22-09-2012, 01:05 PM
FFT Analysis
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What is an FFT
The FFT, or Fast Fourier Transform is a method of
calculating harmonics not one at a time, but as a group,
using a special algorithm.
Advantage
The FFT requires much less processing power than a DFT
for the same number of harmonic results.
Disadvantage
Requires that the number of samples being analysed are N
to a power of 2, such as 64, 128, 256 etc
FFT Disadvatages
• Frequency Resolution
• Harmonic Leakage
• Requires synchronisation
Comments
• Although the frequency spectrum correctly shows a spike at 10Hz, the spike is not
infinitesimally narrow. In fact, it appears from the frequency spectrum that there is a
significant component of the signal at frequencies near 10Hz (specifically within about
5Hz to either side of 10Hz). This unphysical error in the FFT is called leakage, which
appears when the discrete data acquisition does not stop exactly in the same phase of the
sine wave as it started, it is not synchronised.
• In principle if an infinite number of discrete samples are taken, leakage would not be a
problem. However, any real data acquisition system performing FFT’s uses a finite
(rather than infinite) number of discrete samples, and there will always be some leakage.
• The maximum amplitude of the FFT is not exactly 1 (in fact it is less than 1), even
though the amplitude of the original signal was exactly 1. This is another consequence
of leakage – namely some of the energy at 10Hz is distributed among frequencies near
10Hz, thereby reducing the calculated amplitude at 10 Hz
• This can be reduced using a correction window function such as a Hanning window
which modifies the result of the FFT to mathematically
• The frequency range is from 0 to 100Hz, half the sampling frequency (Nyquist criterion)
Synchronisation
It is theoretically possible to obtain a perfect FFT from
sampled data.
However this is only possible if the sampled data begins
and ends in the same phase of the signal
10Hz sine wave with an amplitude of 1
Number of samples = 512
Sampling frequency (Fs)is 25.6Hz
Sample time = N/Fs =512/25.6 equals 20 seconds
This T corresponds to exactly 200 complete cycles of the
10Hz sine wave.