The Zoom FFT is a way of speeding up some computations. Despite the
name, it doesn't allow you to zoom in on a signal any closer than
you could with a regular FFT. But, it does allow you to zoom in on
a signal with a lot less computation than you might otherwise. Let
us take as an example a waveform sampled at 10,000 samples/s which
we suspect (for some reason) has two components, one at 4000 Hz and
one at 3999 Hz. In order to distinguish these two components, we will
need a frequency resolution of at least 0.5 Hz, which calls for a
2 second window. That requires a 20,000 point FFT. Actually, with
a modern PC, that really doesn't take that long. But maybe we want
to do a short time fourier transform 8.1over
a few minutes worth of data (this happens quite frequently), or maybe
we have an embedded device that has limited computational capabilities
(this also happens quite frequently). Let us assume that we need to
be able to use a 2000 point FFT. Then we employ the zoom FFT. There
are three steps: frequency shift, low pass filter, and decimate. First,
a frequency shift. We multiply the signal by the complex phasor
exp(- i
t)
where is choosen to be 3900 Hz. It should be easy to see
that this will shift the signal at 4000 Hz down to 100 Hz (remember,
when two exponential expressions are multiplied, their exponents add).
Of course it has also made the signal complex, but that's okay. Then,
we low pass filter with a cutoff filter of, say, 300 Hz (this is 3700
- 4300 Hz in the original signal, remember). Now, since we have no
frequency content above 300 Hz, a sampling rate of 10,000 Hz is unnecessary.
So we decimate by a factor of 10. Now the signal is sampled at 1000
Hz. So a 2 second window requires only a 2000 point FFT. After the
FFT, we simply shift the frequencies back up by adding 3900 Hz to
the result. Now we have 0.5 Hz resolution between 3700 Hz and 4300
hz with 10 times less computation.
Remember, we still need 2 seconds worth of data to get a 0.5 Hz resolution. This is no better than we could do without a zoom. All we did was save computation time.
