665 lines
20 KiB
Python
665 lines
20 KiB
Python
# Author: Travis Oliphant
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# 2003
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#
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# Feb. 2010: Updated by Warren Weckesser:
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# Rewrote much of chirp()
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# Added sweep_poly()
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import numpy as np
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from numpy import asarray, zeros, place, nan, mod, pi, extract, log, sqrt, \
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exp, cos, sin, polyval, polyint
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__all__ = ['sawtooth', 'square', 'gausspulse', 'chirp', 'sweep_poly',
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'unit_impulse']
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def sawtooth(t, width=1):
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"""
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Return a periodic sawtooth or triangle waveform.
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The sawtooth waveform has a period ``2*pi``, rises from -1 to 1 on the
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interval 0 to ``width*2*pi``, then drops from 1 to -1 on the interval
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``width*2*pi`` to ``2*pi``. `width` must be in the interval [0, 1].
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Note that this is not band-limited. It produces an infinite number
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of harmonics, which are aliased back and forth across the frequency
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spectrum.
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Parameters
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----------
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t : array_like
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Time.
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width : array_like, optional
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Width of the rising ramp as a proportion of the total cycle.
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Default is 1, producing a rising ramp, while 0 produces a falling
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ramp. `width` = 0.5 produces a triangle wave.
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If an array, causes wave shape to change over time, and must be the
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same length as t.
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Returns
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-------
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y : ndarray
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Output array containing the sawtooth waveform.
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Examples
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--------
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A 5 Hz waveform sampled at 500 Hz for 1 second:
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>>> from scipy import signal
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>>> import matplotlib.pyplot as plt
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>>> t = np.linspace(0, 1, 500)
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>>> plt.plot(t, signal.sawtooth(2 * np.pi * 5 * t))
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"""
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t, w = asarray(t), asarray(width)
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w = asarray(w + (t - t))
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t = asarray(t + (w - w))
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if t.dtype.char in ['fFdD']:
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ytype = t.dtype.char
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else:
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ytype = 'd'
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y = zeros(t.shape, ytype)
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# width must be between 0 and 1 inclusive
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mask1 = (w > 1) | (w < 0)
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place(y, mask1, nan)
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# take t modulo 2*pi
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tmod = mod(t, 2 * pi)
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# on the interval 0 to width*2*pi function is
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# tmod / (pi*w) - 1
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mask2 = (1 - mask1) & (tmod < w * 2 * pi)
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tsub = extract(mask2, tmod)
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wsub = extract(mask2, w)
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place(y, mask2, tsub / (pi * wsub) - 1)
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# on the interval width*2*pi to 2*pi function is
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# (pi*(w+1)-tmod) / (pi*(1-w))
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mask3 = (1 - mask1) & (1 - mask2)
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tsub = extract(mask3, tmod)
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wsub = extract(mask3, w)
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place(y, mask3, (pi * (wsub + 1) - tsub) / (pi * (1 - wsub)))
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return y
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def square(t, duty=0.5):
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"""
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Return a periodic square-wave waveform.
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The square wave has a period ``2*pi``, has value +1 from 0 to
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``2*pi*duty`` and -1 from ``2*pi*duty`` to ``2*pi``. `duty` must be in
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the interval [0,1].
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Note that this is not band-limited. It produces an infinite number
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of harmonics, which are aliased back and forth across the frequency
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spectrum.
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Parameters
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----------
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t : array_like
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The input time array.
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duty : array_like, optional
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Duty cycle. Default is 0.5 (50% duty cycle).
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If an array, causes wave shape to change over time, and must be the
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same length as t.
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Returns
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-------
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y : ndarray
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Output array containing the square waveform.
