# How Best to Characterise a Window Function

How do I characterise my window function?

Do please forgive me here as I am more a practical than theoretical person. I have invented a window function which I use prior to discrete Fourier transforms. My field is sound, and the window is designed to make accurate measurements from adjacent frequency bins.

For example, an asynchronous 1,000.1 Hz sine wave sampled at 48kHz for a 240 sample DFT will read with a level accuracy of better than 0.02dB in the 1,000 Hz bin but bleeds less than -60dB in adjacent bins (800 & 1,200). While a 1,001 Hz sine wave bleeds less than -40dB into the adjacent bins. A synchronous sine wave at 1,000 Hz bleeds less than -150dB into either adjacent bin.

I would like to describe its performance to compare it to other windows. Wikipedia provides graphs of bins against amplitude for window functions. How would I create such graphs? Are there better, more revealing ways of measuring how effective a window function is? I need a simple and patient explanation I'm afraid because as I said my mathematical knowledge is not great. Your help and patience is greatly appreciated.

• Sounds like you're doing sinusoidal modeling. You might wanna look at the this. I found that the gaussian window is sorta useful in determining both the frequency and amplitude of each frequency component but also the rate of change of both the frequency and amplitude. Commented Sep 30, 2023 at 4:34
• Er, maybe. To be honest I don't know! I wanted to measure audio systems more precisely. I was looking for a window with good frequency discrimination (low spectral leakage) but good amplitude accuracy. Mine seems to do both very well. I'll try the Gaussian window for comparison though, and to see if it offers inspiration. A very big thanks for that, hugely appreciated. Commented Sep 30, 2023 at 13:56

An oldie but a goodie it fred harris's "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform". It has a nice table showing different figures of merit for different windows.

The figures are:

• Highest Side Lobe Level (dB)
• Side Lobe Fall Off (dB / Octave)
• Coherent Gain
• Equivalent Noise Bandwidth (bins)
• 3.0-dB Bandwidth (bins)
• Scallop Loss (dB)
• Worst Case Process Loss (dB)
• 6.0-dB Bandwidth (bins)
• Overlap Correlation for 75% and 50% overlap.
• Thanks enormously. A very useful reference that looks like it is the go-to tome on specifying windows. It's going to take me some time to work out how to measure all these though. Really appreciated. Commented Sep 30, 2023 at 13:34