555 lines
32 KiB
Plaintext
555 lines
32 KiB
Plaintext
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†R jv ã ó„ — d Z ddlZddlmZ ddlZddlmZmZm Z d„ Z
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d„ Z d$d„Z d%d„Z
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d%d „Zd
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„ Z d&d „Z d'd „Zej$ j' d
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dd¬« ej$ d(d„« Zej$ d(d„« ZdZ ej. ed« Z ej2 d)ddiej$ j4 ¤Že_ ej. ed« Z ej2 d)ddiej$ j4 ¤Že_ ej. ed« Z ej2 d)ddiej$ j4 ¤Že _ ej. ed« Z ej2 d)dd iej$ j4 ¤Že _ ej$ d*d „« Z ej$ d d ee
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ddd dfd!„« Z G d"„ d#« Z y)+a¶
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Numerical Python functions written for compatibility with MATLAB
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commands with the same names. Most numerical Python functions can be found in
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the `NumPy`_ and `SciPy`_ libraries. What remains here is code for performing
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spectral computations and kernel density estimations.
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.. _NumPy: https://numpy.org
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.. _SciPy: https://www.scipy.org
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Spectral functions
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------------------
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`cohere`
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Coherence (normalized cross spectral density)
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`csd`
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Cross spectral density using Welch's average periodogram
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`detrend`
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Remove the mean or best fit line from an array
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`psd`
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Power spectral density using Welch's average periodogram
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`specgram`
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Spectrogram (spectrum over segments of time)
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`complex_spectrum`
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Return the complex-valued frequency spectrum of a signal
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`magnitude_spectrum`
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Return the magnitude of the frequency spectrum of a signal
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`angle_spectrum`
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Return the angle (wrapped phase) of the frequency spectrum of a signal
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`phase_spectrum`
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Return the phase (unwrapped angle) of the frequency spectrum of a signal
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`detrend_mean`
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Remove the mean from a line.
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`detrend_linear`
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Remove the best fit line from a line.
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`detrend_none`
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Return the original line.
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é N)ÚNumber)Ú_apiÚ
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_docstringÚcbookc óD — t j t | « « | z S )z‹
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Return *x* times the Hanning (or Hann) window of len(*x*).
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See Also
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--------
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window_none : Another window algorithm.
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)ÚnpÚhanningÚlen©Úxs úR/home/antigravity/intellecton/venv/lib/python3.12/site-packages/matplotlib/mlab.pyÚwindow_hanningr : s € ô �:‰:”c˜!“fÓ ˜aÑ Ð ó c ó — | S )zz
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No window function; simply return *x*.
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See Also
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--------
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window_hanning : Another window algorithm.
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© r s r
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Ú window_noner E s € ð
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€Hr c óê — |�|dv rt | t |¬« S |dk( rt | t |¬« S |dk( rt | t |¬« S t |« rmt j | « } |�!|dz | j kD rt d|› d�« ‚|€| j dk( s|s| j dk( r || « S || |¬ « S t d |›d �« ‚# t $ r t j ||| ¬
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« cY S w xY w)
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al
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Return *x* with its trend removed.
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Parameters
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----------
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x : array or sequence
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Array or sequence containing the data.
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key : {'default', 'constant', 'mean', 'linear', 'none'} or function
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The detrending algorithm to use. 'default', 'mean', and 'constant' are
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the same as `detrend_mean`. 'linear' is the same as `detrend_linear`.
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'none' is the same as `detrend_none`. The default is 'mean'. See the
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corresponding functions for more details regarding the algorithms. Can
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also be a function that carries out the detrend operation.
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axis : int
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The axis along which to do the detrending.
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See Also
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--------
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detrend_mean : Implementation of the 'mean' algorithm.
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detrend_linear : Implementation of the 'linear' algorithm.
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detrend_none : Implementation of the 'none' algorithm.