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Examples
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--------
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A 5 Hz waveform sampled at 500 Hz for 1 second:
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>>> from scipy import signal
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>>> import matplotlib.pyplot as plt
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>>> t = np.linspace(0, 1, 500, endpoint=False)
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>>> plt.plot(t, signal.square(2 * np.pi * 5 * t))
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>>> plt.ylim(-2, 2)
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A pulse-width modulated sine wave:
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>>> plt.figure()
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>>> sig = np.sin(2 * np.pi * t)
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>>> pwm = signal.square(2 * np.pi * 30 * t, duty=(sig + 1)/2)
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>>> plt.subplot(2, 1, 1)
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>>> plt.plot(t, sig)
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>>> plt.subplot(2, 1, 2)
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>>> plt.plot(t, pwm)
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>>> plt.ylim(-1.5, 1.5)
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"""
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t, w = asarray(t), asarray(duty)
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w = asarray(w + (t - t))
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t = asarray(t + (w - w))
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if t.dtype.char in ['fFdD']:
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ytype = t.dtype.char
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else:
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ytype = 'd'
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y = zeros(t.shape, ytype)
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# width must be between 0 and 1 inclusive
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mask1 = (w > 1) | (w < 0)
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place(y, mask1, nan)
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# on the interval 0 to duty*2*pi function is 1
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tmod = mod(t, 2 * pi)
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mask2 = (1 - mask1) & (tmod < w * 2 * pi)
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place(y, mask2, 1)
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# on the interval duty*2*pi to 2*pi function is
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# (pi*(w+1)-tmod) / (pi*(1-w))
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mask3 = (1 - mask1) & (1 - mask2)
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place(y, mask3, -1)
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return y
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def gausspulse(t, fc=1000, bw=0.5, bwr=-6, tpr=-60, retquad=False,
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retenv=False):
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"""
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Return a Gaussian modulated sinusoid:
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``exp(-a t^2) exp(1j*2*pi*fc*t).``
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If `retquad` is True, then return the real and imaginary parts
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(in-phase and quadrature).
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If `retenv` is True, then return the envelope (unmodulated signal).
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Otherwise, return the real part of the modulated sinusoid.
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Parameters
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----------
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t : ndarray or the string 'cutoff'
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Input array.
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fc : float, optional
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Center frequency (e.g. Hz). Default is 1000.
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bw : float, optional
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Fractional bandwidth in frequency domain of pulse (e.g. Hz).
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Default is 0.5.
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bwr : float, optional
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Reference level at which fractional bandwidth is calculated (dB).
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Default is -6.
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tpr : float, optional
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If `t` is 'cutoff', then the function returns the cutoff
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time for when the pulse amplitude falls below `tpr` (in dB).
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Default is -60.
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retquad : bool, optional
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If True, return the quadrature (imaginary) as well as the real part
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of the signal. Default is False.
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retenv : bool, optional
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If True, return the envelope of the signal. Default is False.
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Returns
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-------
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yI : ndarray
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Real part of signal. Always returned.
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yQ : ndarray
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Imaginary part of signal. Only returned if `retquad` is True.
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yenv : ndarray
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Envelope of signal. Only returned if `retenv` is True.
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See Also
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--------
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scipy.signal.morlet
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Examples
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--------
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Plot real component, imaginary component, and envelope for a 5 Hz pulse,
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sampled at 100 Hz for 2 seconds:
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>>> from scipy import signal
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>>> import matplotlib.pyplot as plt
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>>> t = np.linspace(-1, 1, 2 * 100, endpoint=False)
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>>> i, q, e = signal.gausspulse(t, fc=5, retquad=True, retenv=True)
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>>> plt.plot(t, i, t, q, t, e, '--')
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"""
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if fc < 0:
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raise ValueError("Center frequency (fc=%.2f) must be >=0." % fc)
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if bw <= 0:
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raise ValueError("Fractional bandwidth (bw=%.2f) must be > 0." % bw)
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if bwr >= 0:
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raise ValueError("Reference level for bandwidth (bwr=%.2f) must "
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"be < 0 dB" % bwr)
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# exp(-a t^2) <-> sqrt(pi/a) exp(-pi^2/a * f^2) = g(f)
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ref = pow(10.0, bwr / 20.0)
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# fdel = fc*bw/2: g(fdel) = ref --- solve this for a
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#
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# pi^2/a * fc^2 * bw^2 /4=-log(ref)
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a = -(pi * fc * bw) ** 2 / (4.0 * log(ref))
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if isinstance(t, str):
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if t == 'cutoff': # compute cut_off point
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# Solve exp(-a tc**2) = tref for tc
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# tc = sqrt(-log(tref) / a) where tref = 10^(tpr/20)
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if tpr >= 0:
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raise ValueError("Reference level for time cutoff must "
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"be < 0 dB")
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tref = pow(10.0, tpr / 20.0)
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return sqrt(-log(tref) / a)
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else:
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raise ValueError("If `t` is a string, it must be 'cutoff'")
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yenv = exp(-a * t * t)
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yI = yenv * cos(2 * pi * fc * t)
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yQ = yenv * sin(2 * pi * fc * t)
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if not retquad and not retenv:
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return yI
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if not retquad and retenv:
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return yI, yenv
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if retquad and not retenv:
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return yI, yQ
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if retquad and retenv:
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return yI, yQ, yenv
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def chirp(t, f0, t1, f1, method='linear', phi=0, vertex_zero=True):
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"""Frequency-swept cosine generator.
