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Peak detection in Python
Musings about the peakdetect functions by Sixten Bergman
Note that this code should work with both python 2.7 and python3.x.
All the peak detection functions in __all__ of peakdetect.py will work on
consistent waveforms, but only peakdetect.peakdetect can properly handle
offsets.
The most accurate method for pure sine seems to be peakdetect_parabola,
which for a 50Hz sine wave lasting 0.1s with 10k samples has an error in
the order of 1e-10, whilst a naive most extreme sample will have an error
in the order of 7e-5 for the position and 4e-7 for the amplitude
Do note that this accuracy most likely doesn't stay true for any real world
data where you'll have noise and harmonics in the signal which may produce
errors in the functions, which may be smaller or larger then the error of
naively using the highest/lowest point in a local maxima/minima.
The sine fit function seem to perform even worse than a just retrieving the
highest or lowest data point and is as such not recommended. The reason for
this as far as I can tell is that the scipy.optimize.curve_fit can't optimize
the variables.
For parabola fit to function well, it must be fitted to a small section of the
peak as the curvature will start to mismatch with the function, but this also
means that the parabola should be quite sensitive to noise
FFT interpolation has between 0 to 2 orders of magnitude improvement over a
raw peak fit. To obtain this improvement the wave needs to be heavily padded
in length
Spline seems to have similar performance to a FFT interpolation of the time
domain. Spline does however seem to be better at estimating amplitude than the
FFT method, but is unknown if this will hold true for wave-shapes that are
noisy.
It should also be noted that the errors as given in "Missmatch data.txt"
generated by the test routine are for pure functions with no noise, so the only
error being reduced by the "non-raw" peakdetect functions are errors stemming
low time resolution and are in no way an indication of how the functions can
handle any kind of noise that real signals will have.
Automatic tests for sine fitted peak detection is disabled due to it's problems
Avoid using the following functions as they're questionable in performance:
peakdetect_sine
peakdetect_sine_locked
#!/usr/bin/python2
# Copyright (C) 2016 Sixten Bergman
# License WTFPL
#
# This program is free software. It comes without any warranty, to the extent
# permitted by applicable law.
# You can redistribute it and/or modify it under the terms of the Do What The
# Fuck You Want To Public License, Version 2, as published by Sam Hocevar. See
# http://www.wtfpl.net/ for more details.
#
import numpy as np
from math import pi, sqrt
__all__ = [
'ACV_A1',
'ACV_A2',
'ACV_A3',
'ACV_A4',
'ACV_A5',
'ACV_A6',
'ACV_A7',
'ACV_A8'
]
#Heavyside step function
H_num = lambda t: 1 if t > 0 else 0
H = lambda T: np.asarray([1 if t > 0 else 0 for t in T])
# pure sine
def ACV_A1(T, Hz=50):
"""
Generate a pure sine wave at a specified frequency
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
T = np.asarray(T, dtype=np.float64)
return ampl * sqrt(2) * np.sin(2*pi*Hz * T)
def ACV_A2(T, Hz=50):
"""
Generate a pure sine wave with a DC offset at a specified frequency
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
offset = 500
T = np.asarray(T, dtype=np.float64)
return ampl * sqrt(2) * np.sin(2*pi*Hz * T) + offset
def ACV_A3(T, Hz=50):
"""
Generate a fundamental with a 3rd overtone
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
T = np.asarray(T, dtype=np.float64)
main_wave = np.sin(2*pi*Hz * T)
harmonic_wave = 0.05 * np.sin(2*pi*Hz * T * 4 + pi * 2 / 3)
return ampl * sqrt(2) * (main_wave + harmonic_wave)
def ACV_A4(T, Hz=50):
"""
Generate a fundamental with a 4th overtone
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
T = np.asarray(T, dtype=np.float64)
main_wave = np.sin(2*pi*Hz * T)
harmonic_wave = 0.07 * np.sin(2*pi*Hz * T * 5 + pi * 22 / 18)
return ampl * sqrt(2) * (main_wave + harmonic_wave)
def ACV_A5(T, Hz=50):
"""
Generate a realistic triangle wave
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
T = np.asarray(T, dtype=np.float64)
wave_1 = np.sin(2*pi*Hz * T)
wave_2 = 0.05 * np.sin(2*pi*Hz * T * 3 - pi)
wave_3 = 0.05 * np.sin(2*pi*Hz * T * 5)
wave_4 = 0.02 * np.sin(2*pi*Hz * T * 7 - pi)
wave_5 = 0.01 * np.sin(2*pi*Hz * T * 9)
return ampl * sqrt(2) * (wave_1 + wave_2 + wave_3 + wave_4 + wave_5)
def ACV_A6(T, Hz=50):
"""
Generate a realistic triangle wave
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
T = np.asarray(T, dtype=np.float64)
wave_1 = np.sin(2*pi*Hz * T)
wave_2 = 0.02 * np.sin(2*pi*Hz * T * 3 - pi)
wave_3 = 0.02 * np.sin(2*pi*Hz * T * 5)
wave_4 = 0.0015 * np.sin(2*pi*Hz * T * 7 - pi)
wave_5 = 0.009 * np.sin(2*pi*Hz * T * 9)
return ampl * sqrt(2) * (wave_1 + wave_2 + wave_3 + wave_4 + wave_5)
def ACV_A7(T, Hz=50):
"""
Generate a growing sine wave, where the wave starts at 0 and reaches 0.9 of
full amplitude at 250 cycles. Thereafter it will linearly increase to full
amplitude at 500 cycles and terminate to 0
Frequency locked to 50Hz and = 0 at t>10
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
Hz = 50
T = np.asarray(T, dtype=np.float64)
wave_main = np.sin(2*pi*Hz * T)
step_func = (0.9 * T / 5 * H(5-T) + H(T-5) * H(10-T) * (0.9 + 0.1 * (T-5) / 5))
return ampl * sqrt(2) * wave_main * step_func
def ACV_A8(T, Hz=50):
"""
Generate a growing sine wave, which reaches 100 times the amplitude at
500 cycles
frequency not implemented and signal = 0 at t>1000*pi
signal frequency = 0.15915494309189535 Hz?