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)ÚconstantÚmeanÚdefault)ÚkeyÚaxisÚlinearÚnoneé zaxis(=z) out of boundsr ©r )r ÚarrzUnknown value for key: zH, must be one of: 'default', 'constant', 'mean', 'linear', or a function) ÚdetrendÚ detrend_meanÚdetrend_linearÚ detrend_noneÚcallabler ÚasarrayÚndimÚ
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ValueErrorÚ TypeErrorÚapply_along_axis)r r r s r
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r r P s € ð2 €{�cÐ <Ñ<Ü�qœl°Ô6Ð6Ø �Š Ü�qœn°4Ô8Ð8Ø �Š Ü�qœl°Ô6Ð6Ü �#Œ Ü �J‰J�q‹MˆØ Ð q¡¨1¯6©6Ò 1Ü ˜v d V¨?Ð ;Ó<Ð <Ø ˆL˜QŸV™V qš[±$¸1¿6¹6ÀQº;Ù�q“6ˆMð >Ù�q˜tÔ$Ð $ô Ø% c Wð -:ð
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;ó<ð <øô ò >Ü×&Ñ& s°¸1Ô=Ò =ð >ús Â5 C Ã!C2Ã1C2c óœ — t j | « } |� |dz | j kD rt d|z « ‚| | j |d¬« z
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S )aå
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Return *x* minus the mean(*x*).
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Parameters
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----------
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x : array or sequence
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Array or sequence containing the data
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Can have any dimensionality
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axis : int
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The axis along which to take the mean. See `numpy.mean` for a
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description of this argument.
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See Also
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--------
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detrend_linear : Another detrend algorithm.
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detrend_none : Another detrend algorithm.
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detrend : A wrapper around all the detrend algorithms.
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r zaxis(=%s) out of boundsT)Úkeepdims)r r# r$ r% r ©r r s r
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r r � sN € ô( �
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‰
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�1‹
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€Aà ИD ™F Q§V¡VšOÜÐ2°TÑ9Ó:Ð:à ˆq�v‰v�d TˆvÓ*Ñ *Ð*r c ó — | S )až
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Return *x*: no detrending.
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Parameters
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----------
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x : any object
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An object containing the data
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axis : int
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This parameter is ignored.
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It is included for compatibility with detrend_mean
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See Also
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--------
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detrend_mean : Another detrend algorithm.
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detrend_linear : Another detrend algorithm.
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detrend : A wrapper around all the detrend algorithms.
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r r* s r
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r! r! � s € ð&
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€Hr c ó¨ — t j | « } | j dkD r t d« ‚| j s!t j d| j
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¬« S t j | j t ¬« }t j || d¬« }|d |d z }| j « ||j « z z
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}| ||z |z z
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S )ab
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Return *x* minus best fit line; 'linear' detrending.
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Parameters
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----------
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y : 0-D or 1-D array or sequence
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Array or sequence containing the data
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See Also
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--------
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detrend_mean : Another detrend algorithm.
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detrend_none : Another detrend algorithm.
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detrend : A wrapper around all the detrend algorithms.
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r zy cannot have ndim > 1g )Údtype)Úbias)r r )r r ) r r# r$ r% Úarrayr- ÚarangeÚsizeÚfloatÚcovr )Úyr ÚCÚbÚas r
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dk7 rd} n| €d} |d k( r|}
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|dz r |dz
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dz dz }n|dz }d}n |d
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k( r|dz r |dz dz }
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n|dz dz }
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d}t j |« s& |t j || j « « }t |« |k7 r t d« ‚t
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j j j | |d¬« dd||z
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… j }t# ||d¬« }||j% d« z }t
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j& j' ||d¬« d
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j& j) |d|z « d|
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j j j ||d¬« dd||z
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j& j' ||d¬« d|
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…dd…f }t j* |« |z }n‚|
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dk( rt j* |« |z }nd|
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dk( r't j, |« |j/ « z }n8|
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dk( s|
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dk( rt j0 |« }n|
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dk( r||j/ « z }|
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dk( ra|dz st3 dd d« }n
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t3 ddd« }||xx z cc< | r ||z }||dz j/ « z }n||j/ « dz z }t j4 |dz t | « |dz z
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dz ||z
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« |z }|d k( r3t j6 | d¬« }t j6 || d¬« }n|dz s
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|d xx d z cc< |
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dk( rt j8 |d¬« }|||fS ) z�
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Private helper implementing the common parts between the psd, csd,
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spectrogram and complex, magnitude, angle, and phase spectrums.