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In the following, 'Hz' should be interpreted as 'cycles per unit';
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there is no requirement here that the unit is one second. The
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important distinction is that the units of rotation are cycles, not
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radians. Likewise, `t` could be a measurement of space instead of time.
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Parameters
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----------
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t : array_like
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Times at which to evaluate the waveform.
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f0 : float
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Frequency (e.g. Hz) at time t=0.
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t1 : float
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Time at which `f1` is specified.
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f1 : float
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Frequency (e.g. Hz) of the waveform at time `t1`.
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method : {'linear', 'quadratic', 'logarithmic', 'hyperbolic'}, optional
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Kind of frequency sweep. If not given, `linear` is assumed. See
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Notes below for more details.
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phi : float, optional
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Phase offset, in degrees. Default is 0.
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vertex_zero : bool, optional
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This parameter is only used when `method` is 'quadratic'.
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It determines whether the vertex of the parabola that is the graph
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of the frequency is at t=0 or t=t1.
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Returns
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-------
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y : ndarray
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A numpy array containing the signal evaluated at `t` with the
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requested time-varying frequency. More precisely, the function
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returns ``cos(phase + (pi/180)*phi)`` where `phase` is the integral
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(from 0 to `t`) of ``2*pi*f(t)``. ``f(t)`` is defined below.
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See Also
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--------
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sweep_poly
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Notes
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-----
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There are four options for the `method`. The following formulas give
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the instantaneous frequency (in Hz) of the signal generated by
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`chirp()`. For convenience, the shorter names shown below may also be
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used.
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linear, lin, li:
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``f(t) = f0 + (f1 - f0) * t / t1``
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quadratic, quad, q:
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The graph of the frequency f(t) is a parabola through (0, f0) and
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(t1, f1). By default, the vertex of the parabola is at (0, f0).
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If `vertex_zero` is False, then the vertex is at (t1, f1). The
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formula is:
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if vertex_zero is True:
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``f(t) = f0 + (f1 - f0) * t**2 / t1**2``
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else:
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``f(t) = f1 - (f1 - f0) * (t1 - t)**2 / t1**2``
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To use a more general quadratic function, or an arbitrary
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polynomial, use the function `scipy.signal.sweep_poly`.
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logarithmic, log, lo:
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``f(t) = f0 * (f1/f0)**(t/t1)``
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f0 and f1 must be nonzero and have the same sign.
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This signal is also known as a geometric or exponential chirp.
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hyperbolic, hyp:
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``f(t) = f0*f1*t1 / ((f0 - f1)*t + f1*t1)``
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f0 and f1 must be nonzero.
