### Example filterbank usage

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Demonstrates how to initialize a Gammatone filterbank, analyze an impulse, and plot the results.

```python
def example_filterbank():
    from pylab import plt
    import numpy as np

    x = _create_impulse(2000)
    gfb = GammatoneFilterbank(density=1)

    analyse = gfb.analyze(x)
    imax, slopes = gfb.estimate_max_indices_and_slopes()
    fig, axs = plt.subplots(len(gfb.centerfrequencies), 1)
    for (band, state), imx, ax in zip(analyse, imax, axs):
        ax.plot(np.real(band))
        ax.plot(np.imag(band))
        ax.plot(np.abs(band))
        ax.plot(imx, 0, 'o')
        ax.set_yticklabels([])
        [ax.set_xticklabels([]) for ax in axs[:-1]]

    axs[0].set_title('Impulse responses of gammatone bands')

    fig, ax = plt.subplots()

    def plotfun(x, y):
        ax.semilogx(x, 20*np.log10(np.abs(y)**2))
        plt.hold(True)

    gfb.freqz(nfft=2*4096, plotfun=plotfun)
    plt.grid(True)
    plt.title('Absolute spectra of gammatone bands.')
    plt.xlabel('Normalized Frequency (log)')
    plt.ylabel('Attenuation /dB(FS)')
    plt.axis('Tight')
    plt.ylim([-90, 1])
    plt.xlim([0.001, 0.1])
    # plt.show()
    return gfb
```

--------------------------------

### Initialize CFFI interface

Source: https://siggigue.github.io/pyfilterbank/_modules/sosfiltering.html

Setup of the CFFI interface and loading of the shared library based on the operating system.

```python
import os
from sys import platform
from platform import architecture

import numpy as np
from cffi import FFI
from scipy.signal import lfilter


ffi = FFI()
ffi.cdef("""
void sosfilter(float*, int, float*, int, float*);
void sosfilter_double(double*, int, double*, int, double*);
void sosfilter_double_mimo(double*, int, int, double*, int, int, double*);
""")

if platform == 'win32' and architecture()[0] == '64bit':
    _dl = 'sosfilt64.dll'
elif platform == 'win32' and architecture()[0] == '32bit':
    _dl = 'sosfilt32.dll'
else:
    _dl = 'sosfilt.so'


if __name__ != '__main__':
    _mylibpath = os.path.join('.', os.path.dirname(__file__))
else:
    _mylibpath = os.curdir

_mylibpath = os.path.join(_mylibpath, _dl)

_c = ffi.dlopen(_mylibpath)
```

--------------------------------

### Visualize Gammatone Filter Examples

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Demonstrates usage of the gammatone filter and filterbank examples via plot directives.

```python
import pyfilterbank.gammatone
pyfilterbank.gammatone.example_filter()
```

```python
import pyfilterbank.gammatone
pyfilterbank.gammatone.example_filterbank()
```

--------------------------------

### Example plot function

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Demonstrates how to initialize a FractionalOctaveFilterbank and use the freqz function to generate frequency response data.

```python
def example_plot():
    """Creates a plot with :func:`freqz` of the default
    :class:`FractionalOctaveFilterbank`.
    """
    ofb = FractionalOctaveFilterbank()
    x, y, f, Y = freqz(ofb)
```

--------------------------------

### Plotting an example filter bank

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Visualizes the filter bank response using matplotlib.

```python
from pylab import plt
import octbank
octbank.example_plot()
plt.show()
```

--------------------------------

### Visualize Mel Filter Bank

Source: https://siggigue.github.io/pyfilterbank/_modules/melbank.html

Example usage of compute_melmat to generate and plot a mel filter bank matrix.

```python
from pylab import plt
import melbank

f1, f2 = 1000, 8000
melmat, (melfreq, fftfreq) = melbank.compute_melmat(6, f1, f2, num_fft_bands=4097)
fig, ax = plt.subplots(figsize=(8, 3))
ax.plot(fftfreq, melmat.T)
ax.grid(True)
ax.set_ylabel('Weight')
ax.set_xlabel('Frequency / Hz')
ax.set_xlim((f1, f2))
ax2 = ax.twiny()
ax2.xaxis.set_ticks_position('top')
ax2.set_xlim((f1, f2))
ax2.xaxis.set_ticks(melbank.mel_to_hertz(melfreq))
ax2.xaxis.set_ticklabels(['{:.0f}'.format(mf) for mf in melfreq])
ax2.set_xlabel('Frequency / mel')
plt.tight_layout()

fig, ax = plt.subplots()
ax.matshow(melmat)
plt.axis('equal')
plt.axis('tight')
plt.title('Mel Matrix')
plt.tight_layout()
```

--------------------------------

### Get frequency response

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Returns frequency and impulse responses, supporting custom plotting functions.

```python
def freqz(self, nfft=4096, plotfun=None):
    """Returns frequency and impulse responses.
    Accepts `plotfun` function for cusomized plotting.
    """
    def gen_freqz():
        for b, a in self._coeffs:
            yield freqz_fos(b, a, self.order, nfft, plotfun)
    return list(gen_freqz())
```

