This page has been under construction.

Theory

To be expanded sometime in the future.

Joint time-frequency analysis

What is a spectrogram?

Horizontal is time-axis; Vertical is frequency axis; Graph intensity is signal power
Similar to musical score

Fourier analysis

The Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT): basically the same, but the latter is much faster to calculate
Short-term Fourier transform (STFT)

Trade-off between time resolution and frequency resolution

Uncertainty principle
Window functions for the STFT : Gaussian window offers lowest uncertainty product
Choosing the length of the FFT : as big as possible!
Choosing the window width : appropriate to screen dimensions and sample length
Make the uncertainty expressed in number of pixels the same in both time and frequency

spectrogram example

Time resolution vs. frequency resolution

Below I've shown the spectrogram of a single call from one of my own recordings. The program used is CoolEdit.
To illustrate the effect of varying window size, it plotted the spectrogram for various FFT sizes, using a Gaussian window:
16 frequency bands

The spectrogram for 16 bands has good time resolution (horizontal direction), but poor frequency resolution (vertical direction). Also the picture is very blocky. The spectrogram for 128 bands has good frequency resolution, but poor time resolution (it looks 'smeared' in the horizontal direction). Also note the 'noisy bits' in the picture, which are mostly vertical for the 16-band spectrogram and mostly horizontal in the 128-band spectrogram.
The spectrograms for 32 and 64 bands give the best of both worlds, fair time and fair frequency resolution, but are still quite blocky.

How do we remove the blockiness?

The answer is simple: use a bigger FFT, but keep the window size small !!!
Fortunately, Cooledit allows us to use a window function that is smaller than the FFT size. For example, a 256-band spectogram with a 25% window size has the same time and frequency resolution as a 64-band spectrogram with a 100% window size, but it is not blocky at all! 256-band spectrogram, window size 25%

Two conclusions can be drawn from these pictures:

The FFT size determines the number of pixels in the vertical direction
The window size determines the trade-off between horizontal and vertical resolution

free bat call analysis software

There's a lot of places on the net about sound analysis, here are just a few of them:

Spectrogram by R.S. Horne, who also has a lot of links to other sound analysis software
Lots of links at Steven L. Hopp's page about sound analysis software for bioacoustics
A 'light' version of AviSoft SASLab
A demo version of CoolEdit Pro, a very powerful sound editor, that also has a versatile spectral viewing mode

This page was last updated August 7th, 2005