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# Author: Travis Oliphant
# 2003
#
# Feb. 2010: Updated by Warren Weckesser:
# Rewrote much of chirp()
# Added sweep_poly()
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy import asarray, zeros, place, nan, mod, pi, extract, log, sqrt, \
exp, cos, sin, polyval, polyint
from scipy._lib.six import string_types
__all__ = ['sawtooth', 'square', 'gausspulse', 'chirp', 'sweep_poly',
'unit_impulse']
def sawtooth(t, width=1):
"""
Return a periodic sawtooth or triangle waveform.
The sawtooth waveform has a period ``2*pi``, rises from -1 to 1 on the
interval 0 to ``width*2*pi``, then drops from 1 to -1 on the interval
``width*2*pi`` to ``2*pi``. `width` must be in the interval [0, 1].
Note that this is not band-limited. It produces an infinite number
of harmonics, which are aliased back and forth across the frequency
spectrum.
Parameters
----------
t : array_like
Time.
width : array_like, optional
Width of the rising ramp as a proportion of the total cycle.
Default is 1, producing a rising ramp, while 0 produces a falling
ramp. `width` = 0.5 produces a triangle wave.
If an array, causes wave shape to change over time, and must be the
same length as t.
Returns
-------
y : ndarray
Output array containing the sawtooth waveform.
Examples
--------
A 5 Hz waveform sampled at 500 Hz for 1 second:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> t = np.linspace(0, 1, 500)
>>> plt.plot(t, signal.sawtooth(2 * np.pi * 5 * t))
"""
t, w = asarray(t), asarray(width)
w = asarray(w + (t - t))
t = asarray(t + (w - w))
if t.dtype.char in ['fFdD']:
ytype = t.dtype.char
else:
ytype = 'd'
y = zeros(t.shape, ytype)
# width must be between 0 and 1 inclusive
mask1 = (w > 1) | (w < 0)
place(y, mask1, nan)
# take t modulo 2*pi
tmod = mod(t, 2 * pi)
# on the interval 0 to width*2*pi function is
# tmod / (pi*w) - 1
mask2 = (1 - mask1) & (tmod < w * 2 * pi)
tsub = extract(mask2, tmod)
wsub = extract(mask2, w)
place(y, mask2, tsub / (pi * wsub) - 1)
# on the interval width*2*pi to 2*pi function is
# (pi*(w+1)-tmod) / (pi*(1-w))
mask3 = (1 - mask1) & (1 - mask2)
tsub = extract(mask3, tmod)
wsub = extract(mask3, w)
place(y, mask3, (pi * (wsub + 1) - tsub) / (pi * (1 - wsub)))
return y
def square(t, duty=0.5):
"""
Return a periodic square-wave waveform.
The square wave has a period ``2*pi``, has value +1 from 0 to
``2*pi*duty`` and -1 from ``2*pi*duty`` to ``2*pi``. `duty` must be in
the interval [0,1].
Note that this is not band-limited. It produces an infinite number
of harmonics, which are aliased back and forth across the frequency
spectrum.
Parameters
----------
t : array_like
The input time array.
duty : array_like, optional
Duty cycle. Default is 0.5 (50% duty cycle).
If an array, causes wave shape to change over time, and must be the
same length as t.
Returns
-------
y : ndarray
Output array containing the square waveform.