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NTé r é z noverlap must be less than NFFTr Úpsd)r r; ÚcomplexÚ magnitudeÚangleÚphase©Úmodez*x and y must be equal if mode is not 'psd'ÚtwosidedÚonesided)r rC rB )ÚsidesFr g ð?ç @z7The window length must match the data's first dimensionr )éÿÿÿÿr )Únr r= r> r? r< rF ) r! r r% r Ú
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check_in_listr r# Ú iscomplexobjr
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scale_by_freqrA Ú same_datarG ÚnumFreqsÚ
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freqcenterÚscaling_factorÚresultÚfreqsÚresultYÚslcÚts r
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zu
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Private helper implementing the commonality between the complex, magnitude,
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angle, and phase spectrums.
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The sampling frequency (samples per time unit). It is used to calculate
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the Fourier frequencies, *freqs*, in cycles per time unit.
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window : callable or ndarray, default: `.window_hanning`
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A function or a vector of length *NFFT*. To create window vectors see
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`.window_hanning`, `.window_none`, `numpy.blackman`, `numpy.hamming`,
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`numpy.bartlett`, `scipy.signal`, `scipy.signal.get_window`, etc. If a
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function is passed as the argument, it must take a data segment as an
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argument and return the windowed version of the segment.
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sides : {'default', 'onesided', 'twosided'}, optional
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Which sides of the spectrum to return. 'default' is one-sided for real
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data and two-sided for complex data. 'onesided' forces the return of a
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one-sided spectrum, while 'twosided' forces two-sided.aÚ pad_to : int, optional
|
|||
|
|
The number of points to which the data segment is padded when performing
|
|||
|
|
the FFT. While not increasing the actual resolution of the spectrum (the
|
|||
|
|
minimum distance between resolvable peaks), this can give more points in
|
|||
|
|
the plot, allowing for more detail. This corresponds to the *n* parameter
|
|||
|
|
in the call to `~numpy.fft.fft`. The default is None, which sets *pad_to*
|
|||
|
|
equal to the length of the input signal (i.e. no padding).af pad_to : int, optional
|
|||
|
|
The number of points to which the data segment is padded when performing
|
|||
|
|
the FFT. This can be different from *NFFT*, which specifies the number
|
|||
|
|
of data points used. While not increasing the actual resolution of the
|
|||
|
|
spectrum (the minimum distance between resolvable peaks), this can give
|
|||
|
|
more points in the plot, allowing for more detail. This corresponds to
|
|||
|
|
the *n* parameter in the call to `~numpy.fft.fft`. The default is None,
|
|||
|
|
which sets *pad_to* equal to *NFFT*
|
|||
|
|
|
|||
|
|
NFFT : int, default: 256
|
|||
|
|
The number of data points used in each block for the FFT. A power 2 is
|
|||
|
|
most efficient. This should *NOT* be used to get zero padding, or the
|
|||
|
|
scaling of the result will be incorrect; use *pad_to* for this instead.
|
|||
|
|
|
|||
|
|
detrend : {'none', 'mean', 'linear'} or callable, default: 'none'
|
|||
|
|
The function applied to each segment before fft-ing, designed to remove
|
|||
|
|
the mean or linear trend. Unlike in MATLAB, where the *detrend* parameter
|
|||
|
|
is a vector, in Matplotlib it is a function. The :mod:`~matplotlib.mlab`
|
|||
|
|
module defines `.detrend_none`, `.detrend_mean`, and `.detrend_linear`,
|
|||
|
|
but you can use a custom function as well. You can also use a string to
|
|||
|
|
choose one of the functions: 'none' calls `.detrend_none`. 'mean' calls
|
|||
|
|
`.detrend_mean`. 'linear' calls `.detrend_linear`.
