# scipy/scipy

3929 lines (3207 sloc) 126 KB
 """Filter design. """ from __future__ import division, print_function, absolute_import import warnings import numpy from numpy import (atleast_1d, poly, polyval, roots, real, asarray, allclose, resize, pi, absolute, logspace, r_, sqrt, tan, log10, arctan, arcsinh, sin, exp, cosh, arccosh, ceil, conjugate, zeros, sinh, append, concatenate, prod, ones, array) from numpy import mintypecode import numpy as np from scipy import special, optimize from scipy.special import comb from scipy.misc import factorial from numpy.polynomial.polynomial import polyval as npp_polyval import math __all__ = ['findfreqs', 'freqs', 'freqz', 'tf2zpk', 'zpk2tf', 'normalize', 'lp2lp', 'lp2hp', 'lp2bp', 'lp2bs', 'bilinear', 'iirdesign', 'iirfilter', 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel', 'band_stop_obj', 'buttord', 'cheb1ord', 'cheb2ord', 'ellipord', 'buttap', 'cheb1ap', 'cheb2ap', 'ellipap', 'besselap', 'BadCoefficients', 'tf2sos', 'sos2tf', 'zpk2sos', 'sos2zpk', 'group_delay', 'sosfreqz', 'iirnotch', 'iirpeak'] class BadCoefficients(UserWarning): """Warning about badly conditioned filter coefficients""" pass abs = absolute def findfreqs(num, den, N): """ Find array of frequencies for computing the response of an analog filter. Parameters ---------- num, den : array_like, 1-D The polynomial coefficients of the numerator and denominator of the transfer function of the filter or LTI system. The coefficients are ordered from highest to lowest degree. N : int The length of the array to be computed. Returns ------- w : (N,) ndarray A 1-D array of frequencies, logarithmically spaced. Examples -------- Find a set of nine frequencies that span the "interesting part" of the frequency response for the filter with the transfer function H(s) = s / (s^2 + 8s + 25) >>> from scipy import signal >>> signal.findfreqs([1, 0], [1, 8, 25], N=9) array([ 1.00000000e-02, 3.16227766e-02, 1.00000000e-01, 3.16227766e-01, 1.00000000e+00, 3.16227766e+00, 1.00000000e+01, 3.16227766e+01, 1.00000000e+02]) """ ep = atleast_1d(roots(den)) + 0j tz = atleast_1d(roots(num)) + 0j if len(ep) == 0: ep = atleast_1d(-1000) + 0j ez = r_['-1', numpy.compress(ep.imag >= 0, ep, axis=-1), numpy.compress((abs(tz) < 1e5) & (tz.imag >= 0), tz, axis=-1)] integ = abs(ez) < 1e-10 hfreq = numpy.around(numpy.log10(numpy.max(3 * abs(ez.real + integ) + 1.5 * ez.imag)) + 0.5) lfreq = numpy.around(numpy.log10(0.1 * numpy.min(abs(real(ez + integ)) + 2 * ez.imag)) - 0.5) w = logspace(lfreq, hfreq, N) return w def freqs(b, a, worN=None, plot=None): """ Compute frequency response of analog filter. Given the M-order numerator b and N-order denominator a of an analog filter, compute its frequency response:: b[0]*(jw)**M + b[1]*(jw)**(M-1) + ... + b[M] H(w) = ---------------------------------------------- a[0]*(jw)**N + a[1]*(jw)**(N-1) + ... + a[N] Parameters ---------- b : array_like Numerator of a linear filter. a : array_like Denominator of a linear filter. worN : {None, int, array_like}, optional If None, then compute at 200 frequencies around the interesting parts of the response curve (determined by pole-zero locations). If a single integer, then compute at that many frequencies. Otherwise, compute the response at the angular frequencies (e.g. rad/s) given in worN. plot : callable, optional A callable that takes two arguments. If given, the return parameters w and h are passed to plot. Useful for plotting the frequency response inside freqs. Returns ------- w : ndarray The angular frequencies at which h was computed. h : ndarray The frequency response. See Also -------- freqz : Compute the frequency response of a digital filter. Notes ----- Using Matplotlib's "plot" function as the callable for plot produces unexpected results, this plots the real part of the complex transfer function, not the magnitude. Try lambda w, h: plot(w, abs(h)). Examples -------- >>> from scipy.signal import freqs, iirfilter >>> b, a = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1') >>> w, h = freqs(b, a, worN=np.logspace(-1, 2, 1000)) >>> import matplotlib.pyplot as plt >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.xlabel('Frequency') >>> plt.ylabel('Amplitude response [dB]') >>> plt.grid() >>> plt.show() """ if worN is None: w = findfreqs(b, a, 200) elif isinstance(worN, int): N = worN w = findfreqs(b, a, N) else: w = worN w = atleast_1d(w) s = 1j * w h = polyval(b, s) / polyval(a, s) if plot is not None: plot(w, h) return w, h def freqz(b, a=1, worN=None, whole=False, plot=None): """ Compute the frequency response of a digital filter. Given the M-order numerator b and N-order denominator a of a digital filter, compute its frequency response:: jw -jw -jwM jw B(e ) b[0] + b[1]e + .... + b[M]e H(e ) = ---- = ----------------------------------- jw -jw -jwN A(e ) a[0] + a[1]e + .... + a[N]e Parameters ---------- b : array_like numerator of a linear filter a : array_like denominator of a linear filter worN : {None, int, array_like}, optional If None (default), then compute at 512 frequencies equally spaced around the unit circle. If a single integer, then compute at that many frequencies. If an array_like, compute the response at the frequencies given (in radians/sample). whole : bool, optional Normally, frequencies are computed from 0 to the Nyquist frequency, pi radians/sample (upper-half of unit-circle). If whole is True, compute frequencies from 0 to 2*pi radians/sample. plot : callable A callable that takes two arguments. If given, the return parameters w and h are passed to plot. Useful for plotting the frequency response inside freqz. Returns ------- w : ndarray The normalized frequencies at which h was computed, in radians/sample. h : ndarray The frequency response, as complex numbers. See Also -------- sosfreqz Notes ----- Using Matplotlib's "plot" function as the callable for plot produces unexpected results, this plots the real part of the complex transfer function, not the magnitude. Try lambda w, h: plot(w, abs(h)). Examples -------- >>> from scipy import signal >>> b = signal.firwin(80, 0.5, window=('kaiser', 8)) >>> w, h = signal.freqz(b) >>> import matplotlib.pyplot as plt >>> fig = plt.figure() >>> plt.title('Digital filter frequency response') >>> ax1 = fig.add_subplot(111) >>> plt.plot(w, 20 * np.log10(abs(h)), 'b') >>> plt.ylabel('Amplitude [dB]', color='b') >>> plt.xlabel('Frequency [rad/sample]') >>> ax2 = ax1.twinx() >>> angles = np.unwrap(np.angle(h)) >>> plt.plot(w, angles, 'g') >>> plt.ylabel('Angle (radians)', color='g') >>> plt.grid() >>> plt.axis('tight') >>> plt.show() """ b, a = map(atleast_1d, (b, a)) if whole: lastpoint = 2 * pi else: lastpoint = pi if worN is None: N = 512 w = numpy.linspace(0, lastpoint, N, endpoint=False) elif isinstance(worN, int): N = worN w = numpy.linspace(0, lastpoint, N, endpoint=False) else: w = worN w = atleast_1d(w) zm1 = exp(-1j * w) h = polyval(b[::-1], zm1) / polyval(a[::-1], zm1) if plot is not None: plot(w, h) return w, h def group_delay(system, w=None, whole=False): r"""Compute the group delay of a digital filter. The group delay measures by how many samples amplitude envelopes of various spectral components of a signal are delayed by a filter. It is formally defined as the derivative of continuous (unwrapped) phase:: d jw D(w) = - -- arg H(e) dw Parameters ---------- system : tuple of array_like (b, a) Numerator and denominator coefficients of a filter transfer function. w : {None, int, array-like}, optional If None (default), then compute at 512 frequencies equally spaced around the unit circle. If a single integer, then compute at that many frequencies. If array, compute the delay at the frequencies given (in radians/sample). whole : bool, optional Normally, frequencies are computed from 0 to the Nyquist frequency, pi radians/sample (upper-half of unit-circle). If whole is True, compute frequencies from 0 to 2*pi radians/sample. Returns ------- w : ndarray The normalized frequencies at which the group delay was computed, in radians/sample. gd : ndarray The group delay. Notes ----- The similar function in MATLAB is called grpdelay. If the transfer function :math:H(z) has zeros or poles on the unit circle, the group delay at corresponding frequencies is undefined. When such a case arises the warning is raised and the group delay is set to 0 at those frequencies. For the details of numerical computation of the group delay refer to [1]_. .. versionadded: 0.16.0 See Also -------- freqz : Frequency response of a digital filter References ---------- .. [1] Richard G. Lyons, "Understanding Digital Signal Processing, 3rd edition", p. 830. Examples -------- >>> from scipy import signal >>> b, a = signal.iirdesign(0.1, 0.3, 5, 50, ftype='cheby1') >>> w, gd = signal.group_delay((b, a)) >>> import matplotlib.pyplot as plt >>> plt.title('Digital filter group delay') >>> plt.plot(w, gd) >>> plt.ylabel('Group delay [samples]') >>> plt.xlabel('Frequency [rad/sample]') >>> plt.show() """ if w is None: w = 512 if isinstance(w, int): if whole: w = np.linspace(0, 2 * pi, w, endpoint=False) else: w = np.linspace(0, pi, w, endpoint=False) w = np.atleast_1d(w) b, a = map(np.atleast_1d, system) c = np.convolve(b, a[::-1]) cr = c * np.arange(c.size) z = np.exp(-1j * w) num = np.polyval(cr[::-1], z) den = np.polyval(c[::-1], z) singular = np.absolute(den) < 10 * EPSILON if np.any(singular): warnings.warn( "The group delay is singular at frequencies [{0}], setting to 0". format(", ".join("{0:.3f}".format(ws) for ws in w[singular])) ) gd = np.zeros_like(w) gd[~singular] = np.real(num[~singular] / den[~singular]) - a.size + 1 return w, gd def _validate_sos(sos): """Helper to validate a SOS input""" sos = np.atleast_2d(sos) if sos.ndim != 2: raise ValueError('sos array must be 2D') n_sections, m = sos.shape if m != 6: raise ValueError('sos array must be shape (n_sections, 6)') if not (sos[:, 3] == 1).all(): raise ValueError('sos[:, 3] should be all ones') return sos, n_sections def sosfreqz(sos, worN=None, whole=False): """ Compute the frequency response of a digital filter in SOS format. Given sos, an array with shape (n, 6) of second order sections of a digital filter, compute the frequency response of the system function:: B0(z) B1(z) B{n-1}(z) H(z) = ----- * ----- * ... * --------- A0(z) A1(z) A{n-1}(z) for z = exp(omega*1j), where B{k}(z) and A{k}(z) are numerator and denominator of the transfer function of the k-th second order section. Parameters ---------- sos : array_like Array of second-order filter coefficients, must have shape (n_sections, 6). Each row corresponds to a second-order section, with the first three columns providing the numerator coefficients and the last three providing the denominator coefficients. worN : {None, int, array_like}, optional If None (default), then compute at 512 frequencies equally spaced around the unit circle. If a single integer, then compute at that many frequencies. If an array_like, compute the response at the frequencies given (in radians/sample). whole : bool, optional Normally, frequencies are computed from 0 to the Nyquist frequency, pi radians/sample (upper-half of unit-circle). If whole is True, compute frequencies from 0 to 2*pi radians/sample. Returns ------- w : ndarray The normalized frequencies at which h was computed, in radians/sample. h : ndarray The frequency response, as complex numbers. See Also -------- freqz, sosfilt Notes ----- .. versionadded:: 0.19.0 Examples -------- Design a 15th-order bandpass filter in SOS format. >>> from scipy import signal >>> sos = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass', ... output='sos') Compute the frequency response at 1500 points from DC to Nyquist. >>> w, h = signal.sosfreqz(sos, worN=1500) Plot the response. >>> import matplotlib.pyplot as plt >>> plt.subplot(2, 1, 1) >>> db = 20*np.log10(np.abs(h)) >>> plt.plot(w/np.pi, db) >>> plt.ylim(-75, 5) >>> plt.grid(True) >>> plt.yticks([0, -20, -40, -60]) >>> plt.ylabel('Gain [dB]') >>> plt.title('Frequency Response') >>> plt.subplot(2, 1, 2) >>> plt.plot(w/np.pi, np.angle(h)) >>> plt.grid(True) >>> plt.yticks([-np.pi, -0.5*np.pi, 0, 0.5*np.pi, np.pi], ... ['$-\pi$', '$-\pi/2$', '0', '$\pi/2$', '$\pi$']) >>> plt.ylabel('Phase [rad]') >>> plt.xlabel('Normalized frequency (1.0 = Nyquist)') >>> plt.show() If the same filter is implemented as a single transfer function, numerical error corrupts the frequency response: >>> b, a = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass', ... output='ba') >>> w, h = signal.freqz(b, a, worN=1500) >>> plt.subplot(2, 1, 1) >>> db = 20*np.log10(np.abs(h)) >>> plt.plot(w/np.pi, db) >>> plt.subplot(2, 1, 2) >>> plt.plot(w/np.pi, np.angle(h)) >>> plt.show() """ sos, n_sections = _validate_sos(sos) if n_sections == 0: raise ValueError('Cannot compute frequencies with no sections') h = 1. for row in sos: w, rowh = freqz(row[:3], row[3:], worN=worN, whole=whole) h *= rowh return w, h def _cplxreal(z, tol=None): """ Split into complex and real parts, combining conjugate pairs. The 1D input vector z is split up into its complex (zc) and real (zr) elements. Every complex element must be part of a complex-conjugate pair, which are combined into a single number (with positive imaginary part) in the output. Two complex numbers are considered a conjugate pair if their real and imaginary parts differ in magnitude by less than tol * abs(z). Parameters ---------- z : array_like Vector of complex numbers to be sorted and split tol : float, optional Relative tolerance for testing realness and conjugate equality. Default is 100 * spacing(1) of z's data type (i.e. 2e-14 for float64) Returns ------- zc : ndarray Complex elements of z, with each pair represented by a single value having positive imaginary part, sorted first by real part, and then by magnitude of imaginary part. The pairs are averaged when combined to reduce error. zr : ndarray Real elements of z (those having imaginary part less than tol times their magnitude), sorted by value. Raises ------ ValueError If there are any complex numbers in z for which a conjugate cannot be found. See Also -------- _cplxpair Examples -------- >>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j] >>> zc, zr = _cplxreal(a) >>> print zc [ 1.+1.j 2.+1.j 2.+1.j 2.+2.j] >>> print zr [ 1. 3. 4.] """ z = atleast_1d(z) if z.size == 0: return z, z elif z.ndim != 1: raise ValueError('_cplxreal only accepts 1D input') if tol is None: # Get tolerance from dtype of input tol = 100 * np.finfo((1.0 * z).dtype).eps # Sort by real part, magnitude of imaginary part (speed up further sorting) z = z[np.lexsort((abs(z.imag), z.real))] # Split reals from conjugate pairs real_indices = abs(z.imag) <= tol * abs(z) zr = z[real_indices].real if len(zr) == len(z): # Input is entirely real return array([]), zr # Split positive and negative halves of conjugates z = z[~real_indices] zp = z[z.imag > 0] zn = z[z.imag < 0] if len(zp) != len(zn): raise ValueError('Array contains complex value with no matching ' 'conjugate.') # Find runs of (approximately) the same real part same_real = np.diff(zp.real) <= tol * abs(zp[:-1]) diffs = numpy.diff(concatenate(([0], same_real, [0]))) run_starts = numpy.where(diffs > 0)[0] run_stops = numpy.where(diffs < 0)[0] # Sort each run by their imaginary parts for i in range(len(run_starts)): start = run_starts[i] stop = run_stops[i] + 1 for chunk in (zp[start:stop], zn[start:stop]): chunk[...] = chunk[np.lexsort([abs(chunk.imag)])] # Check that negatives match positives if any(abs(zp - zn.conj()) > tol * abs(zn)): raise ValueError('Array contains complex value with no matching ' 'conjugate.') # Average out numerical inaccuracy in real vs imag parts of pairs zc = (zp + zn.conj()) / 2 return zc, zr def _cplxpair(z, tol=None): """ Sort into pairs of complex conjugates. Complex conjugates in z are sorted by increasing real part. In each pair, the number with negative imaginary part appears first. If pairs have identical real parts, they are sorted by increasing imaginary magnitude. Two complex numbers are considered a conjugate pair if their real and imaginary parts differ in magnitude by less than tol * abs(z). The pairs are forced to be exact complex conjugates by averaging the positive and negative values. Purely real numbers are also sorted, but placed after the complex conjugate pairs. A number is considered real if its imaginary part is smaller than tol times the magnitude of the number. Parameters ---------- z : array_like 1-dimensional input array to be sorted. tol : float, optional Relative tolerance for testing realness and conjugate equality. Default is 100 * spacing(1) of z's data type (i.e. 2e-14 for float64) Returns ------- y : ndarray Complex conjugate pairs followed by real numbers. Raises ------ ValueError If there are any complex numbers in z for which a conjugate cannot be found. See Also -------- _cplxreal Examples -------- >>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j] >>> z = _cplxpair(a) >>> print(z) [ 1.-1.j 1.+1.j 2.-1.j 2.+1.j 2.-1.j 2.+1.j 2.-2.j 2.+2.j 1.+0.j 3.+0.j 4.+0.j] """ z = atleast_1d(z) if z.size == 0 or np.isrealobj(z): return np.sort(z) if z.ndim != 1: raise ValueError('z must be 1-dimensional') zc, zr = _cplxreal(z, tol) # Interleave complex values and their conjugates, with negative imaginary # parts first in each pair zc = np.dstack((zc.conj(), zc)).flatten() z = np.append(zc, zr) return z def tf2zpk(b, a): r"""Return zero, pole, gain (z, p, k) representation from a numerator, denominator representation of a linear filter. Parameters ---------- b : array_like Numerator polynomial coefficients. a : array_like Denominator polynomial coefficients. Returns ------- z : ndarray Zeros of the transfer function. p : ndarray Poles of the transfer function. k : float System gain. Notes ----- If some values of b are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted. The b and a arrays are interpreted as coefficients for positive, descending powers of the transfer function variable. So the inputs :math:b = [b_0, b_1, ..., b_M] and :math:a =[a_0, a_1, ..., a_N] can represent an analog filter of the form: .. math:: H(s) = \frac {b_0 s^M + b_1 s^{(M-1)} + \cdots + b_M} {a_0 s^N + a_1 s^{(N-1)} + \cdots + a_N} or a discrete-time filter of the form: .. math:: H(z) = \frac {b_0 z^M + b_1 z^{(M-1)} + \cdots + b_M} {a_0 z^N + a_1 z^{(N-1)} + \cdots + a_N} This "positive powers" form is found more commonly in controls engineering. If M and N are equal (which is true for all filters generated by the bilinear transform), then this happens to be equivalent to the "negative powers" discrete-time form preferred in DSP: .. math:: H(z) = \frac {b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}} {a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}} Although this is true for common filters, remember that this is not true in the general case. If M and N are not equal, the discrete-time transfer function coefficients must first be converted to the "positive powers" form before finding the poles and zeros. """ b, a = normalize(b, a) b = (b + 0.0) / a[0] a = (a + 0.0) / a[0] k = b[0] b /= b[0] z = roots(b) p = roots(a) return z, p, k def zpk2tf(z, p, k): """ Return polynomial transfer function representation from zeros and poles Parameters ---------- z : array_like Zeros of the transfer function. p : array_like Poles of the transfer function. k : float System gain. Returns ------- b : ndarray Numerator polynomial coefficients. a : ndarray Denominator polynomial coefficients. """ z = atleast_1d(z) k = atleast_1d(k) if len(z.shape) > 1: temp = poly(z[0]) b = zeros((z.shape[0], z.shape[1] + 1), temp.dtype.char) if len(k) == 1: k = [k[0]] * z.shape[0] for i in range(z.shape[0]): b[i] = k[i] * poly(z[i]) else: b = k * poly(z) a = atleast_1d(poly(p)) # Use real output if possible. Copied from numpy.poly, since # we can't depend on a specific version of numpy. if issubclass(b.dtype.type, numpy.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = numpy.asarray(z, complex) pos_roots = numpy.compress(roots.imag > 0, roots) neg_roots = numpy.conjugate(numpy.compress(roots.imag < 0, roots)) if len(pos_roots) == len(neg_roots): if numpy.all(numpy.sort_complex(neg_roots) == numpy.sort_complex(pos_roots)): b = b.real.copy() if issubclass(a.dtype.type, numpy.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = numpy.asarray(p, complex) pos_roots = numpy.compress(roots.imag > 0, roots) neg_roots = numpy.conjugate(numpy.compress(roots.imag < 0, roots)) if len(pos_roots) == len(neg_roots): if numpy.all(numpy.sort_complex(neg_roots) == numpy.sort_complex(pos_roots)): a = a.real.copy() return b, a def tf2sos(b, a, pairing='nearest'): """ Return second-order sections from transfer function representation Parameters ---------- b : array_like Numerator polynomial coefficients. a : array_like Denominator polynomial coefficients. pairing : {'nearest', 'keep_odd'}, optional The method to use to combine pairs of poles and zeros into sections. See zpk2sos. Returns ------- sos : ndarray Array of second-order filter coefficients, with shape (n_sections, 6). See sosfilt for the SOS filter format specification. See Also -------- zpk2sos, sosfilt Notes ----- It is generally discouraged to convert from TF to SOS format, since doing so usually will not improve numerical precision errors. Instead, consider designing filters in ZPK format and converting directly to SOS. TF is converted to SOS by first converting to ZPK format, then converting ZPK to SOS. .. versionadded:: 0.16.0 """ return zpk2sos(*tf2zpk(b, a), pairing=pairing) def sos2tf(sos): """ Return a single transfer function from a series of second-order sections Parameters ---------- sos : array_like Array of second-order filter coefficients, must have shape (n_sections, 6). See sosfilt for the SOS filter format specification. Returns ------- b : ndarray Numerator polynomial coefficients. a : ndarray Denominator polynomial coefficients. Notes ----- .. versionadded:: 0.16.0 """ sos = np.asarray(sos) b = [1.] a = [1.] n_sections = sos.shape[0] for section in range(n_sections): b = np.polymul(b, sos[section, :3]) a = np.polymul(a, sos[section, 3:]) return b, a def sos2zpk(sos): """ Return zeros, poles, and gain of a series of second-order sections Parameters ---------- sos : array_like Array of second-order filter coefficients, must have shape (n_sections, 6). See sosfilt for the SOS filter format specification. Returns ------- z : ndarray Zeros of the transfer function. p : ndarray Poles of the transfer function. k : float System gain. Notes ----- .. versionadded:: 0.16.0 """ sos = np.asarray(sos) n_sections = sos.shape[0] z = np.empty(n_sections*2, np.complex128) p = np.empty(n_sections*2, np.complex128) k = 1. for section in range(n_sections): zpk = tf2zpk(sos[section, :3], sos[section, 3:]) z[2*section:2*(section+1)] = zpk[0] p[2*section:2*(section+1)] = zpk[1] k *= zpk[2] return z, p, k def _nearest_real_complex_idx(fro, to, which): """Get the next closest real or complex element based on distance""" assert which in ('real', 'complex') order = np.argsort(np.abs(fro - to)) mask = np.isreal(fro[order]) if which == 'complex': mask = ~mask return order[np.where(mask)[0][0]] def zpk2sos(z, p, k, pairing='nearest'): """ Return second-order sections from zeros, poles, and gain of a system Parameters ---------- z : array_like Zeros of the transfer function. p : array_like Poles of the transfer function. k : float System gain. pairing : {'nearest', 'keep_odd'}, optional The method to use to combine pairs of poles and zeros into sections. See Notes below. Returns ------- sos : ndarray Array of second-order filter coefficients, with shape (n_sections, 6). See sosfilt for the SOS filter format specification. See Also -------- sosfilt Notes ----- The algorithm used to convert ZPK to SOS format is designed to minimize errors due to numerical precision issues. The pairing algorithm attempts to minimize the peak gain of each biquadratic section. This is done by pairing poles with the nearest zeros, starting with the poles closest to the unit circle. *Algorithms* The current algorithms are designed specifically for use with digital filters. (The output coefficents are not correct for analog filters.) The steps in the pairing='nearest' and pairing='keep_odd' algorithms are mostly shared. The nearest algorithm attempts to minimize the peak gain, while 'keep_odd' minimizes peak gain under the constraint that odd-order systems should retain one section as first order. The algorithm steps and are as follows: As a pre-processing step, add poles or zeros to the origin as necessary to obtain the same number of poles and zeros for pairing. If pairing == 'nearest' and there are an odd number of poles, add an additional pole and a zero at the origin. The following steps are then iterated over until no more poles or zeros remain: 1. Take the (next remaining) pole (complex or real) closest to the unit circle to begin a new filter section. 2. If the pole is real and there are no other remaining real poles [#]_, add the closest real zero to the section and leave it as a first order section. Note that after this step we are guaranteed to be left with an even number of real poles, complex poles, real zeros, and complex zeros for subsequent pairing iterations. 3. Else: 1. If the pole is complex and the zero is the only remaining real zero*, then pair the pole with the *next* closest zero (guaranteed to be complex). This is necessary to ensure that there will be a real zero remaining to eventually create a first-order section (thus keeping the odd order). 2. Else pair the pole with the closest remaining zero (complex or real). 3. Proceed to complete the second-order section by adding another pole and zero to the current pole and zero in the section: 1. If the current pole and zero are both complex, add their conjugates. 2. Else if the pole is complex and the zero is real, add the conjugate pole and the next closest real zero. 3. Else if the pole is real and the zero is complex, add the conjugate zero and the real pole closest to those zeros. 4. Else (we must have a real pole and real zero) add the next real pole closest to the unit circle, and then add the real zero closest to that pole. .. [#] This conditional can only be met for specific odd-order inputs with the pairing == 'keep_odd' method. .. versionadded:: 0.16.0 Examples -------- Design a 6th order low-pass elliptic digital filter for a system with a sampling rate of 8000 Hz that has a pass-band corner frequency of 1000 Hz. The ripple in the pass-band should not exceed 0.087 dB, and the attenuation in the stop-band should be at least 90 dB. In the following call to signal.ellip, we could use output='sos', but for this example, we'll use output='zpk', and then convert to SOS format with zpk2sos: >>> from scipy import signal >>> z, p, k = signal.ellip(6, 0.087, 90, 1000/(0.5*8000), output='zpk') Now convert to SOS format. >>> sos = signal.zpk2sos(z, p, k) The coefficients of the numerators of the sections: >>> sos[:, :3] array([[ 0.0014154 , 0.00248707, 0.0014154 ], [ 1. , 0.72965193, 1. ], [ 1. , 0.17594966, 1. ]]) The symmetry in the coefficients occurs because all the zeros are on the unit circle. The coefficients of the denominators of the sections: >>> sos[:, 3:] array([[ 1. , -1.32543251, 0.46989499], [ 1. , -1.26117915, 0.6262586 ], [ 1. , -1.25707217, 0.86199667]]) The next example shows the effect of the pairing option. We have a system with three poles and three zeros, so the SOS array will have shape (2, 6). The means there is, in effect, an extra pole and an extra zero at the origin in the SOS representation. >>> z1 = np.array([-1, -0.5-0.5j, -0.5+0.5j]) >>> p1 = np.array([0.75, 0.8+0.1j, 0.8-0.1j]) With pairing='nearest' (the default), we obtain >>> signal.zpk2sos(z1, p1, 1) array([[ 1. , 1. , 0.5 , 1. , -0.75, 0. ], [ 1. , 1. , 0. , 1. , -1.6 , 0.65]]) The first section has the zeros {-0.5-0.05j, -0.5+0.5j} and the poles {0, 0.75}, and the second section has the zeros {-1, 0} and poles {0.8+0.1j, 0.8-0.1j}. Note that the extra pole and zero at the origin have been assigned to different sections. With pairing='keep_odd', we obtain: >>> signal.zpk2sos(z1, p1, 1, pairing='keep_odd') array([[ 1. , 1. , 0. , 1. , -0.75, 0. ], [ 1. , 1. , 0.5 , 1. , -1.6 , 0.65]]) The extra pole and zero at the origin are in the same section. The first section is, in effect, a first-order section. """ # TODO in the near future: # 1. Add SOS capability to filtfilt, freqz, etc. somehow (#3259). # 2. Make decimate use sosfilt instead of lfilter. # 3. Make sosfilt automatically simplify sections to first order # when possible. Note this might make sosfiltfilt a bit harder (ICs). # 4. Further optimizations of the section ordering / pole-zero pairing. # See the wiki for other potential issues. valid_pairings = ['nearest', 'keep_odd'] if pairing not in valid_pairings: raise ValueError('pairing must be one of %s, not %s' % (valid_pairings, pairing)) if len(z) == len(p) == 0: return array([[k, 0., 0., 1., 0., 0.]]) # ensure we have the same number of poles and zeros, and make copies p = np.concatenate((p, np.zeros(max(len(z) - len(p), 0)))) z = np.concatenate((z, np.zeros(max(len(p) - len(z), 0)))) n_sections = (max(len(p), len(z)) + 1) // 2 sos = zeros((n_sections, 6)) if len(p) % 2 == 1 and pairing == 'nearest': p = np.concatenate((p, [0.])) z = np.concatenate((z, [0.])) assert len(p) == len(z) # Ensure we have complex conjugate pairs # (note that _cplxreal only gives us one element of each complex pair): z = np.concatenate(_cplxreal(z)) p = np.concatenate(_cplxreal(p)) p_sos = np.zeros((n_sections, 2), np.complex128) z_sos = np.zeros_like(p_sos) for si in range(n_sections): # Select the next "worst" pole p1_idx = np.argmin(np.abs(1 - np.abs(p))) p1 = p[p1_idx] p = np.delete(p, p1_idx) # Pair that pole with a zero if np.isreal(p1) and np.isreal(p).sum() == 0: # Special case to set a first-order section z1_idx = _nearest_real_complex_idx(z, p1, 'real') z1 = z[z1_idx] z = np.delete(z, z1_idx) p2 = z2 = 0 else: if not np.isreal(p1) and np.isreal(z).sum() == 1: # Special case to ensure we choose a complex zero to pair # with so later (setting up a first-order section) z1_idx = _nearest_real_complex_idx(z, p1, 'complex') assert not np.isreal(z[z1_idx]) else: # Pair the pole with the closest zero (real or complex) z1_idx = np.argmin(np.abs(p1 - z)) z1 = z[z1_idx] z = np.delete(z, z1_idx) # Now that we have p1 and z1, figure out what p2 and z2 need to be if not np.isreal(p1): if not np.isreal(z1): # complex pole, complex zero p2 = p1.conj() z2 = z1.conj() else: # complex pole, real zero p2 = p1.conj() z2_idx = _nearest_real_complex_idx(z, p1, 'real') z2 = z[z2_idx] assert np.isreal(z2) z = np.delete(z, z2_idx) else: if not np.isreal(z1): # real pole, complex zero z2 = z1.conj() p2_idx = _nearest_real_complex_idx(p, z1, 'real') p2 = p[p2_idx] assert np.isreal(p2) else: # real pole, real zero # pick the next "worst" pole to use idx = np.where(np.isreal(p))[0] assert len(idx) > 0 p2_idx = idx[np.argmin(np.abs(np.abs(p[idx]) - 1))] p2 = p[p2_idx] # find a real zero to match the added pole assert np.isreal(p2) z2_idx = _nearest_real_complex_idx(z, p2, 'real') z2 = z[z2_idx] assert np.isreal(z2) z = np.delete(z, z2_idx) p = np.delete(p, p2_idx) p_sos[si] = [p1, p2] z_sos[si] = [z1, z2] assert len(p) == len(z) == 0 # we've consumed all poles and zeros del p, z # Construct the system, reversing order so the "worst" are last p_sos = np.reshape(p_sos[::-1], (n_sections, 2)) z_sos = np.reshape(z_sos[::-1], (n_sections, 2)) gains = np.ones(n_sections) gains[0] = k for si in range(n_sections): x = zpk2tf(z_sos[si], p_sos[si], gains[si]) sos[si] = np.concatenate(x) return sos def _align_nums(nums): """Aligns the shapes of multiple numerators. Given an array of numerator coefficient arrays [[a_1, a_2,..., a_n],..., [b_1, b_2,..., b_m]], this function pads shorter numerator arrays with zero's so that all numerators have the same length. Such alignment is necessary for functions like 'tf2ss', which needs the alignment when dealing with SIMO transfer functions. Parameters ---------- nums: array_like Numerator or list of numerators. Not necessarily with same length. Returns ------- nums: array The numerator. If nums input was a list of numerators then a 2d array with padded zeros for shorter numerators is returned. Otherwise returns np.asarray(nums). """ try: # The statement can throw a ValueError if one # of the numerators is a single digit and another # is array-like e.g. if nums = [5, [1, 2, 3]] nums = asarray(nums) if not np.issubdtype(nums.dtype, np.number): raise ValueError("dtype of numerator is non-numeric") return nums except ValueError: nums = [np.atleast_1d(num) for num in nums] max_width = max(num.size for num in nums) # pre-allocate aligned_nums = np.zeros((len(nums), max_width)) # Create numerators with padded zeros for index, num in enumerate(nums): aligned_nums[index, -num.size:] = num return aligned_nums def normalize(b, a): """Normalize numerator/denominator of a continuous-time transfer function. If values of b are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted. Parameters ---------- b: array_like Numerator of the transfer function. Can be a 2d array to normalize multiple transfer functions. a: array_like Denominator of the transfer function. At most 1d. Returns ------- num: array The numerator of the normalized transfer function. At least a 1d array. A 2d-array if the input num is a 2d array. den: 1d-array The denominator of the normalized transfer function. Notes ----- Coefficients for both the numerator and denominator should be specified in descending exponent order (e.g., s^2 + 3s + 5 would be represented as [1, 3, 5]). """ num, den = b, a den = np.atleast_1d(den) num = np.atleast_2d(_align_nums(num)) if den.ndim != 1: raise ValueError("Denominator polynomial must be rank-1 array.") if num.ndim > 2: raise ValueError("Numerator polynomial must be rank-1 or" " rank-2 array.") if np.all(den == 0): raise ValueError("Denominator must have at least on nonzero element.") # Trim leading zeros in denominator, leave