keyword arguments:
T -- time points to generate the waveform given in seconds
Hz -- The desired frequency of the signal (default:50)
"""
ampl = 1000
Hz = 50
T = np.asarray(T, dtype=np.float64)
wave_main = np.sin(T)
step_func = T / (10 * pi) * H(10 - T / (2*pi*Hz))
return ampl * sqrt(2) * wave_main * step_func
_ACV_A1_L = lambda T, Hz = 50: 1000 * sqrt(2) * np.sin(2*pi*Hz * T)
#
_ACV_A2_L = lambda T, Hz = 50: 1000 * sqrt(2) * np.sin(2*pi*Hz * T) + 500
#
_ACV_A3_L = lambda T, Hz = 50: 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) +
0.05 * np.sin(2*pi*Hz * T * 4 + pi * 2 / 3))
#
_ACV_A4_L = lambda T, Hz = 50:( 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) +
0.07 * np.sin(2*pi*Hz * T * 5 + pi * 22 / 18)))
# Realistic triangle
_ACV_A5_L = lambda T, Hz = 50:( 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) +
0.05 * np.sin(2*pi*Hz * T * 3 - pi) +
0.05 * np.sin(2*pi*Hz * T * 5) +
0.02 * np.sin(2*pi*Hz * T * 7 - pi) +
0.01 * np.sin(2*pi*Hz * T * 9)))
#
_ACV_A6_L = lambda T, Hz = 50:( 1000 * sqrt(2) * (np.sin(2*pi*Hz * T) +
0.02 * np.sin(2*pi*Hz * T * 3 - pi) +
0.02 * np.sin(2*pi*Hz * T * 5) +
0.0015 * np.sin(2*pi*Hz * T * 7 - pi) +
0.009 * np.sin(2*pi*Hz * T * 9)))
#A7 & A8 convert so that a input of 16*pi corresponds to a input 0.25 in the current version
_ACV_A7_OLD = lambda T: [1000 * sqrt(2) * np.sin(100 * pi * t) *
(0.9 * t / 5 * H_num(5-t) + H_num(t-5) * H_num(10-t) * (0.9 + 0.1 * (t-5) / 5)) for t in T]
_ACV_A8_OLD = lambda T: [1000 * sqrt(2) * np.sin(t) *
t / (10 * pi) * H_num(10 - t / (100 * pi)) for t in T]
if __name__ == "__main__":
#create 1 period triangle
x = np.linspace(0, 0.02, 4000)
y = ACV_A5(x)
#!/usr/bin/python2
# Copyright (C) 2016 Sixten Bergman
# License WTFPL
#
# This program is free software. It comes without any warranty, to the extent
# permitted by applicable law.
# You can redistribute it and/or modify it under the terms of the Do What The
# Fuck You Want To Public License, Version 2, as published by Sam Hocevar. See
# http://www.wtfpl.net/ for more details.
#
# note that the function peakdetect is derived from code which was released to
# public domain see: http://billauer.co.il/peakdet.html
#
import logging
from math import pi, log
import numpy as np
import pylab
from scipy import fft, ifft
from scipy.optimize import curve_fit
from scipy.signal import cspline1d_eval, cspline1d
__all__ = [
"peakdetect",
"peakdetect_fft",
"peakdetect_parabola",
"peakdetect_sine",
"peakdetect_sine_locked",
"peakdetect_spline",
"peakdetect_zero_crossing",
"zero_crossings",
"zero_crossings_sine_fit"
]
def _datacheck_peakdetect(x_axis, y_axis):
if x_axis is None:
x_axis = range(len(y_axis))
if len(y_axis) != len(x_axis):
raise ValueError(
"Input vectors y_axis and x_axis must have same length")
#needs to be a numpy array
y_axis = np.array(y_axis)
x_axis = np.array(x_axis)
return x_axis, y_axis
def _pad(fft_data, pad_len):
"""
Pads fft data to interpolate in time domain
keyword arguments:
fft_data -- the fft
pad_len -- By how many times the time resolution should be increased by
return: padded list
"""
l = len(fft_data)
n = _n(l * pad_len)
fft_data = list(fft_data)
return fft_data[:l // 2] + [0] * (2**n-l) + fft_data[l // 2:]
def _n(x):
"""
Find the smallest value for n, which fulfils 2**n >= x
keyword arguments:
x -- the value, which 2**n must surpass
return: the integer n
"""
return int(log(x)/log(2)) + 1
def _peakdetect_parabola_fitter(raw_peaks, x_axis, y_axis, points):
"""
Performs the actual parabola fitting for the peakdetect_parabola function.
keyword arguments:
raw_peaks -- A list of either the maxima or the minima peaks, as given
by the peakdetect functions, with index used as x-axis
x_axis -- A numpy array of all the x values
y_axis -- A numpy array of all the y values
points -- How many points around the peak should be used during curve
fitting, must be odd.
return: A list giving all the peaks and the fitted waveform, format:
[[x, y, [fitted_x, fitted_y]]]
"""
func = lambda x, a, tau, c: a * ((x - tau) ** 2) + c
fitted_peaks = []
distance = abs(x_axis[raw_peaks[1][0]] - x_axis[raw_peaks[0][0]]) / 4
for peak in raw_peaks:
index = peak[0]
x_data = x_axis[index - points // 2: index + points // 2 + 1]
y_data = y_axis[index - points // 2: index + points // 2 + 1]
# get a first approximation of tau (peak position in time)
tau = x_axis[index]
# get a first approximation of peak amplitude
c = peak[1]
a = np.sign(c) * (-1) * (np.sqrt(abs(c))/distance)**2
"""Derived from ABC formula to result in a solution where A=(rot(c)/t)**2"""
# build list of approximations
p0 = (a, tau, c)
popt, pcov = curve_fit(func, x_data, y_data, p0)
# retrieve tau and c i.e x and y value of peak
x, y = popt[1:3]
# create a high resolution data set for the fitted waveform
x2 = np.linspace(x_data[0], x_data[-1], points * 10)
y2 = func(x2, *popt)
fitted_peaks.append([x, y, [x2, y2]])
return fitted_peaks
def peakdetect_parabole(*args, **kwargs):
"""
Misspelling of peakdetect_parabola
function is deprecated please use peakdetect_parabola
"""
logging.warn("peakdetect_parabole is deprecated due to misspelling use: peakdetect_parabola")
return peakdetect_parabola(*args, **kwargs)
def peakdetect(y_axis, x_axis = None, lookahead = 200, delta=0):
"""
Converted from/based on a MATLAB script at:
http://billauer.co.il/peakdet.html
function for detecting local maxima and minima in a signal.
Discovers peaks by searching for values which are surrounded by lower
or larger values for maxima and minima respectively
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list and is used
in the return to specify the position of the peaks. If omitted an
index of the y_axis is used.
(default: None)
lookahead -- distance to look ahead from a peak candidate to determine if
it is the actual peak
(default: 200)
'(samples / period) / f' where '4 >= f >= 1.25' might be a good value
delta -- this specifies a minimum difference between a peak and
the following points, before a peak may be considered a peak. Useful
to hinder the function from picking up false peaks towards to end of
the signal. To work well delta should be set to delta >= RMSnoise * 5.
(default: 0)
When omitted delta function causes a 20% decrease in speed.
