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
Chapter 8 Problem solution 8.22 In Figure(a), a system is shown with inputx()and output y() The input signal has the Fourier transform (jo) shown in Figure(b) Determine and sketch y(o) H,GjO) 73 H Go) y 5W-3W 3W 5W 3W 0 3W 0 cos 5wt cos 3wt Figure(a) xGa) -2W0 2N0 Figure(b)6
6 8.22 In Figure (a) ,a system is shown with input and output The input signal has the Fourier transform shown in Figure (b) Determine and sketch . Chapter 8 Problem Solution X(j) Y(j) x(t) y(t) 0 1 −3W 3W H (j) 2 x(t) r(t) 1 r (t) 2 −3W 3W 1 H (j) 1 −5W 5W r (t) y(t) 3 cos5Wt cos3Wt X(j) − 2W 0 2W Figure (a) 1 Figure (b)
Problems for Fourier analysis Example 1 f() Determine the fourier A period of sin nt transform off(t) 2 t 2jo Fgo 丌- 7
7 Problems for Fourier Analysis ( ) 2 2 2 − − = − j e F j f (t) 0 1 2 1 t A period of sin t Example 1 Determine the Fourier transform of f (t)
Problems for Fourier analysis Example 2 A real continuous-time signal f(with Fourier transform F(a), and In FGo)=Ho 1. If f(is even, determine f() 2. Iff()is odd, determine f(o) 1.∫()= ±1 丌(1+t 2.∫()=千 丌(1+t
8 Example 2 A real continuous-time signal with Fourier transform , and 1. If is even, determine . 2. If is odd, determine . f (t) f (t) f (t) f (t) f (t) F(j) ln F(j) = − Problems for Fourier Analysis ( ) ( ) 2 1 1. 1 f t t = + ( ) ( ) 2 2. 1 t f t t = +
Problems for Fourier analysis 例3试计算下列无穷积分 ∫ o sint dt=? too sin t dt=? ()( sint dt= t )=(m)le? 2
9 例3 试计算下列无穷积分 ( ) sin a ? t dt t + − = ( ) 2 sin b ? t dt t + − = ( ) 3 sin c ? t dt t + − = ( ) 2 sin d cos ? t t dt t + − = Problems for Fourier Analysis
Problems for Fourier analysis 例4试计算下列卷积积分 a)x(t)=e“u(t)米cost 3 rt cos3丌t-sin3t 32t2 cos(2t+0) 其中,6任意常数
10 例4 试计算下列卷积积分 ( ) ( ) ( ) 4 a cos t x t e u t t − = ( ) ( ) 2 2 3 cos 3 sin 3 b cos 2 3 t t t t t − + 其中，θ为任意常数。 Problems for Fourier Analysis