2 dY1dY2
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and noting that here , we obtain the bit error bound
(2.229) Figures 2.108 to 2.111 show these bounds for various values of Kand diver- sity mtogether with the corresponding exact bit error probabilities given by (2.221) and (2.222).
Next choose the value of diversity mthat minimizes the bound in (2.228).
This is shown in Figure 2.112. Figure 2.113 shows the same bound with m 2K2c4m1mK1Eb>NJ22
12mK1Eb>NJ222 dm. Pb
1 2M M1 Ps EKEb
Slow Fading Uniform Channels 609
Figure 2.110. Rayleigh fading 16FSK—diversity.
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Kwhere M2K. For fixed data rate, this case results in the bandwidth of each FH/MFSK chip signal being the same for M2, 4, 8, 16, and 32.
Regarding each FH/MFSK chip as a coded M-ary symbol, a coding chan- nel (see Figure 4.1 in Chapter 4, Part 1) with output
(2.230) where ylis the cosine and sine components of the channel output signal at the l-th chip tone is obtained. In a coded system, a sequence of coded chips is transmitted and the channel outputs are the corresponding vectors of these cosine and sine components. Any coding system uses a metric m(y,l) which assigns a value to each channel output corresponding to each possible input.
y1y1, y2,p, yM2
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Figure 2.111. Rayleigh fading 32FSK—diversity.
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Slow Fading Uniform Channels 611
Figure 2.113.Pb-FH/MFSK,mKdiversity.Figure 2.112.Pb-FH/MFSK optimal diversity.
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The ML energy metric we have considered previously is
(2.231) For the ML metric and simple diversity, the expresion for the bit error probability given by (2.221) and (2.222) is quite complex. Obtaining an exact bit error probability expression is much more difficult with other metrics.
Hence we examine bounds on the coded bit error probabilities.
Based on the general approach of Chapter 4, Part 1, for an arbitrary met- ric m(y,l), the general coded bit error bound is given by
(2.232) where for FH/MFSK the cutoff rate R0is given by (2.180) where
(2.233) with l l. For the energy metric (2.232),Dis given by
(2.234) Also, for the special case of an mdiversity code, the function G() is
(2.235) with code rate RK/mbits per chip and . This is the result given by (2.229).
Rather than the energy metric, if a hard decision is made for each MFSK chip then
(2.236) where
(2.237) which is the chip error probability given by (2.208). Figures 2.114 and 2.115 show the performance for optimum diversity and diversity mKwhere M 2Kfor the hard decision metric.