When used Correctly it can double the speed of the function
return: two lists [max_peaks, min_peaks] containing the positive and
negative peaks respectively. Each cell of the lists contains a tuple
of: (position, peak_value)
to get the average peak value do: np.mean(max_peaks, 0)[1] on the
results to unpack one of the lists into x, y coordinates do:
x, y = zip(*max_peaks)
"""
max_peaks = []
min_peaks = []
dump = [] #Used to pop the first hit which almost always is false
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
# store data length for later use
length = len(y_axis)
#perform some checks
if lookahead < 1:
raise ValueError("Lookahead must be '1' or above in value")
if not (np.isscalar(delta) and delta >= 0):
raise ValueError("delta must be a positive number")
#maxima and minima candidates are temporarily stored in
#mx and mn respectively
mn, mx = np.Inf, -np.Inf
#Only detect peak if there is 'lookahead' amount of points after it
for index, (x, y) in enumerate(zip(x_axis[:-lookahead],
y_axis[:-lookahead])):
if y > mx:
mx = y
mxpos = x
if y < mn:
mn = y
mnpos = x
####look for max####
if y < mx-delta and mx != np.Inf:
#Maxima peak candidate found
#look ahead in signal to ensure that this is a peak and not jitter
if y_axis[index:index+lookahead].max() < mx:
max_peaks.append([mxpos, mx])
dump.append(True)
#set algorithm to only find minima now
mx = np.Inf
mn = np.Inf
if index+lookahead >= length:
#end is within lookahead no more peaks can be found
break
continue
#else: #slows shit down this does
# mx = ahead
# mxpos = x_axis[np.where(y_axis[index:index+lookahead]==mx)]
####look for min####
if y > mn+delta and mn != -np.Inf:
#Minima peak candidate found
#look ahead in signal to ensure that this is a peak and not jitter
if y_axis[index:index+lookahead].min() > mn:
min_peaks.append([mnpos, mn])
dump.append(False)
#set algorithm to only find maxima now
mn = -np.Inf
mx = -np.Inf
if index+lookahead >= length:
#end is within lookahead no more peaks can be found
break
#else: #slows shit down this does
# mn = ahead
# mnpos = x_axis[np.where(y_axis[index:index+lookahead]==mn)]
#Remove the false hit on the first value of the y_axis
try:
if dump[0]:
max_peaks.pop(0)
else:
min_peaks.pop(0)
del dump
except IndexError:
#no peaks were found, should the function return empty lists?
pass
return [max_peaks, min_peaks]
def peakdetect_fft(y_axis, x_axis, pad_len = 20):
"""
Performs a FFT calculation on the data and zero-pads the results to
increase the time domain resolution after performing the inverse fft and
send the data to the 'peakdetect' function for peak
detection.
Omitting the x_axis is forbidden as it would make the resulting x_axis
value silly if it was returned as the index 50.234 or similar.
Will find at least 1 less peak then the 'peakdetect_zero_crossing'
function, but should result in a more precise value of the peak as
resolution has been increased. Some peaks are lost in an attempt to
minimize spectral leakage by calculating the fft between two zero
crossings for n amount of signal periods.
The biggest time eater in this function is the ifft and thereafter it's
the 'peakdetect' function which takes only half the time of the ifft.
Speed improvements could include to check if 2**n points could be used for
fft and ifft or change the 'peakdetect' to the 'peakdetect_zero_crossing',
which is maybe 10 times faster than 'peakdetct'. The pro of 'peakdetect'
is that it results in one less lost peak. It should also be noted that the
time used by the ifft function can change greatly depending on the input.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list and is used
in the return to specify the position of the peaks.
pad_len -- By how many times the time resolution should be
increased by, e.g. 1 doubles the resolution. The amount is rounded up
to the nearest 2**n amount
(default: 20)
return: two lists [max_peaks, min_peaks] containing the positive and
negative peaks respectively. Each cell of the lists contains a tuple
of: (position, peak_value)
to get the average peak value do: np.mean(max_peaks, 0)[1] on the
results to unpack one of the lists into x, y coordinates do:
x, y = zip(*max_peaks)
"""
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
zero_indices = zero_crossings(y_axis, window_len = 11)
#select a n amount of periods
last_indice = - 1 - (1 - len(zero_indices) & 1)
###
# Calculate the fft between the first and last zero crossing
# this method could be ignored if the beginning and the end of the signal
# are unnecessary as any errors induced from not using whole periods
# should mainly manifest in the beginning and the end of the signal, but
# not in the rest of the signal
# this is also unnecessary if the given data is an amount of whole periods
###
fft_data = fft(y_axis[zero_indices[0]:zero_indices[last_indice]])
padd = lambda x, c: x[:len(x) // 2] + [0] * c + x[len(x) // 2:]
n = lambda x: int(log(x)/log(2)) + 1
# pads to 2**n amount of samples
fft_padded = padd(list(fft_data), 2 **
n(len(fft_data) * pad_len) - len(fft_data))
# There is amplitude decrease directly proportional to the sample increase
sf = len(fft_padded) / float(len(fft_data))
# There might be a leakage giving the result an imaginary component
# Return only the real component
y_axis_ifft = ifft(fft_padded).real * sf #(pad_len + 1)
x_axis_ifft = np.linspace(
x_axis[zero_indices[0]], x_axis[zero_indices[last_indice]],
len(y_axis_ifft))
# get the peaks to the interpolated waveform
max_peaks, min_peaks = peakdetect(y_axis_ifft, x_axis_ifft, 500,
delta = abs(np.diff(y_axis).max() * 2))
#max_peaks, min_peaks = peakdetect_zero_crossing(y_axis_ifft, x_axis_ifft)
# store one 20th of a period as waveform data
data_len = int(np.diff(zero_indices).mean()) / 10
data_len += 1 - data_len & 1
return [max_peaks, min_peaks]
def peakdetect_parabola(y_axis, x_axis, points = 31):
"""
Function for detecting local maxima and minima in a signal.
Discovers peaks by fitting the model function: y = k (x - tau) ** 2 + m
to the peaks. The amount of points used in the fitting is set by the
points argument.
Omitting the x_axis is forbidden as it would make the resulting x_axis
value silly, if it was returned as index 50.234 or similar.
will find the same amount of peaks as the 'peakdetect_zero_crossing'
function, but might result in a more precise value of the peak.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list and is used
in the return to specify the position of the peaks.
points -- How many points around the peak should be used during curve
fitting (default: 31)
return: two lists [max_peaks, min_peaks] containing the positive and
negative peaks respectively. Each cell of the lists contains a tuple
of: (position, peak_value)
to get the average peak value do: np.mean(max_peaks, 0)[1] on the
results to unpack one of the lists into x, y coordinates do:
x, y = zip(*max_peaks)
"""
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
# make the points argument odd
points += 1 - points % 2
#points += 1 - int(points) & 1 slower when int conversion needed
# get raw peaks
max_raw, min_raw = peakdetect_zero_crossing(y_axis)
# define output variable
max_peaks = []
min_peaks = []
max_ = _peakdetect_parabola_fitter(max_raw, x_axis, y_axis, points)
min_ = _peakdetect_parabola_fitter(min_raw, x_axis, y_axis, points)
max_peaks = map(lambda x: [x[0], x[1]], max_)
max_fitted = map(lambda x: x[-1], max_)
min_peaks = map(lambda x: [x[0], x[1]], min_)
min_fitted = map(lambda x: x[-1], min_)
return [max_peaks, min_peaks]
def peakdetect_sine(y_axis, x_axis, points = 31, lock_frequency = False):
"""
Function for detecting local maxima and minima in a signal.
