**High resolution bat sonar analysis**

Full-spectrum analysis of bat sonar typically makes use of short-time discrete Fourier transforms (STFT, more commonly known as DFT). There are quite a few alternative analysis methods, but none are as generally robust with short time-varying signals, or as computationally efficient, as DFT. [Zero-crossing analysis can be a useful tool for identification, but any harmonic information is discarded.] The digitised ultrasound signal is analysed by DFT in short samples, and the result is a frequency *vs.* time spectrogram. For DFT the frequency resolution is roughly 1 part in N (the sample size). Frequency resolution can be refined by increasing the length of the sample, but then the sample is acquired, and then averaged, over a longer time. Furthermore, processing time increases proportionally to Nlog_{2}N. Faster A-to-D converters help, but at the expense of higher power consumption (important in hand-held devices) and larger data files.

Typically, bat sonar is analysed with DFT sample sizes of 256 and sampling rates of about 400 kS/s, giving spectrogram frequency resolutions of about a kHz, and a time resolution of around a millisecond. This is almost good enough for showing sonar from many European and US bats, but inadequate for detailed study of sonar from some tropical bats operating with very high frequencies, or the many common bats producing very short sonar pulses.

Standard DFT algorithms discard phase information, but this can be used to produce dramatic improvements in time and frequency resolution, so that DFT sample size can be much reduced. For instance, a 64-point DFT incorporating this “phase integrated” technique produces frequency and time resolutions of better than ± 0.1 kHz and 50 µs (with a 12-bit AtoD converter sampling at 600 kS/s). Surprisingly, this technique has been ignored by bat researchers, although it has been applied in other research fields for over 50 years.

The bat sonar gallery shows spectrograms produced by a hand-held analyser using this technique. Here the frequency resolution is limited by the 320x240 pixel screen; the analyser is actually calculating frequencies internally to within 0.1 kHz with a DFT size of 64.