U md6@sddlZddlmZddlmZddlmZddddd d d gZd dZ d dZ dddZ dddZ dd Z ddd Zddd ZdS)N)eigcomb)convolvedaubqmfcascademorletrickermorlet2cwtc stj}dkrtddkr8d|d}t||gSdkrz|dd}|d}|td|d|d|d|gSdkrNd|d}d|d |d d |d ||d d }t|}|dd}td|d|}t||}dt|} ||t|d|| d|d| d|d| dd| dgSd krvd krfd dtDddd} t| } n.fddtDddd} t| d} tddg}tdg} tdD]Z} | | }d|||d}dd|}||}t |dkr0||}| d| g} q|t| } | t | |d} | j dddStddS)aT The coefficients for the FIR low-pass filter producing Daubechies wavelets. p>=1 gives the order of the zero at f=1/2. There are 2p filter coefficients. Parameters ---------- p : int Order of the zero at f=1/2, can have values from 1 to 34. Returns ------- daub : ndarray Return zp must be at least 1. g??#cs"g|]}td||ddqS)r exactr.0kpO/home/sam/Atlas/atlas_env/lib/python3.8/site-packages/scipy/signal/_wavelets.py 2szdaub..Ncs*g|]"}td||ddd|qS)r rg@rrrrrr 5sz.Nr!)lenr)r#r&)hkNZasgnrrrrNs cs:t|d}|dt|dkr*td|dkr:tdtjd|d|f\}}td}tj|df}t|}tj|df}td||d|d} td||dd|d} t dd||fd } t || d| d <t || d| d <t || d| d <t || d| d <| |9} tj d|d|>t dd|>} d| } d| }t | d \}}tt|d}t|dd|f}t|}|dkr| }| }d||i}t| d |d|d<d|>}|d| dd|<|d| d|d>d|<t| d |d|dd|<t| d |d|d|d>d|<dgtd|dD]}fddd D}d||>}|D]}d}t|D](}||dkr|d|d|>7}q||dd}t|d}t| d|f|}|||<|| ||d|<t| d|f||||d|<q~|qX| | |fS)a Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients. Parameters ---------- hk : array_like Coefficients of low-pass filter. J : int, optional Values will be computed at grid points ``K/2**J``. Default is 7. Returns ------- x : ndarray The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where ``len(hk) = len(gk) = N+1``. phi : ndarray The scaling function ``phi(x)`` at `x`: ``phi(x) = sum(hk * phi(2x-k))``, where k is from 0 to N. psi : ndarray, optional The wavelet function ``psi(x)`` at `x`: ``phi(x) = sum(gk * phi(2x-k))``, where k is from 0 to N. `psi` is only returned if `gk` is not None. Notes ----- The algorithm uses the vector cascade algorithm described by Strang and Nguyen in "Wavelets and Filter Banks". It builds a dictionary of values and slices for quick reuse. Then inserts vectors into final vector at the end. r zToo many levels.zToo few levels.Nrrr!d)rrr4)r r)r r dtype01cs"g|]}D]}d||fq qS)z%d%sr)rZxxyyZprevkeysrrr szcascade..)r5r#log2r%Zogridr$Zr_rZclipemptyZtakearangefloatrZargminabsoluter(r,dotr)int)r6Jr7nnkks2ZthkZgkZtgkZindx1Zindx2mxphipsiZlamvindsmZbitdicsteplevelZnewkeysZfackeynumposZpastphiiitemprr@rrbsh      &   &@?TcCstt| dtj|dtj|}td||}|rP|td|d8}|td|dtjd9}|S)a) Complex Morlet wavelet. Parameters ---------- M : int Length of the wavelet. w : float, optional Omega0. Default is 5 s : float, optional Scaling factor, windowed from ``-s*2*pi`` to ``+s*2*pi``. Default is 1. complete : bool, optional Whether to use the complete or the standard version. Returns ------- morlet : (M,) ndarray See Also -------- morlet2 : Implementation of Morlet wavelet, compatible with `cwt`. scipy.signal.gausspulse Notes ----- The standard version:: pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2)) This commonly used wavelet is often referred to simply as the Morlet wavelet. Note that this simplified version can cause admissibility problems at low values of `w`. The complete version:: pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2)) This version has a correction term to improve admissibility. For `w` greater than 5, the correction term is negligible. Note that the energy of the return wavelet is not normalised according to `s`. The fundamental frequency of this wavelet in Hz is given by ``f = 2*s*w*r / M`` where `r` is the sampling rate. Note: This function was created before `cwt` and is not compatible with it. Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> M = 100 >>> s = 4.0 >>> w = 2.0 >>> wavelet = signal.morlet(M, s, w) >>> plt.plot(wavelet) >>> plt.show() rrп)r#Zlinspacepiexp)MwsZcompleterMoutputrrrr s @$ c Cstdtd|tjd}|d}td||dd}|d}d||}t| d|}|||}|S)a Return a Ricker wavelet, also known as the "Mexican hat wavelet". It models the function: ``A * (1 - (x/a)**2) * exp(-0.5*(x/a)**2)``, where ``A = 2/(sqrt(3*a)*(pi**0.25))``. Parameters ---------- points : int Number of points in `vector`. Will be centered around 0. a : scalar Width parameter of the wavelet. Returns ------- vector : (N,) ndarray Array of length `points` in shape of ricker curve. Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> points = 100 >>> a = 4.0 >>> vec2 = signal.ricker(points, a) >>> print(len(vec2)) 100 >>> plt.plot(vec2) >>> plt.show() rrg?rr[r )r#r$r^rCr_) ZpointsaAZwsqZvecZxsqmodgausstotalrrrr s%  cCsdtd||dd}||}td||td|dtjd}td||}|S)aR Complex Morlet wavelet, designed to work with `cwt`. Returns the complete version of morlet wavelet, normalised according to `s`:: exp(1j*w*x/s) * exp(-0.5*(x/s)**2) * pi**(-0.25) * sqrt(1/s) Parameters ---------- M : int Length of the wavelet. s : float Width parameter of the wavelet. w : float, optional Omega0. Default is 5 Returns ------- morlet : (M,) ndarray See Also -------- morlet : Implementation of Morlet wavelet, incompatible with `cwt` Notes ----- .. versionadded:: 1.4.0 This function was designed to work with `cwt`. Because `morlet2` returns an array of complex numbers, the `dtype` argument of `cwt` should be set to `complex128` for best results. Note the difference in implementation with `morlet`. The fundamental frequency of this wavelet in Hz is given by:: f = w*fs / (2*s*np.pi) where ``fs`` is the sampling rate and `s` is the wavelet width parameter. Similarly we can get the wavelet width parameter at ``f``:: s = w*fs / (2*f*np.pi) Examples -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> M = 100 >>> s = 4.0 >>> w = 2.0 >>> wavelet = signal.morlet2(M, s, w) >>> plt.plot(abs(wavelet)) >>> plt.show() This example shows basic use of `morlet2` with `cwt` in time-frequency analysis: >>> t, dt = np.linspace(0, 1, 200, retstep=True) >>> fs = 1/dt >>> w = 6. >>> sig = np.cos(2*np.pi*(50 + 10*t)*t) + np.sin(40*np.pi*t) >>> freq = np.linspace(1, fs/2, 100) >>> widths = w*fs / (2*freq*np.pi) >>> cwtm = signal.cwt(sig, signal.morlet2, widths, w=w) >>> plt.pcolormesh(t, freq, np.abs(cwtm), cmap='viridis', shading='gouraud') >>> plt.show() rr[rrr\r]r )r#rCr_r^r$)r`rbrarMwaveletrcrrrr Fs H.c Ks|dkr6t|d|df|jjdkr0tj}ntj}tjt|t|f|d}t|D]N\}}t d|t|g}t |||f|ddd} t || dd ||<qX|S) a Continuous wavelet transform. Performs a continuous wavelet transform on `data`, using the `wavelet` function. A CWT performs a convolution with `data` using the `wavelet` function, which is characterized by a width parameter and length parameter. The `wavelet` function is allowed to be complex. Parameters ---------- data : (N,) ndarray data on which to perform the transform. wavelet : function Wavelet function, which should take 2 arguments. The first argument is the number of points that the returned vector will have (len(wavelet(length,width)) == length). The second is a width parameter, defining the size of the wavelet (e.g. standard deviation of a gaussian). See `ricker`, which satisfies these requirements. widths : (M,) sequence Widths to use for transform. dtype : data-type, optional The desired data type of output. Defaults to ``float64`` if the output of `wavelet` is real and ``complex128`` if it is complex. .. versionadded:: 1.4.0 kwargs Keyword arguments passed to wavelet function. .. versionadded:: 1.4.0 Returns ------- cwt: (M, N) ndarray Will have shape of (len(widths), len(data)). Notes ----- .. versionadded:: 1.4.0 For non-symmetric, complex-valued wavelets, the input signal is convolved with the time-reversed complex-conjugate of the wavelet data [1]. :: length = min(10 * width[ii], len(data)) cwt[ii,:] = signal.convolve(data, np.conj(wavelet(length, width[ii], **kwargs))[::-1], mode='same') References ---------- .. [1] S. Mallat, "A Wavelet Tour of Signal Processing (3rd Edition)", Academic Press, 2009. Examples -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> t = np.linspace(-1, 1, 200, endpoint=False) >>> sig = np.cos(2 * np.pi * 7 * t) + signal.gausspulse(t - 0.4, fc=2) >>> widths = np.arange(1, 31) >>> cwtmatr = signal.cwt(sig, signal.ricker, widths) .. note:: For cwt matrix plotting it is advisable to flip the y-axis >>> cwtmatr_yflip = np.flipud(cwtmatr) >>> plt.imshow(cwtmatr_yflip, extent=[-1, 1, 1, 31], cmap='PRGn', aspect='auto', ... vmax=abs(cwtmatr).max(), vmin=-abs(cwtmatr).max()) >>> plt.show() Nr rZFDGr;rr!Zsame)mode) r#Zasarrayr<charZ complex128Zfloat64rBr5 enumerateminr'r) datarjwidthsr<kwargsrcrQwidthr7Z wavelet_datarrrr sL )r8)rZr[T)ri)N)numpyr#Z scipy.linalgrZ scipy.specialrZ scipy.signalr__all__rrrr r r r rrrrs   E j K/ O