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

# 422a t he pulse j t is t he gate pulse rect in fig

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Unformatted text preview: of modulation is known as a mplitude m odulation. T he sinusoid cos wot i s called t he c arrier, t he signal f (t) is t he m odulating s ignal, a nd the signal f (t) c os wot is t he m odulated s ignal. F urther discussion of modulation when cos wot = 1 when cos wot = - 1 Therefore, f (t) cos wot touches f (t) when t he sinusoid cos wot is a t i ts positive peaks and touches - f(t) when cos wot is a t i ts negative peaks. This means t hat f (t) a nd - f(t) a ct as envelopes for the signal f (t) cos wot (see Fig. 4.23). T he signal - f(t) is a mirror image of f (t) a bout t he horizontal axis. Figure 4.23 shows t he signal f (t), f (t) cos wot a nd their spectra. • E xample 4 .12 F ind a nd s ketch t he F ourier t ransform o f t he m odulated s ignal f (t) cos lOt in which f (t) is a g ate p ulse rect (~) as i llustrated i n F igure 4.24a. From p air 17 (Table 4.1) we find rect (~) 4 sine (2w), w hich is d epicted in Fig. 4.24b. From Eq. (4.41) it follows t hat = j (t) cos lOt = ~[F(w + 10) + F (w - 10)J I n t his case, F(w) = 4 sinc (2w). T herefore j (t) cos lOt = 2 sine [2(w + 10)J + 2 sine [2(w - lO)J T he s pectrum o f f (t) cos lOt is o btained b y shifting F(w) i n Fig. 4.24b t o t he left by 10 a nd also t o t he r ight by 10, a nd t hen m ultiplying i t by one-half, as depicted in Fig. 4.24d. • f::,. E xercise E 4.7 Sketch signal e - Itl cos lOt. Find the Fourier transform of this signal and sketch its spectrum. Answer: F(w) = (W-l~)2+1 + (W+l~)2+1' T he spectrum is t hat in Fig. 4.19b (with a = 1), shifted to ±10 and multiplied b y one-half. 'V Application t o Modulation Modulation is used t o shift signal spectra. Some of the situations where spectrum shifting is necessary are presented next. 262 4 Continuous-Time Signal Analysis: T he Fourier Transform 4.3 Some P roperties of the Fourier Transform 263 then (time c onvolution) He claims this method i ncorpol8tes the b est o f t wo o lder methods. h (t) * h (t) {=} (4.42) Fl(W)F2(W) a nd (frequency c onvolution) h (t)h(t) 1 {=} 27T Fl(W) * F2(W) (4.43) Proof: By definition F lh(t) * h (t)1 = = [I: 1: [1: [ : e- jwt h er) h (r)h(t - r)dr] dt e -jwth(t - r)dt] dr The inner integral is t he Fourier transform of 12 (t - r), given by [time-shifting property in Eq. (4.37)] F2(w)e- jwT . Hence 1. I f several signals, each occupying the same frequency band, are transmitted simultaneously over t he same transmission medium, they will all interfere; it will be impossible to separate or retrieve them a t a receiver. For example, if all radio s tations decide t o b roadcast audio signals simultaneously, the receiver will not b e able to separate them. This problem is solved by using modulation, whereby each radio station is assigned a distinct carrier frequency. Each station t ransmits a m odulated signal. This procedure shifts the signal spectrum to its allocated band, which is n ot occupied by any other station. A radio receiver c an pick up any s tation by tuning t o t he band of the desired station. T he receiver must now demodulate the received...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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