## Fourier Series in Differential Equations

We’ve been seeing some really interesting stuff in my Differential Equations class. This week, we started seeing Fourier series, which, in some cases, are great to approximate functions. If the function is smooth enough, adding up the series actually equals the function.

## Cepstrum and Quefrency

A cepstrum (pronounced /ˈkɛpstrəm/) is the result of taking the Fourier transform (FT) of the decibel spectrum as if it were a signal. Its name was derived by reversing the first four letters of “spectrum”. There is a complex cepstrum, a real cepstrum, a power cepstrum, and phase cepstrum.

The independent variable of a cepstral graph is called the quefrency. The quefrency is a measure of time, though not in the sense of a signal in the time domain. For example, if the sampling rate of an audio signal is 44100 Hz and there is a large peak in the cepstrum whose quefrency is 100 samples, the peak indicates the presence of a pitch that is 44100/100 = 441 Hz. This peak occurs in the cepstrum because the harmonics in the spectrum are periodic, and the period corresponds to the pitch.

Some really cool math words I learned today. Ceptstrum and quefrency! Yeah!

(info via wiki, idea via 360)