--------------------------------

### Initialize delay estimation

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Sets up delay variables and memory buffers to optimize the impulse response.

```python
def init_delay(self, desired_delay_sec):
    """Initializes delay estimation and variables for delay bands
    to achieve a optimized impulse response.
    """
    self.desired_delay_sec = desired_delay_sec
    self.desired_delay_samples = int(self.samplerate*desired_delay_sec)
    self.max_indices, self.slopes = self.estimate_max_indices_and_slopes(
        delay_samples=self.desired_delay_samples)
    self.delay_samples = self.desired_delay_samples - self.max_indices
    self.delay_memory = np.zeros((len(self.centerfrequencies),
                                  np.max(self.delay_samples)))
```

--------------------------------

### Initialize ThirdOctFFTLevel class

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Sets up the frequency bands and FFT mapping matrix for third-octave level estimation.

```python
class ThirdOctFFTLevel:

    """Third octave levels by fft.
    TODO: rename variables
    TODO: Write Documentation
    """

    def __init__(self, fmin=30, fmax=17000, nfft=16384, fs=44100, flag_mean=False):
        self.nfft = nfft
        self.fs = fs

        # following should go into some functions:
        kmin = 11 + int(10*np.log10(fmin))
        kmax = 11 + int(10*np.log10(fmax))
        f_terz = standardized_nominal_frequencies[kmin:kmax]
        n = int(1 + kmax - kmin)
        halfbw = 2**(1.0/6)
        df = fs/nfft
        idx_lower = np.zeros(n)
        idx_lower[0] = 10 + np.round((standardized_nominal_frequencies[kmin]/halfbw)/df)

        idx_upper = 10 + np.round(halfbw*standardized_nominal_frequencies[kmin:kmax]/df)
        idx_lower[1:n] = idx_upper[0:n-1] + 1

        upperedge = halfbw *standardized_nominal_frequencies[kmax]
        print(idx_upper[0]-idx_lower[0])
        #if idx_upper(1) - idx_lower(1) < 4:
        #    raise ValueError('Too few FFT lines per frequency band')

        M = np.zeros((n, nfft/2+1))

        for cc in range(n-1):
            kk = range(int(idx_lower[cc]), int(idx_upper[cc]))
            if not flag_mean:
                M[cc, kk] = 1.0
            else:
                M[cc, kk] = 1.0 / len(kk)

        self.M = M
        self.f_terz = f_terz
```

--------------------------------

### Initialize and Apply Fractional Octave Filter Bank

Source: https://siggigue.github.io/pyfilterbank/octbank.html

Demonstrates creating a filter bank object, processing a random noise signal, and plotting the resulting band levels.

```python
>>> from pyfilterbank import FractionalOctaveFilterbank
>>> from pylab import plt, np
>>>
>>> sample_rate = 44100
>>> ofb = FractionalOctaveFilterbank(sample_rate, order=4)
>>>
>>> x = np.random.randn(4*sample_rate)
>>> y, states = ofb.filter(x)
>>> L = 10 * np.log10(np.sum(y*y,axis=0))
>>> plt.plot(L)
```

--------------------------------

### example_plot()

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Creates a plot illustrating the frequency response of the default FractionalOctaveFilterbank.

```APIDOC
## Function: example_plot()

### Description
Creates a plot using `freqz` for the default `FractionalOctaveFilterbank` instance. This is a utility function for visualizing filter characteristics.
```

--------------------------------

### Execute Plotting Script

Source: https://siggigue.github.io/pyfilterbank/_modules/splweighting.html

Entry point to generate and display the weighting plots.

```python
if __name__ == '__main__':
    fig, ax = plot_weightings()
    fig.show()
```

--------------------------------

### Initialize band gains

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Initializes gains for weighting bands to achieve an equalized frequency response during synthesis.

```python
def init_gains(self):
    """Initializes gains for weighting bands leading to a equalized freqresponse
    when summing (synthesizing) bands.
    """
    self.gains = np.ones(len(self.centerfrequencies))
    # not correct until now:
    # x, s = list(zip(*self.analyze(_create_impulse(self.samplerate/10))))
    # rss = [np.sqrt(np.sum(np.real(b)**2)) for b in x]
    # self.gains = 1/np.array(rss)
```

--------------------------------

### Design and Test Gammatone Filter

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Demonstrates designing a 4th-order gammatone filter and plotting its impulse response using pylab.

```python
def example_filter():
    from pylab import plt, np
    sample_rate = 44100
    order = 4
    b, a = design_filter(
        sample_rate=sample_rate,
        order=order,
        centerfrequency=1000.0,
        attenuation_half_bandwidth_db=-3,
        band_width_factor=1.0)

    x = _create_impulse(1000)
    y, states = fosfilter(b, a, order, x)
    y = y[:800]
    plt.plot(np.real(y), label='Re(z)')
    plt.plot(np.imag(y), label='Im(z)')
    plt.plot(np.abs(y), label='|z|')
    plt.legend()
    # plt.show()
    return y, b, a


if __name__ == '__main__':
    from pylab import plt
    gfb = example_filterbank()
    y = example_filter()
    plt.show()
```

