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Applying a window function makes the signal smoothly approach zero at both ends. This affects the total signal power:
Because the window function attenuates the signal at both ends, it reduces the overall signal power. This reduction in signal power is called the Coherent Power Gain. Its result is that the amplitude you measure at the FFT bin is not the same as the 'real' amplitude of the signal's frequency component at that frequency. The contribution from the signal's frequency component at the FFT bin is reduced by the Coherent Power Gain. This is one reason why amplitudes measured from a Fourier Transform never quite seem to be as expected.
Don't blame me for calling a reduction in signal power a Coherent Power Gain - I don't make up these terms, I just have to live with them.
For an ideal, single discrete line frequency component, the 'noiseless' signal contribution to the FFT bin is proportional to the signal amplitude. The proportionality factor is the area under the window function's kernel - or in a sampled system, the sum of the amplitudes of the window function. The Coherent Power Gain is the square of this, or in other words the Coherent Power Gain is the square of the sum of the amplitudes of the window function's kernel.
You may notice that the Coherent Power Gain is just the DC gain of the window function.
For a rectangular window, where every amplitude is 1, the DC gain is N - the number of terms - and the Coherent Power Gain is N^2: but for any other window function the DC gain will be reduced because the window function goes smoothly to zero at its ends and so reduces the signal power.
Coherent Power Gain is important because it represents a definite scaling of the amplitudes of the measured frequency spectrum which requires correction for any absolute measurements to be correct.
Coherent Power Gain is usually normalised by dividing by the number of terms N, so that the Coherent Power Gain of a rectangular window function would be normalised to 1.
| Last updated: 29th January 1998 | http://www.bores.com/courses/advanced/windows/10_cpg.htm