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Unformatted text preview: erty (dual of t he t ime differentiation): P rove t he following results, which are duals of each other: f (t) sin wat { =} h[F(w  wa)  F(w  h[t(t + T )  f(t  T)] { =}  jtf(t) + wo)] Using t he frequencyshifting property a nd T able 4.1, find t he inverse Fourier transform
o fthe s pectra d epicted in Fig. P4.37.
4 .38 Using t he t ime convolution property, prove pairs 2, 4, 13 a nd 14 in Table 2.1 (assume
>. < 0 in p air 2, >'1 a nd >'2 < 0 in pair 4, >'1 < 0 a nd .\2 > 0 in pair 13, a nd >'1 a nd
>'2 > 0 i n p air 14). Hint: You will need p artial fraction expansion. For pair 2, you
need t o a pply t he r esult in Eq. (1.23). ! F(W) ( b) Using t his p roperty a nd p air 1 (Table 4.1), determine t he Fourier transform of
t eatu(t). F(w) sin T w Using t he l atter r esult a nd Table 4.1, find t he Fourier transform of t he signal in Fig.
P4.35.
4 .36 T he s ignals i n Fig. P4.36 are modulated signals with carrier cos lOt. F ind t he Fourier
t ransforms o f these signals using t he a ppropriate properties of t he F ourier transform
a nd T able 4.1. Sketch t he a mplitude a nd p hase spectra for p arts ( a) a nd ( b). { =} 4.41 For an LTIC s ystem with transfer function
1 H (s) = s + 1
find t he ( zerostate) response if t he i nput f (t) is ( a) e  2t u(t) ( b) e tu(t)
( c) e tu(t) ( d) u (t)
Hint: For p art ( d), you need t o a pply t he r esult in Eq. (1.23). 4 .37 4.42 A stable LTIC s ystem is specified by t he t ransfer function
1 H (w)=. JW  2 314 4 C ontinuousTime S ignal A nalysis: T he F ourier T ransform 315 P roblems specifies t he real p art, t he i maginary p art c annot b e specified independently. T he
i maginary p art is p redetermined by t he real p art, a nd vice versa. This result also
leads t o t he conclusion t hat t he m agnitude a nd angle of H (w) a re r elated provided
all t he poles a nd zeros of H (w) lie in t he LHP. f,( t )
4 .51 Consider a filter w ith t he t ransfer function H (w) = F ig. P 4.43 F ind t he i mpulse response of this system a nd show t hat t his is a n oncausal system.
F ind t he ( zerostate) response of this system if t he i nput j (t) is
( a) e tu(t) ( b)etu(  t).
4 .43 4 .44 4 .45 4 .46 Signals / I(t) = 104 rect(10 4 t) a nd h (t) = 6(t) are applied a t t he i npnts o f t he ideal
lowpass filters Hl(W) = rect(40.~00") a nd H2(W) = rect(20.~") (Fig. P4.43). T he
o utputs Y l(t) a nd Y2(t) o f these filters are multiplied to o btain t he signal y (t) =
Y l(t)Y2(t).
( a) S ketch Fl(W) a nd F2(W).
( b) S ketch Hl(W) a nd H2(W),
( e) S ketch Yl(W) a nd Y2(W),
( d) F ind t he b andwidths o f Y l(t),Y2(t), a nd y (t).
H int for p art ( d): Use t he convolution property a nd t he w idth property of convolution
t o d etermine t he b andwidth o f Yl(t)Y2(t).
A low pass system time constant is often defined as t he w idth of its u nit impulse
response h (t) (see Sec. 2.72). An i nput pulse p(t) t o this system acts like an impulse
of s trength e qual t o t he a rea of p(t) if t he w idth of p(t) is much smaller t han t he s ystem
time c onstant. Assume p( t ) t o b e a lo...
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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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