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 onehalf, 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) = (Wl~)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 onehalf. '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 ContinuousTime 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 [timeshifting
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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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