Examples
--------
A 5 Hz waveform sampled at 500 Hz for 1 second:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> t = np.linspace(0, 1, 500, endpoint=False)
>>> plt.plot(t, signal.square(2 * np.pi * 5 * t))
>>> plt.ylim(-2, 2)
A pulse-width modulated sine wave:
>>> plt.figure()
>>> sig = np.sin(2 * np.pi * t)
>>> pwm = signal.square(2 * np.pi * 30 * t, duty=(sig + 1)/2)
>>> plt.subplot(2, 1, 1)
>>> plt.plot(t, sig)
>>> plt.subplot(2, 1, 2)
>>> plt.plot(t, pwm)
>>> plt.ylim(-1.5, 1.5)
"""
t, w = asarray(t), asarray(duty)
w = asarray(w + (t - t))
t = asarray(t + (w - w))
if t.dtype.char in ['fFdD']:
ytype = t.dtype.char
else:
ytype = 'd'
y = zeros(t.shape, ytype)
# width must be between 0 and 1 inclusive
mask1 = (w > 1) | (w < 0)
place(y, mask1, nan)
# on the interval 0 to duty*2*pi function is 1
tmod = mod(t, 2 * pi)
mask2 = (1 - mask1) & (tmod < w * 2 * pi)
place(y, mask2, 1)
# on the interval duty*2*pi to 2*pi function is
# (pi*(w+1)-tmod) / (pi*(1-w))
mask3 = (1 - mask1) & (1 - mask2)
place(y, mask3, -1)
return y
def gausspulse(t, fc=1000, bw=0.5, bwr=-6, tpr=-60, retquad=False,
retenv=False):
"""
Return a Gaussian modulated sinusoid:
``exp(-a t^2) exp(1j*2*pi*fc*t).``
If `retquad` is True, then return the real and imaginary parts
(in-phase and quadrature).
If `retenv` is True, then return the envelope (unmodulated signal).
Otherwise, return the real part of the modulated sinusoid.
Parameters
----------
t : ndarray or the string 'cutoff'
Input array.
fc : int, optional
Center frequency (e.g. Hz). Default is 1000.
bw : float, optional
Fractional bandwidth in frequency domain of pulse (e.g. Hz).
Default is 0.5.
bwr : float, optional
Reference level at which fractional bandwidth is calculated (dB).
Default is -6.
tpr : float, optional
If `t` is 'cutoff', then the function returns the cutoff
time for when the pulse amplitude falls below `tpr` (in dB).
Default is -60.
retquad : bool, optional
If True, return the quadrature (imaginary) as well as the real part
of the signal. Default is False.
retenv : bool, optional
If True, return the envelope of the signal. Default is False.
Returns
-------
yI : ndarray
Real part of signal. Always returned.
yQ : ndarray
Imaginary part of signal. Only returned if `retquad` is True.
yenv : ndarray
Envelope of signal. Only returned if `retenv` is True.
See Also
--------
scipy.signal.morlet
Examples
--------
Plot real component, imaginary component, and envelope for a 5 Hz pulse,
sampled at 100 Hz for 2 seconds:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> t = np.linspace(-1, 1, 2 * 100, endpoint=False)
>>> i, q, e = signal.gausspulse(t, fc=5, retquad=True, retenv=True)
>>> plt.plot(t, i, t, q, t, e, '--')
"""
if fc < 0:
raise ValueError("Center frequency (fc=%.2f) must be >=0." % fc)
if bw <= 0:
raise ValueError("Fractional bandwidth (bw=%.2f) must be > 0." % bw)
if bwr >= 0:
raise ValueError("Reference level for bandwidth (bwr=%.2f) must "
"be < 0 dB" % bwr)
# exp(-a t^2) <-> sqrt(pi/a) exp(-pi^2/a * f^2) = g(f)
ref = pow(10.0, bwr / 20.0)
# fdel = fc*bw/2: g(fdel) = ref --- solve this for a
#
# pi^2/a * fc^2 * bw^2 /4=-log(ref)
a = -(pi * fc * bw) ** 2 / (4.0 * log(ref))
if isinstance(t, string_types):
if t == 'cutoff': # compute cut_off point
# Solve exp(-a tc**2) = tref for tc
# tc = sqrt(-log(tref) / a) where tref = 10^(tpr/20)
if tpr >= 0:
raise ValueError("Reference level for time cutoff must be < 0 dB")
tref = pow(10.0, tpr / 20.0)
return sqrt(-log(tref) / a)
else:
raise ValueError("If `t` is a string, it must be 'cutoff'")
yenv = exp(-a * t * t)
yI = yenv * cos(2 * pi * fc * t)
yQ = yenv * sin(2 * pi * fc * t)
if not retquad and not retenv:
return yI
if not retquad and retenv:
return yI, yenv
if retquad and not retenv:
return yI, yQ
if retquad and retenv:
return yI, yQ, yenv
def chirp(t, f0, t1, f1, method='linear', phi=0, vertex_zero=True):
"""Frequency-swept cosine generator.