Discovers peaks by fitting the model function:
y = A * sin(2 * pi * f * (x - tau)) to the peaks. The amount of points used
in the fitting is set by the points argument.
Omitting the x_axis is forbidden as it would make the resulting x_axis
value silly if it was returned as index 50.234 or similar.
will find the same amount of peaks as the 'peakdetect_zero_crossing'
function, but might result in a more precise value of the peak.
The function might have some problems if the sine wave has a
non-negligible total angle i.e. a k*x component, as this messes with the
internal offset calculation of the peaks, might be fixed by fitting a
y = k * x + m function to the peaks for offset calculation.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list and is used
in the return to specify the position of the peaks.
points -- How many points around the peak should be used during curve
fitting (default: 31)
lock_frequency -- Specifies if the frequency argument of the model
function should be locked to the value calculated from the raw peaks
or if optimization process may tinker with it.
(default: False)
return: two lists [max_peaks, min_peaks] containing the positive and
negative peaks respectively. Each cell of the lists contains a tuple
of: (position, peak_value)
to get the average peak value do: np.mean(max_peaks, 0)[1] on the
results to unpack one of the lists into x, y coordinates do:
x, y = zip(*max_peaks)
"""
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
# make the points argument odd
points += 1 - points % 2
#points += 1 - int(points) & 1 slower when int conversion needed
# get raw peaks
max_raw, min_raw = peakdetect_zero_crossing(y_axis)
# define output variable
max_peaks = []
min_peaks = []
# get global offset
offset = np.mean([np.mean(max_raw, 0)[1], np.mean(min_raw, 0)[1]])
# fitting a k * x + m function to the peaks might be better
#offset_func = lambda x, k, m: k * x + m
# calculate an approximate frequency of the signal
Hz_h_peak = np.diff(zip(*max_raw)[0]).mean()
Hz_l_peak = np.diff(zip(*min_raw)[0]).mean()
Hz = 1 / np.mean([Hz_h_peak, Hz_l_peak])
# model function
# if cosine is used then tau could equal the x position of the peak
# if sine were to be used then tau would be the first zero crossing
if lock_frequency:
func = lambda x_ax, A, tau: A * np.sin(
2 * pi * Hz * (x_ax - tau) + pi / 2)
else:
func = lambda x_ax, A, Hz, tau: A * np.sin(
2 * pi * Hz * (x_ax - tau) + pi / 2)
#func = lambda x_ax, A, Hz, tau: A * np.cos(2 * pi * Hz * (x_ax - tau))
#get peaks
fitted_peaks = []
for raw_peaks in [max_raw, min_raw]:
peak_data = []
for peak in raw_peaks:
index = peak[0]
x_data = x_axis[index - points // 2: index + points // 2 + 1]
y_data = y_axis[index - points // 2: index + points // 2 + 1]
# get a first approximation of tau (peak position in time)
tau = x_axis[index]
# get a first approximation of peak amplitude
A = peak[1]
# build list of approximations
if lock_frequency:
p0 = (A, tau)
else:
p0 = (A, Hz, tau)
# subtract offset from wave-shape
y_data -= offset
popt, pcov = curve_fit(func, x_data, y_data, p0)
# retrieve tau and A i.e x and y value of peak
x = popt[-1]
y = popt[0]
# create a high resolution data set for the fitted waveform
x2 = np.linspace(x_data[0], x_data[-1], points * 10)
y2 = func(x2, *popt)
# add the offset to the results
y += offset
y2 += offset
y_data += offset
peak_data.append([x, y, [x2, y2]])
fitted_peaks.append(peak_data)
# structure date for output
max_peaks = map(lambda x: [x[0], x[1]], fitted_peaks[0])
max_fitted = map(lambda x: x[-1], fitted_peaks[0])
min_peaks = map(lambda x: [x[0], x[1]], fitted_peaks[1])
min_fitted = map(lambda x: x[-1], fitted_peaks[1])
return [max_peaks, min_peaks]
def peakdetect_sine_locked(y_axis, x_axis, points = 31):
"""
Convenience function for calling the 'peakdetect_sine' function with
the lock_frequency argument as True.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list and is used
in the return to specify the position of the peaks.
points -- How many points around the peak should be used during curve
fitting (default: 31)
return: see the function 'peakdetect_sine'
"""
return peakdetect_sine(y_axis, x_axis, points, True)
def peakdetect_spline(y_axis, x_axis, pad_len=20):
"""
Performs a b-spline interpolation on the data to increase resolution and
send the data to the 'peakdetect_zero_crossing' function for peak
detection.
Omitting the x_axis is forbidden as it would make the resulting x_axis
value silly if it was returned as the index 50.234 or similar.
will find the same amount of peaks as the 'peakdetect_zero_crossing'
function, but might result in a more precise value of the peak.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list and is used
in the return to specify the position of the peaks.
x-axis must be equally spaced.
pad_len -- By how many times the time resolution should be increased by,
e.g. 1 doubles the resolution.
(default: 20)
return: two lists [max_peaks, min_peaks] containing the positive and
negative peaks respectively. Each cell of the lists contains a tuple
of: (position, peak_value)
to get the average peak value do: np.mean(max_peaks, 0)[1] on the
results to unpack one of the lists into x, y coordinates do:
x, y = zip(*max_peaks)
"""
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
# could perform a check if x_axis is equally spaced
#if np.std(np.diff(x_axis)) > 1e-15: raise ValueError
# perform spline interpolations
dx = x_axis[1] - x_axis[0]
x_interpolated = np.linspace(x_axis.min(), x_axis.max(), len(x_axis) * (pad_len + 1))
cj = cspline1d(y_axis)
y_interpolated = cspline1d_eval(cj, x_interpolated, dx=dx,x0=x_axis[0])
# get peaks
max_peaks, min_peaks = peakdetect_zero_crossing(y_interpolated, x_interpolated)
return [max_peaks, min_peaks]
def peakdetect_zero_crossing(y_axis, x_axis = None, window = 11):
"""
Function for detecting local maxima and minima in a signal.
Discovers peaks by dividing the signal into bins and retrieving the
maximum and minimum value of each the even and odd bins respectively.
Division into bins is performed by smoothing the curve and finding the
zero crossings.
Suitable for repeatable signals, where some noise is tolerated. Executes
faster than 'peakdetect', although this function will break if the offset
of the signal is too large. It should also be noted that the first and
last peak will probably not be found, as this function only can find peaks
between the first and last zero crossing.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list
and is used in the return to specify the position of the peaks. If
omitted an index of the y_axis is used.
