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

# A nd demodulation appears in secs 4 7 a nd 48 to

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Unformatted text preview: signal (undo the effect of modulation). Demodulation therefore consists of another spectral shift required to restore t he signal t o its original band. Note t hat b oth modulation and demodulation implement spectral shifting; consequently, demodulation operation is similar t o m odulation (see Sec. 4.7). This m ethod o f transmitting several signals simultaneously over a channel by sharing i ts frequency band is known as f requency-division m ultiplexing ( FDM). 2. For effective radiation of power over a radio link, the a ntenna size must be of t he o rder of t he wavelength of the signal t o be radiated. Audio signal frequencies are so low (wavelengths are so large) t hat impracticably large antennas will be required for radiation. Here, shifting the spectrum t o a higher frequency (a smaller wavelength) by modulation solves the problem. We have demonstrated in Eq. (2.48) t hat t he transfer function H (w) is t he Fourier transform of the unit impulse response h (t). T hus h (t) {=} H(w) (4.44a) Application of t he t ime convolution property t o yet) = J (t) b oth J (t) a nd h (t) are Fourier transformable) * h(t) y ew) = F (w)H(w) yields (assuming (4.44b) This is precisely the result proved earlier in Eq. (4.19).t T he frequency convolution property (4.43) can be proved in exactly the same way by reversing t he roles of J (t) a nd F (w ). • E xample 4 .13 Using t he t ime c onvolution property, show t hat i f J(t) &lt;==* F(w) then l' J (r)dr &lt;==* F~w) + 7TF(O)8(w) (4.45) JW - 00 B ecause u(t-r)={~ r ::ot r &gt;t it follows t hat 4.3-6 Convolution T he time convolution property and its dual, the frequency convolution property, s tate t hat if and J(t) * u(t) = [ : J(r)u(t - r)dr = loo J(r) dr t In Eq. (4.44b), h(t) ~ H(w). ' lb understand finer points of Eq. (4.44b), see footnote on p. 243. 4 2 64 C ontinuous-Time S ignal A nalysis: T he F ourier T ransform 4.3 S ome P roperties o f t he F ourier T ransform T able 4 .2 Now, from t he t ime convolution property [Eq. 4.42], it follows t hat f (t) * u(t) loo = f (r) dr = F(w) 265 F ourier T ransform O perations [j~ + 7r&lt;5(W)] =F(w) + 1T F(O)8(w) O peration f (t) F (w) A ddition h (t) S calar m ultiplication k f(t) k F(w) S ymmetry F (t) 2 1Tf(-w) S caling (a r eal) f (at) I~IF (~) T ime s hift JW In deriving the l ast result, we used Eq. (1.23a) • 6 E xercise E 4.8 Using the time convolution property, show t hat f (t) * 6(t) = f (t) \l + h (t) FJ(w) 6 E xercise E 4.9 Using the time convolution property, show t hat + F2(W) F (w)e- iwto F requency s hift (wa r eal) T ime c onvolution h (t) F requency c onvolution h (t)h(t) 1 - FJ(w) 21T T ime d ifferentiation d nf dtn ( jw)nF(w) T ime i ntegration Hint: Use property (4.42) to find the Fourier transform of e -atu(t) * e -btu(t). Then use partial fraction expansion t o find its inverse Fourier transform. \ l f (t - ta) f (t)e iwot [ oof(X)dX - .- + 1T F(O)6(w) 4.3-7 Time Differentiation and Time Integration If f (t) ~ F (w) F (w - wa) * h (t) FJ(w)F2(W) *...
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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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