def firwin(N, cutoff, width=None, window='hamming'):
"""
FIR Filter Design using windowed ideal filter method.
Parameters
----------
N -- order of filter (number of taps)
cutoff -- cutoff frequency of filter (normalized so that 1 corresponds to
Nyquist or pi radians / sample)
width -- if width is not None, then assume it is the approximate width of
the transition region (normalized so that 1 corresonds to pi)
for use in kaiser FIR filter design.
window -- desired window to use. See get_window for a list
of windows and required parameters.
Returns
-------
h -- coefficients of length N fir filter.
"""
from signaltools import get_window
if isinstance(width,float):
A = 2.285*N*width + 8
if (A < 21): beta = 0.0
elif (A <= 50): beta = 0.5842*(A-21)**0.4 + 0.07886*(A-21)
else: beta = 0.1102*(A-8.7)
window=('kaiser',beta)
win = get_window(window,N,fftbins=1)
alpha = N//2
m = numpy.arange(0,N)
h = win*special.sinc(cutoff*(m-alpha))
return h / numpy.sum(h,axis=0)
def firwin2(numtaps, freq, gain, nfreqs=None, window='hamming', nyq=1.0):
"""FIR filter design using the window method.
From the given frequencies `freq` and corresponding gains `gain`,
this function constructs an FIR filter with linear phase and
(approximately) the given frequency response.
Parameters
----------
numtaps : int
The number of taps in the FIR filter. `numtaps` must be less than
`nfreqs`. If the gain at the Nyquist rate, `gain[-1]`, is not 0,
then `numtaps` must be odd.
freq : array-like, 1D
The frequency sampling points. Typically 0.0 to 1.0 with 1.0 being
Nyquist. The Nyquist frequency can be redefined with the argument
`nyq`.
The values in `freq` must be nondecreasing. A value can be repeated
once to implement a discontinuity. The first value in `freq` must
be 0, and the last value must be `nyq`.
gain : array-like
The filter gains at the frequency sampling points.
nfreqs : int, optional
The size of the interpolation mesh used to construct the filter.
For most efficient behavior, this should be a power of 2 plus 1
(e.g, 129, 257, etc). The default is one more than the smallest
power of 2 that is not less than `numtaps`. `nfreqs` must be greater
than `numtaps`.
window : string or (string, float) or float, or None, optional
Window function to use. Default is "hamming". See
`scipy.signal.get_window` for the complete list of possible values.
If None, no window function is applied.
nyq : float
Nyquist frequency. Each frequency in `freq` must be between 0 and
`nyq` (inclusive).
Returns
-------
taps : numpy 1D array of length `numtaps`
The filter coefficients of the FIR filter.
Example
-------
A lowpass FIR filter with a response that is 1 on [0.0, 0.5], and
that decreases linearly on [0.5, 1.0] from 1 to 0:
>>> taps = firwin2(150, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0])
>>> print(taps[72:78])
[-0.02286961 -0.06362756 0.57310236 0.57310236 -0.06362756 -0.02286961]
See also
--------
scipy.signal.firwin
Notes
-----
From the given set of frequencies and gains, the desired response is
constructed in the frequency domain. The inverse FFT is applied to the
desired response to create the associated convolution kernel, and the
first `numtaps` coefficients of this kernel, scaled by `window`, are
returned.
The FIR filter will have linear phase. The filter is Type I if `numtaps`
is odd and Type II if `numtaps` is even. Because Type II filters always
have a zero at the Nyquist frequency, `numtaps` must be odd if `gain[-1]`
is not zero.
.. versionadded:: 0.9.0
References
----------
.. [1] Oppenheim, A. V. and Schafer, R. W., "Discrete-Time Signal
Processing", Prentice-Hall, Englewood Cliffs, New Jersey (1989).
(See, for example, Section 7.4.)
.. [2] Smith, Steven W., "The Scientist and Engineer's Guide to Digital
Signal Processing", Ch. 17. http://www.dspguide.com/ch17/1.htm
"""
if len(freq) != len(gain):
raise ValueError('freq and gain must be of same length.')
if nfreqs is not None and numtaps >= nfreqs:
raise ValueError('ntaps must be less than nfreqs, but firwin2 was '
'called with ntaps=%d and nfreqs=%s' % (numtaps, nfreqs))
if freq[0] != 0 or freq[-1] != nyq:
raise ValueError('freq must start with 0 and end with `nyq`.')
d = np.diff(freq)
if (d < 0).any():
raise ValueError('The values in freq must be nondecreasing.')
d2 = d[:-1] + d[1:]
if (d2 == 0).any():
raise ValueError('A value in freq must not occur more than twice.')
if numtaps % 2 == 0 and gain[-1] != 0.0:
raise ValueError("A filter with an even number of coefficients must "
"have zero gain at the Nyquist rate.")
if nfreqs is None:
nfreqs = 1 + 2 ** int(ceil(log(numtaps,2)))
# Tweak any repeated values in freq so that interp works.
eps = np.finfo(float).eps
for k in range(len(freq)):
if k < len(freq)-1 and freq[k] == freq[k+1]:
freq[k] = freq[k] - eps
freq[k+1] = freq[k+1] + eps
# Linearly interpolate the desired response on a uniform mesh `x`.
x = np.linspace(0.0, nyq, nfreqs)
fx = np.interp(x, freq, gain)
# Adjust the phases of the coefficients so that the first `ntaps` of the
# inverse FFT are the desired filter coefficients.
shift = np.exp(-(numtaps-1)/2. * 1.j * np.pi * x / nyq)
fx2 = fx * shift
# Use irfft to compute the inverse FFT.
out_full = irfft(fx2)
if window is not None:
# Create the window to apply to the filter coefficients.
from signaltools import get_window
wind = get_window(window, numtaps, fftbins=False)
else:
wind = 1
# Keep only the first `numtaps` coefficients in `out`, and multiply by
# the window.
out = out_full[:numtaps] * wind
return out
def firwin(numtaps, cutoff, width=None, window='hamming', pass_zero=True,
scale=True, nyq=1.0):
"""
FIR filter design using the window method.