--------------------------------

### OctBank Class Initialization

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

The constructor sets up the filter bank parameters and initializes the filter structure.

```python
def __init__(self,
             sample_rate=44100,
             order=4,
             nth_oct=3.0,
             norm_freq=1000.0,
             start_band=-19,
             end_band=13,
             edge_correction_percent=0.01,
             filterfun='cffi'):
    self._sample_rate = sample_rate
    self._order = order
    self._nth_oct = nth_oct
    self._norm_freq = norm_freq
    self._start_band = start_band
    self._end_band = end_band
    self._edge_correction_percent = edge_correction_percent
    self._initialize_filter_bank()
    self.set_filterfun(filterfun)
```

--------------------------------

### Filter method implementation

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Calculates the filter output using rfft and a dot product with the filter matrix M.

```python
def filter(self, x):
    X = rfft(x, self.nfft/2+1)
    return 20*np.log10(np.dot(self.M, np.abs(X)))
```

--------------------------------

### Apply delay to bands

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Returns delayed bands for an optimized impulse response at synthesis.

```python
def delay(self, bands):
    """Returns delayed bands for an optimized impulseresponse at synthesis.
    """
    self.phase_factors = np.abs(self.slopes)*1j / self.slopes
    for i, band in enumerate(bands):
        phase_factor = self.phase_factors[i]
        delay_samples = self.delay_samples[i]
        if delay_samples == 0:
            yield np.real(band) * phase_factor
        else:
            yield np.concatenate(
                (self.delay_memory[i, :delay_samples],
                 np.real(band[:-delay_samples])),
                axis=0)
            self.delay_memory[i, :delay_samples] = np.real(
                band[-delay_samples:])
```

--------------------------------

### Compiling the C source for SOS filtering

Source: https://siggigue.github.io/pyfilterbank/sosfiltering.html

Commands to compile the C source code into a shared library for use with Python CFFI on Linux or Windows.

```bash
$ gcc -c -std=c99 -O3 sosfilter.c
$ gcc -shared -o sosfilter.so sosfilter.o
$ or the last line for windows users:
$ gcc -shared -o sosfilter.dll sosfilter.o
```

--------------------------------

### Calculate Gammatone Filter Coefficients

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Calculates filter coefficients based on bandwidth and attenuation parameters.

```python
    else:
        raise ValueError(
            'You need to specify either `band_width` or `band_width_factor!`')

    alpha = 10**(0.1 * attenuation_half_bandwidth_db / order)
    p = (-2 + 2 * alpha * cos(phi)) / (1 - alpha)
    lambda_ = -p/2 - sqrt(p*p/4 - 1)
    beta = 2*pi * centerfrequency / sample_rate
    coef = lambda_ * exp(1j*beta)
    factor = 2 * (1 - abs(coef))**order
    b, a = array([factor]), array([1., -coef])
    return b, a
```

--------------------------------

### Synthesize signal from bands

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Returns the summed, delayed, and weighted signal bands.

```python
def synthesize(self, bands):
    """Returns summed dleayed and weighted bands.
    """
    return np.array(list(self.delay(
        [b*g for b, g in zip(bands, self.gains)]))).sum(axis=0)
```

--------------------------------

### Plotting Weighting Filter Responses

Source: https://siggigue.github.io/pyfilterbank/_modules/splweighting.html

Visualize the frequency response of the implemented weighting filters.

```python
import splweighting
fig, ax = splweighting.plot_weightings()
fig.show()
```

--------------------------------

### FractionalOctaveFilterbank

Source: https://siggigue.github.io/pyfilterbank/octbank.html

Constructor for creating a fractional octave filter bank object.

```APIDOC
## class FractionalOctaveFilterbank(sample_rate=44100, order=4, nth_oct=3.0, norm_freq=1000.0, start_band=-19, end_band=13, edge_correction_percent=0.01, filterfun='cffi')

### Description
Initializes a fractional octave filter bank with second-order section Butterworth band passes.

### Parameters
- **sample_rate** (int) - Sampling rate of the signals to be filtered.
- **order** (int) - Filter order of the bands (must be even).
- **nth_oct** (scalar) - Number of bands per octave.
- **norm_freq** (scalar) - Reference frequency for all fractional octaves.
- **start_band** (int) - First band number below norm_freq.
- **end_band** (int) - Last band number above norm_freq.
- **edge_correction_percent** (scalar) - Percentage of widening or narrowing the bands.
- **filterfun** ({'cffi', 'py', 'cprototype'}) - Function used by the filter() method.
```

