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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online