"""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) from numpy import mintypecode from scipy import special, optimize from scipy.misc import comb __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', 'filter_dict', 'band_dict', 'BadCoefficients'] class BadCoefficients(UserWarning): pass abs = absolute def findfreqs(num, den, N): """ Find an array of frequencies for computing the response of a 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) >>> 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 numerator b and denominator a of a filter, compute its frequency response:: b[0]*(jw)**(nb-1) + b[1]*(jw)**(nb-2) + ... + b[nb-1] H(w) = ------------------------------------------------------- a[0]*(jw)**(na-1) + a[1]*(jw)**(na-2) + ... + a[na-1] Parameters ---------- b : ndarray Numerator of a linear filter. a : ndarray Denominator of a linear filter. worN : {None, int}, 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 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 not plot is None: plot(w, h) return w, h def freqz(b, a=1, worN=None, whole=0, plot=None): """ Compute the frequency response of a digital filter. Given the numerator b and denominator a of a digital filter, compute its frequency response:: jw -jw -jmw jw B(e) b[0] + b[1]e + .... + b[m]e H(e) = ---- = ------------------------------------ jw -jw -jnw A(e) a[0] + a[1]e + .... + a[n]e Parameters ---------- b : ndarray numerator of a linear filter a : ndarray 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. 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 not plot is None: plot(w, h) return w, h def tf2zpk(b, a): r"""Return zero, pole, gain (z,p,k) representation from a numerator, denominator representation of a linear filter. Parameters ---------- b : ndarray Numerator polynomial. a : ndarray Denominator polynomial. 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 : ndarray Zeros of the transfer function. p : ndarray Poles of the transfer function. k : float System gain. Returns ------- b : ndarray Numerator polynomial. a : ndarray Denominator polynomial. """ 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 normalize(b, a): """Normalize polynomial representation of a transfer function. If values of b are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted. """ b, a = map(atleast_1d, (b, a)) if len(a.shape) != 1: raise ValueError("Denominator polynomial must be rank-1 array.") if len(b.shape) > 2: raise ValueError("Numerator polynomial must be rank-1 or" " rank-2 array.") if len(b.shape) == 1: b = asarray([b], b.dtype.char) while a[0] == 0.0 and len(a) > 1: a = a[1:] outb = b * (1.0) / a[0] outa = a * (1.0) / a[0] if allclose(0, outb[:, 0], atol=1e-14): warnings.warn("Badly conditioned filter coefficients (numerator): the " "results may be meaningless", BadCoefficients) while allclose(0, outb[:, 0], atol=1e-14) and (outb.shape[-1] > 1): outb = outb[:, 1:] if outb.shape[0] == 1: outb = outb[0] return outb, outa 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') or pole-zero ('zpk') 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). 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'. 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 """ 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 in (B,A) (numerator, denominator) or (Z,P,K) form. 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). Default is 'ba'. 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 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.plot(w, 20 * np.log10(abs(h))) >>> ax.set_xscale('log') >>> 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']: 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 in [buttap, besselap]: z, p, k = typefunc(N) 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) 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 : ndarray Zeros of the analog IIR filter transfer function. p : ndarray 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): """ 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 : ndarray Zeros of the analog IIR filter transfer function. p : ndarray 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 np.