This function computes the coefficients of a finite impulse response filter.
The filter will have linear phase; it will be Type I if `numtaps` is odd and
Type II if `numtaps` is even.
Type II filters always have zero response at the Nyquist rate, so a
ValueError exception is raised if firwin is called with `numtaps` even and
having a passband whose right end is at the Nyquist rate.
Parameters
----------
numtaps : int
Length of the filter (number of coefficients, i.e. the filter
order + 1). `numtaps` must be even if a passband includes the
Nyquist frequency.
cutoff : float or 1D array_like
Cutoff frequency of filter (expressed in the same units as `nyq`)
OR an array of cutoff frequencies (that is, band edges). In the
latter case, the frequencies in `cutoff` should be positive and
monotonically increasing between 0 and `nyq`. The values 0 and
`nyq` must not be included in `cutoff`.
width : float or None
If `width` is not None, then assume it is the approximate width
of the transition region (expressed in the same units as `nyq`)
for use in Kaiser FIR filter design. In this case, the `window`
argument is ignored.
window : string or tuple of string and parameter values
Desired window to use. See `scipy.signal.get_window` for a list
of windows and required parameters.
pass_zero : bool
If True, the gain at the frequency 0 (i.e. the "DC gain") is 1.
Otherwise the DC gain is 0.
scale : bool
Set to True to scale the coefficients so that the frequency
response is exactly unity at a certain frequency.
That frequency is either:
0 (DC) if the first passband starts at 0 (i.e. pass_zero
is True);
`nyq` (the Nyquist rate) if the first passband ends at
`nyq` (i.e the filter is a single band highpass filter);
center of first passband otherwise.
nyq : float
Nyquist frequency. Each frequency in `cutoff` must be between 0
and `nyq`.
Returns
-------
h : 1D ndarray
Coefficients of length `numtaps` FIR filter.
Raises
------
ValueError
If any value in `cutoff` is less than or equal to 0 or greater
than or equal to `nyq`, if the values in `cutoff` are not strictly
monotonically increasing, or if `numtaps` is even but a passband
includes the Nyquist frequency.
Examples
--------
Low-pass from 0 to f::
>>> firwin(numtaps, f)
Use a specific window function::
>>> firwin(numtaps, f, window='nuttall')
High-pass ('stop' from 0 to f)::
>>> firwin(numtaps, f, pass_zero=False)
Band-pass::
>>> firwin(numtaps, [f1, f2], pass_zero=False)
Band-stop::
>>> firwin(numtaps, [f1, f2])
Multi-band (passbands are [0, f1], [f2, f3] and [f4, 1])::
>>>firwin(numtaps, [f1, f2, f3, f4])
Multi-band (passbands are [f1, f2] and [f3,f4])::
>>> firwin(numtaps, [f1, f2, f3, f4], pass_zero=False)
See also
--------
scipy.signal.firwin2
"""
# The major enhancements to this function added in November 2010 were
# developed by Tom Krauss (see ticket #902).
cutoff = np.atleast_1d(cutoff) / float(nyq)
# Check for invalid input.
if cutoff.ndim > 1:
raise ValueError("The cutoff argument must be at most one-dimensional.")
if cutoff.size == 0:
raise ValueError("At least one cutoff frequency must be given.")
if cutoff.min() <= 0 or cutoff.max() >= 1:
raise ValueError("Invalid cutoff frequency: frequencies must be greater than 0 and less than nyq.")
if np.any(np.diff(cutoff) <= 0):
raise ValueError("Invalid cutoff frequencies: the frequencies must be strictly increasing.")
if width is not None:
# A width was given. Find the beta parameter of the Kaiser window
# and set `window`. This overrides the value of `window` passed in.
atten = kaiser_atten(numtaps, float(width)/nyq)
beta = kaiser_beta(atten)
window = ('kaiser', beta)
pass_nyquist = bool(cutoff.size & 1) ^ pass_zero
if pass_nyquist and numtaps % 2 == 0:
raise ValueError("A filter with an even number of coefficients must "
"have zero response at the Nyquist rate.")
# Insert 0 and/or 1 at the ends of cutoff so that the length of cutoff is even,
# and each pair in cutoff corresponds to passband.
cutoff = np.hstack(([0.0]*pass_zero, cutoff, [1.0]*pass_nyquist))
# `bands` is a 2D array; each row gives the left and right edges of a passband.
bands = cutoff.reshape(-1,2)
# Build up the coefficients.
alpha = 0.5 * (numtaps-1)
m = np.arange(0, numtaps) - alpha
h = 0
for left, right in bands:
h += right * sinc(right * m)
h -= left * sinc(left * m)
# Get and apply the window function.
from signaltools import get_window
win = get_window(window, numtaps, fftbins=False)
h *= win
# Now handle scaling if desired.
if scale:
# Get the first passband.
left, right = bands[0]
if left == 0:
scale_frequency = 0.0
elif right == 1:
scale_frequency = 1.0
else:
scale_frequency = 0.5 * (left + right)
c = np.cos(np.pi * m * scale_frequency)
s = np.sum(h * c)
h /= s
return h