Third-Octave Filter Banks

Third-octave filter banks have historically been popular in audio analysis, as the bandwidths of these types of banks have been shown to loosely approximate the measured bandwidths of the auditory filters. Third-octave banks have also been internationally standardized for use in audio analysis [1]. In a third-octave filter bank, the center frequencies of the various bands $f_c[k]$ are defined relative to a bandpass filter centered at $f_c[0] = 1000$ Hz, by the following formula:

$$f_c[k] = 2^{k/3} 1000 \mathrm{Hz.}$$ (3)

The upper and lower band edges in the $k$th band are further given by the geometric means

$$f_{ch}[k] = \sqrt{ f_c[k] f_c[k+1] },$$ (4)

and

$$f_{cl}[k] = \sqrt{ f_c[k-1] f_c[k] },$$ (5)

respectively. From the above equations, it may be shown that the bandwidth of the $k$th band is given by

$$BW[k] = f_c[k] \frac{2^{1/3} - 1}{2^{1/6}}.$$ (6)

It may be shown that as the bandwidth in Equation (6) above is proportional to center frequency, the quality factor of each third-octave band filter is independent of $k$. As a result, filter banks such as the third-octave bank are referred to as constant-Q filter banks.