--------------------------------

### Python Prototype for MIMO SOS Filter

Source: https://siggigue.github.io/pyfilterbank/_modules/sosfiltering.html

A pure Python implementation of the MIMO IIR DF-II biquad filter structure for prototyping.

```python
[docs]def sosfilter_mimo_cprototype_py(signal_in, sos_in, states_in=None):
    """Prototype for the mimo c-filter function.
    Implements a IIR DF-II biquad filter strucure. But with multiple
    input und multiple bands."""
    signal = signal_in.copy().flatten('F')
    shape_signal = signal_in.shape
    print(len(signal))
    nframes = int(signal_in.shape[0])
    print(nframes)
    nchan = int(signal_in.shape[2])
    print(nchan)
    sos = np.tile(sos_in.copy().flatten('F'), (nchan))
    ksos = int(sos_in.shape[0]/6)
    print(ksos)
    kbands = int(sos_in.shape[1])
    print(kbands)
    if not states_in:
        states = np.zeros(nchan*ksos*kbands*2)
    else:
        states = states_in

    ii = 0
    for c in range(nchan):
        for b in range(kbands):
            for k in range(ksos):
                w1 = states[c*ksos*kbands*2 + b*ksos*2 + k*2]
                w2 = states[c*ksos*kbands*2 + b*ksos*2 + k*2 + 1]
                b0 = sos[ii]
                ii += 1
                b1 = sos[ii]
                ii += 1
                b2 = sos[ii]
                ii += 1
                a0 = sos[ii]
                ii += 1
                a1 = sos[ii]
                ii += 1
                a2 = sos[ii]
                ii += 1

                for n in range(nframes):
                    w0 = signal[c*nframes*kbands + b*nframes + n]
                    w0 = w0 - a1*w1 - a2*w2
                    yn = b0*w0 + b1*w1 + b2*w2
                    w2 = w1
                    w1 = w0
                    signal[c*nframes*kbands + b*nframes + n] = yn
```

--------------------------------

### Applying Weighting Filters to Signals

Source: https://siggigue.github.io/pyfilterbank/_modules/splweighting.html

Use the weight_signal function to apply A, B, or C weighting to an input signal.

```python
def weight_signal(data, sample_rate=44100, weighting='A'):
    """Returns filtered signal with a weighting filter.

    Parameters
    ----------
    data : ndarray
        Input signal to be filtered.
    sample_rate : int
        Sample rate of the signal.
    weighting : {'A', 'B', 'C'}
        Specify the weighting function by a string.

    Returns
    -------
    outdata : ndarray
        Filtered output signal. The output will be weighted by
        the specified filter function.
    """
    b, a = _weighting_coeff_design_funsd[weighting](sample_rate)
    return lfilter(b, a, x)
```

--------------------------------

### Calculating fractional octave frequencies

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Computes center and band edge frequencies for fractional octave bands.

```python
def frequencies_fractional_octaves(start_band, end_band, norm_freq, nth_oct):
    """Return center and band edge frequencies of fractional octaves.

    Parameters
    ----------
    start_band : int
        The starting center frequency at `norm_freq`*2^(`start_band`/`nth_oct`).
    end_band : int
        The last center frequency at `norm_freq`*2^(`end_band`/`nth_oct`).
    norm_freq : scalar
        The center frequency of the band number 0.
    nth_oct : scalar
        The distance between the center frequencies.
        For third octaves `nth_oct=3`.

    Returns
    -------
    center_frequencies : ndarray
        Frequencies spaced in `nth_oct` from `start_band` to `end_band`
        with the `norm_freq` at band number 0.
    band_edges : ndarray
        Edge frequencies (-3 dB points) of the fractional octave bands.
        With constant relative Bandwidth.

    """
    k = arange(start_band-1, end_band+2)
    frequencies = norm_freq * 2.0**(k/nth_oct)
    band_edges = sqrt(frequencies[:-1] * frequencies[1:])
    center_frequencies = frequencies[1:-1]
    return center_frequencies, band_edges
```

--------------------------------

### Design band-pass filter matrix

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Generates a matrix of second-order section coefficients for band-pass filters based on specified order and band edges.

```python
def design_sosmat_band_passes(order, band_edges, sample_rate,
                           edge_correction_percent=0.0):
    """Return matrix containig sos coeffs of bandpasses.

    Parameters
    ----------
    order : int
        Order of the band pass filters.
    band_edges : ndarray
        Band edge frequencies for the bandpasses.
    sample_rate : scalar
        Sample frequency.
    edge_correction_percent : scalar
        Percentage for the correction of the bandedges.
        Float between -100 % and 100 %.
        It can be helpfull dependent on the used filter order.
        p > 0 widens the band passes.

    Returns
    -------
    sosmat : ndarray
        Second order section coefficients.
        Each column is one band pass cascade of coefficients.
    """
    num_coeffs_biquad_bandpass = 6
    num_coeffs_cascade = order * num_coeffs_biquad_bandpass
    num_bands = len(band_edges) - 1
    sosmat = zeros((num_coeffs_cascade, num_bands))

    band_edges_normalized = to_normalized_frequencies(band_edges, sample_rate)
    p_lower = (1 - edge_correction_percent*1e-2)
    p_upper = (1 + edge_correction_percent*1e-2)

    for i, (lower_freq, upper_freq) in enumerate(zip(
            band_edges_normalized[:-1],
            band_edges_normalized[1:])):
        sos = butter_sos('bandpass',
                         order,
                         p_lower*lower_freq,
                         p_upper*upper_freq)
        sosmat[:, i] = sos.flatten()
    return sosmat
```

--------------------------------

### Generate Gammatone Bank Frequencies

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Calculates center frequencies and auditory bandwidths for a range of filters.

```python
def frequencies_gammatone_bank(start_band, end_band, norm_freq, density):
    norm_erb = hertz_to_erbscale(norm_freq)
    centerfrequencies = erbscale_to_hertz(
        arange(start_band, end_band, density) + norm_erb)
    return centerfrequencies
```

