# 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:

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!

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