|
|||
|
|
|
|||
|
|
scale_by_freq : bool, default: True
|
|||
|
|
Whether the resulting density values should be scaled by the scaling
|
|||
|
|
frequency, which gives density in units of 1/Hz. This allows for
|
|||
|
|
integration over the returned frequency values. The default is True for
|
|||
|
|
MATLAB compatibility.)ÚSpectralÚSingle_SpectrumÚPSDc óN — t | d||||||||¬«
|
|||
|
|
\ } }
|
|||
|
|
| j |
|
|||
|
|
fS )a
|
|||
|
|
Compute the power spectral density.
|
|||
|
|
|
|||
|
|
The power spectral density :math:`P_{xx}` by Welch's average
|
|||
|
|
periodogram method. The vector *x* is divided into *NFFT* length
|
|||
|
|
segments. Each segment is detrended by function *detrend* and
|
|||
|
|
windowed by function *window*. *noverlap* gives the length of
|
|||
|
|
the overlap between segments. The :math:`|\mathrm{fft}(i)|^2`
|
|||
|
|
of each segment :math:`i` are averaged to compute :math:`P_{xx}`.
|
|||
|
|
|
|||
|
|
If len(*x*) < *NFFT*, it will be zero padded to *NFFT*.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x : 1-D array or sequence
|
|||
|
|
Array or sequence containing the data
|
|||
|
|
|
|||
|
|
%(Spectral)s
|
|||
|
|
|
|||
|
|
%(PSD)s
|
|||
|
|
|
|||
|
|
noverlap : int, default: 0 (no overlap)
|
|||
|
|
The number of points of overlap between segments.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
Pxx : 1-D array
|
|||
|
|
The values for the power spectrum :math:`P_{xx}` (real valued)
|
|||
|
|
|
|||
|
|
freqs : 1-D array
|
|||
|
|
The frequencies corresponding to the elements in *Pxx*
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
Bendat & Piersol -- Random Data: Analysis and Measurement Procedures, John
|
|||
|
|
Wiley & Sons (1986)
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
specgram
|
|||
|
|
`specgram` differs in the default overlap; in not returning the mean of
|
|||
|
|
the segment periodograms; and in returning the times of the segments.
|
|||
|
|
|
|||
|
|
magnitude_spectrum : returns the magnitude spectrum.
|
|||
|
|
|
|||
|
|
csd : returns the spectral density between two signals.
|
|||
|
|
N)
|
|||
|
|
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|||
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|
|||
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|
|||
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|
|||
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|||
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|
|||
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|
|||
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|||
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|
|||
|
|
dd…df }
|
|||
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|
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|
|||
|
|
| fS ) a
|
|||
|
|
Compute the cross-spectral density.
|
|||
|
|
|
|||
|
|
The cross spectral density :math:`P_{xy}` by Welch's average
|
|||
|
|
periodogram method. The vectors *x* and *y* are divided into
|
|||
|
|
*NFFT* length segments. Each segment is detrended by function
|
|||
|
|
*detrend* and windowed by function *window*. *noverlap* gives
|
|||
|
|
the length of the overlap between segments. The product of
|
|||
|
|
the direct FFTs of *x* and *y* are averaged over each segment
|
|||
|
|
to compute :math:`P_{xy}`, with a scaling to correct for power
|
|||
|
|
loss due to windowing.
|
|||
|
|
|
|||
|
|
If len(*x*) < *NFFT* or len(*y*) < *NFFT*, they will be zero
|
|||
|
|
padded to *NFFT*.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x, y : 1-D arrays or sequences
|
|||
|
|
Arrays or sequences containing the data
|
|||
|
|
|
|||
|
|
%(Spectral)s
|
|||
|
|
|
|||
|
|
%(PSD)s
|
|||
|
|
|
|||
|
|
noverlap : int, default: 0 (no overlap)
|
|||
|
|
The number of points of overlap between segments.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
Pxy : 1-D array
|
|||
|
|
The values for the cross spectrum :math:`P_{xy}` before scaling (real
|
|||
|
|
valued)
|
|||
|
|
|
|||
|
|
freqs : 1-D array
|
|||
|
|
The frequencies corresponding to the elements in *Pxy*
|
|||
|
|
|
|||
|
|
References
|
|||
|
|
----------
|
|||
|
|
Bendat & Piersol -- Random Data: Analysis and Measurement Procedures, John
|
|||
|
|
Wiley & Sons (1986)
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
psd : equivalent to setting ``y = x``.