--------------------------------

### Bilinear Transformation for SOS

Source: https://siggigue.github.io/pyfilterbank/_modules/sosfiltering.html

Performs bilinear transformation of analog filter weights to digital weights in cascade form.

```python
def bilinear_sos(d, c):
    """Bilinear transformation of analog weights to digital weights.
    >>>>>>> 8d01abb1e1f252834c0666d50c645dd3d35a1f52

    Bilinear transformation of analog weights to digital weights.
    Weights of IIR digital filter in cascade form with
    2-pole sections; H(z)=H(z,1)H(z,2)...H(z,L/2) where
    L is the number of poles and each section is a ratio of quadratics.

    Parameters
    ----------
    d : ndarray
        Numerator weights of analog filter in 1-pole
        sections. d is dimensioned (L/2 x 2).
    c : ndarray
        Denominator weights, dimensioned same as d.

    Returns
    -------
    b : ndarray
        Digital numerator weights, dimensioned (L/2 x 3).
    a : ndarray
        Digital denominator weights, dimensioned the same.

    """
    L2, ncd = d.shape
    nr, ncc = c.shape

    # Check for errors.
    if(nr != L2 or ncd != 2 or ncc != 2):
        raise Exception('Inputs d and c must both be L/2 x 2 arrays.')

    # Bilinear transformation of H(s) to H(z) using z and p vectors.
    a = np.zeros((L2, 3), dtype=np.double)
    a[:, 0] = np.abs(c[:, 0] + c[:, 1])**2

    if np.min(a[:, 0]) == 0:
        raise Exception('"c" should not have a row of zeros.')
    a[:, 1] = 2*np.real((c[:, 0] + c[:, 1]) * np.conj(c[:, 1] - c[:, 0]))
    a[:, 2] = np.abs(c[:, 1] - c[:, 0])**2

    b = np.zeros((L2, 3), dtype=np.double)
    b[:, 0] = np.abs(d[:, 0] + d[:, 1])**2
    b[:, 1] = 2*np.real((d[:, 0] + d[:, 1]) * np.conj(d[:, 1] - d[:, 0]))
    b[:, 2] = np.abs(d[:, 1] - d[:, 0])**2

    # Scale H(z) so a(:,1)=1:
    sa = np.kron(np.ones((3, 1)), a[:, 0]).T
    a = a / sa
    b = b / sa
    return b, a
```

--------------------------------

### splweighting.weight_signal(data, sample_rate=44100, weighting='A')

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Applies a specified weighting filter to an input signal.

```APIDOC
## splweighting.weight_signal(data, sample_rate=44100, weighting='A')

### Description
Returns a filtered signal using the specified weighting filter.

### Parameters
- **data** (ndarray) - Required - Input signal to be filtered.
- **sample_rate** (int) - Optional - Sample rate of the signal (default: 44100).
- **weighting** (string) - Optional - Specify the weighting function: 'A', 'B', or 'C' (default: 'A').

### Returns
- **outdata** (ndarray) - Filtered output signal.
```

--------------------------------

### Calculate Frequency Response of FOS

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Computes the frequency and impulse response for first-order section coefficients.

```python
[docs]def freqz_fos(b, a, order, nfft, plotfun=None):
    """Returns Frequency Response and impulse response
    of first order section coeffs.

    Parameters
    ----------
    b : arraylike
    a : arraylike
    order : int
    nfft : int
    plotfun : function(frequencies, response)

    Returns
    -------
    freqresponse : arraylike
    frequencies : arraylike
    response : arraylike

    """
    impulse = _create_impulse(nfft)
    response, states = fosfilter(b, a, order, impulse)
    freqresponse = rfft(np.real(response))
    frequencies = rfftfreq(nfft)
    if plotfun:
        plotfun(frequencies, freqresponse)
    return freqresponse, frequencies, response
```

--------------------------------

### Apply Weighting Filter to Signal

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Use scipy.signal.lfilter to apply the generated filter coefficients to an input signal.

```python
>>> from scipy.signal import lfilter
>>> y = lfilter(b, a, x)
```

--------------------------------

### Design A-Weighting Filter Coefficients

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Generate filter coefficients for A-weighting based on the provided sample rate.

```python
>>> b, a = a_weighting_coeff_design(sample_rate)
```

--------------------------------

### Calculate ERB Properties

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Functions to calculate ERB counts, bandwidths, and frequency conversions.

```python
def erb_count(centerfrequency):
    return 21.4 * log10(4.37 * 0.001 * centerfrequency + 1)
```

```python
def erb_aud(centerfrequency):
    return _ERB_L + centerfrequency / _ERB_Q
```

```python
def hertz_to_erbscale(frequency):
    return _ERB_Q * np.log(1 + frequency / (_ERB_L * _ERB_Q))
```

```python
def erbscale_to_hertz(erb):
    return (exp(erb/_ERB_Q) - 1) * _ERB_L * _ERB_Q
```

--------------------------------

### Design C-Weighting Filter Coefficients

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Generate filter coefficients for C-weighting based on the provided sample rate.

```python
b, a = c_weighting_coeffs_design(sample_rate)
To Filter a signal use scipy lfilter:
from scipy.signal import lfilter y = lfilter(b, a, x) 
```