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Examples
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--------
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The following will be used in the examples:
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>>> from scipy.signal import chirp, spectrogram
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>>> import matplotlib.pyplot as plt
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For the first example, we'll plot the waveform for a linear chirp
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from 6 Hz to 1 Hz over 10 seconds:
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>>> t = np.linspace(0, 10, 1500)
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>>> w = chirp(t, f0=6, f1=1, t1=10, method='linear')
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>>> plt.plot(t, w)
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>>> plt.title("Linear Chirp, f(0)=6, f(10)=1")
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>>> plt.xlabel('t (sec)')
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>>> plt.show()
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For the remaining examples, we'll use higher frequency ranges,
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and demonstrate the result using `scipy.signal.spectrogram`.
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We'll use a 4 second interval sampled at 7200 Hz.
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>>> fs = 7200
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>>> T = 4
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>>> t = np.arange(0, int(T*fs)) / fs
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We'll use this function to plot the spectrogram in each example.
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>>> def plot_spectrogram(title, w, fs):
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... ff, tt, Sxx = spectrogram(w, fs=fs, nperseg=256, nfft=576)
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... plt.pcolormesh(tt, ff[:145], Sxx[:145], cmap='gray_r', shading='gouraud')
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... plt.title(title)
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... plt.xlabel('t (sec)')
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... plt.ylabel('Frequency (Hz)')
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... plt.grid()
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...
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Quadratic chirp from 1500 Hz to 250 Hz
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(vertex of the parabolic curve of the frequency is at t=0):
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>>> w = chirp(t, f0=1500, f1=250, t1=T, method='quadratic')
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>>> plot_spectrogram(f'Quadratic Chirp, f(0)=1500, f({T})=250', w, fs)
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>>> plt.show()
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Quadratic chirp from 1500 Hz to 250 Hz
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(vertex of the parabolic curve of the frequency is at t=T):
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>>> w = chirp(t, f0=1500, f1=250, t1=T, method='quadratic',
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... vertex_zero=False)
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>>> plot_spectrogram(f'Quadratic Chirp, f(0)=1500, f({T})=250\\n' +
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... '(vertex_zero=False)', w, fs)
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>>> plt.show()
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Logarithmic chirp from 1500 Hz to 250 Hz:
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>>> w = chirp(t, f0=1500, f1=250, t1=T, method='logarithmic')
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>>> plot_spectrogram(f'Logarithmic Chirp, f(0)=1500, f({T})=250', w, fs)
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>>> plt.show()
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Hyperbolic chirp from 1500 Hz to 250 Hz:
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>>> w = chirp(t, f0=1500, f1=250, t1=T, method='hyperbolic')
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>>> plot_spectrogram(f'Hyperbolic Chirp, f(0)=1500, f({T})=250', w, fs)
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>>> plt.show()
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"""
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# 'phase' is computed in _chirp_phase, to make testing easier.
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phase = _chirp_phase(t, f0, t1, f1, method, vertex_zero)
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# Convert phi to radians.
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phi *= pi / 180
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return cos(phase + phi)
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def _chirp_phase(t, f0, t1, f1, method='linear', vertex_zero=True):
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"""
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Calculate the phase used by `chirp` to generate its output.
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See `chirp` for a description of the arguments.
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"""
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t = asarray(t)
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f0 = float(f0)
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t1 = float(t1)
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f1 = float(f1)
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if method in ['linear', 'lin', 'li']:
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beta = (f1 - f0) / t1
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phase = 2 * pi * (f0 * t + 0.5 * beta * t * t)
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elif method in ['quadratic', 'quad', 'q']:
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beta = (f1 - f0) / (t1 ** 2)
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if vertex_zero:
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phase = 2 * pi * (f0 * t + beta * t ** 3 / 3)
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else:
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phase = 2 * pi * (f1 * t + beta * ((t1 - t) ** 3 - t1 ** 3) / 3)
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elif method in ['logarithmic', 'log', 'lo']:
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if f0 * f1 <= 0.0:
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raise ValueError("For a logarithmic chirp, f0 and f1 must be "
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"nonzero and have the same sign.")
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if f0 == f1:
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phase = 2 * pi * f0 * t
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else:
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beta = t1 / log(f1 / f0)
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phase = 2 * pi * beta * f0 * (pow(f1 / f0, t / t1) - 1.0)
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elif method in ['hyperbolic', 'hyp']:
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if f0 == 0 or f1 == 0:
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raise ValueError("For a hyperbolic chirp, f0 and f1 must be "
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"nonzero.")