(default: None)
window -- the dimension of the smoothing window; should be an odd integer
(default: 11)
return: two lists [max_peaks, min_peaks] containing the positive and
negative peaks respectively. Each cell of the lists contains a tuple
of: (position, peak_value)
to get the average peak value do: np.mean(max_peaks, 0)[1] on the
results to unpack one of the lists into x, y coordinates do:
x, y = zip(*max_peaks)
"""
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
zero_indices = zero_crossings(y_axis, window_len = window)
period_lengths = np.diff(zero_indices)
bins_y = [y_axis[index:index + diff] for index, diff in
zip(zero_indices, period_lengths)]
bins_x = [x_axis[index:index + diff] for index, diff in
zip(zero_indices, period_lengths)]
even_bins_y = bins_y[::2]
odd_bins_y = bins_y[1::2]
even_bins_x = bins_x[::2]
odd_bins_x = bins_x[1::2]
hi_peaks_x = []
lo_peaks_x = []
#check if even bin contains maxima
if abs(even_bins_y[0].max()) > abs(even_bins_y[0].min()):
hi_peaks = [bin.max() for bin in even_bins_y]
lo_peaks = [bin.min() for bin in odd_bins_y]
# get x values for peak
for bin_x, bin_y, peak in zip(even_bins_x, even_bins_y, hi_peaks):
hi_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
for bin_x, bin_y, peak in zip(odd_bins_x, odd_bins_y, lo_peaks):
lo_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
else:
hi_peaks = [bin.max() for bin in odd_bins_y]
lo_peaks = [bin.min() for bin in even_bins_y]
# get x values for peak
for bin_x, bin_y, peak in zip(odd_bins_x, odd_bins_y, hi_peaks):
hi_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
for bin_x, bin_y, peak in zip(even_bins_x, even_bins_y, lo_peaks):
lo_peaks_x.append(bin_x[np.where(bin_y==peak)[0][0]])
max_peaks = [[x, y] for x,y in zip(hi_peaks_x, hi_peaks)]
min_peaks = [[x, y] for x,y in zip(lo_peaks_x, lo_peaks)]
return [max_peaks, min_peaks]
def _smooth(x, window_len=11, window="hanning"):
"""
smooth the data using a window of the requested size.
This method is based on the convolution of a scaled window on the signal.
The signal is prepared by introducing reflected copies of the signal
(with the window size) in both ends so that transient parts are minimized
in the beginning and end part of the output signal.
keyword arguments:
x -- the input signal
window_len -- the dimension of the smoothing window; should be an odd
integer (default: 11)
window -- the type of window from 'flat', 'hanning', 'hamming',
'bartlett', 'blackman', where flat is a moving average
(default: 'hanning')
return: the smoothed signal
example:
t = linspace(-2,2,0.1)
x = sin(t)+randn(len(t))*0.1
y = _smooth(x)
see also:
numpy.hanning, numpy.hamming, numpy.bartlett, numpy.blackman,
numpy.convolve, scipy.signal.lfilter
"""
if x.ndim != 1:
raise ValueError("smooth only accepts 1 dimension arrays.")
if x.size < window_len:
raise ValueError("Input vector needs to be bigger than window size.")
if window_len<3:
return x
#declare valid windows in a dictionary
window_funcs = {
"flat": lambda _len: np.ones(_len, "d"),
"hanning": np.hanning,
"hamming": np.hamming,
"bartlett": np.bartlett,
"blackman": np.blackman
}
s = np.r_[x[window_len-1:0:-1], x, x[-1:-window_len:-1]]
try:
w = window_funcs[window](window_len)
except KeyError:
raise ValueError(
"Window is not one of '{0}', '{1}', '{2}', '{3}', '{4}'".format(
*window_funcs.keys()))
y = np.convolve(w / w.sum(), s, mode = "valid")
return y
def zero_crossings(y_axis, window_len = 11,
window_f="hanning", offset_corrected=False):
"""
Algorithm to find zero crossings. Smooths the curve and finds the
zero-crossings by looking for a sign change.
keyword arguments:
y_axis -- A list containing the signal over which to find zero-crossings
window_len -- the dimension of the smoothing window; should be an odd
integer (default: 11)
window_f -- the type of window from 'flat', 'hanning', 'hamming',
'bartlett', 'blackman' (default: 'hanning')
offset_corrected -- Used for recursive calling to remove offset when needed
return: the index for each zero-crossing
"""
# smooth the curve
length = len(y_axis)
# discard tail of smoothed signal
y_axis = _smooth(y_axis, window_len, window_f)[:length]
indices = np.where(np.diff(np.sign(y_axis)))[0]
# check if zero-crossings are valid
diff = np.diff(indices)
if diff.std() / diff.mean() > 0.1:
#Possibly bad zero crossing, see if it's offsets
if ((diff[::2].std() / diff[::2].mean()) < 0.1 and
(diff[1::2].std() / diff[1::2].mean()) < 0.1 and
not offset_corrected):
#offset present attempt to correct by subtracting the average
offset = np.mean([y_axis.max(), y_axis.min()])
return zero_crossings(y_axis-offset, window_len, window_f, True)
#Invalid zero crossings and the offset has been removed
print(diff.std() / diff.mean())
print(np.diff(indices))
raise ValueError(
"False zero-crossings found, indicates problem {0!s} or {1!s}".format(
"with smoothing window", "unhandled problem with offset"))
# check if any zero crossings were found
if len(indices) < 1:
raise ValueError("No zero crossings found")
#remove offset from indices due to filter function when returning
return indices - (window_len // 2 - 1)
# used this to test the fft function's sensitivity to spectral leakage
#return indices + np.asarray(30 * np.random.randn(len(indices)), int)
############################Frequency calculation#############################
# diff = np.diff(indices)
# time_p_period = diff.mean()
#
# if diff.std() / time_p_period > 0.1:
# raise ValueError(
# "smoothing window too small, false zero-crossing found")
#
# #return frequency
# return 1.0 / time_p_period
##############################################################################
def zero_crossings_sine_fit(y_axis, x_axis, fit_window = None, smooth_window = 11):
"""
Detects the zero crossings of a signal by fitting a sine model function
around the zero crossings:
y = A * sin(2 * pi * Hz * (x - tau)) + k * x + m
Only tau (the zero crossing) is varied during fitting.
Offset and a linear drift of offset is accounted for by fitting a linear
function the negative respective positive raw peaks of the wave-shape and
the amplitude is calculated using data from the offset calculation i.e.
the 'm' constant from the negative peaks is subtracted from the positive
one to obtain amplitude.
Frequency is calculated using the mean time between raw peaks.
Algorithm seems to be sensitive to first guess e.g. a large smooth_window
will give an error in the results.
keyword arguments:
y_axis -- A list containing the signal over which to find peaks
x_axis -- A x-axis whose values correspond to the y_axis list
and is used in the return to specify the position of the peaks. If
omitted an index of the y_axis is used. (default: None)
fit_window -- Number of points around the approximate zero crossing that
should be used when fitting the sine wave. Must be small enough that
no other zero crossing will be seen. If set to none then the mean
distance between zero crossings will be used (default: None)
smooth_window -- the dimension of the smoothing window; should be an odd
integer (default: 11)
return: A list containing the positions of all the zero crossings.
