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I've been studying the time-frequency uncertainty principle of Dennis Gabor, and the tradeoff of the STFT window size in representing the tonal and transient characteristics of the musical signal being studied.

I understand that by observing the spectrogram, one can see "blurry" temporal events (vertical lines) with a long window, and sharp frequency lines. The opposite is true, with blurry frequencies with a short window, and sharp temporal events.

Is there an objective measurement that can output a numerical value for "sharp vs. blurry" spectrogram visual comparison? Such that we can definitively say "for this signal x[n], the best STFT uses a window size of XYZ, which is the sparsest/least blurry/best representation". Is the term for this "sparsity"?

Here are some examples of different window sizes of STFT spectrogram of the glockenspiel signal: enter image description here enter image description here enter image description here

Sevag
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2 Answers2

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This is subject of ridge analysis. The "quality" of a representation can be quantified as follows:

Component extraction

Ability to separate intrinsic modes / independent time-frequency components. This depends on

  1. Time-frequency atom used; for wavelets, additionally on the scale-to-frequency mapping.
  2. Frequency tiling scaling (linear/log): STFT (linear) can track LFM (linear chirp) everywhere, CWT (log) can track exponential and hyperbolic chirps. See Section 4.4.

Redundancy

We wish to characterize the input with as few points as possible. Redundancy can be quantified with the reproducing kernel equation (4.40), or simply frequency-domain overlap (compute_filter_redundancy()).

Capturing multi-scale structures

Audio structures can vary greatly in scales, from seconds to milliseconds; this makes a fixed resolution kernel (STFT) not suitable - see this paper.

A simple way to measure both redundancy and ability for component extraction is to apply a sparsity measure to a time-frequency reassignment. If signal characteristics are approximately known, we can run the transform against synthetic test signals and compare qualitatively (then design numeric measures).

All of the above is only as far as "sharp vs blurry" goes. A full description involves many more factors - see "Properties summary".

OverLordGoldDragon
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  • Audio structures can vary greatly in scales, from seconds to milliseconds; this makes a fixed resolution kernel (STFT) not suitable - see this paper.

    On this note, I'm also interested in the NSGT (Nonstationary Gabor Transform) these days. I would like to perform a similar analysis of the STFT to the NSGT, which is much more configurable (e.g. arbitrary frequency scales like mel, Bark, logarithmic) with varying windows: https://github.com/sevagh/nsgt

     – Sevag Oct 05 '21 at 12:15
  • By "similar analysis" I mean this ridge analysis idea, to choose the best NSGT for a given musical signal. – Sevag Oct 05 '21 at 12:19
  • @Sevag Sure, these measures are universal to time-frequency. A practical approach for some frequency axis scaling schemes is to treat it piecewise - then linear will inherit STFT properties, log will inherit CWT. – OverLordGoldDragon Oct 05 '21 at 13:47
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There is a fundamental tradeoff between frequency resolution and time resolution. The more granular you want your measurements in time, the less frequency distinction there is in each time bin. Conversely, longer time bins allow for very precise frequency distinctions. Higher sample rates will benefit your resolution in either dimension, but there will always be a tradeoff at a given sampling rate. So no, there is no objective optimization for frequency versus time resolution until you define requirements or objectives for one or the other.

Keegs
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