--------------------------------

### Compute Mel Transformation Matrix

Source: https://siggigue.github.io/pyfilterbank/_modules/melbank.html

Generates a transformation matrix to convert FFT spectra to mel spectra.

```python
def compute_melmat(num_mel_bands=12, freq_min=64, freq_max=8000,
                   num_fft_bands=513, sample_rate=16000):
    """Returns tranformation matrix for mel spectrum.

    Parameters
    ----------
    num_mel_bands : int
        Number of mel bands. Number of rows in melmat.
        Default: 24
    freq_min : scalar
        Minimum frequency for the first band.
        Default: 64
    freq_max : scalar
        Maximum frequency for the last band.
        Default: 8000
    num_fft_bands : int
        Number of fft-frequenc bands. This ist NFFT/2+1 !
        number of columns in melmat.
        Default: 513   (this means NFFT=1024)
    sample_rate : scalar
        Sample rate for the signals that will be used.
        Default: 44100

    Returns
    -------
    melmat : ndarray
        Transformation matrix for the mel spectrum.
        Use this with fft spectra of num_fft_bands_bands length
        and multiply the spectrum with the melmat
        this will tranform your fft-spectrum
        to a mel-spectrum.

    frequencies : tuple (ndarray <num_mel_bands>, ndarray <num_fft_bands>)
        Center frequencies of the mel bands, center frequencies of fft spectrum.

    """
    center_frequencies_mel, lower_edges_mel, upper_edges_mel =  \
        melfrequencies_mel_filterbank(
            num_mel_bands,
            freq_min,
            freq_max,
            num_fft_bands
    )

    len_fft = float(num_fft_bands) / sample_rate
    center_frequencies_hz = mel_to_hertz(center_frequencies_mel)
    lower_edges_hz = mel_to_hertz(lower_edges_mel)
    upper_edges_hz = mel_to_hertz(upper_edges_mel)
    freqs = linspace(0.0, sample_rate/2.0, num_fft_bands)
    melmat = zeros((num_mel_bands, num_fft_bands))

    for imelband, (center, lower, upper) in enumerate(zip(
            center_frequencies_hz, lower_edges_hz, upper_edges_hz)):

        left_slope = (freqs >= lower)  == (freqs <= center)
        melmat[imelband, left_slope] = (
            (freqs[left_slope] - lower) / (center - lower)
        )

        right_slope = (freqs >= center) == (freqs <= upper)
        melmat[imelband, right_slope] = (
            (upper - freqs[right_slope]) / (upper - center)
        )

    return melmat, (center_frequencies_mel, freqs)
```

--------------------------------

### Design Gammatone Filter

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Returns filter coefficients for a gammatone filter based on Hohmann (2002).

```python
def design_filter(
        sample_rate=44100,
        order=4,
        centerfrequency=1000.0,
        band_width=None,
        band_width_factor=1.0,
        attenuation_half_bandwidth_db=-3):
    if band_width:
        phi = pi * band_width / sample_rate

    elif band_width_factor:
        erb_audiological = band_width_factor * erb_aud(centerfrequency)
        phi = pi * erb_audiological / sample_rate
```

--------------------------------

### splweighting.c_weighting_coeffs_design(sample_rate)

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Returns the b and a coefficients for a C-weighting filter based on the provided sample rate.

```APIDOC
## splweighting.c_weighting_coeffs_design(sample_rate)

### Description
Returns b and a coefficients of a C-weighting filter.

### Parameters
- **sample_rate** (scalar) - Required - Sample rate of the signals that will be filtered.

### Returns
- **b, a** (ndarray) - Filter coefficients for a digital weighting filter.
```

--------------------------------

### frequencies_fractional_octaves

Source: https://siggigue.github.io/pyfilterbank/octbank.html

Calculates center and band edge frequencies for fractional octave bands.

```APIDOC
## frequencies_fractional_octaves(start_band, end_band, norm_freq, nth_oct)

### Description
Returns center and band edge frequencies of fractional octaves based on the provided band range and reference frequency.

### Parameters
- **start_band** (int) - Required - The starting center frequency index.
- **end_band** (int) - Required - The last center frequency index.
- **norm_freq** (scalar) - Required - The center frequency of the band number 0.
- **nth_oct** (scalar) - Required - The distance between the center frequencies (e.g., 3 for third octaves).

### Returns
- **center_frequencies** (ndarray) - Frequencies spaced in nth_oct from start_band to end_band.
- **band_edges** (ndarray) - Edge frequencies (-3 dB points) of the fractional octave bands.
```

--------------------------------

### SOS Filter C-style Prototype

Source: https://siggigue.github.io/pyfilterbank/_modules/sosfiltering.html

Implements an IIR DF-II biquad filter structure as a Python prototype.

```python
def sosfilter_cprototype_py(signal, sos, states):
    """Prototype for second order section filtering c function.
    Implements a IIR DF-II biquad filter strucure.
    """
    N = int(len(signal))
    K = int(sos.size/6)
    if isinstance(states, type(None)):
        states = np.zeros(K*2).astype(np.double)
    signal = signal.copy()  # only python specific
    sos = sos.copy().flatten()
    yn = 0.0  # buffer for output
    w0 = 0.0  # signal states

    for k in range(K):
        # get coefficients of current biquad
        w1 = states[k*2]
        w2 = states[k*2+1]
        b0 = sos[k*6]
        b1 = sos[k*6+1]
        b2 = sos[k*6+2]
        a0 = sos[k*6+3]
        a1 = sos[k*6+4]
        a2 = sos[k*6+5]

        for n in range(N):
            # get a sample
            w0 = signal[n].copy()
            # recursive path
            w0 = w0 - a1*w1 - a2*w2
            # transversal path
            yn = b0*w0 + b1*w1 + b2*w2
            # delays
            w2 = w1
            w1 = w0
            # write output signal
            signal[n] = yn

    states[k*2] = w1
    states[k*2+1] = w2

    return signal, states
```