"""
# check input data
x_axis, y_axis = _datacheck_peakdetect(x_axis, y_axis)
#get first guess
zero_indices = zero_crossings(y_axis, window_len = smooth_window)
#modify fit_window to show distance per direction
if fit_window == None:
fit_window = np.diff(zero_indices).mean() // 3
else:
fit_window = fit_window // 2
#x_axis is a np array, use the indices to get a subset with zero crossings
approx_crossings = x_axis[zero_indices]
#get raw peaks for calculation of offsets and frequency
raw_peaks = peakdetect_zero_crossing(y_axis, x_axis)
#Use mean time between peaks for frequency
ext = lambda x: list(zip(*x)[0])
_diff = map(np.diff, map(ext, raw_peaks))
Hz = 1 / np.mean(map(np.mean, _diff))
#Hz = 1 / np.diff(approx_crossings).mean() #probably bad precision
#offset model function
offset_func = lambda x, k, m: k * x + m
k = []
m = []
amplitude = []
for peaks in raw_peaks:
#get peak data as nparray
x_data, y_data = map(np.asarray, zip(*peaks))
#x_data = np.asarray(x_data)
#y_data = np.asarray(y_data)
#calc first guess
A = np.mean(y_data)
p0 = (0, A)
popt, pcov = curve_fit(offset_func, x_data, y_data, p0)
#append results
k.append(popt[0])
m.append(popt[1])
amplitude.append(abs(A))
#store offset constants
p_offset = (np.mean(k), np.mean(m))
A = m[0] - m[1]
#define model function to fit to zero crossing
#y = A * sin(2*pi * Hz * (x - tau)) + k * x + m
func = lambda x, tau: A * np.sin(2 * pi * Hz * (x - tau)) + offset_func(x, *p_offset)
#get true crossings
true_crossings = []
for indice, crossing in zip(zero_indices, approx_crossings):
p0 = (crossing, )
subset_start = max(indice - fit_window, 0.0)
subset_end = min(indice + fit_window + 1, len(x_axis) - 1.0)
x_subset = np.asarray(x_axis[subset_start:subset_end])
y_subset = np.asarray(y_axis[subset_start:subset_end])
#fit
popt, pcov = curve_fit(func, x_subset, y_subset, p0)
true_crossings.append(popt[0])
return true_crossings
def _test_zero():
_max, _min = peakdetect_zero_crossing(y,x)
def _test():
_max, _min = peakdetect(y,x, delta=0.30)
def _test_graph():
i = 10000
x = np.linspace(0,3.7*pi,i)
y = (0.3*np.sin(x) + np.sin(1.3 * x) + 0.9 * np.sin(4.2 * x) + 0.06 *
np.random.randn(i))
y *= -1
x = range(i)
_max, _min = peakdetect(y,x,750, 0.30)
xm = [p[0] for p in _max]
ym = [p[1] for p in _max]
xn = [p[0] for p in _min]
yn = [p[1] for p in _min]
plot = pylab.plot(x,y)
pylab.hold(True)
pylab.plot(xm, ym, "r+")
pylab.plot(xn, yn, "g+")
_max, _min = peak_det_bad.peakdetect(y, 0.7, x)
xm = [p[0] for p in _max]
ym = [p[1] for p in _max]
xn = [p[0] for p in _min]
yn = [p[1] for p in _min]
pylab.plot(xm, ym, "y*")
pylab.plot(xn, yn, "k*")
pylab.show()
def _test_graph_cross(window = 11):
i = 10000
x = np.linspace(0,8.7*pi,i)
y = (2*np.sin(x) + 0.006 *
np.random.randn(i))
y *= -1
pylab.plot(x,y)
#pylab.show()
crossings = zero_crossings_sine_fit(y,x, smooth_window = window)
y_cross = [0] * len(crossings)
plot = pylab.plot(x,y)
pylab.hold(True)
pylab.plot(crossings, y_cross, "b+")
pylab.show()
if __name__ == "__main__":
from math import pi
import pylab
i = 10000
x = np.linspace(0,3.7*pi,i)
y = (0.3*np.sin(x) + np.sin(1.3 * x) + 0.9 * np.sin(4.2 * x) + 0.06 *
np.random.randn(i))
y *= -1
_max, _min = peakdetect(y, x, 750, 0.30)
xm = [p[0] for p in _max]
ym = [p[1] for p in _max]
xn = [p[0] for p in _min]
yn = [p[1] for p in _min]
plot = pylab.plot(x, y)
pylab.hold(True)
pylab.plot(xm, ym, "r+")
pylab.plot(xn, yn, "g+")
pylab.show()
#!/usr/bin/python2
# Copyright (C) 2016 Sixten Bergman
# License WTFPL
#
# This program is free software. It comes without any warranty, to the extent
# permitted by applicable law.
# You can redistribute it and/or modify it under the terms of the Do What The
# Fuck You Want To Public License, Version 2, as published by Sam Hocevar. See
# http://www.wtfpl.net/ for more details.
#
import analytic_wfm
import numpy as np
import peakdetect
import unittest
import pdb
#generate time axis for 5 cycles @ 50 Hz
linspace_standard = np.linspace(0, 0.10, 1000)
linspace_peakdetect = np.linspace(0, 0.10, 10000)
def prng():
"""
A numpy random number generator with a known starting state
return: a random number generator
"""
return np.random.RandomState(773889874)
def _write_log(file, header, message):
with open(file, "ab") as f:
f.write(header)
f.write("\n")
f.writelines(message)
f.write("\n")
f.write("\n")
def _calculate_missmatch(received, expected):
"""
Calculates the mean mismatch between received and expected data
keyword arguments:
received -- [[time of peak], [ampl of peak]]
expected -- [[time of peak], [ampl of peak]]
return (time mismatch, ampl mismatch)
"""
#t_diff = np.abs(np.asarray(received[0]) - expected[0])
t_diff = np.asarray(received[0]) - expected[0]
a_diff = np.abs(np.asarray(received[1]) - expected[1])
#t_diff /= np.abs(expected[0]) time error in absolute terms
a_diff /= np.abs(expected[1])
return (t_diff, a_diff)
def _log_diff(t_max, y_max,
t_min, y_min,
t_max_expected, y_max_expected,
t_min_expected, y_min_expected,
file, name
):
"""
keyword arguments:
t_max -- time of maxima
y_max -- amplitude of maxima
t_min -- time of minima
y_min -- amplitude of maxima
t_max_expected -- expected time of maxima
y_max_expected -- expected amplitude of maxima
t_min_expected -- expected time of minima
y_min_expected -- expected amplitude of maxima
file -- log file to write to
name -- name of the test performed
"""
t_diff_h, a_diff_h = _calculate_missmatch([t_max, y_max],
[t_max_expected, y_max_expected])
t_diff_l, a_diff_l = _calculate_missmatch([t_min, y_min],
[t_min_expected, y_min_expected])
#data = ["\t{0:.2e}\t{1:.2e}\t{2:.2e}\t{3:.2e}".format(*d) for d in