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if f0 == f1:
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# Degenerate case: constant frequency.
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phase = 2 * pi * f0 * t
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else:
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# Singular point: the instantaneous frequency blows up
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# when t == sing.
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sing = -f1 * t1 / (f0 - f1)
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phase = 2 * pi * (-sing * f0) * log(np.abs(1 - t/sing))
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else:
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raise ValueError("method must be 'linear', 'quadratic', 'logarithmic',"
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" or 'hyperbolic', but a value of %r was given."
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% method)
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return phase
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def sweep_poly(t, poly, phi=0):
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"""
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Frequency-swept cosine generator, with a time-dependent frequency.
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This function generates a sinusoidal function whose instantaneous
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frequency varies with time. The frequency at time `t` is given by
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the polynomial `poly`.
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Parameters
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----------
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t : ndarray
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Times at which to evaluate the waveform.
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poly : 1-D array_like or instance of numpy.poly1d
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The desired frequency expressed as a polynomial. If `poly` is
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a list or ndarray of length n, then the elements of `poly` are
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the coefficients of the polynomial, and the instantaneous
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frequency is
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``f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ... + poly[n-1]``
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If `poly` is an instance of numpy.poly1d, then the
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instantaneous frequency is
|
|
|
|
``f(t) = poly(t)``
|
|
|
|
phi : float, optional
|
|
Phase offset, in degrees, Default: 0.
|
|
|
|
Returns
|
|
-------
|
|
sweep_poly : ndarray
|
|
A numpy array containing the signal evaluated at `t` with the
|
|
requested time-varying frequency. More precisely, the function
|
|
returns ``cos(phase + (pi/180)*phi)``, where `phase` is the integral
|
|
(from 0 to t) of ``2 * pi * f(t)``; ``f(t)`` is defined above.
|
|
|
|
See Also
|
|
--------
|
|
chirp
|
|
|
|
Notes
|
|
-----
|
|
.. versionadded:: 0.8.0
|
|
|
|
If `poly` is a list or ndarray of length `n`, then the elements of
|
|
`poly` are the coefficients of the polynomial, and the instantaneous
|
|
frequency is:
|
|
|
|
``f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ... + poly[n-1]``
|
|
|
|
If `poly` is an instance of `numpy.poly1d`, then the instantaneous
|
|
frequency is:
|
|
|
|
``f(t) = poly(t)``
|
|
|
|
Finally, the output `s` is:
|
|
|
|
``cos(phase + (pi/180)*phi)``
|
|
|
|
where `phase` is the integral from 0 to `t` of ``2 * pi * f(t)``,
|
|
``f(t)`` as defined above.
|
|
|
|
Examples
|
|
--------
|
|
Compute the waveform with instantaneous frequency::
|
|
|
|
f(t) = 0.025*t**3 - 0.36*t**2 + 1.25*t + 2
|
|
|
|
over the interval 0 <= t <= 10.
|
|
|
|
>>> from scipy.signal import sweep_poly
|
|
>>> p = np.poly1d([0.025, -0.36, 1.25, 2.0])
|
|
>>> t = np.linspace(0, 10, 5001)
|
|
>>> w = sweep_poly(t, p)
|
|
|
|
Plot it:
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> plt.subplot(2, 1, 1)
|
|
>>> plt.plot(t, w)
|
|
>>> plt.title("Sweep Poly\\nwith frequency " +
|
|
... "$f(t) = 0.025t^3 - 0.36t^2 + 1.25t + 2$")
|
|
>>> plt.subplot(2, 1, 2)
|
|
>>> plt.plot(t, p(t), 'r', label='f(t)')
|
|
>>> plt.legend()
|
|
>>> plt.xlabel('t')
|
|
>>> plt.tight_layout()
|
|
>>> plt.show()
|
|
|
|
"""
|
|
# 'phase' is computed in _sweep_poly_phase, to make testing easier.
|
|
phase = _sweep_poly_phase(t, poly)
|
|
# Convert to radians.
|
|
phi *= pi / 180
|
|
return cos(phase + phi)
|
|
|
|
|
|
def _sweep_poly_phase(t, poly):
|
|
"""
|
|
Calculate the phase used by sweep_poly to generate its output.