In the following, 'Hz' should be interpreted as 'cycles per unit';
there is no requirement here that the unit is one second. The
important distinction is that the units of rotation are cycles, not
radians. Likewise, `t` could be a measurement of space instead of time.
Parameters
----------
t : array_like
Times at which to evaluate the waveform.
f0 : float
Frequency (e.g. Hz) at time t=0.
t1 : float
Time at which `f1` is specified.
f1 : float
Frequency (e.g. Hz) of the waveform at time `t1`.
method : {'linear', 'quadratic', 'logarithmic', 'hyperbolic'}, optional
Kind of frequency sweep. If not given, `linear` is assumed. See
Notes below for more details.
phi : float, optional
Phase offset, in degrees. Default is 0.
vertex_zero : bool, optional
This parameter is only used when `method` is 'quadratic'.
It determines whether the vertex of the parabola that is the graph
of the frequency is at t=0 or t=t1.
Returns
-------
y : 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 below.
See Also
--------
sweep_poly
Notes
-----
There are four options for the `method`. The following formulas give
the instantaneous frequency (in Hz) of the signal generated by
`chirp()`. For convenience, the shorter names shown below may also be
used.
linear, lin, li:
``f(t) = f0 + (f1 - f0) * t / t1``
quadratic, quad, q:
The graph of the frequency f(t) is a parabola through (0, f0) and
(t1, f1). By default, the vertex of the parabola is at (0, f0).
If `vertex_zero` is False, then the vertex is at (t1, f1). The
formula is:
if vertex_zero is True:
``f(t) = f0 + (f1 - f0) * t**2 / t1**2``
else:
``f(t) = f1 - (f1 - f0) * (t1 - t)**2 / t1**2``
To use a more general quadratic function, or an arbitrary
polynomial, use the function `scipy.signal.waveforms.sweep_poly`.
logarithmic, log, lo:
``f(t) = f0 * (f1/f0)**(t/t1)``
f0 and f1 must be nonzero and have the same sign.
This signal is also known as a geometric or exponential chirp.
hyperbolic, hyp:
``f(t) = f0*f1*t1 / ((f0 - f1)*t + f1*t1)``
f0 and f1 must be nonzero.
Examples
--------
The following will be used in the examples:
>>> from scipy.signal import chirp, spectrogram
>>> import matplotlib.pyplot as plt
For the first example, we'll plot the waveform for a linear chirp
from 6 Hz to 1 Hz over 10 seconds:
>>> t = np.linspace(0, 10, 5001)
>>> w = chirp(t, f0=6, f1=1, t1=10, method='linear')
>>> plt.plot(t, w)
>>> plt.title("Linear Chirp, f(0)=6, f(10)=1")
>>> plt.xlabel('t (sec)')
>>> plt.show()
For the remaining examples, we'll use higher frequency ranges,
and demonstrate the result using `scipy.signal.spectrogram`.
We'll use a 10 second interval sampled at 8000 Hz.