|
|||
|
|
Nr: r; rl r9 r r r )rj r$ rn r )
|
|||
|
|
r r4 rZ r[ r r] r^ r_ rD r` ÚPxyrf rp s
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|||
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|
r
|
|||
|
|
rv rv s‹ € ð` €|؈Ü$ q¨A°D¸RØ29À&Ø.6¸vØ+0À
|
|||
|
|
Ø*/ô 1�M€Cˆ�ð ‡x�x�1‚}Ø �9‰9�Q‰<˜!Ò Ø—(‘( �(Ó"ˆCð �ˆ:Ðð ’a˜�d‘)ˆCØ �ˆ:Ðr a9 Compute the {quantity} of *x*.
|
|||
|
|
Data is padded to a length of *pad_to* and the windowing function *window* is
|
|||
|
|
applied to the signal.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x : 1-D array or sequence
|
|||
|
|
Array or sequence containing the data
|
|||
|
|
|
|||
|
|
{Spectral}
|
|||
|
|
|
|||
|
|
{Single_Spectrum}
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
spectrum : 1-D array
|
|||
|
|
The {quantity}.
|
|||
|
|
freqs : 1-D array
|
|||
|
|
The frequencies corresponding to the elements in *spectrum*.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
psd
|
|||
|
|
Returns the power spectral density.
|
|||
|
|
complex_spectrum
|
|||
|
|
Returns the complex-valued frequency spectrum.
|
|||
|
|
magnitude_spectrum
|
|||
|
|
Returns the absolute value of the `complex_spectrum`.
|
|||
|
|
angle_spectrum
|
|||
|
|
Returns the angle of the `complex_spectrum`.
|
|||
|
|
phase_spectrum
|
|||
|
|
Returns the phase (unwrapped angle) of the `complex_spectrum`.
|
|||
|
|
specgram
|
|||
|
|
Can return the complex spectrum of segments within the signal.
|
|||
|
|
r< Úquantityz!complex-valued frequency spectrumr= z4magnitude (absolute value) of the frequency spectrumr> z8angle of the frequency spectrum (wrapped phase spectrum)r? z:phase of the frequency spectrum (unwrapped phase spectrum)c
|
|||
|
|
|
|||
|
|
óØ — |€d}|€d}t | « |k r%t j d|› dt | « › d�« t | d||||||||| ¬« \ }
|
|||
|
|
} } | dk7 r |
|
|||
|
|
j }
|
|||
|
|
|
|
|||
|
|
| | fS ) a
|
|||
|
|
Compute a spectrogram.
|
|||
|
|
|
|||
|
|
Compute and plot a spectrogram of data in *x*. Data are split into
|
|||
|
|
*NFFT* length segments and the spectrum of each section is
|
|||
|
|
computed. The windowing function *window* is applied to each
|
|||
|
|
segment, and the amount of overlap of each segment is
|
|||
|
|
specified with *noverlap*.
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x : array-like
|
|||
|
|
1-D array or sequence.
|
|||
|
|
|
|||
|
|
%(Spectral)s
|
|||
|
|
|
|||
|
|
%(PSD)s
|
|||
|
|
|
|||
|
|
noverlap : int, default: 128
|
|||
|
|
The number of points of overlap between blocks.
|
|||
|
|
mode : str, default: 'psd'
|
|||
|
|
What sort of spectrum to use:
|
|||
|
|
'psd'
|
|||
|
|
Returns the power spectral density.
|
|||
|
|
'complex'
|
|||
|
|
Returns the complex-valued frequency spectrum.
|
|||
|
|
'magnitude'
|
|||
|
|
Returns the magnitude spectrum.
