Energy loss of spectra associated with Window Functions

[greydient]

I'm doing research on window functions (such as: rectangular, Hanning, Blackman, etc), but am having trouble with respect to the energy loss associated with each. I know that applying the window causes energy loss in the spectra of interest, and for the Hanning Window and Hamming window, multiplying the FFT by 1.633 and 1.586 respectively offsets this a bit.

How were these two correction factors calculated, and what are the correction factors for other window types?

 

[rbj]

consider the signal: $x(t)$. they like to classify signals in two broad classes:

finite energy signals:

$$\int_{-\infty}^{+\infty} \left( x(t) \right)^2 dt = E < \infty$$

finite power signals:

$$\lim_{T \to +\infty} \frac{1}{T} \int_{-T/2}^{+T/2} \left( x(t) \right)^2 dt = P < \infty$$sinusoids, periodic signals, and stocastic signals (some kind of noise) are finite power, but infinite energy (because they are turned on forever). now, whether it is a finite power signal or a finite energy signal, when you window it, it becomes a finite energy signal:

$$\int_{-\infty}^{+\infty} \left( x(t) w(t) \right)^2 dt = E_w < \infty$$

when you multiply a time-domain signal with a window, $w(t)$, that has the effect of smoothing the spectrum of $x(t)$ using the Fourier transform of the window. now, if the window function is always less than 1 in magnitude, $|w(t)|<1$, then it will only reduce $x(t)$ and if it was a finite energy signal, the energy would also be reduced. sometimes they like to normalize window functions by scaling them so that

$$\int_{-\infty}^{+\infty} w(t) dt =1$$

that will insure that the window will not reduce the average smoothed values of the spectrum of $x(t)$. sometimes they like:$$\int_{-\infty}^{+\infty} \left( w(t) \right)^2 dt =1$$

this will normalize the power spectrum of the window and, i think that means the smoothing done to the power spectrum of $x(t)$ will unscaled, just smoothed.