Digital Signal Processing Reference
In-Depth Information
1
0
−1
0
2
4
6
8
(a) Sample
2
1
0
0
2
4
6
8
(b) Bin
2
1
0
0
200
400
600
800
1000
(c) Bin
Figure 3.41:
(a) Eight-sample impulse response; (b) Magnitude of eight-point DFT; (c) Magnitude of
1024-point DFT (i.e., 1024 samples of the DTFT), solid line, with eight-point DFT magnitude values
plotted as stars.
= 0,1,2,3,and 4, while the 128-pt DFT gives us 65 frequencies, i.e., k = 0:1:64. Consequently, there is a
much finer gradation of frequency with the longer DFT. The 65 unique bins (Bin 0, positive Bins, and
Bin
N/
2) of the 128-pt DFT are shown in Fig. 3.44.
Example 3.36.
Estimate the harmonic content of the output of a zero-order hold DAC converting one
cycle of a sine wave to an analog signal, prior to any post-conversion lowpass filtering.
Plot (a) of Fig. 3.45 shows the theoretical output of a zero-order-hold-reconstructed 1 Hz sine
wave sampled at a rate of 11 Hz. Plot (b) shows a 150 sample segment of the sampled representation of the
waveform at (a) (which was simulated with 1000 samples), from samples 250 to 400, which correspond
to the waveform at (a) from time 0.25 second to 0.4 second (since the samples represent times 1/1000 of
a second apart). A DFT is then performed on the sampled sequence, and a portion of the result (the first
60 bins) is plotted at (c). The 1000-point DFT of the sequence at (b) thus yields 1000 samples of the
DTFT of the simulated 11 Hz-sampled zero-order-hold-reconstructed 1 Hz sine wave at (a). Note that
the one-cycle 11-stairstep sine wave's spectrum has a high amplitude component at Bin 1, and harmonics
at Bins 10 and 12, 21 and 23, 32 and 34, etc. Since the sample rate is 11 Hz, the Nyquist limit is 5.5 Hz,
and this lies well below the lowest harmonics shown in plot (c) - so, with a good lowpass filter cutting
off sharply at the Nyquist limit, it should be possible to eliminate all of the harmonics and have a smooth
sine wave without any stairstep characteristic.
Example 3.37.
. Figure 3.46 shows, in plots (a)-(c), the real
part, imaginary part, and magnitude of the 4-pt DFT of the sequence, while plots (d)-(f ) show the
Consider the sequence
[−
0
.
1
,
1
,
1
,
−
0
.
1
]
