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: [itex] x(t) [/itex]. they like to classify signals in two broad classes:

finite energy signals:

[tex] \int_{-\infty}^{+\infty} \left( x(t) \right)^2 dt = E < \infty [/tex]

finite power signals:

[tex] \lim_{T \to +\infty} \frac{1}{T} \int_{-T/2}^{+T/2} \left( x(t) \right)^2 dt = P < \infty [/tex]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:

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

when you multiply a time-domain signal with a window, [itex] w(t) [/itex], that has the effect of smoothing the spectrum of [itex] x(t) [/itex] using the Fourier transform of the window. now, if the window function is always less than 1 in magnitude, [itex] |w(t)|<1 [/itex], then it will only reduce [itex] x(t) [/itex] 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

[tex] \int_{-\infty}^{+\infty} w(t) dt =1 [/tex]

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

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

 

[ravenprp]

The energy loss associated with window functions is a well-known phenomenon in signal processing. It occurs because window functions are used to reduce the leakage of spectral energy from one frequency component to another, which can distort the true spectrum of a signal.

The correction factors for the Hanning and Hamming windows were calculated based on the mathematical properties of these specific window functions. These factors are known as the side lobe attenuation and main lobe width, which determine the amount of energy loss and the shape of the window's frequency response. The specific values for these correction factors were derived through mathematical analysis and simulations.

For other window types, the correction factors may vary depending on the specific characteristics of the window function. Some common window functions, such as the rectangular window, do not have correction factors as they do not have any side lobes and therefore do not cause energy loss. Other window functions, such as the Blackman window, have more complex frequency responses and may require different correction factors.

In general, the choice of window function should be based on the specific requirements of your signal and the trade-off between reducing spectral leakage and minimizing energy loss. It is important to carefully consider the effects of window functions on your signal and choose the most appropriate one for your application. Additionally, you can also experiment with different window functions and their corresponding correction factors to find the best fit for your specific needs.