|
|
|
|
See `sweep_poly` for a description of the arguments.
|
|
|
|
"""
|
|
# polyint handles lists, ndarrays and instances of poly1d automatically.
|
|
intpoly = polyint(poly)
|
|
phase = 2 * pi * polyval(intpoly, t)
|
|
return phase
|
|
|
|
|
|
def unit_impulse(shape, idx=None, dtype=float):
|
|
"""
|
|
Unit impulse signal (discrete delta function) or unit basis vector.
|
|
|
|
Parameters
|
|
----------
|
|
shape : int or tuple of int
|
|
Number of samples in the output (1-D), or a tuple that represents the
|
|
shape of the output (N-D).
|
|
idx : None or int or tuple of int or 'mid', optional
|
|
Index at which the value is 1. If None, defaults to the 0th element.
|
|
If ``idx='mid'``, the impulse will be centered at ``shape // 2`` in
|
|
all dimensions. If an int, the impulse will be at `idx` in all
|
|
dimensions.
|
|
dtype : data-type, optional
|
|
The desired data-type for the array, e.g., ``numpy.int8``. Default is
|
|
``numpy.float64``.
|
|
|
|
Returns
|
|
-------
|
|
y : ndarray
|
|
Output array containing an impulse signal.
|
|
|
|
Notes
|
|
-----
|
|
The 1D case is also known as the Kronecker delta.
|
|
|
|
.. versionadded:: 0.19.0
|
|
|
|
Examples
|
|
--------
|
|
An impulse at the 0th element (:math:`\\delta[n]`):
|
|
|
|
>>> from scipy import signal
|
|
>>> signal.unit_impulse(8)
|
|
array([ 1., 0., 0., 0., 0., 0., 0., 0.])
|
|
|
|
Impulse offset by 2 samples (:math:`\\delta[n-2]`):
|
|
|
|
>>> signal.unit_impulse(7, 2)
|
|
array([ 0., 0., 1., 0., 0., 0., 0.])
|
|
|
|
2-dimensional impulse, centered:
|
|
|
|
>>> signal.unit_impulse((3, 3), 'mid')
|
|
array([[ 0., 0., 0.],
|
|
[ 0., 1., 0.],
|
|
[ 0., 0., 0.]])
|
|
|
|
Impulse at (2, 2), using broadcasting:
|
|
|
|
>>> signal.unit_impulse((4, 4), 2)
|
|
array([[ 0., 0., 0., 0.],
|
|
[ 0., 0., 0., 0.],
|
|
[ 0., 0., 1., 0.],
|
|
[ 0., 0., 0., 0.]])
|
|
|
|
Plot the impulse response of a 4th-order Butterworth lowpass filter:
|
|
|
|
>>> imp = signal.unit_impulse(100, 'mid')
|
|
>>> b, a = signal.butter(4, 0.2)
|
|
>>> response = signal.lfilter(b, a, imp)
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> plt.plot(np.arange(-50, 50), imp)
|
|
>>> plt.plot(np.arange(-50, 50), response)
|
|
>>> plt.margins(0.1, 0.1)
|
|
>>> plt.xlabel('Time [samples]')
|
|
>>> plt.ylabel('Amplitude')
|
|
>>> plt.grid(True)
|
|
>>> plt.show()
|
|
|
|
"""
|
|
out = zeros(shape, dtype)
|
|
|
|
shape = np.atleast_1d(shape)
|
|
|
|
if idx is None:
|
|
idx = (0,) * len(shape)
|
|
elif idx == 'mid':
|
|
idx = tuple(shape // 2)
|
|
elif not hasattr(idx, "__iter__"):
|
|
idx = (idx,) * len(shape)
|
|
|
|
out[idx] = 1
|
|
return out
|