# [t_diff_h, t_diff_l, a_diff_h, a_diff_l]
# ]
frt = "val:{0} error:{1:.2e}"
data = ["\t{0}".format("\t".join(map(frt.format, val, err))) for val, err in
[(t_max, t_diff_h),
(t_min, t_diff_l),
(y_max, a_diff_h),
(y_min, a_diff_l)]
]
_write_log(file, name, "\n".join(data))
def _is_close(max_p, min_p,
expected_max, expected_min,
atol_time, tol_ampl,
file, name):
"""
Determines if the peaks are within the given tolerance
keyword arguments:
max_p -- location and value of maxima
min_p -- location and value of minima
expected_max -- expected location and value of maxima
expected_min -- expected location and value of minima
atol_time -- absolute tolerance of location of vertex
tol_ampl -- relative tolerance of value of vertex
file -- log file to write to
name -- name of the test performed
"""
if len(max_p) == 5:
t_max_expected, y_max_expected = zip(*expected_max)
else:
if abs(max_p[0][0] - expected_max[0][0]) > 0.001:
t_max_expected, y_max_expected = zip(*expected_max[1:])
else:
t_max_expected, y_max_expected = zip(*expected_max[:-1])
if len(min_p) == 5:
t_min_expected, y_min_expected = zip(*expected_min)
else:
t_min_expected, y_min_expected = zip(*expected_min[:-1])
t_max, y_max = zip(*max_p)
t_min, y_min = zip(*min_p)
t_max_close = np.isclose(t_max, t_max_expected, atol=atol_time, rtol=1e-12)
y_max_close = np.isclose(y_max, y_max_expected, tol_ampl)
t_min_close = np.isclose(t_min, t_min_expected, atol=atol_time, rtol=1e-12)
y_min_close = np.isclose(y_min, y_min_expected, tol_ampl)
_log_diff(t_max, y_max, t_min, y_min,
t_max_expected, y_max_expected,
t_min_expected, y_min_expected,
file, name)
return(t_max_close, y_max_close, t_min_close, y_min_close)
class Test_analytic_wfm(unittest.TestCase):
def test_ACV1(self):
#compare with previous lambda implementation
old = analytic_wfm._ACV_A1_L(linspace_standard)
acv = analytic_wfm.ACV_A1(linspace_standard)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV2(self):
#compare with previous lambda implementation
old = analytic_wfm._ACV_A2_L(linspace_standard)
acv = analytic_wfm.ACV_A2(linspace_standard)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV3(self):
#compare with previous lambda implementation
old = analytic_wfm._ACV_A3_L(linspace_standard)
acv = analytic_wfm.ACV_A3(linspace_standard)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV4(self):
#compare with previous lambda implementation
old = analytic_wfm._ACV_A4_L(linspace_standard)
acv = analytic_wfm.ACV_A4(linspace_standard)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV5(self):
#compare with previous lambda implementation
old = analytic_wfm._ACV_A5_L(linspace_standard)
acv = analytic_wfm.ACV_A5(linspace_standard)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV6(self):
#compare with previous lambda implementation
old = analytic_wfm._ACV_A6_L(linspace_standard)
acv = analytic_wfm.ACV_A6(linspace_standard)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV7(self):
num = np.linspace(0, 20, 1000)
old = analytic_wfm._ACV_A7_OLD(num)
acv = analytic_wfm.ACV_A7(num)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
def test_ACV8(self):
num = np.linspace(0, 3150, 10000)
old = analytic_wfm._ACV_A8_OLD(num)
acv = analytic_wfm.ACV_A8(num)
self.assertTrue(np.allclose(acv, old, rtol=1e-9))
class _Test_peakdetect_template(unittest.TestCase):
func = None
file = "Mismatch data.txt"
name = "template"
args = []
kwargs = {}
msg_t = "Time of {0!s} not within tolerance:\n\t{1}"
msg_y = "Amplitude of {0!s} not within tolerance:\n\t{1}"
def _test_peak_template(self, waveform,
expected_max, expected_min,
wav_name,
atol_time = 1e-5, tol_ampl = 1e-5):
"""
keyword arguments:
waveform -- a function that given x can generate a test waveform
expected_max -- position and amplitude where maxima are expected
expected_min -- position and amplitude where minima are expected
wav_name -- Name of the test waveform
atol_time -- absolute tolerance for position of vertex (default: 1e-5)
tol_ampl -- relative tolerance for position of vertex (default: 1e-5)
"""
y = waveform(linspace_peakdetect)
max_p, min_p = self.func(y, linspace_peakdetect,
*self.args, **self.kwargs
)
#check if the correct amount of peaks were discovered
self.assertIn(len(max_p), [4,5])
self.assertIn(len(min_p), [4,5])
#
# check if position and amplitude is within 0.001% which is approx the
# numeric uncertainty from the amount of samples used
#
t_max_close, y_max_close, t_min_close, y_min_close = _is_close(max_p,
min_p,
expected_max,
expected_min,
atol_time, tol_ampl,
self.file, "{0}: {1}".format(wav_name, self.name))
#assert if values are outside of tolerance
self.assertTrue(np.all(t_max_close),
msg=self.msg_t.format("maxima", t_max_close))
self.assertTrue(np.all(y_max_close),
msg=self.msg_y.format("maxima", y_max_close))
self.assertTrue(np.all(t_min_close),
msg=self.msg_t.format("minima", t_min_close))
self.assertTrue(np.all(y_min_close),
msg=self.msg_y.format("minima", y_min_close))
def test_peak_ACV1(self):
peak_pos = 1000*np.sqrt(2) #1414.2135623730951
peak_neg = -peak_pos
expected_max = [
(0.005, peak_pos),
(0.025, peak_pos),
(0.045, peak_pos),
(0.065, peak_pos),
(0.085, peak_pos)
]
expected_min = [
(0.015, peak_neg),
(0.035, peak_neg),
(0.055, peak_neg),
(0.075, peak_neg),
(0.095, peak_neg)
]
atol_time = 1e-5
tol_ampl = 1e-6
self._test_peak_template(analytic_wfm.ACV_A1,
expected_max, expected_min,
"ACV1",
atol_time, tol_ampl
)
def test_peak_ACV2(self):
peak_pos = 1000*np.sqrt(2) + 500 #1414.2135623730951 + 500
peak_neg = (-1000*np.sqrt(2)) + 500 #-914.2135623730951
expected_max = [
(0.005, peak_pos),
(0.025, peak_pos),
(0.045, peak_pos),