at least one. den = np.trim_zeros(den, 'f') # Normalize transfer function num, den = num / den[0], den / den[0] # Count numerator columns that are all zero leading_zeros = 0 for col in num.T: if np.allclose(col, 0, atol=1e-14): leading_zeros += 1 else: break # Trim leading zeros of numerator if leading_zeros > 0: warnings.warn("Badly conditioned filter coefficients (numerator): the " "results may be meaningless", BadCoefficients) # Make sure at least one column remains if leading_zeros == num.shape[1]: leading_zeros -= 1 num = num[:, leading_zeros:] # Squeeze first dimension if singular if num.shape[0] == 1: num = num[0, :] return num, den def lp2lp(b, a, wo=1.0): """ Transform a lowpass filter prototype to a different frequency. Return an analog low-pass filter with cutoff frequency wo from an analog low-pass filter prototype with unity cutoff frequency, in transfer function ('ba') representation. """ a, b = map(atleast_1d, (a, b)) try: wo = float(wo) except TypeError: wo = float(wo[0]) d = len(a) n = len(b) M = max((d, n)) pwo = pow(wo, numpy.arange(M - 1, -1, -1)) start1 = max((n - d, 0)) start2 = max((d - n, 0)) b = b * pwo[start1] / pwo[start2:] a = a * pwo[start1] / pwo[start1:] return normalize(b, a) def lp2hp(b, a, wo=1.0): """ Transform a lowpass filter prototype to a highpass filter. Return an analog high-pass filter with cutoff frequency wo from an analog low-pass filter prototype with unity cutoff frequency, in transfer function ('ba') representation. """ a, b = map(atleast_1d, (a, b)) try: wo = float(wo) except TypeError: wo = float(wo[0]) d = len(a) n = len(b) if wo != 1: pwo = pow(wo, numpy.arange(max((d, n)))) else: pwo = numpy.ones(max((d, n)), b.dtype.char) if d >= n: outa = a[::-1] * pwo outb = resize(b, (d,)) outb[n:] = 0.0 outb[:n] = b[::-1] * pwo[:n] else: outb = b[::-1] * pwo outa = resize(a, (n,)) outa[d:] = 0.0 outa[:d] = a[::-1] * pwo[:d] return normalize(outb, outa) def lp2bp(b, a, wo=1.0, bw=1.0): """ Transform a lowpass filter prototype to a bandpass filter. Return an analog band-pass filter with center frequency wo and bandwidth bw from an analog low-pass filter prototype with unity cutoff frequency, in transfer function ('ba') representation. """ a, b = map(atleast_1d, (a, b)) D = len(a) - 1 N = len(b) - 1 artype = mintypecode((a, b)) ma = max([N, D]) Np = N + ma Dp = D + ma bprime = numpy.zeros(Np + 1, artype) aprime = numpy.zeros(Dp + 1, artype) wosq = wo * wo for j in range(Np + 1): val = 0.0 for i in range(0, N + 1): for k in range(0, i + 1): if ma - i + 2 * k == j: val += comb(i, k) * b[N - i] * (wosq) ** (i - k) / bw ** i bprime[Np - j] = val for j in range(Dp + 1): val = 0.0 for i in range(0, D + 1): for k in range(0, i + 1): if ma - i + 2 * k == j: val += comb(i, k) * a[D - i] * (wosq) ** (i - k) / bw ** i aprime[Dp - j] = val return normalize(bprime, aprime) def lp2bs(b, a, wo=1.0, bw=1.0): """ Transform a lowpass filter prototype to a bandstop filter. Return an analog band-stop filter with center frequency wo and bandwidth bw from an analog low-pass filter prototype with unity cutoff frequency, in transfer function ('ba') representation. """ a, b = map(atleast_1d, (a, b)) D = len(a) - 1 N = len(b) - 1 artype = mintypecode((a, b)) M = max([N, D]) Np = M + M Dp = M + M bprime = numpy.zeros(Np + 1, artype) aprime = numpy.zeros(Dp + 1, artype) wosq = wo * wo for j in range(Np + 1): val = 0.0 for i in range(0, N + 1): for k in range(0, M - i + 1): if i + 2 * k == j: val += (comb(M - i, k) * b[N - i] * (wosq) ** (M - i - k) * bw ** i) bprime[Np - j] = val for j in range(Dp + 1): val = 0.0 for i in range(0, D + 1): for k in range(0, M - i + 1): if i + 2 * k == j: val += (comb(M - i, k) * a[D - i] * (wosq) ** (M - i - k) * bw ** i) aprime[Dp - j] = val return normalize(bprime, aprime) def bilinear(b, a, fs=1.0): """Return a digital filter from an analog one using a bilinear transform. The bilinear transform substitutes (z-1) / (z+1) for s. """ fs = float(fs) a, b = map(atleast_1d, (a, b)) D = len(a) - 1 N = len(b) - 1 artype = float M = max([N, D]) Np = M Dp = M bprime = numpy.zeros(Np + 1, artype) aprime = numpy.zeros(Dp + 1, artype) for j in range(Np + 1): val = 0.0 for i in range(N + 1): for k in range(i + 1): for l in range(M - i + 1): if k + l == j: val += (comb(i, k) * comb(M - i, l) * b[N - i] * pow(2 * fs, i) * (-1) ** k) bprime[j] = real(val) for j in range(Dp + 1): val = 0.0 for i in range(D + 1): for k in range(i + 1): for l in range(M - i + 1): if k + l == j: val += (comb(i, k) * comb(M - i, l) * a[D - i] * pow(2 * fs, i) * (-1) ** k) aprime[j] = real(val) return normalize(bprime, aprime) def iirdesign(wp, ws, gpass, gstop, analog=False, ftype='ellip', output='ba'): """Complete IIR digital and analog filter design. Given passband and stopband frequencies and gains, construct an analog or digital IIR filter of minimum order for a given basic type. Return the output in numerator, denominator ('ba'), pole-zero ('zpk') or second order sections ('sos') form. Parameters ---------- wp, ws : float Passband and stopband edge frequencies. For digital filters, these are normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (wp and ws are thus in half-cycles / sample.) For example: - Lowpass: wp = 0.2, ws = 0.3 - Highpass: wp = 0.3, ws = 0.2 - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] For analog filters, wp and ws are angular frequencies (e.g. rad/s). gpass : float The maximum loss in the passband (dB). gstop : float The minimum attenuation in the stopband (dB). analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. ftype : str, optional The type of IIR filter to design: - Butterworth : 'butter' - Chebyshev I : 'cheby1' - Chebyshev II : 'cheby2' - Cauer/elliptic: 'ellip' - Bessel/Thomson: 'bessel' output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. See Also -------- butter : Filter design using order and critical points cheby1, cheby2, ellip, bessel buttord : Find order and critical points from passband and stopband spec cheb1ord, cheb2ord, ellipord iirfilter : General filter design using order and critical frequencies Notes ----- The 'sos' output parameter was added in 0.16.0. """ try: ordfunc = filter_dict[ftype][1] except KeyError: raise ValueError("Invalid IIR filter type: %s" % ftype) except IndexError: raise ValueError(("%s does not have order selection. Use " "iirfilter function.") % ftype) wp = atleast_1d(wp) ws = atleast_1d(ws) band_type = 2 * (len(wp) - 1) band_type += 1 if wp[0] >= ws[0]: band_type += 1 btype = {1: 'lowpass', 2: 'highpass', 3: 'bandstop', 4: 'bandpass'}[band_type] N, Wn = ordfunc(wp, ws, gpass, gstop, analog=analog) return iirfilter(N, Wn, rp=gpass, rs=gstop, analog=analog, btype=btype, ftype=ftype, output=output) def iirfilter(N, Wn, rp=None, rs=None, btype='band', analog=False, ftype='butter', output='ba'): """ IIR digital and analog filter design given order and critical points. Design an Nth-order digital or analog filter and return the filter coefficients. Parameters ---------- N : int The order of the filter. Wn : array_like A scalar or length-2 sequence giving the critical frequencies. For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s). rp : float, optional For Chebyshev and elliptic filters, provides the maximum ripple in the passband. (dB) rs : float, optional For Chebyshev and elliptic filters, provides the minimum attenuation in the stop band. (dB) btype : {'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional The type of filter. Default is 'bandpass'. analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. ftype : str, optional The type of IIR filter to design: - Butterworth : 'butter' - Chebyshev I : 'cheby1' - Chebyshev II : 'cheby2' - Cauer/elliptic: 'ellip' - Bessel/Thomson: 'bessel' output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. See Also -------- butter : Filter design using order and critical points cheby1, cheby2, ellip, bessel buttord : Find order and critical points from passband and stopband spec cheb1ord, cheb2ord, ellipord iirdesign : General filter design using passband and stopband spec Notes ----- The 'sos' output parameter was added in 0.16.0. Examples -------- Generate a 17th-order Chebyshev II bandpass filter and plot the frequency response: >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> b, a = signal.iirfilter(17, [50, 200], rs=60, btype='band', ... analog=True, ftype='cheby2') >>> w, h = signal.freqs(b, a, 1000) >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> ax.semilogx(w, 20 * np.log10(abs(h))) >>> ax.set_title('Chebyshev Type II bandpass frequency response') >>> ax.set_xlabel('Frequency [radians / second]') >>> ax.set_ylabel('Amplitude [dB]') >>> ax.axis((10, 1000, -100, 10)) >>> ax.grid(which='both', axis='both') >>> plt.show() """ ftype, btype, output = [x.lower() for x in (ftype, btype, output)] Wn = asarray(Wn) try: btype = band_dict[btype] except KeyError: raise ValueError("'%s' is an invalid bandtype for filter." % btype) try: typefunc = filter_dict[ftype][0] except KeyError: raise ValueError("'%s' is not a valid basic IIR filter." % ftype) if output not in ['ba', 'zpk', 'sos']: raise ValueError("'%s' is not a valid output form." % output) if rp is not None and rp < 0: raise ValueError("passband ripple (rp) must be positive") if rs is not None and rs < 0: raise ValueError("stopband attenuation (rs) must be positive") # Get analog lowpass prototype if typefunc == buttap: z, p, k = typefunc(N) elif typefunc == besselap: z, p, k = typefunc(N, norm=bessel_norms[ftype]) elif typefunc == cheb1ap: if rp is None: raise ValueError("passband ripple (rp) must be provided to " "design a Chebyshev I filter.") z, p, k = typefunc(N, rp) elif typefunc == cheb2ap: if rs is None: raise ValueError("stopband attenuation (rs) must be provided to " "design an Chebyshev II filter.") z, p, k = typefunc(N, rs) elif typefunc == ellipap: if rs is None or rp is None: raise ValueError("Both rp and rs must be provided to design an " "elliptic filter.") z, p, k = typefunc(N, rp, rs) else: raise NotImplementedError("'%s' not implemented in iirfilter." % ftype) # Pre-warp frequencies for digital filter design if not analog: if numpy.any(Wn < 0) or numpy.any(Wn > 1): raise ValueError("Digital filter critical frequencies " "must be 0 <= Wn <= 1") fs = 2.0 warped = 2 * fs * tan(pi * Wn / fs) else: warped = Wn # transform to lowpass, bandpass, highpass, or bandstop if btype in ('lowpass', 'highpass'): if numpy.size(Wn) != 1: raise ValueError('Must specify a single critical frequency Wn') if btype == 'lowpass': z, p, k = _zpklp2lp(z, p, k, wo=warped) elif btype == 'highpass': z, p, k = _zpklp2hp(z, p, k, wo=warped) elif btype in ('bandpass', 'bandstop'): try: bw = warped[1] - warped[0] wo = sqrt(warped[0] * warped[1]) except IndexError: raise ValueError('Wn must specify start and stop frequencies') if btype == 'bandpass': z, p, k = _zpklp2bp(z, p, k, wo=wo, bw=bw) elif btype == 'bandstop': z, p, k = _zpklp2bs(z, p, k, wo=wo, bw=bw) else: raise NotImplementedError("'%s' not implemented in iirfilter." % btype) # Find discrete equivalent if necessary if not analog: z, p, k = _zpkbilinear(z, p, k, fs=fs) # Transform to proper out type (pole-zero, state-space, numer-denom) if output == 'zpk': return z, p, k elif output == 'ba': return zpk2tf(z, p, k) elif output == 'sos': return zpk2sos(z, p, k) def _relative_degree(z, p): """ Return