Page 377 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

Hint: Use the result from Prob. 6.77.
7T
Ans. (a) a,,,= -
2

6.79. Determine the DFT of the sequence

I -aN
Ans. X[k]= k =0.1, ..., N- 1
1 - ae-i(2r/N)k

6.80. Evaluate the circular convolution

where

(a) Assuming N = 4.
(b) Assuming N = 8.
Ans. (a) y[n]=(3,3,3,3)
(b) y[nI=~l,2,3,3,2,l,O,O)

6.81. Consider the sequences x[nl and h[nl in Prob. 6.80.
(a) Find the 4-point DFT of x[nl, hln], and y[n].
(b) Find y[n] by taking the IDFT of Y[k].
Ans. (a) [ X[Ol, X[11, X[21, X[311 = [4,O, 0,Ol
[H[Ol, H[11, HI21, H[311= [3, -j, 1, jl
[ Y[Ol, Y[ 11, YPl, Y[311 = [ 12,0,0,01
(6) y[nI= {3,3,3,3)

6.82. Consider a continuous-time signal At) that has been prefiltered by a low-pass filter with a
cutoff frequency of 10 kHz. The spectrum of x(t) is estimated by use of the N-point DFT. The
desired frequency resolution is 0.1 Hz. Determine the required value of N (assuming a power
of 2) and the necessary data length TI.
Ans. N = 2'' and T, = 13.1072 s