Making other windows seem a little wasteful
A proverb that’s perhaps 2,000 years old describes the mills of the gods as “grinding slowly but exceedingly fine.” I’d like to flatter myself that it applies to my thinking on some matters but, alas, the only relevant part appears be the slowness. Witness how long it’s taken me to get back to FFT window functions and IF filters for RF and microwave measurements.
In both signal analysis and demodulation, flexibility in windows and related filtering operations is increasingly important. As I described in my earlier post, windows are time-varying amplitude-weighting functions that force the signal samples in a time record to be periodic within each block of sampled data. This removes discontinuity errors at the ends of the records that would foul up the spectrum results. To do this, the weighting coefficients are generally zero at either end, with a value of one in the middle, and a smooth range of increasing and decreasing values on either side.
It’s ironic that window functions actually discard or de-emphasize information (e.g., some data samples) to improve measurements. But of course the vital thing is that they trade away this information for desirable frequency-domain filter characteristics such as flatness or selectivity. For example, here’s a Gaussian window in the time and frequency domains.
When compared to a uniform weighting of one, the Gaussian window reduces leakage and improves dynamic range by de-emphasizing a large portion of the sampled data. The weighting coefficient (left) is greater than 0.9 for only about 1/5 of the samples. (image from Wikimedia Commons)
I have always been surprised at the amount of sampled data in each time record that windows remove from the spectrum calculation, as they improve dynamic range (reducing leakage or sidelobes) or improve amplitude accuracy (by reducing scalloping error).
As with so many things in engineering, it’s a matter of understanding requirements and cleverly optimizing tradeoffs. Consider the “confined” version of the Gaussian window below.
Modest time-domain changes in the confined version of the Gaussian window (left) reduce sidelobes dramatically (right) and improve dynamic range. However, even more signal samples are de-emphasized, with a weighting coefficient greater than 0.9 for only about 1/8 of the samples. (image from Wikimedia Commons)
From the standpoint of dynamic range, at least, it appears that selectively removing information improves spectrum characteristics. Of course, dynamic range is not the only important aspect of spectrum measurements, and another important tradeoff can be illustrated with the Tukey window.
The Tukey window is not impressive in the frequency domain, but is remarkable for how much of the sampled data it retains in the spectrum calculation. Its weighting coefficients are greater than 0.9 for about 5/8 of the signal samples. (image from Wikimedia Commons)
Many important signals in RF measurements are noise-like or noisy, and accurate measurements can demand some way to minimize the variance of results. One very good example is ACPR: the larger amount of data retained by the Tukey window means that fewer time records and FFTs will be necessary to reach the variance required for a valid measurement. Thus, the Tukey window’s combination of reasonable dynamic range and efficient use of samples translates to speed and accuracy in ACPR measurements.
Unfortunately, I can’t say if these are the characteristics and tradeoffs the inventor of the Tukey window had in mind. I had assumed the window’s creator was John Tukey, one of the two modern-day discoverers of the FFT algorithm (with J.W. Cooley in 1965). My online research didn’t clarify whether the window was discovered by him or was named after him.
If you have a few minutes to spare, it’s worth browsing available window functions as an example of intelligent tradeoffs. Because you know a lot about the signals you are trying to measure and what’s most important to you, this can be another example of adding information to a measurement to get better results faster.