relative degree of transfer function from zeros and poles """ degree = len(p) - len(z) if degree < 0: raise ValueError("Improper transfer function. " "Must have at least as many poles as zeros.") else: return degree # TODO: merge these into existing functions or make public versions def _zpkbilinear(z, p, k, fs): """ Return a digital filter from an analog one using a bilinear transform. Transform a set of poles and zeros from the analog s-plane to the digital z-plane using Tustin's method, which substitutes (z-1) / (z+1) for s, maintaining the shape of the frequency response. Parameters ---------- z : array_like Zeros of the analog IIR filter transfer function. p : array_like Poles of the analog IIR filter transfer function. k : float System gain of the analog IIR filter transfer function. fs : float Sample rate, as ordinary frequency (e.g. hertz). No prewarping is done in this function. Returns ------- z : ndarray Zeros of the transformed digital filter transfer function. p : ndarray Poles of the transformed digital filter transfer function. k : float System gain of the transformed digital filter. """ z = atleast_1d(z) p = atleast_1d(p) degree = _relative_degree(z, p) fs2 = 2*fs # Bilinear transform the poles and zeros z_z = (fs2 + z) / (fs2 - z) p_z = (fs2 + p) / (fs2 - p) # Any zeros that were at infinity get moved to the Nyquist frequency z_z = append(z_z, -ones(degree)) # Compensate for gain change k_z = k * real(prod(fs2 - z) / prod(fs2 - p)) return z_z, p_z, k_z def _zpklp2lp(z, p, k, wo=1.0): r""" Transform a lowpass filter prototype to a different frequency. Return an analog low-pass filter with cutoff frequency wo from an analog low-pass filter prototype with unity cutoff frequency, using zeros, poles, and gain ('zpk') representation. Parameters ---------- z : array_like Zeros of the analog IIR filter transfer function. p : array_like Poles of the analog IIR filter transfer function. k : float System gain of the analog IIR filter transfer function. wo : float Desired cutoff, as angular frequency (e.g. rad/s). Defaults to no change. Returns ------- z : ndarray Zeros of the transformed low-pass filter transfer function. p : ndarray Poles of the transformed low-pass filter transfer function. k : float System gain of the transformed low-pass filter. Notes ----- This is derived from the s-plane substitution .. math:: s \rightarrow \frac{s}{\omega_0} """ z = atleast_1d(z) p = atleast_1d(p) wo = float(wo) # Avoid int wraparound degree = _relative_degree(z, p) # Scale all points radially from origin to shift cutoff frequency z_lp = wo * z p_lp = wo * p # Each shifted pole decreases gain by wo, each shifted zero increases it. # Cancel out the net change to keep overall gain the same k_lp = k * wo**degree return z_lp, p_lp, k_lp def _zpklp2hp(z, p, k, wo=1.0): r""" Transform a lowpass filter prototype to a highpass filter. Return an analog high-pass filter with cutoff frequency wo from an analog low-pass filter prototype with unity cutoff frequency, using zeros, poles, and gain ('zpk') representation. Parameters ---------- z : array_like Zeros of the analog IIR filter transfer function. p : array_like Poles of the analog IIR filter transfer function. k : float System gain of the analog IIR filter transfer function. wo : float Desired cutoff, as angular frequency (e.g. rad/s). Defaults to no change. Returns ------- z : ndarray Zeros of the transformed high-pass filter transfer function. p : ndarray Poles of the transformed high-pass filter transfer function. k : float System gain of the transformed high-pass filter. Notes ----- This is derived from the s-plane substitution .. math:: s \rightarrow \frac{\omega_0}{s} This maintains symmetry of the lowpass and highpass responses on a logarithmic scale. """ z = atleast_1d(z) p = atleast_1d(p) wo = float(wo) degree = _relative_degree(z, p) # Invert positions radially about unit circle to convert LPF to HPF # Scale all points radially from origin to shift cutoff frequency z_hp = wo / z p_hp = wo / p # If lowpass had zeros at infinity, inverting moves them to origin. z_hp = append(z_hp, zeros(degree)) # Cancel out gain change caused by inversion k_hp = k * real(prod(-z) / prod(-p)) return z_hp, p_hp, k_hp def _zpklp2bp(z, p, k, wo=1.0, bw=1.0): r""" Transform a lowpass filter prototype to a bandpass filter. Return an analog band-pass filter with center frequency wo and bandwidth bw from an analog low-pass filter prototype with unity cutoff frequency, using zeros, poles, and gain ('zpk') representation. Parameters ---------- z : array_like Zeros of the analog IIR filter transfer function. p : array_like Poles of the analog IIR filter transfer function. k : float System gain of the analog IIR filter transfer function. wo : float Desired passband center, as angular frequency (e.g. rad/s). Defaults to no change. bw : float Desired passband width, as angular frequency (e.g. rad/s). Defaults to 1. Returns ------- z : ndarray Zeros of the transformed band-pass filter transfer function. p : ndarray Poles of the transformed band-pass filter transfer function. k : float System gain of the transformed band-pass filter. Notes ----- This is derived from the s-plane substitution .. math:: s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}} This is the "wideband" transformation, producing a passband with geometric (log frequency) symmetry about wo. """ z = atleast_1d(z) p = atleast_1d(p) wo = float(wo) bw = float(bw) degree = _relative_degree(z, p) # Scale poles and zeros to desired bandwidth z_lp = z * bw/2 p_lp = p * bw/2 # Square root needs to produce complex result, not NaN z_lp = z_lp.astype(complex) p_lp = p_lp.astype(complex) # Duplicate poles and zeros and shift from baseband to +wo and -wo z_bp = concatenate((z_lp + sqrt(z_lp**2 - wo**2), z_lp - sqrt(z_lp**2 - wo**2))) p_bp = concatenate((p_lp + sqrt(p_lp**2 - wo**2), p_lp - sqrt(p_lp**2 - wo**2))) # Move degree zeros to origin, leaving degree zeros at infinity for BPF z_bp = append(z_bp, zeros(degree)) # Cancel out gain change from frequency scaling k_bp = k * bw**degree return z_bp, p_bp, k_bp def _zpklp2bs(z, p, k, wo=1.0, bw=1.0): r""" Transform a lowpass filter prototype to a bandstop filter. Return an analog band-stop filter with center frequency wo and stopband width bw from an analog low-pass filter prototype with unity cutoff frequency, using zeros, poles, and gain ('zpk') representation. Parameters ---------- z : array_like Zeros of the analog IIR filter transfer function. p : array_like Poles of the analog IIR filter transfer function. k : float System gain of the analog IIR filter transfer function. wo : float Desired stopband center, as angular frequency (e.g. rad/s). Defaults to no change. bw : float Desired stopband width, as angular frequency (e.g. rad/s). Defaults to 1. Returns ------- z : ndarray Zeros of the transformed band-stop filter transfer function. p : ndarray Poles of the transformed band-stop filter transfer function. k : float System gain of the transformed band-stop filter. Notes ----- This is derived from the s-plane substitution .. math:: s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2} This is the "wideband" transformation, producing a stopband with geometric (log frequency) symmetry about wo. """ z = atleast_1d(z) p = atleast_1d(p) wo = float(wo) bw = float(bw) degree = _relative_degree(z, p) # Invert to a highpass filter with desired bandwidth z_hp = (bw/2) / z p_hp = (bw/2) / p # Square root needs to produce complex result, not NaN z_hp = z_hp.astype(complex) p_hp = p_hp.astype(complex) # Duplicate poles and zeros and shift from baseband to +wo and -wo z_bs = concatenate((z_hp + sqrt(z_hp**2 - wo**2), z_hp - sqrt(z_hp**2 - wo**2))) p_bs = concatenate((p_hp + sqrt(p_hp**2 - wo**2), p_hp - sqrt(p_hp**2 - wo**2))) # Move any zeros that were at infinity to the center of the stopband z_bs = append(z_bs, +1j*wo * ones(degree)) z_bs = append(z_bs, -1j*wo * ones(degree)) # Cancel out gain change caused by inversion k_bs = k * real(prod(-z) / prod(-p)) return z_bs, p_bs, k_bs def butter(N, Wn, btype='low', analog=False, output='ba'): """ Butterworth digital and analog filter design. Design an Nth-order digital or analog Butterworth filter and return the filter coefficients. Parameters ---------- N : int The order of the filter. Wn : array_like A scalar or length-2 sequence giving the critical frequencies. For a Butterworth filter, this is the point at which the gain drops to 1/sqrt(2) that of the passband (the "-3 dB point"). For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s). btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional The type of filter. Default is 'lowpass'. analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. See Also -------- buttord, buttap Notes ----- The Butterworth filter has maximally flat frequency response in the passband. The 'sos' output parameter was added in 0.16.0. Examples -------- Plot the filter's frequency response, showing the critical points: >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> b, a = signal.butter(4, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Butterworth filter frequency response') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.show() """ return iirfilter(N, Wn, btype=btype, analog=analog, output=output, ftype='butter') def cheby1(N, rp, Wn, btype='low', analog=False, output='ba'): """ Chebyshev type I digital and analog filter design. Design an Nth-order digital or analog Chebyshev type I filter and return the filter coefficients. Parameters ---------- N : int The order of the filter. rp : float The maximum ripple allowed below unity gain in the passband. Specified in decibels, as a positive number. Wn : array_like A scalar or length-2 sequence giving the critical frequencies. For Type I filters, this is the point in the transition band at which the gain first drops below -rp. For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s). btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional The type of filter. Default is 'lowpass'. analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. See Also -------- cheb1ord, cheb1ap Notes ----- The Chebyshev type I filter maximizes the rate of cutoff between the frequency response's passband and stopband, at the expense of ripple in the passband and increased ringing in the step response. Type I filters roll off faster than Type II (cheby2), but Type II filters do not have any ripple in the passband. The equiripple passband has N maxima or minima (for example, a 5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is unity for odd-order filters, or -rp dB for even-order filters. The 'sos' output parameter was added in 0.16.0. Examples -------- Plot the filter's frequency response, showing the critical points: >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> b, a = signal.cheby1(4, 5, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Chebyshev Type I frequency response (rp=5)') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.axhline(-5, color='green') # rp >>> plt.show() """ return iirfilter(N, Wn, rp=rp, btype=btype, analog=analog, output=output, ftype='cheby1') def cheby2(N, rs, Wn, btype='low', analog=False, output='ba'): """ Chebyshev type II digital and analog filter design. Design an Nth-order digital or analog Chebyshev type II filter and return the filter coefficients. Parameters ---------- N : int The order of the filter. rs : float The minimum attenuation required in the stop band. Specified in decibels, as a positive number. Wn : array_like A scalar or length-2 sequence giving the critical frequencies. For Type II filters, this is the point in the transition band at which the gain first reaches -rs. For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s). btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional The type of filter. Default is 'lowpass'. analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. See Also -------- cheb2ord, cheb2ap Notes ----- The Chebyshev type II filter maximizes the rate of cutoff between the frequency response's passband and stopband, at the expense of ripple in the stopband and increased ringing in the step response. Type II filters do not roll off as fast as Type I (cheby1). The 'sos' output parameter was added in 0.16.0. Examples -------- Plot the filter's frequency response, showing the critical points: >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> b, a = signal.cheby2(4, 40, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Chebyshev Type II frequency response (rs=40)') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.axhline(-40, color='green') # rs >>> plt.show() """ return iirfilter(N, Wn, rs=rs, btype=btype, analog=analog, output=output, ftype='cheby2') def ellip(N, rp, rs, Wn, btype='low', analog=False, output='ba'): """ Elliptic (Cauer) digital and analog filter design. Design an Nth-order digital or analog elliptic filter and return the filter coefficients. Parameters ---------- N : int The order of the filter. rp : float The maximum ripple allowed below unity gain in the passband. Specified in decibels, as a positive number. rs : float The minimum attenuation required in the stop band. Specified in decibels, as a positive number. Wn : array_like A scalar or length-2 sequence giving the critical frequencies. For elliptic filters, this is the point in the transition band at which the gain first drops below -rp. For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s). btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional The type of filter. Default is 'lowpass'. analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. See Also -------- ellipord, ellipap Notes ----- Also known as Cauer or Zolotarev filters, the elliptical filter maximizes the rate of transition between the frequency response's passband and stopband, at the expense of ripple in both, and increased ringing in the step response. As rp approaches 0, the elliptical filter becomes a Chebyshev type II filter (cheby2). As rs approaches 0, it becomes a Chebyshev type I filter (cheby1). As both approach 0, it becomes a Butterworth filter (butter). The equiripple passband has N maxima or minima (for example, a 5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is unity for odd-order filters, or -rp dB for even-order filters. The 'sos' output parameter was added in 0.16.0. Examples -------- Plot the filter's frequency response, showing the critical points: >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> b, a = signal.ellip(4, 5, 40, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Elliptic filter frequency response (rp=5, rs=40)') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.axhline(-40, color='green') # rs >>> plt.axhline(-5, color='green') # rp >>> plt.show() """ return iirfilter(N, Wn, rs=rs, rp=rp, btype=btype, analog=analog, output=output, ftype='elliptic') def bessel(N, Wn, btype='low', analog=False, output='ba', norm='phase'): """ Bessel/Thomson digital and analog filter design. Design an Nth-order digital or analog Bessel filter and return the filter coefficients. Parameters ---------- N : int The order of the filter. Wn : array_like A scalar or length-2 sequence giving the critical frequencies (defined by the norm parameter). For analog filters, Wn is an angular frequency (e.g. rad/s). For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional The type of filter. Default is 'lowpass'. analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. (See Notes.) output : {'ba', 'zpk', 'sos'}, optional Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or second-order sections ('sos'). Default is 'ba'. norm : {'phase', 'delay', 'mag'}, optional Critical frequency normalization: phase The filter is normalized such that the phase response reaches its midpoint at angular (e.g. rad/s) frequency Wn. This happens for both low-pass and high-pass filters, so this is the "phase-matched" case. The magnitude response asymptotes are the same as a Butterworth filter of the same order with a cutoff of Wn. This is the default, and matches MATLAB's implementation. delay The filter is normalized such that the group delay in the passband is 1/Wn (e.g. seconds). This is the "natural" type obtained by solving Bessel polynomials. mag The filter is normalized such that the gain magnitude is -3 dB at angular frequency Wn. .. versionadded:: 0.18.0 Returns ------- b, a : ndarray, ndarray Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if output='ba'. z, p, k : ndarray, ndarray, float Zeros, poles, and system gain of the IIR filter transfer function. Only returned if output='zpk'. sos : ndarray Second-order sections representation of the IIR filter. Only returned if output=='sos'. Notes ----- Also known as a Thomson filter, the analog Bessel filter has maximally flat group delay and maximally linear phase response, with very little ringing in the step response. [1]_ The Bessel is inherently an analog filter. This function generates digital Bessel filters using the bilinear transform, which does not preserve the phase response of the analog filter. As such, it is only approximately correct at frequencies below about fs/4. To get maximally-flat group delay at higher frequencies, the analog Bessel filter must be transformed using phase-preserving techniques. See besselap for implementation details and references. The 'sos' output parameter was added in 0.16.0. Examples -------- Plot the phase-normalized frequency response, showing the relationship to the Butterworth's cutoff frequency (green): >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> b, a = signal.butter(4, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed') >>> b, a = signal.bessel(4, 100, 'low', analog=True, norm='phase') >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(np.abs(h))) >>> plt.title('Bessel filter magnitude response (with Butterworth)') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.show() and the phase midpoint: >>> plt.figure() >>> plt.semilogx(w, np.unwrap(np.angle(h))) >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.axhline(-np.pi, color='red') # phase midpoint >>> plt.title('Bessel filter phase response') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Phase [radians]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.show() Plot the magnitude-normalized frequency response, showing the -3 dB cutoff: >>> b, a = signal.bessel(3, 10, 'low', analog=True, norm='mag') >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(np.abs(h))) >>> plt.axhline(-3, color='red') # -3 dB magnitude >>> plt.axvline(10, color='green') # cutoff frequency >>> plt.title('Magnitude-normalized Bessel filter frequency response') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.show() Plot the delay-normalized filter, showing the maximally-flat group delay at 0.1 seconds: >>> b, a = signal.bessel(5, 1/0.1, 'low', analog=True, norm='delay') >>> w, h = signal.freqs(b, a) >>> plt.figure() >>> plt.semilogx(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w)) >>> plt.axhline(0.1, color='red') # 0.1 seconds group delay >>> plt.title('Bessel filter group delay') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Group delay [seconds]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.show() References ---------- .. [1] Thomson, W.E., "Delay Networks having Maximally Flat Frequency Characteristics", Proceedings of the Institution of Electrical Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490. """ return iirfilter(N, Wn, btype=btype, analog=analog, output=output, ftype='bessel_'+norm) def maxflat(): pass def yulewalk(): pass def band_stop_obj(wp, ind, passb, stopb, gpass, gstop, type): """ Band Stop Objective Function for order minimization. Returns the non-integer order for an analog band stop filter. Parameters ---------- wp : scalar Edge of passband passb. ind : int, {0, 1} Index specifying which passb edge to vary (0 or 1). passb : ndarray Two element sequence of fixed passband edges. stopb : ndarray Two element sequence of fixed stopband edges. gstop : float Amount of attenuation in stopband in dB. gpass : float Amount of ripple in the passband in dB. type : {'butter', 'cheby', 'ellip'} Type of filter. Returns ------- n : scalar Filter order (possibly non-integer). """ passbC = passb.copy() passbC[ind] = wp nat = (stopb * (passbC[0] - passbC[1]) / (stopb ** 2 - passbC[0] * passbC[1])) nat = min(abs(nat)) if type == 'butter': GSTOP = 10 ** (0.1 * abs(gstop)) GPASS = 10 ** (0.1 * abs(gpass)) n = (log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat))) elif type == 'cheby': GSTOP = 10 ** (0.1 * abs(gstop)) GPASS = 10 ** (0.1 * abs(gpass)) n = arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat) elif type == 'ellip': GSTOP = 10 ** (0.1 * gstop) GPASS = 10 ** (0.1 * gpass) arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0)) arg0 = 1.0 / nat d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2]) d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2]) n = (d0[0] * d1[1] / (d0[1] * d1[0])) else: raise ValueError("Incorrect type: %s" % type) return n def buttord(wp, ws, gpass, gstop, analog=False): """Butterworth filter order selection. Return the order of the lowest order digital or analog Butterworth filter that loses no more than gpass dB in the passband and has at least gstop dB attenuation in the stopband. Parameters ---------- wp, ws : float Passband and stopband edge frequencies. For digital filters, these are normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (wp and ws are thus in half-cycles / sample.) For example: - Lowpass: wp = 0.2, ws = 0.3 - Highpass: wp = 0.3, ws = 0.2 - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] For analog filters, wp and ws are angular frequencies (e.g. rad/s). gpass : float The maximum loss in the passband (dB). gstop : float The minimum attenuation in the stopband (dB). analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. Returns ------- ord : int The lowest order for a Butterworth filter which meets specs. wn : ndarray or float The Butterworth natural frequency (i.e. the "3dB frequency"). Should be used with butter to give filter results. See Also -------- butter : Filter design using order and critical points cheb1ord : Find order and critical points from passband and stopband spec cheb2ord, ellipord iirfilter : General filter design using order and critical frequencies iirdesign : General filter design using passband and stopband spec Examples -------- Design an analog bandpass filter with passband within 3 dB from 20 to 50 rad/s, while rejecting at least -40 dB below 14 and above 60 rad/s. Plot its frequency response, showing the passband and stopband constraints in gray. >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> N, Wn = signal.buttord([20, 50], [14, 60], 3, 40, True) >>> b, a = signal.butter(N, Wn, 'band', True) >>> w, h = signal.freqs(b, a, np.logspace(1, 2, 500)) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Butterworth bandpass filter fit to constraints') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.grid(which='both', axis='both') >>> plt.fill([1, 14, 14, 1], [-40, -40, 99, 99], '0.9', lw=0) # stop >>> plt.fill([20, 20, 50, 50], [-99, -3, -3, -99], '0.9', lw=0) # pass >>> plt.fill([60, 60, 1e9, 1e9], [99, -40, -40, 99], '0.9', lw=0) # stop >>> plt.axis([10, 100, -60, 3]) >>> plt.show() """ wp = atleast_1d(wp) ws = atleast_1d(ws) filter_type = 2 * (len(wp) - 1) filter_type += 1 if wp[0] >= ws[0]: filter_type += 1 # Pre-warp frequencies for digital filter design if not analog: passb = tan(pi * wp / 2.0) stopb = tan(pi * ws / 2.0) else: passb = wp * 1.0 stopb = ws * 1.0 if filter_type == 1: # low nat = stopb / passb elif filter_type == 2: # high nat = passb / stopb elif filter_type == 3: # stop wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12, args=(0, passb, stopb, gpass, gstop, 'butter'), disp=0) passb[0] = wp0 wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1], args=(1, passb, stopb, gpass, gstop, 'butter'), disp=0) passb[1] = wp1 nat = ((stopb * (passb[0] - passb[1])) / (stopb ** 2 - passb[0] * passb[1])) elif filter_type == 4: # pass nat = ((stopb ** 2 - passb[0] * passb[1]) / (stopb * (passb[0] - passb[1]))) nat = min(abs(nat)) GSTOP = 10 ** (0.1 * abs(gstop)) GPASS = 10 ** (0.1 * abs(gpass)) ord = int(ceil(log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat)))) # Find the Butterworth natural frequency WN (or the "3dB" frequency") # to give exactly gpass at passb. try: W0 = (GPASS - 1.0) ** (-1.0 / (2.0 * ord)) except ZeroDivisionError: W0 = 1.0 print("Warning, order is zero...check input parameters.") # now convert this frequency back from lowpass prototype # to the original analog filter if filter_type == 1: # low WN = W0 * passb elif filter_type == 2: # high WN = passb / W0 elif filter_type == 3: # stop WN = numpy.zeros(2, float) discr = sqrt((passb[1] - passb[0]) ** 2 + 4 * W0 ** 2 * passb[0] * passb[1]) WN[0] = ((passb[1] - passb[0]) + discr) / (2 * W0) WN[1] = ((passb[1] - passb[0]) - discr) / (2 * W0) WN = numpy.sort(abs(WN)) elif filter_type == 4: # pass W0 = numpy.array([-W0, W0], float) WN = (-W0 * (passb[1] - passb[0]) / 2.0 + sqrt(W0 ** 2 / 4.0 * (passb[1] - passb[0]) ** 2 + passb[0] * passb[1])) WN = numpy.sort(abs(WN)) else: raise ValueError("Bad type: %s" % filter_type) if not analog: wn = (2.0 / pi) * arctan(WN) else: wn = WN if len(wn) == 1: wn = wn[0] return ord, wn def cheb1ord(wp, ws, gpass, gstop, analog=False): """Chebyshev type I filter order selection. Return the order of the lowest order digital or analog Chebyshev Type I filter that loses no more than gpass dB in the passband and has at least gstop dB attenuation in the stopband. Parameters ---------- wp, ws : float Passband and stopband edge frequencies. For digital filters, these are normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (wp and ws are thus in half-cycles / sample.) For example: - Lowpass: wp = 0.2, ws = 0.3 - Highpass: wp = 0.3, ws = 0.2 - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] For analog filters, wp and ws are angular frequencies (e.g. rad/s). gpass : float The maximum loss in the passband (dB). gstop : float The minimum attenuation in the stopband (dB). analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. Returns ------- ord : int The lowest order for a Chebyshev type I filter that meets specs. wn : ndarray or float The Chebyshev natural frequency (the "3dB frequency") for use with cheby1 to give filter results. See Also -------- cheby1 : Filter design using order and critical points buttord : Find order and critical points from passband and stopband spec cheb2ord, ellipord iirfilter : General filter design using order and critical frequencies iirdesign : General filter design using passband and stopband spec Examples -------- Design a digital lowpass filter such that the passband is within 3 dB up to 0.2*(fs/2), while rejecting at least -40 dB above 0.3*(fs/2). Plot its frequency response, showing the passband and stopband constraints in gray. >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> N, Wn = signal.cheb1ord(0.2, 0.3, 3, 40) >>> b, a = signal.cheby1(N, 3, Wn, 'low') >>> w, h = signal.freqz(b, a) >>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h))) >>> plt.title('Chebyshev I lowpass filter fit to constraints') >>> plt.xlabel('Normalized frequency') >>> plt.ylabel('Amplitude [dB]') >>> plt.grid(which='both', axis='both') >>> plt.fill([.01, 0.2, 0.2, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop >>> plt.fill([0.3, 0.3, 2, 2], [ 9, -40, -40, 9], '0.9', lw=0) # pass >>> plt.axis([0.08, 1, -60, 3]) >>> plt.show() """ wp = atleast_1d(wp) ws = atleast_1d(ws) filter_type = 2 * (len(wp) - 1) if wp[0] < ws[0]: filter_type += 1 else: filter_type += 2 # Pre-warp frequencies for digital filter design if not analog: passb = tan(pi * wp / 2.0) stopb = tan(pi * ws / 2.0) else: passb = wp * 1.0 stopb = ws * 1.0 if filter_type == 1: # low nat = stopb / passb elif filter_type == 2: # high nat = passb / stopb elif filter_type == 3: # stop wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12, args=(0, passb, stopb, gpass, gstop, 'cheby'), disp=0) passb[0] = wp0 wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1], args=(1, passb, stopb, gpass, gstop, 'cheby'), disp=0) passb[1] = wp1 nat = ((stopb * (passb[0] - passb[1])) / (stopb ** 2 - passb[0] * passb[1])) elif filter_type == 4: # pass nat = ((stopb ** 2 - passb[0] * passb[1]) / (stopb * (passb[0] - passb[1]))) nat = min(abs(nat)) GSTOP = 10 ** (0.1 * abs(gstop)) GPASS = 10 ** (0.1 * abs(gpass)) ord = int(ceil(arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat))) # Natural frequencies are just the passband edges if not analog: wn = (2.0 / pi) * arctan(passb) else: wn = passb if len(wn) == 1: wn = wn[0] return ord, wn def cheb2ord(wp, ws, gpass, gstop, analog=False): """Chebyshev type II filter order selection. Return the order of the lowest order digital or analog Chebyshev Type II filter that loses no more than gpass dB in the passband and has at least gstop dB attenuation in the stopband. Parameters ---------- wp, ws : float Passband and stopband edge frequencies. For digital filters, these are normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (wp and ws are thus in half-cycles / sample.) For example: - Lowpass: wp = 0.2, ws = 0.3 - Highpass: wp = 0.3, ws = 0.2 - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] For analog filters, wp and ws are angular frequencies (e.g. rad/s). gpass : float The maximum loss in the passband (dB). gstop : float The minimum attenuation in the stopband (dB). analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. Returns ------- ord : int The lowest order for a Chebyshev type II filter that meets specs. wn : ndarray or float The Chebyshev natural frequency (the "3dB frequency") for use with cheby2 to give filter results. See Also -------- cheby2 : Filter design using order and critical points buttord : Find order and critical points from passband and stopband spec cheb1ord, ellipord iirfilter : General filter design using order and critical frequencies iirdesign : General filter design using passband and stopband spec Examples -------- Design a digital bandstop filter which rejects -60 dB from 0.2*(fs/2) to 0.5*(fs/2), while staying within 3 dB below 0.1*(fs/2) or above 0.6*(fs/2). Plot its frequency response, showing the passband and stopband constraints in gray. >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> N, Wn = signal.cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60) >>> b, a = signal.cheby2(N, 60, Wn, 'stop') >>> w, h = signal.freqz(b, a) >>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h))) >>> plt.title('Chebyshev II bandstop filter fit to constraints') >>> plt.xlabel('Normalized frequency') >>> plt.ylabel('Amplitude [dB]') >>> plt.grid(which='both', axis='both') >>> plt.fill([.01, .1, .1, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop >>> plt.fill([.2, .2, .5, .5], [ 9, -60, -60, 9], '0.9', lw=0) # pass >>> plt.fill([.6, .6, 2, 2], [-99, -3, -3, -99], '0.9', lw=0) # stop >>> plt.axis([0.06, 1, -80, 3]) >>> plt.show() """ wp = atleast_1d(wp) ws = atleast_1d(ws) filter_type = 2 * (len(wp) - 1) if wp[0] < ws[0]: filter_type += 1 else: filter_type += 2 # Pre-warp frequencies for digital filter design if not analog: passb = tan(pi * wp / 2.0) stopb = tan(pi * ws / 2.0) else: passb = wp * 1.0 stopb = ws * 1.0 if filter_type == 1: # low nat = stopb / passb elif filter_type == 2: # high nat = passb / stopb elif filter_type == 3: # stop wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12, args=(0, passb, stopb, gpass, gstop, 'cheby'), disp=0) passb[0] = wp0 wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1], args=(1, passb, stopb, gpass, gstop, 'cheby'), disp=0) passb[1] = wp1 nat = ((stopb * (passb[0] - passb[1])) / (stopb ** 2 - passb[0] * passb[1])) elif filter_type == 4: # pass nat = ((stopb ** 2 - passb[0] * passb[1]) / (stopb * (passb[0] - passb[1]))) nat = min(abs(nat)) GSTOP = 10 ** (0.1 * abs(gstop)) GPASS = 10 ** (0.1 * abs(gpass)) ord = int(ceil(arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat))) # Find frequency where analog response is -gpass dB. # Then convert back from low-pass prototype to the original filter. new_freq = cosh(1.0 / ord * arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0)))) new_freq = 1.0 / new_freq if filter_type == 1: nat = passb / new_freq elif filter_type == 2: nat = passb * new_freq elif filter_type == 3: nat = numpy.zeros(2, float) nat[0] = (new_freq / 2.0 * (passb[0] - passb[1]) + sqrt(new_freq ** 2 * (passb[1] - passb[0]) ** 2 / 4.0 + passb[1] * passb[0])) nat[1] = passb[1] * passb[0] / nat[0] elif filter_type == 4: nat = numpy.zeros(2, float) nat[0] = (1.0 / (2.0 * new_freq) * (passb[0] - passb[1]) + sqrt((passb[1] - passb[0]) ** 2 / (4.0 * new_freq ** 2) + passb[1] * passb[0])) nat[1] = passb[0] * passb[1] / nat[0] if not analog: wn = (2.0 / pi) * arctan(nat) else: wn = nat if len(wn) == 1: wn = wn[0] return ord, wn def ellipord(wp, ws, gpass, gstop, analog=False): """Elliptic (Cauer) filter order selection. Return the order of the lowest order digital or analog elliptic filter that loses no more than gpass dB in the passband and has at least gstop dB attenuation in the stopband. Parameters ---------- wp, ws : float Passband and stopband edge frequencies. For digital filters, these are normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (wp and ws are thus in half-cycles / sample.) For example: - Lowpass: wp = 0.2, ws = 0.3 - Highpass: wp = 0.3, ws = 0.2 - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] For analog filters, wp and ws are angular frequencies (e.g. rad/s). gpass : float The maximum loss in the passband (dB). gstop : float The minimum attenuation in the stopband (dB). analog : bool, optional When True, return an analog filter, otherwise a digital filter is returned. Returns ------- ord : int The lowest order for an Elliptic (Cauer) filter that meets specs. wn : ndarray or float The Chebyshev natural frequency (the "3dB frequency") for use with ellip to give filter results. See Also -------- ellip : Filter design using order and critical points buttord : Find order and critical points from passband and stopband spec cheb1ord, cheb2ord iirfilter : General filter design using order and critical frequencies iirdesign : General filter design using passband and stopband spec Examples -------- Design an analog highpass filter such that the passband is within 3 dB above 30 rad/s, while rejecting -60 dB at 10 rad/s. Plot its frequency response, showing the passband and stopband constraints in gray. >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> N, Wn = signal.ellipord(30, 10, 3, 60, True) >>> b, a = signal.ellip(N, 3, 60, Wn, 'high', True) >>> w, h = signal.freqs(b, a, np.logspace(0, 3, 500)) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Elliptical highpass filter fit to constraints') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.grid(which='both', axis='both') >>> plt.fill([.1, 10, 10, .1], [1e4, 1e4, -60, -60], '0.9', lw=0) # stop >>> plt.fill([30, 30, 1e9, 1e9], [-99, -3, -3, -99], '0.9', lw=0) # pass >>> plt.axis([1, 300, -80, 3]) >>> plt.show() """ wp = atleast_1d(wp) ws = atleast_1d(ws) filter_type = 2 * (len(wp) - 1) filter_type += 1 if wp[0] >= ws[0]: filter_type += 1 # Pre-warp frequencies for digital filter design if not analog: passb = tan(pi * wp / 2.0) stopb = tan(pi * ws / 2.0) else: passb = wp * 1.0 stopb = ws * 1.0 if filter_type == 1: # low nat = stopb / passb elif filter_type == 2: # high nat = passb / stopb elif filter_type == 3: # stop wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12, args=(0, passb, stopb, gpass, gstop, 'ellip'), disp=0) passb[0] = wp0 wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1], args=(1, passb, stopb, gpass, gstop, 'ellip'), disp=0) passb[1] = wp1 nat = ((stopb * (passb[0] - passb[1])) / (stopb ** 2 - passb[0] * passb[1])) elif filter_type == 4: # pass nat = ((stopb ** 2 - passb[0] * passb[1]) / (stopb * (passb[0] - passb[1]))) nat = min(abs(nat)) GSTOP = 10 ** (0.1 * gstop) GPASS = 10 ** (0.1 * gpass) arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0)) arg0 = 1.0 / nat d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2]) d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2]) ord = int(ceil(d0[0] * d1[1] / (d0[1] * d1[0]))) if not analog: wn = arctan(passb) * 2.0 / pi else: wn = passb if len(wn) == 1: wn = wn[0] return ord, wn def buttap(N): """Return (z,p,k) for analog prototype of Nth-order Butterworth filter. The filter will have an angular (e.g. rad/s) cutoff frequency of 1. See Also -------- butter : Filter design function using this prototype """ if abs(int(N)) != N: raise ValueError("Filter order must be a nonnegative integer") z = numpy.array([]) m = numpy.arange(-N+1, N, 2) # Middle value is 0 to ensure an exactly real pole p = -numpy.exp(1j * pi * m / (2 * N)) k = 1 return z, p, k def cheb1ap(N, rp): """ Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter. The returned filter prototype has rp decibels of ripple in the passband. The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1, defined as the point at which the gain first drops below -rp. See Also -------- cheby1 : Filter design function using this prototype """ if abs(int(N)) != N: raise ValueError("Filter order must be a nonnegative integer") elif N == 0: # Avoid divide-by-zero error # Even order filters have DC gain of -rp dB return numpy.array([]), numpy.array([]), 10**(-rp/20) z = numpy.array([]) # Ripple factor (epsilon) eps = numpy.sqrt(10 ** (0.1 * rp) - 1.0) mu = 1.0 / N * arcsinh(1 / eps) # Arrange poles in an ellipse on the left half of the S-plane m = numpy.arange(-N+1, N, 2) theta = pi * m / (2*N) p = -sinh(mu + 1j*theta) k = numpy.prod(-p, axis=0).real if N % 2 == 0: k = k / sqrt((1 + eps * eps)) return z, p, k def cheb2ap(N, rs): """ Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter. The returned filter prototype has rs decibels of ripple in the stopband. The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1, defined as the point at which the gain first reaches -rs. See Also -------- cheby2 : Filter design function using this prototype """ if abs(int(N)) != N: raise ValueError("Filter order must be a nonnegative integer") elif N == 0: # Avoid divide-by-zero warning return numpy.array([]), numpy.array([]), 1 # Ripple factor (epsilon) de = 1.0 / sqrt(10 ** (0.1 * rs) - 1) mu = arcsinh(1.0 / de) / N if N % 2: m = numpy.concatenate((numpy.arange(-N+1, 0, 2), numpy.arange(2, N, 2))) else: m = numpy.arange(-N+1, N, 2) z = -conjugate(1j / sin(m * pi / (2.0 * N))) # Poles around the unit circle like Butterworth p = -exp(1j * pi * numpy.arange(-N+1, N, 2) / (2 * N)) # Warp into Chebyshev II p = sinh(mu) * p.real + 1j * cosh(mu) * p.imag p = 1.0 / p k = (numpy.prod(-p, axis=0) / numpy.prod(-z, axis=0)).real return z, p, k EPSILON = 2e-16 def _vratio(u, ineps, mp): [s, c, d, phi] = special.ellipj(u, mp) ret = abs(ineps - s / c) return ret def _kratio(m, k_ratio): m = float(m) if m < 0: m = 0.0 if m > 1: m = 1.0 if abs(m) > EPSILON and (abs(m) + EPSILON) < 1: k = special.ellipk([m, 1 - m]) r = k[0] / k[1] - k_ratio elif abs(m) > EPSILON: r = -k_ratio else: r = 1e20 return abs(r) def ellipap(N, rp, rs): """Return (z,p,k) of Nth-order elliptic analog lowpass filter. The filter is a normalized prototype that has rp decibels of ripple in the passband and a stopband rs decibels down. The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1, defined as the point at which the gain first drops below -rp. See Also -------- ellip : Filter design function using this prototype References ---------- .. [1] Lutova, Tosic, and Evans, "Filter Design for Signal Processing", Chapters 5 and 12. """ if abs(int(N)) != N: raise ValueError("Filter order must be a nonnegative integer") elif N == 0: # Avoid divide-by-zero warning # Even order filters have DC gain of -rp dB return numpy.array([]), numpy.array([]), 10**(-rp/20) elif N == 1: p = -sqrt(1.0 / (10 ** (0.1 * rp) - 1.0)) k = -p z = [] return asarray(z), asarray(p), k eps = numpy.sqrt(10 ** (0.1 * rp) - 1) ck1 = eps / numpy.sqrt(10 ** (0.1 * rs) - 1) ck1p = numpy.sqrt(1 - ck1 * ck1) if ck1p == 1: raise ValueError("Cannot design a filter with given rp and rs" " specifications.") val = special.ellipk([ck1 * ck1, ck1p * ck1p]) if abs(1 - ck1p * ck1p) < EPSILON: krat = 0 else: krat = N * val[0] / val[1] m = optimize.fmin(_kratio, [0.5], args=(krat,), maxfun=250, maxiter=250, disp=0) if m < 0 or m > 1: m = optimize.fminbound(_kratio, 0, 1, args=(krat,), maxfun=250, maxiter=250, disp=0) capk = special.ellipk(m) j = numpy.arange(1 - N % 2, N, 2) jj = len(j) [s, c, d, phi] = special.ellipj(j * capk / N, m * numpy.ones(jj)) snew = numpy.compress(abs(s) > EPSILON, s, axis=-1) z = 1.0 / (sqrt(m) * snew) z = 1j * z z = numpy.concatenate((z, conjugate(z))) r = optimize.fmin(_vratio, special.ellipk(m), args=(1. / eps, ck1p * ck1p), maxfun=250, maxiter=250, disp=0) v0 = capk * r / (N * val[0]) [sv, cv, dv, phi] = special.ellipj(v0, 1 - m) p = -(c * d * sv * cv + 1j * s * dv) / (1 - (d * sv) ** 2.0) if N % 2: newp = numpy.compress(abs(p.imag) > EPSILON * numpy.sqrt(numpy.sum(p * numpy.conjugate(p), axis=0).real), p, axis=-1) p = numpy.concatenate((p, conjugate(newp))) else: p = numpy.concatenate((p, conjugate(p))) k = (numpy.prod(-p, axis=0) / numpy.prod(-z, axis=0)).real if N % 2 == 0: k = k / numpy.sqrt((1 + eps * eps)) return z, p, k # TODO: Make this a real public function scipy.misc.ff def _falling_factorial(x, n): r""" Return the factorial of x to the n falling. This is defined as: .. math:: x^\underline n = (x)_n = x (x-1) \cdots (x-n+1) This can more efficiently calculate ratios of factorials, since: n!/m! == falling_factorial(n, n-m) where n >= m skipping the factors that cancel out the usual factorial n! == ff(n, n) """ val = 1 for k in range(x - n + 1, x + 1): val *= k return val def _bessel_poly(n, reverse=False): """ Return the coefficients of Bessel polynomial of degree n If reverse is true, a reverse Bessel polynomial is output. Output is a list of coefficients: [1] = 1 [1, 1] = 1*s + 1 [1, 3, 3] = 1*s^2 + 3*s + 3 [1, 6, 15, 15] = 1*s^3 + 6*s^2 + 15*s + 15 [1, 10, 45, 105, 105] = 1*s^4 + 10*s^3 + 45*s^2 + 105*s + 105 etc. Output is a Python list of arbitrary precision long ints, so n is only limited by your hardware's memory. Sequence is http://oeis.org/A001498 , and output can be confirmed to match http://oeis.org/A001498/b001498.txt : >>> i = 0 >>> for n in range(51): ... for x in _bessel_poly(n, reverse=True): ... print(i, x) ... i += 1 """ if abs(int(n)) != n: raise ValueError("Polynomial order must be a nonnegative integer") else: n = int(n) # np.int32 doesn't work, for instance out = [] for k in range(n + 1): num = _falling_factorial(2*n - k, n) den = 2**(n - k) * factorial(k, exact=True) out.append(num // den) if reverse: return out[::-1] else: return out def _campos_zeros(n): """ Return approximate zero locations of Bessel polynomials y_n(x) for order n using polynomial fit (Campos-Calderon 2011) """ if n == 1: return asarray([-1+0j]) s = npp_polyval(n, [0, 0, 2, 0, -3, 1]) b3 = npp_polyval(n, [16, -8]) / s b2 = npp_polyval(n, [-24, -12, 12]) / s b1 = npp_polyval(n, [8, 24, -12, -2]) / s b0 = npp_polyval(n, [0, -6, 0, 5, -1]) / s r = npp_polyval(n, [0, 0, 2, 1]) a1 = npp_polyval(n, [-6, -6]) / r a2 = 6 / r k = np.arange(1, n+1) x = npp_polyval(k, [0, a1, a2]) y = npp_polyval(k, [b0, b1, b2, b3]) return x + 1j*y def _aberth(f, fp, x0, tol=1e-15, maxiter=50): """ Given a function f, its first derivative fp, and a set of initial guesses x0, simultaneously find the roots of the polynomial using the Aberth-Ehrlich method. len(x0) should equal the number of roots of f. (This is not a complete implementation of Bini's algorithm.) """ N = len(x0) x = array(x0, complex) beta = np.empty_like(x0) for iteration in range(maxiter): alpha = -f(x) / fp(x) # Newton's method # Model "repulsion" between zeros for k in range(N): beta[k] = np.sum(1/(x[k] - x[k+1:])) beta[k] += np.sum(1/(x[k] - x[:k])) x += alpha / (1 + alpha * beta) if not all(np.isfinite(x)): raise RuntimeError('Root-finding calculation failed') # Mekwi: The iterative process can be stopped when |hn| has become # less than the largest error one is willing to permit in the root. if all(abs(alpha) <= tol): break else: raise Exception('Zeros failed to converge') return x def _bessel_zeros(N): """ Find zeros of ordinary Bessel polynomial of order N, by root-finding of modified Bessel function of the second kind """ if N == 0: return asarray([]) # Generate starting points x0 = _campos_zeros(N) # Zeros are the same for exp(1/x)*K_{N+0.5}(1/x) and Nth-order ordinary # Bessel polynomial y_N(x) def f(x): return special.kve(N+0.5, 1/x) # First derivative of above def fp(x): return (special.kve(N-0.5, 1/x)/(2*x**2) - special.kve(N+0.5, 1/x)/(x**2) + special.kve(N+1.5, 1/x)/(2*x**2)) # Starting points converge to true zeros x = _aberth(f, fp, x0) # Improve precision using Newton's method on each for i in range(len(x)): x[i] = optimize.newton(f, x[i], fp, tol=1e-15) # Average complex conjugates to make them exactly symmetrical x = np.mean((x, x[::-1].conj()), 0) # Zeros should sum to -1 if abs(np.sum(x) + 1) > 1e-15: raise RuntimeError('Generated zeros are inaccurate') return x def _norm_factor(p, k): """ Numerically find frequency shift to apply to delay-normalized filter such that -3 dB point is at 1 rad/sec.