|
|||
|
|
'angle'
|
|||
|
|
Returns the phase spectrum without unwrapping.
|
|||
|
|
'phase'
|
|||
|
|
Returns the phase spectrum with unwrapping.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
spectrum : array-like
|
|||
|
|
2D array, columns are the periodograms of successive segments.
|
|||
|
|
|
|||
|
|
freqs : array-like
|
|||
|
|
1-D array, frequencies corresponding to the rows in *spectrum*.
|
|||
|
|
|
|||
|
|
t : array-like
|
|||
|
|
1-D array, the times corresponding to midpoints of segments
|
|||
|
|
(i.e the columns in *spectrum*).
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
psd : differs in the overlap and in the return values.
|
|||
|
|
complex_spectrum : similar, but with complex valued frequencies.
|
|||
|
|
magnitude_spectrum : similar single segment when *mode* is 'magnitude'.
|
|||
|
|
angle_spectrum : similar to single segment when *mode* is 'angle'.
|
|||
|
|
phase_spectrum : similar to single segment when *mode* is 'phase'.
|
|||
|
|
|
|||
|
|
Notes
|
|||
|
|
-----
|
|||
|
|
*detrend* and *scale_by_freq* only apply when *mode* is set to 'psd'.
|
|||
|
|
|
|||
|
|
Né€ r: z6Only one segment is calculated since parameter NFFT (=z) >= signal length (=z).rl r< )r
|
|||
|
|
r Ú
|
|||
|
|
warn_externalrj rm )
|
|||
|
|
r rZ r[ r r] r^ r_ rD r` rA ro rf ri s
|
|||
|
|
r
|
|||
|
|
Úspecgramr~ } s� € ð| ÐØˆØ €|؈Ü
|
|||
|
|
ˆ1ƒv�‚~Ü ×Ñð %Ø%) FÐ*?ÄÀAøxÀrð Kô Lô &¨¨T¸ÀØ3:À6Ø/7ÀØ,1Ø4AØ+/ô 1�N€Dˆ%�ð ˆyÒØ�y‰yˆà �˜ˆ>Ðr r: r9 r c
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ó — t | « d|z k r t d« ‚t | |||||||| « \ }
|
|||
|
|
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|
|||
|
|
\ }
|
|||
|
|
} t j
|
|||
|
|
|
|
|||
|
|
« dz |
|
|||
|
|
| z z }|| fS )aÉ
|
|||
|
|
The coherence between *x* and *y*. Coherence is the normalized
|
|||
|
|
cross spectral density:
|
|||
|
|
|
|||
|
|
.. math::
|
|||
|
|
|
|||
|
|
C_{xy} = \frac{|P_{xy}|^2}{P_{xx}P_{yy}}
|
|||
|
|
|
|||
|
|
Parameters
|
|||
|
|
----------
|
|||
|
|
x, y
|
|||
|
|
Array or sequence containing the data
|
|||
|
|
|
|||
|
|
%(Spectral)s
|
|||
|
|
|
|||
|
|
%(PSD)s
|
|||
|
|
|
|||
|
|
noverlap : int, default: 0 (no overlap)
|
|||
|
|
The number of points of overlap between segments.
|
|||
|
|
|
|||
|
|
Returns
|
|||
|
|
-------
|
|||
|
|
Cxy : 1-D array
|
|||
|
|
The coherence vector.
|
|||
|
|
freqs : 1-D array
|
|||
|
|
The frequencies for the elements in *Cxy*.
|
|||
|
|
|
|||
|
|
See Also
|
|||
|
|
--------
|
|||
|
|
:func:`psd`, :func:`csd` :
|
|||
|
|
For information about the methods used to compute :math:`P_{xy}`,
|
|||
|
|
:math:`P_{xx}` and :math:`P_{yy}`.
|
|||
|
|
r9 zvCoherence is calculated by averaging over *NFFT* length segments. Your signal is too short for your choice of *NFFT*.)r
|
|||
|
|
r% r; rv r rU )r r4 rZ r[ r r] r^ r_ rD r` rw ÚfÚPyyry ÚCxys r
|