习题Ch3.doc

Exercise Ch. 3 3.3. For the continuous-time periodic signal Determine the fundamental frequency and the Fourier series coefficients such that . Solution: So, And for elsewhere. 3.4 Use the Fourier series analysis equation (3.39) to calculate the coefficients ak for the continuous-time periodic signal with fundamental frequency ??0 = ?. Solution 3.6 Consider three continuous-time periodic signals whose Fourier series representations are as follows: Use Fourier series properties to help answer the following questions: Which of the three signals is/are real valued? Which of the three signals is/are even? Solution: (a) If , then is real valued. For , we find that: So is not real For , we find that: So is real For , we find that: So is real (b) If , then is even For , we find that: So is not even For , we find that: So is even For , we find that: So is not even 3.8 Suppose we are given the following information about a signal x(t): 1. x(t) is real and odd. 2. x(t) is periodic with period T = 2 and has Fourier coefficients ak. 3. ak = 0 for |k| 1 4. Specify two different signals that satisfy these conditions. Solution : Since x(t) is real and odd (from fact 1), its Fourier series coefficients a are purely imaginary and odd (See Table 3.1). Therefore, a= -a and a= 0. Also, since it is given that a= 0 for |k| 1, the only unknown Fourier series coefficient are a and a. Using Parseval’s relation, 1/T For the given signal we have 1/2 Using the information given in fact (4) along with the above equation, |a|+|a|=1 = 2|a|=1 Therefore, a=- a=1/ or a=- a=-1/ The two possible signals which satisfy the given information are x(t)= -sin(t) and x(t)= sin(t) 3.13 Consider a continuous-time LTI system whose frequency response is If the input to this system is a periodic signal with period T = 8, dete