Rayleigh fading is characteristic of numerous systems, including, along with communications, radar, navigation etc. The deep falls in signal intensity inherent in it are not as a rule neutralized by sporadic rises of Ar when multipath signals arrive nearly in phase. As a result, the overall effect of Rayleigh fading on the system performance appears to be pretty destructive, as the analysis below corroborates.

3.5.4 Performance analysis

Consider as an example binary data transmission over the Rayleigh channel with a slow and flat fading. The first of the attributes means that the interference pattern remains stable during many symbols and the current reference phase may be retrieved from the received signal by averaging over an appropriate time interval. In other words, signal randomness does not exclude the BPSK from the available options. The second term stresses that the delay spread of multipath signals ds is small enough compared to the duration Tb of the individual BPSK bit: ds Tb. As a result, successive BPSK symbols do not overlap with each other, i.e. ISI (see Section 2.15) does not emerge. To understand why the word ‘flat’ is relevant, return to (3.11) and note that an adequate model of a multipath channel is a delay line with taps having delays i and complex weights Ai exp (j i). The transfer function of such a system strongly depends on tap delays, and when ds Tb is rather uniform (flat) within the signal bandwidth so that all signal frequency components are distorted identically and signal shape remains unchanged. The only sort of corruption which the signal undergoes due to multipath propagation in such a case is Rayleigh amplitude fluctuations described by (3.12).

Figure 3.15 enlarges on these definitions. Plots simulated in Matlab show a slow flat fading (a) as opposed to the fast flat one (b) for the case of the bell-shaped symbol pulses. The second of the fading types is characterized by a rapid change of the interference pattern in time so that distortions of successive symbols are practically independent.

Figure 3.15 Slow (a) and fast (b) fading

Let the energy of the received signal corresponding to Ar ¼ 1 be E. Then the energy of a signal with another value of amplitude is E(Ar) ¼ A2r E and the average energy

again equals E due to the normalization adopted above: E(Ar) ¼ A2r E ¼ E. Equation (2.19) may be used to calculate a conditional error probability Pe(Ar), when the received amplitude is assumed fixed and equal to Ar:

p

where SNR qb ¼ 2E/N0 corresponds to the signal of energy E(A) ¼ E, i.e. of amplitude Ar ¼ 1. Actual amplitude Ar is random and fluctuates from one receiving session to another according to the Rayleigh PDF (3.12). It is natural, then, to characterize the performance of data transmission by the value of Pe(Ar) averaged over all Ar. Reserving now the term ‘error probability’, with designation Pe, for this expectation we have:

0Arqb

where definition of the complementary error function Q( ) is used. Reversing the order of the integrals gives:

00

where the first term equals Q(0) ¼ 1/2 and the second is brought to the same form if

01

To assess quantitatively the extent of the harmful influence of the fading, look at Figure 3.16, which presents error probabilities of the BPSK transmission over the AWGN and Rayleigh channels. As is seen, Pe ¼ 10 3 in the AWGN channel may be guaranteed with the bit SNR around 10 dB, while the Rayleigh channel requires bit SNR of at least 27 dB,

Bit SNR (dB)

Figure 3.16 Bit error probability for AWGN and Rayleigh channels