of Microprocessors

SOURCE-SIGNAL ANALYSIS 559
Since the spectrum is averaged over the duration of a record, the time
resolution can be no better than the record duration. And since the FFT gives
harmonic frequencies that are integral multiples of the reciprocal of the
record duration, the frequency resolution can be no better than this either.
Thus, time resolution multiplied by frequency resolution can be no less than
unity. Another consideration is that for maximum efficiency and ease of
programming, the record size should be a power of two.
Continuing the example used earlier (20-ks/s signal sample rate, 50-Hz
frequency resolution, 100-sls spectrum sample rate), we see that the record
size must be at least 400 samples to obtain the required frequency resolution.
The corresponding time resolution is 20 msec, which means that the 100-s/s
spectrum sample rate oversamples the changing spectrum by a factor of two.
We will see later that overlap between successive records is often desirable and
that it results in spectral oversampling as well. After rounding up to the next
power of two, the FFT example used in the following discussion will assume
a record size of 512, which gives a frequency resolution of 39 Hz, and a
spectral sample rate of 78 sis.
Equivalent Bandpass Filter
Since the filterbank and the FFT methods of spectral analysis yield
similar results, it makes sense to talk about an equivalent bandpass filter
corresponding to each of the "harmonics" computed by the FFT. As an
example, let us study the 10th FFT harmonic. Ideally, it should act as a
bandpass filter with a lower cutoff of 9. 5 X 39Hz = 370 Hz and an upper
cutoff of 10.5 X 39 Hz = 409 Hz. Conceptually, it is simple to plot the
o
-5
-10
-15
(\
~ -20
..... -25
~ -30 (\ nf\ f\
~ -35
-40
-45
f\f\
f\{\
-50
-55
- 60 '---'------'--:'----'_'---'------'--:'----'---,_-'----':---'::---'---,"------'------'--:'----'---,"-----..
o 2 3 4 5 6 7 8 9 10 /I 12 13 14 15 16 17 18 19 20
FREOUENCY (HARMONIC NUMBER)
Fig. 16-10. Equivalent bandpass filter response of 10th FFT harmonic