Signal Processing and Linear Systems-B.P.Lathi copy

# 433a t he multiplier output is 1 et 2 cos we wmt

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Unformatted text preview: passing .,p(t) through the inverse linear system, which has the transfer function l /H(s). 290 4 Continuous-Time Signal Analysis: T he Fourier Transform digital communication, use of phase a nd frequency modulation is common, t he socalled b roadcast F M is n ot F M in t he classical sense, b ut is a generalized angle modulation because of t he inclusion of t he p reemphasis filter used t o improve its noise suppressing abilities. I t is called FM for historical reason in t he sense t hat angle m odulation was first conceived a nd i ntroduced in t he form of frequency modulation. T he b roadcast FM, although originating as true FM in the laboratories, was modified in broadcasting for b etter performance. Yet, t he t erm F M c ontinued t o be used t o describe this scheme. In a mplit ude modulation, t he c arrier frequency is c onstant, b ut t he a mplitude changes with m (t). I n contrast, in angle modulation, the carrier amplitude is always constant, b ut t he c arrier frequency varies continuously with t he message m (t). By definition, a sinusoidal signal is e xpected t o have a constant frequency; hence, the variation of frequency with time appears t o be contradictory t o t he conventional definition o f a sinusoidal signal frequency. Therefore, we m ust generalize t he notion of a sinusoid so as t o make allowance for variation of frequency with time. This generalization leads us t o a new concept of i nstantaneous f requency. 291 4.8 Angle Modulation 'P(t) = A cos O(t) where O(t), t he g eneralized a ngle, is a function of t. F igure 4.41 illustrates a hypothetical case of O(t). T he generalized angle for a conventional sinusoid A cos (wet + 4>0) is wet + 4>0' T his plot, a straight line with a slope We a nd intercept 4>0' is also illustrated in Fig. 4.41. T he p lot of O(t) for t he h ypothetical case happens t o be tangential t o t he angle (w et+4>o) a t some i nstant t. T he crucial point is t hat over a small interval t:.t - t 0, t he signal <p (t) = A cos 0 (t) a nd t he sinusoid A cos (wet + 4>0) a re identical; t hat is, <p(t) = A cos (wet + 4>0) We a re certainly justified in saying t hat over this small interval t:.t, t he frequency of 'P(t), is We' Because (wet + 4>0) is t angential t o O(t), t he frequency of 'P(t) is t he slope of its angle O(t) over this small interval. We c an generalize this concept a t every i nstant a nd say t hat t he i nstantaneons frequency Wi a t a ny instant t is t he slope of O(t) a t t. Thus, for 'P(t) in Eq. (4.82), t he i nstantaneous frequency Wi(t) is given by dO Wi(t) = dt O(t) = F ig. 4.41 Concept of instantaneous frequency. (4.82) (4.83a) lex> Wi(a) da (4.83b) For a conventional sinusoid A cos (wet + 4>0)' we have O(t) = wet + 4>0' a nd Wi(t) = dO(t)/dt = We, a c onstant, as desired. Clearly, t he generalized definition of instantaneous frequency does not conflict with our old notion of frequency. Now we can see t he possibility of t ransmitting t he i nformation of m (t) by varying t he angle 0 of a carrier. Two simple pos...
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