FFT RELATIONSHIP BETWEEN SAMPLINNG RATE, BANDWWTH, RESOLUTION, ETC:

By Wayne Pauly, Wayne Pauly, Inc.

All of the numbers in a FFT are related: sampling rate, bandwidth, resolution, data acquisition time, alias frequencies, and computational time. Let's start with a typical example:

Suppose for starters we have a data acquisition system that takes samples at a rate of 1024 Hertz, and we decide to take a 1024 point FFT since that seems to be a popular number. We will need 1024 sample points to perform the 1024 point FFT, so it will take one second to accumulate thc needed samples; longer sample sizes would take longer to* *gather. A 1024 point FFT will appear to give us 1024 spectral lines evenly spaced from DC to the sampling frequency, so each line is spaced 1 Hertz apart in this example.

Now for the rub; only 1/2 of the lines are useful; the upper half are mirror images of the lower half. Nyquist figured out that any frequency present above 1/2 the sampling rate will result in a ghost at a lower frequency that appears real; hence it is necessary to sample at at least twice the desired bandwidth and also to filter the signal so that nothing higher than 1/2 Fs gets through. These ghost images are called aliases and an 'anti-alias" filter is a sharp low-pass filter that limits the signal to 1/2 Fs. Since no filter is perfect a little bandwidth must be given* *up for the filter to work. We started out with 1024 theoretical lines of which only 512 were actually real and we ended up with about 400 to give the filter room to work, thus our 1024 point FFT yielded 400 useful lines spaced 1 Hertz apart from DC-400 Hz.

We could just as well taken 256 or 512 samples and performed 256 or 512 point FFTs, which would result in RESOLUTION of 100 or 200 useful lines; their lines would be spaced 4 and 2 Hertz apart up to the same 400 Hertz, so the BANDWIDTH remained the same but he RESOLUTION changed. The ACQUISITION TIME dropped to 0.25 and 0.50 seconds and the computing times dropped to about 1000 and 2000 steps from the 5000 needed for the 1024 point one. Note that computation time increases with FFT size, but only the resolution is improved, hence it may be better to use smaller FFTs for quick looks or with slower computers.

The BANDIWDH, or maximum frequency is solely a function of the SAMPLING RATE; since the sampling rate must be >2X the bandwidth, we need higher sampling rates for higher bandwidths. If the clock is raised to 10.24 Khz,* *we will get 400 lines spaced 10 Hz apart from a 1024 point FFT up to 4000 Hertz. We can slow the sample rate down to get more resolution at the expense of bandwidth or reduce the FFT size to get less resolution without sacrificing bandwidth.

Since noise may be present on samples, it may be advantageous to take several samples and average the resu1ts. This is done by averaging the spectral lines after the FFT is completed, which may reduce the background noise level. Since this greatly increases the computer time, it should not be used except as needed and with fast processors.