Low Pass Filtering, Welch's Method, ENBW

Question

I am working on using Welch's method (https://www.osti.gov/servlets/purl/5688766/) to estimate a Power Spectral Density (PSD). This algorithm involves scaling a set of averaged FFTs by the equivalent noise bandwidth (ENBW) or the sum of the coefficients a window function applied to the time domain signal, to arrive at an estimate of the PSD. In the examples I've reviewed thus far, the coefficients of the windowing function provide the means for estimating the PD

In my system, I also want to implement a low-pass, antialiasing, FIR filter on the input data. To first-order, I use a specific window (e.g. Blackman, Hamming) to truncate the ideal, infinite impulse response of the anti-aliasing filter. In the link I shared before, the windowing function provides both a means of estimating ENBW and limiting spectral leakage, however anti aliasing filters are not discussed.

1. Can I estimate the ENBW of the FIR filter or W as the coefficients of the window function I use in designing my FIR filter? Do I need to additionally compensate for the FIR filter?

2. If I do need to additionally adjust my estimate of W or ENBW due to the transfer function of the FIR filter (anti aliasing & window), how should I do so?

Answer 1

The concern is with spectral density, which is power per unit bandwidth, and this is why compensating for the effect of the window's resolution bandwidth is important in determining the periodogram.

It may help to distinguish to the very different operations in how the window used in each case effects the measure of power spectral density:

For the case of windowing with the DFT, as done when determining each periodogram. Note how only the rectangular window provides non-overlapping bins of equivalent resolution bandwidth (the equivalent noise bandwidth in a rectangular windowed DFT is one bin), while any other window increases the width of its main lobe in exchange for greater dynamic range, thus each bin in a windowed DFT (with any window other than the rectangular window) will overlap the other bins due to its wider resolution bandwidth. For power spectral density that spans multiple DFT bins (such as white noise), this overlap in the response of each bin will result in an overestimation of the noise density which is compensated for properly by factoring in the resolution bandwidth of the window.

In contrast, for the low pass filter that is first used prior to decimation (which the OP is doing by windowing an ideal low-pass impulse response), the kernel (Fourier Transform) of the window in this case will convolve with the ideal low pass response, spearing its transition band but not otherwise modifying the power spectral density in band - beyond the change in overall coherent gain from the filtering and down-sampling as evidenced by the frequency response of the filter.