--------------------------------

### splweighting.b_weighting_coeffs_design(sample_rate)

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Returns the b and a coefficients for a B-weighting filter based on the provided sample rate.

```APIDOC
## splweighting.b_weighting_coeffs_design(sample_rate)

### Description
Returns b and a coefficients of a B-weighting filter. Note: B-Weighting is no longer described in DIN61672.

### Parameters
- **sample_rate** (scalar) - Required - Sample rate of the signals that will be filtered.

### Returns
- **b, a** (ndarray) - Filter coefficients for a digital weighting filter.
```

--------------------------------

### filter(self, x, ffilt=False, states=None)

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Filters an input signal using the settings of the filterbank object.

```APIDOC
## filter(self, x, ffilt=False, states=None)

### Description
Filters the input signal by the settings of the filterbank object.

### Parameters
- **x** (ndarray) - Required - Input signal (Nx0).
- **ffilt** (bool) - Optional - Forward and backward filtering if True.
- **states** (dict) - Optional - States of all filter sections in the filterbank. Initial state can be None.

### Returns
- **y** (ndarray) - Fractional octave signals of the filtered input x.
- **states** (dict) - Dictionary containing all filter section states.
```

--------------------------------

### GammatoneFilterbank Class

Source: https://siggigue.github.io/pyfilterbank/gammatone.html

A class for initializing and managing a Gammatone filterbank for signal processing tasks.

```APIDOC
## class GammatoneFilterbank(samplerate=44100, order=4, startband=-12, endband=12, normfreq=1000.0, density=1.0, bandwidth_factor=1.0, desired_delay_sec=0.02)

### Description
Initializes a GammatoneFilterbank instance for filtering signals.

### Parameters
- **samplerate** (scalar) - Default: 44100
- **order** (scalar) - Default: 4
- **startband** (scalar) - Default: -12
- **endband** (scalar) - Default: 12
- **normfreq** (scalar) - Default: 1000.0
- **density** (scalar) - Default: 1.0
- **bandwidth_factor** (scalar) - Default: 1.0
- **desired_delay_sec** (scalar) - Default: 0.02
```

--------------------------------

### splweighting.plot_weightings()

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Plots all weighting functions defined in the splweighting module.

```APIDOC
## splweighting.plot_weightings()

### Description
Plots all weighting functions defined in the module.
```

--------------------------------

### Frequency Response Plotting

Source: https://siggigue.github.io/pyfilterbank/_modules/sosfiltering.html

Plots the frequency response of an SOS matrix.

```python
def freqz(sosmat, nsamples=44100, sample_rate=44100, plot=True):
    """Plots Frequency response of sosmat."""
    from pylab import np, plt, fft, fftfreq
    x = np.zeros(nsamples)
    x[nsamples/2] = 0.999
    y, states = sosfilter_double_c(x, sosmat)
    Y = fft(y)
    f = fftfreq(len(x), 1.0/sample_rate)
    if plot:
        plt.grid(True)
        plt.axis([0, sample_rate / 2, -100, 5])
        L = 20*np.log10(np.abs(Y[:len(x)/2]) + 1e-17)
        plt.semilogx(f[:len(x)/2], L, lw=0.5)
        plt.hold(True)
        plt.title(u'freqz sos filter')
        plt.xlabel('Frequency / Hz')
        plt.ylabel(u'Damping /dB(FS)')
        plt.xlim((10, sample_rate/2))
        plt.hold(False)
    return x, y, f, Y
```

--------------------------------

### Multi-Channel C-accelerated SOS Filter

Source: https://siggigue.github.io/pyfilterbank/_modules/sosfiltering.html

Handles multi-channel input signals for second-order section filtering using CFFI.

```python
[docs]def sosfilter_double_mimo_c(signal, sos, states=None):
    """Second order section filter function for  multi channel input
    using cffi, double precision

    signal_out, states = sosfilter_c(signal_in, sos, states=None)

    Parameters
    ----------
    signal : ndarray
        Signal array of shape (N x C).
    sos : ndarray
        Second order section filter coefficients (K*6 x B x C) np-array.

    states : ndarray
        Filter states, initial can be None.
        Otherwise shape is (K*2 x B x C)

    Returns
    -------
    signal : ndarray
        Filtered signal of shape (N x B x C).
        Where N is the number of samples, B is th number of
        filter bands and C is the number of signal channels.
    states : ndarray
        Filter states of shape (K*2 x B x C).

    """

    shape_signal = signal.shape
    nframes = int(shape_signal[0])
    if len(shape_signal) > 1:
        nchan = int(shape_signal[1])
    else:
        nchan = int(1)

    shape_sos = sos.shape
    ksos = int(shape_sos[0]/6)
    if len(shape_sos) > 1:
        kbands = int(shape_sos[1])
        if len(shape_sos) == 2 and nchan > 1:
            sos = np.tile(sos.flatten('F'), (nchan))
    else:
        kbands = int(1)

    if isinstance(states, type(None)):
        states = np.zeros(nchan*kbands*ksos*2).astype(np.double)

    states_c = ffi.new(
        'char[]', np.array(states, dtype=np.double).flatten('F').tostring())

    sos_c = ffi.new(
        'char[]', np.array(sos, dtype=np.double).flatten('F').tostring())

    if nchan > 1:
        signal = np.tile(signal, (kbands, 1))
    else:
        signal = np.tile(signal, (kbands))

    shape_signal = signal.shape
    signal_c = ffi.new(
        'char[]', np.array(signal, dtype=np.double).T.flatten().tostring())

    _c.sosfilter_double_mimo(
        ffi.cast("double*", signal_c),
        nframes,
        nchan,
        ffi.cast("double*", sos_c),
        ksos,
        kbands,
        ffi.cast("double*", states_c)
    )

    out = np.fromstring(
        ffi.buffer(signal_c),
        dtype=np.double,
        count=signal.size
    )
    states = np.fromstring(
        ffi.buffer(states_c),
        dtype=np.double,
        count=len(states)
    )
    return out.reshape((nframes, kbands, nchan), order='F'), states
```

