Same doc with math equations properly displayed can be found in github page

Gammatone-filters

Python implementation of all-pole Gammatone filter. The filtering part of code is written in c.

Basic idea of implementation ¹

Gammatone filter can be regarded as low-pass filter with frequency shitfted by cf(center freuqency of filter). Equalently, we can

1. Shift the frequency of input signal by -cf(center frequency of filter);
2. Filter shifted signal with corresponding lowpass filter
3. Shift the frequency of filtered signal back by cf.

Comparison with other module

Detly has published a python module of gammatone filterbank. Using the sample settings, outputs of Detly module and current module are as follow:

fs=16kHz
cfs = [100 393  948 2000]

Detly
Current

Difference

In Delty's code, given the low and high frequencies($low_freq$, $high_freq$ and the number of bands($n$), $n+1$ frequency points are sampled equally in $ERB scale$, the first $n$ frequency points are used as the center frequency bands. This can be reasonable only if $n$ is appropriate, for example,

n=4	n=8	n=16	n=32

In this module, there are two ways to ways to specified the frequency range:

1. freq_low, freq_high: the frequency range of ERB

2. cf_low, cf_hight: the frequency range of center frequencies

   eg, frequency range [70, 7000], n=4

   

Spectrum of filter

Taking filter with $f_c=4kHz$ as example, the amplitude spectrum $$gain(f)$$ and phase spectrum $$\phi(f)$$ is plotted as follow

amp & phase	amp & delay

As shown in figure,

• $ \phi(f_c)\approx 0 $;
• Amplitude of $$\phi(f)$$ increase as $$f$$ move away from $$f_c$$;

Gain and delay at cf as function of cf

## Gain normalization

Gammatone filter is normalized by scaling filter gain at fc to 1

• IRs before gain normalization

- IRs after gain normalization

## Phase compensation

Phase compensation is actually to align the peaks of all filter impulse response².

The impulse response of Gammatone filter is given as

$$ \begin{equation} \begin{aligned} g(t) = a\frac{t^{n-1}\cos(2\pi f_ct+\phi)}{e^{2\pi b t}} \end{aligned} \end{equation}$$ $$

$$g(t)$$ can be be regarded as production of two parts :

$$ \begin{equation} \begin{aligned} g(t)=g_{amp}(t)\times g_{fine}(t) \end{aligned} \end{equation} $$

• Envelope parts: $$\quad g_{amp}(t) = a\frac{t^{n-1}}{e^{2\pi b t}}$$
• Fine structure part: $$\quad g_{fine}(t) = \cos(2\pi f_ct+\phi)$$

Envelope alignment

The peak position $t_{peak}$ can be obtained by setting first-order derivative of $g_{amp}(t)$ to 0

$$ \begin{equation} \begin{aligned} \frac{\partial g_{amp}(t)}{\partial t} &= \frac{(n-1)t^{n-2}}{e^{2\pi bt}}-\frac{t^{n-1}2\pi b}{e^{2\pi bt}}\ &=\frac{t^{n-2}}{e^{2\pi bt}}(n-1-2\pi bt) \triangleq 0\ \Rightarrow& \quad t_{peak}=\frac{(n-1)}{2\pi b} \end{aligned} \end{equation} $$ Delay $g_{amp}$ by $-t_{peak}$ to align the peaks of filter bank

$$ \begin{equation} \begin{aligned} g_{align}(t) = g_{amp}(t-\tau)g_{fine}(t) \end{aligned} \end{equation} $$

Example of $$g_{align}$$ ($$\phi$$ is set to 0)

### Fine structure alignment Further more, align $$g_{fine}(t)$$

$$ \begin{equation} \begin{aligned} & \cos(2\pi f_ct+\phi)|{t=t{max}} \triangleq 1\ \Rightarrow& \quad \phi = -\frac{(n-1)f_c}{b}+i2\pi, \quad i=0,\pm 1,\cdots \end{aligned} \end{equation} $$

### Illustration of purpose of alignment

For a stimulus of impulse, what if we add up all filter outpus ? Ideally, a impulse is expected

Not aligned	Envelope aligned	Envelop & fine structure aligned

About efficiency

 python .\efficiency.py
 #time consumed(s) for filtering signal with length of 16e3 samples
 #     c  :0.11
 #     python  :11.81

Built-in examples

Corresponding to above-mentioned tests

validate.py
efficiency.py
filter_spectrum.py
gain_normalization.py
phase_compensation.py
stimulus_restore.py

Footnotes

1. Holdsworth, John, Roy Patterson, and Ian Nimmo-Smith. Implementing a GammaTone Filter Bank ↩

2. G. J. Brown and M. P. Cooke (1994) Computational auditory scene analysis. Computer Speech and Language, 8, pp. 297-336 ↩