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): """ 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 : ndarray Zeros of the analog IIR filter transfer function. p : ndarray 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): """ 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 : ndarray Zeros of the analog IIR filter transfer function. p : ndarray 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): """ 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 : ndarray Zeros of the analog IIR filter transfer function. p : ndarray 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 in (B,A) or (Z,P,K) form. 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). 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'. See also -------- buttord Notes ----- The Butterworth filter has maximally flat frequency response in the passband. 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.plot(w, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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 in (B,A) or (Z,P,K) form. 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). 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'. See also -------- cheb1ord 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. 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.plot(w, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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 in (B,A) or (Z,P,K) form. 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). 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'. See also -------- cheb2ord 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). 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.plot(w, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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 in (B,A) or (Z,P,K) form. 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). 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'. See also -------- ellipord 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. 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.plot(w, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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'): """Bessel/Thomson digital and analog filter design. Design an Nth order digital or analog Bessel filter and return the filter coefficients in (B,A) or (Z,P,K) form. Parameters ---------- N : int The order of the filter. Wn : array_like A scalar or length-2 sequence giving the critical frequencies. For a Bessel filter, this is defined as the point at which the asymptotes of the response are the same as a Butterworth filter of the same order. 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'}, optional Type of output: numerator/denominator ('ba') or pole-zero ('zpk'). 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'. 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. As order increases, the Bessel filter approaches a Gaussian filter. The digital Bessel filter is generated 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. For a given Wn, the lowpass and highpass filter have the same phase vs frequency curves; they are "phase-matched". Examples -------- Plot the filter's frequency response, showing the flat group delay and the relationship to the Butterworth's cutoff frequency: >>> 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.plot(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed') >>> b, a = signal.bessel(4, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.plot(w, 20 * np.log10(np.abs(h))) >>> plt.xscale('log') >>> plt.title('Bessel filter frequency 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() >>> plt.figure() >>> plt.plot(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w)) >>> plt.xscale('log') >>> 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() """ return iirfilter(N, Wn, btype=btype, analog=analog, output=output, ftype='bessel') 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.plot(w, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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.plot(w / np.pi, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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.plot(w / np.pi, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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.plot(w, 20 * np.log10(abs(h))) >>> plt.xscale('log') >>> 