Digital Signal Processing Reference
In-Depth Information
Nyquistrate =
2 (max(700 Hz; 450 Hz; 630 Hz)0Hz)
=
2(700 Hz)
=
1400 Hz
Therefore, our sampling rate should be at least 1400 Hz. This process is called
lowpass sampling, since it accurately records all frequencies lower than the highest
frequency component. However, if we do not need all frequencies between 0 and
the maximum frequency (in this case, 700 Hz), section 5.6 shows another way to
perform sampling.
5.6
Bandpass Sampling
It is possible to sample a signal at a rate lower than twice its highest frequency, if
the signal has a limited bandwidth. This is called bandpass sampling.
Let's start with an example. Suppose there is a signal composed of four sinusoids,
with frequencies 1010 Hz, 1020 Hz, 1030 Hz, and 1040 Hz.
x(t) = 4 cos(21010t) + 6 cos(21020t) + 8 cos(21030t) + 10 cos(21040t)
Figure 5.9 shows the spectrum of the sampled signal. The number of simulated
samples is chosen to be a multiple of f
s
, which in this case means that the DFT's
analysis frequencies exactly match up to the frequencies present in the signal. In
other words, there is no DFT leakage (see section 6.8), so the gures look nice. Also
in this gure, only part of the spectrum is shown. The amplitudes below 1000 Hz in
Figure 5.9 are all zero. One nal note about this gure is that a simulated sampling
frequency of 2081 samples/second was chosen because it gives a nice gure. If 2080
samples/second were used, then the frequency component at 1040 Hz would appear
to have double the amplitude.
It would be possible to sample signal x(t) at or above 2080 Hz, as Figure 5.9
shows, but what happens when we sample it at 100 Hz? We would have spectral
content, as shown in Figure 5.10. As we can see from these graphs, the same
information is present. There are advantages to sampling at a lower rate; namely,
that the equipment to do so may be cheaper. But why does this work?
The answer lies in the same analysis that we used with aliasing. When a signal is
sampled (digitized), values are periodically read from the analog signal. Eectively,
this means that we replace t with n=f
s
in the analog signal's mathematical repre-
sentation. We are not likely to have a mathematical representation of our analog
