UploadFile_6417

160 Chapter 5 THE DISCRETE FOURIER TRANSFORM
Magnitude of the DFT: N=8
4
3
|X(k)|
2
1
0
−1
0 1 2 3 4 5 6 7 8
k
Angle of the DFT: N=8
200
100
Degrees
0
−100
−200
0 1 2 3 4 5 6 7 8
k
FIGURE 5.6 The DFT plots of Example 5.7: N =8
which is shown in Figure 5.6. Continuing further, if we treat x(n) asa16-point
sequence by padding 12 zeros, such that
x(n) ={1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
↑
then the frequency resolution is ω 1 =2π/16 = π/8 and W 16 = e −jπ/8 . Therefore
we get a more dense spectrum with spectral samples separated by π/8. The
sketch of X 16 (k) isshown in Figure 5.7.
It should be clear then that if we obtain many more spectral samples by
choosing a large N value then the resulting DFT samples will be very close to
each other and we will obtain plot values similar to those in Figure 5.4. However,
the displayed stem-plots will be dense. In this situation a better approach to
display samples is to either show them using dots or join the sample values using
the plot command (that is, using the FOH studied in Chapter 3). Figure 5.8
shows the magnitude and phase of the 128-point DFT x 128(k) obtained by
padding 120 zeros. The DFT magnitude plot overlaps the DTFT magnitude plot
shown as dotted-line while the phase plot shows discrepancy at discontinuities
due to finite N value, which should be expected.
□
Comments: Based on the last two examples there are several comments
that we can make.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.