Chapter 2 Problems solution ,0≤n≤9 1.0≤n≤N 2.5 0. elsewhere 0. elsewhere =x小]*y4]=5,y4=0 Determine the value of n N=4
1 Chapter 2 Problems Solution N = 4 2.5 = 0, elsewhere 1, 0 n 9 x n = 0, elsewhere 1, 0 n N h n yn= xnhn y4= 5 , y14= 0 Determine the value of N
Chapter 2 Problems solution 2.7 A linear system S has the relationship =∑ xkgn-2 between its input xn] and output[n] gn]=un]-un-4 (a)x[小]=o[n-1]y[小=[n-21-u[n-6 (b)x{]=o8[n-2]y[小=utn-4]-m-8 (c)s is time-varying. uIn+uln =2u 6n-6n
2 Chapter 2 Problems Solution 2.7 A linear system S has the relationship yn xkgn k k = − 2 + =− between its input and output xn yn gn= un−un−4 (a 1 ) x n n = − y n u n u n = − − − 2 6 (b 2 ) x n n = − y n u n u n = − − − 4 8 (c) S is time-varying. (d ) x n u n = y n u n u n = + − 2 = − − − 2 1 u n n n
Chapter 2 Problems solution M0)=0(),0s161.0x1≤1 2.10 Suppose that 0 elsewhere (a)Determine y()=x(t)*h(t) (b)If dy(t/ dt contains only three discontinuities what is the value of a? Solution o a1 l+ t
3 Chapter 2 Problems Solution 2.10 Suppose that ( ) = 0, elsewhere 1, 0 t 1 x t h(t) = x(t / a) , 0 a 1 y(t) = x(t)h(t) d y(t) dt (a) Determine (b) If contains only three discontinuities,what is the value of a? Solution ： y(t) a 0 a 1 1+a t
Chapter 2 Problems solution 212Lety()=e"l()*∑o(-3k) k=-00 Show that y()=Ae' for 0<t<3 Determine the value ofa y()=c2=,1e 0≤t<3 e 4
4 Chapter 2 Problems Solution 2.12 Let ( ) ( ) ( ) =− − = − k t y t e u t t 3k Show that for ( ) t y t Ae− = 0 t 3 Determine the value of A. ( ) 3 3 0 1 0 t 3 1 t k t k y t e e e e + − − − − = = = − 3 1 1 A e − = −
Chapter 2 Problems solution 22(c) ( h() 2 one period of sin ru 米 0 123t 0 2 t 0 t5 5
5 Chapter 2 Problems Solution 0 1 2 2 3 x(t) t sin t h(t) 0 1 2 1 t 2.22(c) one period of ( ) ( ) ( ) 0 t 1 2 1 cos 1 t 3 2 - 1 cos 3 t 5 0 t 5 t y t t + = +
Chapter 2 Problems solution 2.23 h( x()=∑(-k7) 2T-T0 T 2 t Determine and sketch y()=x(t)*h(t) for the following value ofT (a)T=4(b)T=2(c)T=3/2(d)T=1 p()2∑(-An)*()=(-A)
6 Chapter 2 Problems Solution 2.23 − 2T −T 0 T 2T t 1 −1 1 0 1 t h(t) Determine and sketch for the following value of T: y(t) = x(t)h(t) (a) T=4 (b) T=2 (c) T=3/2 (d) T=1 ( ) ( ) =− = − k x t t k T ( ) ( ) ( ) ( ) k k y t t kT h t h t kT =− =− = − = −
Chapter 2 Problems solution 5-4-3 1011 10 (b)T=2 0 0 t (c)T=3/2 (d)T=1 7
7 Chapter 2 Problems Solution 1 y(t) (a) T=4 -5 -4 -3 -1 0 1 3 5 t (d) T=1 y(t) 1 0 t 1 y(t) -3 -1 0 1 3 t (b) T=2 1 y(t) 0 t (c) T=3/2
Chapter 2 Problems solution 2.40(a)an LTI system: y(t)=e-(=x(-2 dr What is the impulse response h()for this system? (b)Determine the response of x(t) 2 (a)h(t)=e(2n(t-2) (b)y()=[1-e](-)[l-c(]a(-4)
8 Chapter 2 Problems Solution 2.40 (a) an LTI system: ( ) ( ) ( ) y t e x d t t = − 2 − − − What is the impulse response for this system? h(t) (b) Determine the response of x(t) −1 0 2 1 x(t) t ( ) ( ) ( ) ( ) 2 a 2 t h t e u t − − = − ( ) ( ) ( ) ( ) ( ) ( ) 1 4 b 1 1 1 4 t t y t e u t e u t − − − − = − − − − −
Chapter 2 Problems solution 2.46 Consider an Lti system s and a signal )=2e3l(t-1) dx() If x()→y() and →-3y()+el( dt Determine the impulse response h(t)ofS ()=e)(t+1) h 2
9 Chapter 2 Problems Solution 2.46 Consider an LTI system S and a signal ( ) 2 ( 1) 3 = − − x t e u t t x(t)→ y(t) and ( ) y(t) e u(t) dt dx t 2t 3 − If → − + Determine the impulse response of S. h(t) ( ) ( ) ( ) 1 3 2 1 1 2 t h t e e u t − + = +
Chapter 2 Problems solution 2.47 An LTI system with impulse response ho(t) x(→y0( ( In each of these cases. determine whether or not we have enough 0 2 Information to determine the output y() (d)x()=xn(-)(t)=h(t) We have not enough information to determine the output (e)x()=x0(-)h()=b1(-) y()=y(-)
10 Chapter 2 Problems Solution 2.47 An LTI system with impulse response h (t) 0 x (t) y (t) 0 → 0 0 2 t 1 y (t) 0 In each of these cases,determine whether or not we have enough Information to determine the output y(t) ( ) ( ) ( ) ( ) ( ) d 0 0 x t x t h t h t = − = We have not enough information to determine the output ( ) ( ) ( ) ( ) ( ) 0 0 e x t x t h t h t = − = − ( ) ( ) 0 y t y t = −