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) zero, pole, gain for analog prototype of an Nth order Butterworth filter with an angular (e.g. rad/s) cutoff frequency of 1. """ 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) zero, pole, gain for Nth order Chebyshev type I lowpass analog filter prototype with 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. """ 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) zero, pole, gain for Nth order Chebyshev type II lowpass analog filter prototype with 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. """ 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) zeros, poles, and gain of an Nth order normalized prototype elliptic analog lowpass filter with 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. References ---------- 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 def besselap(N): """Return (z,p,k) zero, pole, gain for analog prototype of an Nth order Bessel filter. The filter is normalized such that the filter asymptotes are the same as a Butterworth filter of the same order with an angular (e.g. rad/s) cutoff frequency of 1. """ z = [] k = 1 if N == 0: p = [] elif N == 1: p = [-1] elif N == 2: p = [-.8660254037844386467637229 + .4999999999999999999999996j, -.8660254037844386467637229 - .4999999999999999999999996j] elif N == 3: p = [-.9416000265332067855971980, -.7456403858480766441810907 - .7113666249728352680992154j, -.7456403858480766441810907 + .7113666249728352680992154j] elif N == 4: p = [-.6572111716718829545787781 - .8301614350048733772399715j, -.6572111716718829545787788 + .8301614350048733772399715j, -.9047587967882449459642637 - .2709187330038746636700923j, -.9047587967882449459642624 + .2709187330038746636700926j] elif N == 5: p = [-.9264420773877602247196260, -.8515536193688395541722677 - .4427174639443327209850002j, -.8515536193688395541722677 + .4427174639443327209850002j, -.5905759446119191779319432 - .9072067564574549539291747j, -.5905759446119191779319432 + .9072067564574549539291747j] elif N == 6: p = [-.9093906830472271808050953 - .1856964396793046769246397j, -.9093906830472271808050953 + .1856964396793046769246397j, -.7996541858328288520243325 - .5621717346937317988594118j, -.7996541858328288520243325 + .5621717346937317988594118j, -.5385526816693109683073792 - .9616876881954277199245657j, -.5385526816693109683073792 + .9616876881954277199245657j] elif N == 7: p = [-.9194871556490290014311619, -.8800029341523374639772340 - .3216652762307739398381830j, -.8800029341523374639772340 + .3216652762307739398381830j, -.7527355434093214462291616 - .6504696305522550699212995j, -.7527355434093214462291616 + .6504696305522550699212995j, -.4966917256672316755024763 - 1.002508508454420401230220j, -.4966917256672316755024763 + 1.002508508454420401230220j] elif N == 8: p = [-.9096831546652910216327629 - .1412437976671422927888150j, -.9096831546652910216327629 + .1412437976671422927888150j, -.8473250802359334320103023 - .4259017538272934994996429j, -.8473250802359334320103023 + .4259017538272934994996429j, -.7111381808485399250796172 - .7186517314108401705762571j, -.7111381808485399250796172 + .7186517314108401705762571j, -.4621740412532122027072175 - 1.034388681126901058116589j, -.4621740412532122027072175 + 1.034388681126901058116589j] elif N == 9: p = [-.9154957797499037686769223, -.8911217017079759323183848 - .2526580934582164192308115j, -.8911217017079759323183848 + .2526580934582164192308115j, -.8148021112269012975514135 - .5085815689631499483745341j, -.8148021112269012975514135 + .5085815689631499483745341j, -.6743622686854761980403401 - .7730546212691183706919682j, -.6743622686854761980403401 + .7730546212691183706919682j, -.4331415561553618854685942 - 1.060073670135929666774323j, -.4331415561553618854685942 + 1.060073670135929666774323j] elif N == 10: p = [-.9091347320900502436826431 - .1139583137335511169927714j, -.9091347320900502436826431 + .1139583137335511169927714j, -.8688459641284764527921864 - .3430008233766309973110589j, -.8688459641284764527921864 + .3430008233766309973110589j, -.7837694413101441082655890 - .5759147538499947070009852j, -.7837694413101441082655890 + .5759147538499947070009852j, -.6417513866988316136190854 - .8175836167191017226233947j, -.6417513866988316136190854 + .8175836167191017226233947j, -.4083220732868861566219785 - 1.081274842819124562037210j, -.4083220732868861566219785 + 1.081274842819124562037210j] elif N == 11: p = [-.9129067244518981934637318, -.8963656705721166099815744 - .2080480375071031919692341j, -.8963656705721166099815744 + .2080480375071031919692341j, -.8453044014712962954184557 - .4178696917801248292797448j, -.8453044014712962954184557 + .4178696917801248292797448j, -.7546938934722303128102142 - .6319150050721846494520941j, -.7546938934722303128102142 + .6319150050721846494520941j, -.6126871554915194054182909 - .8547813893314764631518509j, -.6126871554915194054182909 + .8547813893314764631518509j, -.3868149510055090879155425 - 1.099117466763120928733632j, -.3868149510055090879155425 + 1.099117466763120928733632j] elif N == 12: p = [-.9084478234140682638817772 - 95506365213450398415258360.0e-27j, -.9084478234140682638817772 + 95506365213450398415258360.0e-27j, -.8802534342016826507901575 - .2871779503524226723615457j, -.8802534342016826507901575 + .2871779503524226723615457j, -.8217296939939077285792834 - .4810212115100676440620548j, -.8217296939939077285792834 + .4810212115100676440620548j, -.7276681615395159454547013 - .6792961178764694160048987j, -.7276681615395159454547013 + .6792961178764694160048987j, -.5866369321861477207528215 - .8863772751320727026622149j, -.5866369321861477207528215 + .8863772751320727026622149j, -.3679640085526312839425808 - 1.114373575641546257595657j, -.3679640085526312839425808 + 1.114373575641546257595657j] elif N == 13: p = [-.9110914665984182781070663, -.8991314665475196220910718 - .1768342956161043620980863j, -.8991314665475196220910718 + .1768342956161043620980863j, -.8625094198260548711573628 - .3547413731172988997754038j, -.8625094198260548711573628 + .3547413731172988997754038j, -.7987460692470972510394686 - .5350752120696801938272504j, -.7987460692470972510394686 + .5350752120696801938272504j, -.7026234675721275653944062 - .7199611890171304131266374j, -.7026234675721275653944062 + .7199611890171304131266374j, -.5631559842430199266325818 - .9135900338325109684927731j, -.5631559842430199266325818 + .9135900338325109684927731j, -.3512792323389821669401925 - 1.127591548317705678613239j, -.3512792323389821669401925 + 1.127591548317705678613239j] elif N == 14: p = [-.9077932138396487614720659 - 82196399419401501888968130.0e-27j, -.9077932138396487614720659 + 82196399419401501888968130.0e-27j, -.8869506674916445312089167 - .2470079178765333183201435j, -.8869506674916445312089167 + .2470079178765333183201435j, -.8441199160909851197897667 - .4131653825102692595237260j, -.8441199160909851197897667 + .4131653825102692595237260j, -.7766591387063623897344648 - .5819170677377608590492434j, -.7766591387063623897344648 + .5819170677377608590492434j, -.6794256425119233117869491 - .7552857305042033418417492j, -.6794256425119233117869491 + .7552857305042033418417492j, -.5418766775112297376541293 - .9373043683516919569183099j, -.5418766775112297376541293 + .9373043683516919569183099j, -.3363868224902037330610040 - 1.139172297839859991370924j, -.3363868224902037330610040 + 1.139172297839859991370924j] elif N == 15: p = [-.9097482363849064167228581, -.9006981694176978324932918 - .1537681197278439351298882j, -.9006981694176978324932918 + .1537681197278439351298882j, -.8731264620834984978337843 - .3082352470564267657715883j, -.8731264620834984978337843 + .3082352470564267657715883j, -.8256631452587146506294553 - .4642348752734325631275134j, -.8256631452587146506294553 + .4642348752734325631275134j, -.7556027168970728127850416 - .6229396358758267198938604j, -.7556027168970728127850416 + .6229396358758267198938604j, -.6579196593110998676999362 - .7862895503722515897065645j, -.6579196593110998676999362 + .7862895503722515897065645j, -.5224954069658330616875186 - .9581787261092526478889345j, -.5224954069658330616875186 + .9581787261092526478889345j, -.3229963059766444287113517 - 1.149416154583629539665297j, -.3229963059766444287113517 + 1.149416154583629539665297j] elif N == 16: p = [-.9072099595087001356491337 - 72142113041117326028823950.0e-27j, -.9072099595087001356491337 + 72142113041117326028823950.0e-27j, -.8911723070323647674780132 - .2167089659900576449410059j, -.8911723070323647674780132 + .2167089659900576449410059j, -.8584264231521330481755780 - .3621697271802065647661080j, -.8584264231521330481755780 + .3621697271802065647661080j, -.8074790293236003885306146 - .5092933751171800179676218j, -.8074790293236003885306146 + .5092933751171800179676218j, -.7356166304713115980927279 - .6591950877860393745845254j, -.7356166304713115980927279 + .6591950877860393745845254j, -.6379502514039066715773828 - .8137453537108761895522580j, -.6379502514039066715773828 + .8137453537108761895522580j, -.5047606444424766743309967 - .9767137477799090692947061j, -.5047606444424766743309967 + .9767137477799090692947061j, -.3108782755645387813283867 - 1.158552841199330479412225j, -.3108782755645387813283867 + 1.158552841199330479412225j] elif N == 17: p = [-.9087141161336397432860029, -.9016273850787285964692844 - .1360267995173024591237303j, -.9016273850787285964692844 + .1360267995173024591237303j, -.8801100704438627158492165 - .2725347156478803885651973j, -.8801100704438627158492165 + .2725347156478803885651973j, -.8433414495836129204455491 - .4100759282910021624185986j, -.8433414495836129204455491 + .4100759282910021624185986j, -.7897644147799708220288138 - .5493724405281088674296232j, -.7897644147799708220288138 + .5493724405281088674296232j, -.7166893842372349049842743 - .6914936286393609433305754j, -.7166893842372349049842743 + .6914936286393609433305754j, -.6193710717342144521602448 - .8382497252826992979368621j, -.6193710717342144521602448 + .8382497252826992979368621j, -.4884629337672704194973683 - .9932971956316781632345466j, -.4884629337672704194973683 + .9932971956316781632345466j, -.2998489459990082015466971 - 1.166761272925668786676672j, -.2998489459990082015466971 + 1.166761272925668786676672j] elif N == 18: p = [-.9067004324162775554189031 - 64279241063930693839360680.0e-27j, -.9067004324162775554189031 + 64279241063930693839360680.0e-27j, -.8939764278132455733032155 - .1930374640894758606940586j, -.8939764278132455733032155 + .1930374640894758606940586j, -.8681095503628830078317207 - .3224204925163257604931634j, -.8681095503628830078317207 + .3224204925163257604931634j, -.8281885016242836608829018 - .4529385697815916950149364j, -.8281885016242836608829018 + .4529385697815916950149364j, -.7726285030739558780127746 - .5852778162086640620016316j, -.7726285030739558780127746 + .5852778162086640620016316j, -.6987821445005273020051878 - .7204696509726630531663123j, -.6987821445005273020051878 + .7204696509726630531663123j, -.6020482668090644386627299 - .8602708961893664447167418j, -.6020482668090644386627299 + .8602708961893664447167418j, -.4734268069916151511140032 - 1.008234300314801077034158j, -.4734268069916151511140032 + 1.008234300314801077034158j, -.2897592029880489845789953 - 1.174183010600059128532230j, -.2897592029880489845789953 + 1.174183010600059128532230j] elif N == 19: p = [-.9078934217899404528985092, -.9021937639390660668922536 - .1219568381872026517578164j, -.9021937639390660668922536 + .1219568381872026517578164j, -.8849290585034385274001112 - .2442590757549818229026280j, -.8849290585034385274001112 + .2442590757549818229026280j, -.8555768765618421591093993 - .3672925896399872304734923j, -.8555768765618421591093993 + .3672925896399872304734923j, -.8131725551578197705476160 - .4915365035562459055630005j, -.8131725551578197705476160 + .4915365035562459055630005j, -.7561260971541629355231897 - .6176483917970178919174173j, -.7561260971541629355231897 + .6176483917970178919174173j, -.6818424412912442033411634 - .7466272357947761283262338j, -.6818424412912442033411634 + .7466272357947761283262338j, -.5858613321217832644813602 - .8801817131014566284786759j, -.5858613321217832644813602 + .8801817131014566284786759j, -.4595043449730988600785456 - 1.021768776912671221830298j, -.4595043449730988600785456 + 1.021768776912671221830298j, -.2804866851439370027628724 - 1.180931628453291873626003j, -.2804866851439370027628724 + 1.180931628453291873626003j] elif N == 20: p = [-.9062570115576771146523497 - 57961780277849516990208850.0e-27j, -.9062570115576771146523497 + 57961780277849516990208850.0e-27j, -.8959150941925768608568248 - .1740317175918705058595844j, -.8959150941925768608568248 + .1740317175918705058595844j, -.8749560316673332850673214 - .2905559296567908031706902j, -.8749560316673332850673214 + .2905559296567908031706902j, -.8427907479956670633544106 - .4078917326291934082132821j, -.8427907479956670633544106 + .4078917326291934082132821j, -.7984251191290606875799876 - .5264942388817132427317659j, -.7984251191290606875799876 + .5264942388817132427317659j, -.7402780309646768991232610 - .6469975237605228320268752j, -.7402780309646768991232610 + .6469975237605228320268752j, -.6658120544829934193890626 - .7703721701100763015154510j, -.6658120544829934193890626 + .7703721701100763015154510j, -.5707026806915714094398061 - .8982829066468255593407161j, -.5707026806915714094398061 + .8982829066468255593407161j, -.4465700698205149555701841 - 1.034097702560842962315411j, -.4465700698205149555701841 + 1.034097702560842962315411j, -.2719299580251652601727704 - 1.187099379810885886139638j, -.2719299580251652601727704 + 1.187099379810885886139638j] elif N == 21: p = [-.9072262653142957028884077, -.9025428073192696303995083 - .1105252572789856480992275j, -.9025428073192696303995083 + .1105252572789856480992275j, -.8883808106664449854431605 - .2213069215084350419975358j, -.8883808106664449854431605 + .2213069215084350419975358j, -.8643915813643204553970169 - .3326258512522187083009453j, -.8643915813643204553970169 + .3326258512522187083009453j, -.8299435470674444100273463 - .4448177739407956609694059j, -.8299435470674444100273463 + .4448177739407956609694059j, -.7840287980408341576100581 - .5583186348022854707564856j, -.7840287980408341576100581 + .5583186348022854707564856j, -.7250839687106612822281339 - .6737426063024382240549898j, -.7250839687106612822281339 + .6737426063024382240549898j, -.6506315378609463397807996 - .7920349342629491368548074j, -.6506315378609463397807996 + .7920349342629491368548074j, -.5564766488918562465935297 - .9148198405846724121600860j, -.5564766488918562465935297 + .9148198405846724121600860j, -.4345168906815271799687308 - 1.045382255856986531461592j, -.4345168906815271799687308 + 1.045382255856986531461592j, -.2640041595834031147954813 - 1.192762031948052470183960j, -.2640041595834031147954813 + 1.192762031948052470183960j] elif N == 22: p = [-.9058702269930872551848625 - 52774908289999045189007100.0e-27j, -.9058702269930872551848625 + 52774908289999045189007100.0e-27j, -.8972983138153530955952835 - .1584351912289865608659759j, -.8972983138153530955952835 + .1584351912289865608659759j, -.8799661455640176154025352 - .2644363039201535049656450j, -.8799661455640176154025352 + .2644363039201535049656450j, -.8534754036851687233084587 - .3710389319482319823405321j, -.8534754036851687233084587 + .3710389319482319823405321j, -.8171682088462720394344996 - .4785619492202780899653575j, -.8171682088462720394344996 + .4785619492202780899653575j, -.7700332930556816872932937 - .5874255426351153211965601j, -.7700332930556816872932937 + .5874255426351153211965601j, -.7105305456418785989070935 - .6982266265924524000098548j, -.7105305456418785989070935 + .6982266265924524000098548j, -.6362427683267827226840153 - .8118875040246347267248508j, -.6362427683267827226840153 + .8118875040246347267248508j, -.5430983056306302779658129 - .9299947824439872998916657j, -.5430983056306302779658129 + .9299947824439872998916657j, -.4232528745642628461715044 - 1.055755605227545931204656j, -.4232528745642628461715044 + 1.055755605227545931204656j, -.2566376987939318038016012 - 1.197982433555213008346532j, -.2566376987939318038016012 + 1.197982433555213008346532j] elif N == 23: p = [-.9066732476324988168207439, -.9027564979912504609412993 - .1010534335314045013252480j, -.9027564979912504609412993 + .1010534335314045013252480j, -.8909283242471251458653994 - .2023024699381223418195228j, -.8909283242471251458653994 + .2023024699381223418195228j, -.8709469395587416239596874 - .3039581993950041588888925j, -.8709469395587416239596874 + .3039581993950041588888925j, -.8423805948021127057054288 - .4062657948237602726779246j, -.8423805948021127057054288 + .4062657948237602726779246j, -.8045561642053176205623187 - .5095305912227258268309528j, -.8045561642053176205623187 + .5095305912227258268309528j, -.7564660146829880581478138 - .6141594859476032127216463j, -.7564660146829880581478138 + .6141594859476032127216463j, -.6965966033912705387505040 - .7207341374753046970247055j, -.6965966033912705387505040 + .7207341374753046970247055j, -.6225903228771341778273152 - .8301558302812980678845563j, -.6225903228771341778273152 + .8301558302812980678845563j, -.5304922463810191698502226 - .9439760364018300083750242j, -.5304922463810191698502226 + .9439760364018300083750242j, -.4126986617510148836149955 - 1.065328794475513585531053j, -.4126986617510148836149955 + 1.065328794475513585531053j, -.2497697202208956030229911 - 1.202813187870697831365338j, -.2497697202208956030229911 + 1.202813187870697831365338j] elif N == 24: p = [-.9055312363372773709269407 - 48440066540478700874836350.0e-27j, -.9055312363372773709269407 + 48440066540478700874836350.0e-27j, -.8983105104397872954053307 - .1454056133873610120105857j, -.8983105104397872954053307 + .1454056133873610120105857j, -.8837358034555706623131950 - .2426335234401383076544239j, -.8837358034555706623131950 + .2426335234401383076544239j, -.8615278304016353651120610 - .3403202112618624773397257j, -.8615278304016353651120610 + .3403202112618624773397257j, -.8312326466813240652679563 - .4386985933597305434577492j, -.8312326466813240652679563 + .4386985933597305434577492j, -.7921695462343492518845446 - .5380628490968016700338001j, -.7921695462343492518845446 + .5380628490968016700338001j, -.7433392285088529449175873 - .6388084216222567930378296j, -.7433392285088529449175873 + .6388084216222567930378296j, -.6832565803536521302816011 - .7415032695091650806797753j, -.6832565803536521302816011 + .7415032695091650806797753j, -.6096221567378335562589532 - .8470292433077202380020454j, -.6096221567378335562589532 + .8470292433077202380020454j, -.5185914574820317343536707 - .9569048385259054576937721j, -.5185914574820317343536707 + .9569048385259054576937721j, -.4027853855197518014786978 - 1.074195196518674765143729j, -.4027853855197518014786978 + 1.074195196518674765143729j, -.2433481337524869675825448 - 1.207298683731972524975429j, -.2433481337524869675825448 + 1.207298683731972524975429j] elif N == 25: p = [-.9062073871811708652496104, -.9028833390228020537142561 - 93077131185102967450643820.0e-27j, -.9028833390228020537142561 + 93077131185102967450643820.0e-27j, -.8928551459883548836774529 - .1863068969804300712287138j, -.8928551459883548836774529 + .1863068969804300712287138j, -.8759497989677857803656239 - .2798521321771408719327250j, -.8759497989677857803656239 + .2798521321771408719327250j, -.8518616886554019782346493 - .3738977875907595009446142j, -.8518616886554019782346493 + .3738977875907595009446142j, -.8201226043936880253962552 - .4686668574656966589020580j, -.8201226043936880253962552 + .4686668574656966589020580j, -.7800496278186497225905443 - .5644441210349710332887354j, -.7800496278186497225905443 + .5644441210349710332887354j, -.7306549271849967721596735 - .6616149647357748681460822j, -.7306549271849967721596735 + .6616149647357748681460822j, -.6704827128029559528610523 - .7607348858167839877987008j, -.6704827128029559528610523 + .7607348858167839877987008j, -.5972898661335557242320528 - .8626676330388028512598538j, -.5972898661335557242320528 + .8626676330388028512598538j, -.5073362861078468845461362 - .9689006305344868494672405j, -.5073362861078468845461362 + .9689006305344868494672405j, -.3934529878191079606023847 - 1.082433927173831581956863j, -.3934529878191079606023847 + 1.082433927173831581956863j, -.2373280669322028974199184 - 1.211476658382565356579418j, -.2373280669322028974199184 + 1.211476658382565356579418j] else: raise ValueError("Bessel Filter not supported for order %s" % N) return asarray(z), asarray(p), k filter_dict = {'butter': [buttap, buttord], 'butterworth': [buttap, buttord], 'cauer': [ellipap, ellipord], 'elliptic': [ellipap, ellipord], 'ellip': [ellipap, ellipord], 'bessel': [besselap], 'cheby1': [cheb1ap, cheb1ord], 'chebyshev1': [cheb1ap, cheb1ord], 'chebyshevi': [cheb1ap, cheb1ord], 'cheby2': [cheb2ap, cheb2ord], 'chebyshev2': [cheb2ap, cheb2ord], 'chebyshevii': [cheb2ap, cheb2ord], } band_dict = {'band': 'bandpass', 'bandpass': 'bandpass', 'pass': 'bandpass', 'bp': 'bandpass', 'bs': 'bandstop', 'bandstop': 'bandstop', 'bands': 'bandstop', 'stop': 'bandstop', 'l': 'lowpass', 'low': 'lowpass', 'lowpass': 'lowpass', 'lp': 'lowpass', 'high': 'highpass', 'highpass': 'highpass', 'h': 'highpass', 'hp': 'highpass', }