Digital Signal Processing Reference
In-Depth Information
Exercise:
w[n] = x[n] + x[n1]
y[n] = x[n]x[n1]
Find and plot the frequency response for w and y.
First, assume x is a unit impulse function, and nd w[n] and y[n]. Pad these
signals, say up to 128 values, so that the result will appear smooth. Next, nd the
DFT (or FFT) of w and y, and plot the magnitudes of the result. The plots should
go to only half of the magnitudes, since the other half will be a mirror image.
6.11
Summary
This chapter covers the Fourier transform, inverse Fourier transform, and related
topics. The Fourier transform gives frequency (spectral) content of a signal. Given
any series of samples, we can create a sum of sinusoids that approximate it, if not
represent it exactly.
6.12
Review Questions
1. Suppose we were to sample a signal at f
s
= 5000 samples per second, and we
were to take 250 samples. After performing the DFT, we nd that the rst
10 results are as follows. What does this say about the frequencies that are
present in our input signal? (Assume that the other values for X[m] up to
m = 125 are 0.)
X[m] = 10; 0; 0; 2 + j4; 0; 1; 0; 0; 0; 2j4
2. An input sequence, x[n], has the Fourier transform performed on it. The result
is:
X[m] =f3, 2+j4, 1, 5-j3, 0, 0, 0, 5+j3, 1, 2-j4g:
a. Find (and plot) the magnitudes and phase angles.
b. You should notice some symmetry in your answer for the rst part. What
