B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

102 ANALYSIS AND TRANSMISSION OF SIGNALS
Figure 3.31
Multipath
transmission.
Transmitted
signal ----i
(a)
(b)
H(f), given by
H(f) = e-j 2 nftd + ae-j 2 nf(td+t,.t)
= e-j 2 nftd (l +ae -j 2 nft,.t )
Delay l d
Delay t d + /J.t
IH(f) I
Received
signal
(3.64a)
= e-j 2 nft d (1 + a cos 2rrf t!..t - ja sin 2rrf M)
✓ [ ( 1 asin 2rrf M
= 1
+ a 2 + 2a cos 2rrf f!..t exp -} 2rrftd + tan - 1 + a cos 2rrf f!..t
) ]
(3.64b)
Both the magnitude and the phase characteristics of H (f) are periodic in f with a period of
1 / t!..t (Fig. 3.31 b ). The multipath channel, therefore, can exhibit nonidealities in the magnitude
and the phase characteristics of the channel and can cause linear distortion (pulse dispersion),
as discussed earlier.
If, for instance, the gains of the two paths are very close, that is, a 1, then the signals
received from the two paths may have opposite phase (rr radians apart) at certain frequencies.
This means that at those frequencies where the two paths happen to result in opposite
phases, the signals from the two paths will almost cancel each other. Equation (3.64b) shows
that at frequencies where f = n/(2t!..t) (n odd), cos 2rrf t!..t = -1, and IH (f) I 0 when
a 1. These frequencies are the multipath null frequencies. At frequencies / = n/(2t!..t)
(n even), the two signals interfere constructively to enhance the gain. Such channels cause
frequency-selective fading of transmitted signals. Such distortion can be partly corrected by
using the tapped delay-line equalizer, as shown in Prob. 3.6-2. These equalizers are useful in
several applications in communications. Their design issues are addressed later in Chapters 7
and 12.