Chapter 6 Problem solution 6.23 Shown in Figure 6.23 is(jo)for a lowpass filter. Determine and sketch the impulse response of the filter for each of the following phase characteristics: HGo (a)∠H(jo)=0h()= sin o t t (b)ZHGjO=oT, where T is a constant sin a(t+T h() (+r) (c)∠H(o) 丌/2>0 丌/2<0 h(t) 2sin(ot 元t
1 Chapter 6 Problem Solution 6.23 Shown in Figure 6.23 is for a lowpass filter. Determine and sketch the impulse response of the filter for each of the following phase characteristics: H(j) (a 0 ) = H j ( ) 0 1 −c H(j) c ( ) t t h t c sin = ( ) ( ) (t T ) t T h t c + + = sin (b) , where H(j)=T T is a constant. ( ) = / 2 0 / 2 0 -π (c) H j ( ) ( ) t t h t c 2sin / 2 2 − =
Chapter 7 Problem solution 7.3 Determine the Nyquist rate corresponding to each of the following signals: a)x(t)=1+cos(2,000x)+sn(4000x) O,,=4.000丌 O.=204=8,000兀 (b)x()= sin(4000m) 丌t O,,=4.000丌 O=201=8.,000兀 sn(40002 丌t O4=8,000兀 O.=20,=16.000丌
2 Chapter 7 Problem Solution 7.3 Determine the Nyquist rate corresponding to each of the following signals: (a) x(t)=1+cos(2,000t)+sin(4,000t) ( ) ( ) ( ) t , t x t sin 4 000 b = ( ) ( ) ( ) 2 sin 4 000 c = t , t x t M = 4,000 s = 2 M = 8,000 M = 4,000 s = 2 M = 8,000 M = 8,000 s = 2 M =16,000
Chapter 7 Problem solution 7.6 X(o)=0,ol≥a aw,( X2(0)=0,l≥a2 x2(t) p()=∑(-n) n=-00 Determine the maximum sampling interval T such that w() is recoverable fromw,(t)through the use of an ideal LPF. 2x兀 sampling interval max 01+O
3 Chapter 7 Problem Solution 7.6 ( ) 1 1 X j = 0 , x (t) 1 x (t) 2 w(t) p(t) (t nT ) n = − + =− w (t) p ( ) 2 2 X j = 0 , w (t) p Determine the maximum sampling interval T such that w(t) is recoverable from through the use of an ideal LPF. maximum sampling interval 1 2 max 2 + = = s T
Chapter 7 Problem solution 7.9 Consider the signal 2 sn50丌t 丌t which we wish to sample with a sampling frequency of =150T to obtain a signal g(t) with Fourier transform G(o). Determine the maximum value of o for which it is guaranteed that G(o)=75X(o)o≤ao x la Gla 50 50个 1507 100x0100兀 △150z -100z0100 100元 0=50元 4
4 Chapter 7 Problem Solution 7.9 Consider the signal ( ) 2 sin 50 = t t x t which we wish to sample with a sampling frequency of to obtain a signal with Fourier transform . Determine the maximum value of for which it is guaranteed that s =150 ( ) ( ) 0 G j = 75X j g(t) G(j) 0 −100 0 50X(j) 100 0 50G(j) −100 100 −150 150 100 0 = 50
Chapter 8 Problem solution 8.3 Determine y(. g(t LPF HGo x(0)=0,|>20077 g()=x()sn(200 H(o)= 2@≤200x 0ol>200x Solution m()=(o.00)=12(0n(4007 Be out of the passband of LPF ()=0 5
5 Chapter 8 Problem Solution m(t) y(t) = 0 m(t) g(t) ( t) x(t)sin (4,000t) 2 1 = cos 2,000 = ( ) = 0 2,000 2 2,000 H j 8.3 Determine . X(j) = 0 , 2,000 g(t) = x(t)sin(2,000t) cos(2000t) g(t) y(t) LPF H(j) y(t) Solution Be out of the passband of LPF