--------------------------------

### Designing B-Weighting Coefficients

Source: https://siggigue.github.io/pyfilterbank/_modules/splweighting.html

Generate filter coefficients for B-weighting and apply them using scipy.signal.lfilter.

```python
>>> b, a = b_weighting_coeff_design(sample_rate)

To Filter a signal use :function: scipy.lfilter:

>>> from scipy.signal import lfilter
>>> y = lfilter(b, a, x)
```

--------------------------------

### splweighting.a_weighting_coeffs_design(sample_rate)

Source: https://siggigue.github.io/pyfilterbank/splweighting.html

Returns the b and a coefficients for an A-weighting filter based on the provided sample rate.

```APIDOC
## splweighting.a_weighting_coeffs_design(sample_rate)

### Description
Returns b and a coefficients of an A-weighting filter.

### Parameters
- **sample_rate** (scalar) - Required - Sample rate of the signals that will be filtered.

### Returns
- **b, a** (ndarray) - Filter coefficients for a digital weighting filter.
```

--------------------------------

### Create impulse signal

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Generates a complex impulse signal of a specified length.

```python
def _create_impulse(num_samples):
    sig = zeros(num_samples) + 0j
    sig[0] = 1.0
    return sig
```

--------------------------------

### Filter input signal with filterbank

Source: https://siggigue.github.io/pyfilterbank/_modules/pyfilterbank/octbank.html

Filters an input signal using the filterbank's center frequencies and SOS matrices. Supports forward-backward filtering via the ffilt parameter.

```python
def filter(self, x, ffilt=False, states=None):
    """Filters the input by the settings of the filterbank object.

    Parameters
    ----------
    x :  ndarray
        Input signal (Nx0)
    ffilt : bool
        Forward and backward filtering, if Ture.
    states : dict
        States of all filter sections in the filterbank.
        Initial you can states=None before block process.

    Returns
    -------
    y : ndarray
        Fractional octave signals of the filtered input x
    states : dict
        Dictionary containing all filter section states.
    """
    # in the next version this will be turne to a multi dimensional np array
    y_data = zeros((len(x), len(self.center_frequencies)))

    if not isinstance(states, dict):
        states_allbands = dict()
        for f in self.center_frequencies: states_allbands[f] = None
    else :
        states_allbands = states

    for i, f in enumerate(self.center_frequencies):
        states = states_allbands[f]
        sos = reshape(self.sosmat[:, i], (self.order, 6))
        if not ffilt:
            y, states = self.sosfilterfun(x.copy(), sos, states)
        elif ffilt:
            y, states = self.sosfilterfun(x.copy()[::-1], sos, states)
            y, states = self.sosfilterfun(y[::-1], sos, states)

        y_data[:, i] = y
        states_allbands[f] = states
    return y_data, states_allbands
```

--------------------------------

### Apply First Order Section Filtering

Source: https://siggigue.github.io/pyfilterbank/_modules/gammatone.html

Filters a signal using cascaded complex gammatone filters by applying first-order sections.

```python
[docs]def fosfilter(b, a, order, signal, states=None):
    """Return signal filtered with `b` and `a` (first order section)
    by filtering the signal `order` times.

    This Function was created for filtering signals by first order section
    cascaded complex gammatone filters.

    Parameters
    ----------
    b, a : ndarray, ndarray
        Filter coefficients of a first order section filter.
        Can be complex valued.
    order : int
        Order of the filter to be applied. This will
        be the count of refiltering the signal order times
        with the given coefficients.
    signal : ndarray
        Input signal to be filtered.
    states : ndarray, default None
        Array with the filter states of length `order`.
        Initial you can set it to None.

    Returns
    -------
    signal : ndarray
        Output signal, that is filtered and complex valued
        (analytical signal).
    states : ndarray
        Array with the filter states of length `order`.
        You need to loop it back into this function when block
        processing.

    """
    if not states:
        states = zeros(order, dtype=np.complex128)

    for i in range(order):
        state = [states[i]]
        signal, state = lfilter(b, a, signal, zi=state)
        states[i] = state[0]
        b = ones_like(b)
    return signal, states
```