>>> fs = 8000
>>> T = 10
>>> t = np.linspace(0, T, T*fs, endpoint=False)
Quadratic chirp from 1500 Hz to 250 Hz over 10 seconds
(vertex of the parabolic curve of the frequency is at t=0):
>>> w = chirp(t, f0=1500, f1=250, t1=10, method='quadratic')
>>> ff, tt, Sxx = spectrogram(w, fs=fs, noverlap=256, nperseg=512,
... nfft=2048)
>>> plt.pcolormesh(tt, ff[:513], Sxx[:513], cmap='gray_r')
>>> plt.title('Quadratic Chirp, f(0)=1500, f(10)=250')
>>> plt.xlabel('t (sec)')
>>> plt.ylabel('Frequency (Hz)')
>>> plt.grid()
>>> plt.show()
Quadratic chirp from 1500 Hz to 250 Hz over 10 seconds
(vertex of the parabolic curve of the frequency is at t=10):
>>> w = chirp(t, f0=1500, f1=250, t1=10, method='quadratic',
... vertex_zero=False)
>>> ff, tt, Sxx = spectrogram(w, fs=fs, noverlap=256, nperseg=512,
... nfft=2048)
>>> plt.pcolormesh(tt, ff[:513], Sxx[:513], cmap='gray_r')
>>> plt.title('Quadratic Chirp, f(0)=2500, f(10)=250\\n' +
... '(vertex_zero=False)')
>>> plt.xlabel('t (sec)')
>>> plt.ylabel('Frequency (Hz)')
>>> plt.grid()
>>> plt.show()
Logarithmic chirp from 1500 Hz to 250 Hz over 10 seconds:
>>> w = chirp(t, f0=1500, f1=250, t1=10, method='logarithmic')
>>> ff, tt, Sxx = spectrogram(w, fs=fs, noverlap=256, nperseg=512,
... nfft=2048)
>>> plt.pcolormesh(tt, ff[:513], Sxx[:513], cmap='gray_r')
>>> plt.title('Logarithmic Chirp, f(0)=1500, f(10)=250')
>>> plt.xlabel('t (sec)')
>>> plt.ylabel('Frequency (Hz)')
>>> plt.grid()
>>> plt.show()
Hyperbolic chirp from 1500 Hz to 250 Hz over 10 seconds:
>>> w = chirp(t, f0=1500, f1=250, t1=10, method='hyperbolic')
>>> ff, tt, Sxx = spectrogram(w, fs=fs, noverlap=256, nperseg=512,
... nfft=2048)
>>> plt.pcolormesh(tt, ff[:513], Sxx[:513], cmap='gray_r')
>>> plt.title('Hyperbolic Chirp, f(0)=1500, f(10)=250')
>>> plt.xlabel('t (sec)')
>>> plt.ylabel('Frequency (Hz)')
>>> plt.grid()
>>> plt.show()
"""
# 'phase' is computed in _chirp_phase, to make testing easier.
phase = _chirp_phase(t, f0, t1, f1, method, vertex_zero)
# Convert phi to radians.
phi *= pi / 180
return cos(phase + phi)
def _chirp_phase(t, f0, t1, f1, method='linear', vertex_zero=True):
"""
Calculate the phase used by chirp_phase to generate its output.
See `chirp` for a description of the arguments.
"""
t = asarray(t)
f0 = float(f0)
t1 = float(t1)
f1 = float(f1)
if method in ['linear', 'lin', 'li']:
beta = (f1 - f0) / t1
phase = 2 * pi * (f0 * t + 0.5 * beta * t * t)
elif method in ['quadratic', 'quad', 'q']:
beta = (f1 - f0) / (t1 ** 2)
if vertex_zero:
phase = 2 * pi * (f0 * t + beta * t ** 3 / 3)
else:
phase = 2 * pi * (f1 * t + beta * ((t1 - t) ** 3 - t1 ** 3) / 3)
elif method in ['logarithmic', 'log', 'lo']:
if f0 * f1 <= 0.0:
raise ValueError("For a logarithmic chirp, f0 and f1 must be "
"nonzero and have the same sign.")
if f0 == f1:
phase = 2 * pi * f0 * t
else:
beta = t1 / log(f1 / f0)
phase = 2 * pi * beta * f0 * (pow(f1 / f0, t / t1) - 1.0)
elif method in ['hyperbolic', 'hyp']:
if f0 == 0 or f1 == 0:
raise ValueError("For a hyperbolic chirp, f0 and f1 must be "
"nonzero.")
if f0 == f1:
# Degenerate case: constant frequency.
phase = 2 * pi * f0 * t
else:
# Singular point: the instantaneous frequency blows up
# when t == sing.
sing = -f1 * t1 / (f0 - f1)
phase = 2 * pi * (-sing * f0) * log(np.abs(1 - t/sing))
else:
raise ValueError("method must be 'linear', 'quadratic', 'logarithmic',"
" or 'hyperbolic', but a value of %r was given."
% method)
return phase
def sweep_poly(t, poly, phi=0):
"""
Frequency-swept cosine generator, with a time-dependent frequency.
This function generates a sinusoidal function whose instantaneous
frequency varies with time. The frequency at time `t` is given by
the polynomial `poly`.
Parameters
----------
t : ndarray
Times at which to evaluate the waveform.
poly : 1-D array_like or instance of numpy.poly1d
The desired frequency expressed as a polynomial. 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)``
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