信号与系统第二版课后习题解答(6_7_9)奥本海姆

Chap 6

6.1 Consider a continuous-time LTI system with frequency response

H(j?)?|H(j?)|eSH(j?)and real impulse response h(t).

Suppose that we apply an input x(t)?cos(?0t??0) to this system .The resulting output can be shown to be of the form

y(t)?Ax(t?t0)

Where A is a nonnegative real number representing an amplitude-scaling factor and t0 is a time delay.

(a)Express A in terms of |H(j?)|. (b)Express t0 in terms of SH(j?0) Solution:

(a) For y(t)?Ax(t?t0)

So Y(j?)?AX(j?)e H(j?)??jt0?

Y(j?)?Ae?j?t0

X(j?)

So A?|H(j?)|

(b) for SH(j?)???t0 So t0??SH(j?)?

6.3 Consider the following frequency response for a causal and stable LTI system:

H(j?)?1?j?

1?j?(a) Show that |H(j?)|?A,and determine the values of A. (b)Determine which of the following statements is true about

?(?),the group delay of the system.(Note

?(?)??d(SH(j?))/d?,where SH(j?)is expressed in a

form that does not contain any discontinuities.) 1.?(?)?0 for ??0 2.?(?)?0 for ??0 3 ?(?)?0 for ??0 Solution:

1??2(a) for |H(j?)|??1 21?? So A=1

?H(j?)??(1?j?)??(1?j?)?(b) for

arctg(??)?arctg(?)??2arctg(?) ?(?)??d?H(j?)2 ?2d?1?? So ?(?)?0 for ??0

6.5 Consider a continuous-time ideal bandpass filter whose frequency

response is

?1,??|?|?3?c H(j?)??c?0,elsewhere(a) If h(t) is the impulse response of this filter, determine a function

g(t) such that

h(t)?sin?ctg(t) ?t(b) As ?c is increased, dose the impulse response of the filter get more concentrated or less concentrated about the origin? Solution

(a) Method 1. Let

1h(t)?x(t)g(t)?H(j?)?X(j?)?G(j?)

2?They are shown in the figures,where

x(t)?sin?ct?t1,???c ?X(j?)?{0,???cSo we can get

g(t)?2cos(2?ct)?G(j?)?2?[?(??2?c)??(??2?c)]

Method 2. Using the inverse FT definition,it is obtained

3?1??j?th(t)?{ed??ej?td?}

?2??3?11?{sin3?ct?sin?ct}?{sin?ct}{2cos2?ct} ?t?t(b) more concentrated.

?cc?cc