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?

Please note: I posted this in the EE forum but it probably belongs here.

 

[Valkarie]

The correction factors for the Hanning and Hamming windows were derived by taking the ratio of the power in an ideal rectangular window (which is 1.0) to the power in the Hanning or Hamming window. By multiplying the FFT by these correction factors, the power in the resulting spectrum is brought back up to the same level as an equivalent rectangular window. The correction factors for other window types vary depending on the specific window type and the number of points used in the FFT. Generally speaking, the larger the number of points used in the FFT, the smaller the correction factor required. For example, the Blackman window typically requires a correction factor of about 0.98 when using 2048 points in the FFT, whereas using 8192 points only requires a correction factor of 0.99. In general, you can calculate the correction factor for any window type by taking the ratio of the power in an ideal rectangular window to the power in your chosen window type, then multiplying the FFT output by this ratio to bring the power back up to the same level as an equivalent rectangular window.

 

[tacman]

The correction factors for the Hanning and Hamming windows were calculated using mathematical analysis and simulation. These factors are specific to these two windows and may not apply to other window types. Other windows may have different correction factors, and these can also vary depending on the specific application and parameters used.

The energy loss associated with window functions is due to the fact that these functions are designed to reduce spectral leakage, which occurs when the signal of interest is not perfectly periodic within the window length. This spectral leakage causes energy to "leak" into neighboring frequency bins, resulting in a distorted spectrum.

To understand how the correction factors were calculated, it is important to first understand the concept of spectral leakage. When a signal is sampled and analyzed using the FFT algorithm, it is essentially being multiplied by a rectangular window function. This rectangular window has a value of 1 within the window length and 0 outside of it. This abrupt change from 1 to 0 causes the signal to appear as if it has discontinuities at the edges of the window, leading to spectral leakage.

To reduce this leakage, different window functions are used that have smoother transitions from 1 to 0 at the edges. The Hanning and Hamming windows are two commonly used examples. These windows have correction factors that were calculated to compensate for the energy loss caused by the smoother transition. The calculation involves taking into account the shape of the window and the frequency response of the windowed signal.

The correction factors for other window types may have been calculated using similar methods, but they may also be specific to the characteristics of each individual window. It is important to note that while these correction factors can help offset energy loss, they cannot completely eliminate it. Therefore, it is important to carefully choose the appropriate window function for your specific application to minimize energy loss and achieve the desired spectral resolution.