Once the waveform has been aquired and digitised, it can be fast-Fourier-transformed to the frequency domain. The FFT results can be either real and imaginary, or magnitude and phase, functions of frequency. The choice of output format belongs to the user.
Since the FFT generates the frequency spectrum for a time domain waveform, some fairly simple applications, e.g., harmonic analysis, distortion analysis, vibration analysis, and modulation measurements, might suggest themselves immediately.
Another important area is that of frequency response estimation. A linear, time-invariant system can be stimulated with an impulse function. Its output, the impulse response, can then be acquired and fast-Fourier-transformed to the frequency domain. The FFT of the impulse response, referred to as the frequency response function, completely characterises the system.
Once a system's frequency response function is known, one can predict how that system will react to any waveform. This is done by Convolution.
An important aspect of the FFT is that convolution can easily be performed through frequency-domain multiplication. Let's say you know a system's impulse response, given by h(t), and an input waveform given by x(t). The output, say y(t), caused by x(t), can be computed in the classical manner by the convolution integral. But this is tedious and slow. An easier and faster approach is to FFT x(t) and h(t) to the frequency domain. Then the product of their frequency domain functions can be formed, giving Y(f) = X(f) H(f). Forming this product corresponds to time domain convolution, and the convolution result can be obtained by inverse-Fourier-transforming (IFT) the Y(f) function back to the time domain.
Correlation is another useful operation that the FFT makes easier. Mathematically, correlation looks and is performed in a manner similar to convolution. The difference is that one of the frequency domain functions is conjugated before the frequency domain product is formed.
Although the operations of convolution and correlation may look similar, their applications are not. Correlation is a sort of searching or looking for similarities between two waveforms. When two waveforms have absolutely no similarity, like uncorrelated noise, their correlation function is zero. On the other hand, correlation two waveforms that are exactly alike produces a perfect correlation function.
This property of finding similarities makes correlation a useful tool for detecting signals that are hidden or masked by other signals.
Another useful property of correlation is its ability to indicate delay. This is particularly useful in measuring things like path delay, path diversity, and echo return times.
Refer to the M-files List for an example of how to perform the FFT using Matlab, and also an m-file which analyses .wav files using the FFT.
When a signal is measured as a time signal it can be converted to a spectrum. This spectral analysis shows the amplitude of the various frequencies contained within the signal. On this spectrum it is usual for a resonance to occur. The resonance is seen by a comparatively large amplitude at a specific frequency. This frequency is of interest in terms of the device operating conditions.
In order to further study the resonance it is possible to employ a band pass filter for a specified frequency range. This filter allows only the frequencies within the given band to pass. This method eliminates the noise which occurs when sampling. It should be noted that the noise present is not only electrical but may be other resonances, aliasing, other components of the machine or even other devices in close proximity.
As mentioned in tutorial 3 , aliasing is a problem when sampling a vibration. Filters can be used to avoid aliasing in signals containing many frequencies by subjecting the analog signal x(t) to an antialiasing filter. An antialiasing filter is a low-pass (i.e., only allows low frequencies through) sharp cutoff filter. The filter effectively cuts off frequencies higher than about half the maximum frequency of interest, which is also called the Nyquist frequency. This means that some prior knowledge of the nature of the spectrum is often required before the exact sampling rate is determined.
This antialiasing filter must be employed before the signal is digitized. It is no good trying to use a low pass filter on the digitized signal because the aliasing effects occur because of the sampling process. Any aliasing effects would already be stored in the digitized signal and cannot be removed by low pass filtering as the effects appear as low frequencies in the signal.
It should be noted that the SoundBlaster sound card, and most of its clones, do not include antialiasing circuitry in their design. One must include a low pass filter in the data acquitision circuit when connecting to a soundcard, or at the very least it must be ensured that no frequencies greater than half the sampling rate appear in the input signal.