(0.065, peak_pos),
(0.085, peak_pos)
]
expected_min = [
(0.015, peak_neg),
(0.035, peak_neg),
(0.055, peak_neg),
(0.075, peak_neg),
(0.095, peak_neg)
]
atol_time = 1e-5
tol_ampl = 2e-6
self._test_peak_template(analytic_wfm.ACV_A2,
expected_max, expected_min,
"ACV2",
atol_time, tol_ampl
)
def test_peak_ACV3(self):
"""
Sine wave with a 3rd overtone
WolframAlpha solution
max{y = sin(100 pi x)+0.05 sin(400 pi x+(2 pi)/3)}~~
sin(6.28319 n+1.51306)-0.05 sin(25.1327 n+5.00505)
at x~~0.00481623+0.02 n for integer n
min{y = sin(100 pi x)+0.05 sin(400 pi x+(2 pi)/3)}~~
0.05 sin(6.55488-25.1327 n)-sin(1.37692-6.28319 n)
at x~~-0.00438287+0.02 n for integer n
Derivative for 50 Hz in 2 alternative forms
y = 100pi*cos(100pi*x) - 25pi*cos(400pi*x)-0.3464*50*pi*sin(400pi*x)
y = 100pi*cos(100pi*x) + 20pi*cos(400pi*x + 2*pi/3)
root 0 = 1/(50 * pi) * (pi*0 - 0.68846026579266880983)
The exact solution according to WolframAlpha - I haven't the foggiest
(tan^(-1)(root of
{#1^2-3&, 11 #2^8-8 #1 #2^7-8 #2^6+56 #1 #2^5+70 #2^4-56 #1 #2^3-48 #2^2+8 #1 #2-9&}(x)
near x = -0.822751)+pi n) / (50 * pi)
root 1 = 1/(50 * pi) * (pi*0 + 0.75653155241276430710)
period = 0.02
"""
base = 1000*np.sqrt(2)
#def peak_pos(n):
# return base * (np.sin(6.28319 * n + 1.51306)
# -0.05*np.sin(25.1327 * n + 5.00505))
#def peak_neg(n):
# return base * (0.05 * np.sin(6.55488 - 25.1327 * n)
# - np.sin(1.37692 - 6.28319 * n))
def peak_pos(n):
return base * (np.sin(2*np.pi * n + 1.51306)
-0.05*np.sin(8*np.pi * n + 5.00505))
def peak_neg(n):
return base * (0.05 * np.sin(6.55488 - 8*np.pi * n)
- np.sin(1.37692 - 2*np.pi * n))
t_max = [
0.75653155241276430710/(50*np.pi)+0.00,#0.004816229446859069
0.75653155241276430710/(50*np.pi)+0.02,#0.024816229446859069
0.75653155241276430710/(50*np.pi)+0.04,#0.044816229446859069
0.75653155241276430710/(50*np.pi)+0.06,#0.064816229446859069
0.75653155241276430710/(50*np.pi)+0.08 #0.084816229446859069
]
t_min = [
-0.68846026579266880983/(50*np.pi)+0.02,#0.015617125823069466
-0.68846026579266880983/(50*np.pi)+0.04,#0.035617125823069466
-0.68846026579266880983/(50*np.pi)+0.06,#0.055617125823069466
-0.68846026579266880983/(50*np.pi)+0.08,#0.075617125823069466
-0.68846026579266880983/(50*np.pi)+0.10 #0.095617125823069466
]
expected_max = [
(t_max[0], analytic_wfm.ACV_A3(t_max[0])),
(t_max[1], analytic_wfm.ACV_A3(t_max[1])),
(t_max[2], analytic_wfm.ACV_A3(t_max[2])),
(t_max[3], analytic_wfm.ACV_A3(t_max[3])),
(t_max[4], analytic_wfm.ACV_A3(t_max[4])),
]
expected_min = [
(t_min[0], analytic_wfm.ACV_A3(t_min[0])),
(t_min[1], analytic_wfm.ACV_A3(t_min[1])),
(t_min[2], analytic_wfm.ACV_A3(t_min[2])),
(t_min[3], analytic_wfm.ACV_A3(t_min[3])),
(t_min[4], analytic_wfm.ACV_A3(t_min[4])),
]
atol_time = 1e-5
tol_ampl = 2e-6
#reduced tolerance since the expected values are only approximated
self._test_peak_template(analytic_wfm.ACV_A3,
expected_max, expected_min,
"ACV3",
atol_time, tol_ampl
)
def test_peak_ACV4(self):
"""
Sine wave with a 4th overtone
Expected data is from a numerical solution using 1e8 samples
The numerical solution used about 2 GB memory and required 64-bit
python
Test is currently disabled as it pushes time index forward enough to
change what peaks are discovers by peakdetect_fft, such that the last
maxima is lost instead of the first one, which is expected from all the
other functions
"""
expected_max = [
(0.0059351920593519207, 1409.2119572886963),
(0.025935191259351911, 1409.2119572887088),
(0.045935191459351918, 1409.2119572887223),
(0.065935191659351911, 1409.2119572887243),
(0.085935191859351917, 1409.2119572887166)
]
expected_min = [
(0.015935191159351911, -1409.2119572886984),
(0.035935191359351915, -1409.2119572887166),
(0.055935191559351914, -1409.2119572887245),
(0.075935191759351914, -1409.2119572887223),
(0.09593519195935192, -1409.2119572887068)
]
atol_time = 1e-5
tol_ampl = 2.5e-6
#reduced tolerance since the expected values are only approximated
self._test_peak_template(analytic_wfm.ACV_A4,
expected_max, expected_min,
"ACV4",
atol_time, tol_ampl
)
def test_peak_ACV5(self):
"""
Realistic triangle wave
Easy enough to solve, but here is the numerical solution from 1e8
samples. Numerical solution used about 2 GB memory and required
64-bit python
expected_max = [
[0.0050000000500000008, 1598.0613254815967]
[0.025000000250000001, 1598.0613254815778],
[0.045000000450000008, 1598.0613254815346],
[0.064999999650000001, 1598.0613254815594],
[0.084999999849999994, 1598.0613254815908]
]
expected_min = [
[0.015000000150000001, -1598.0613254815908],
[0.035000000350000005, -1598.0613254815594],
[0.054999999549999998, -1598.0613254815346],
[0.074999999750000004, -1598.0613254815778],
[0.094999999949999997, -1598.0613254815967]
]
"""
peak_pos = 1130*np.sqrt(2) #1598.0613254815976
peak_neg = -1130*np.sqrt(2) #-1598.0613254815967
expected_max = [
(0.005, peak_pos),
(0.025, peak_pos),
(0.045, peak_pos),
(0.065, peak_pos),
(0.085, peak_pos)
]
expected_min = [
(0.015, peak_neg),
(0.035, peak_neg),
(0.055, peak_neg),
(0.075, peak_neg),
(0.095, peak_neg)
]
atol_time = 1e-5
tol_ampl = 4e-6
self._test_peak_template(analytic_wfm.ACV_A5,
expected_max, expected_min,
"ACV5",
atol_time, tol_ampl
)
def test_peak_ACV6(self):
"""
Realistic triangle wave
Easy enough to solve, but here is the numerical solution from 1e8
samples. Numerical solution used about 2 GB memory and required
64-bit python
expected_max = [
[0.0050000000500000008, 1485.6313472729362],
[0.025000000250000001, 1485.6313472729255],
[0.045000000450000008, 1485.6313472729012],
[0.064999999650000001, 1485.6313472729153],
[0.084999999849999994, 1485.6313472729323]
]
expected_min = [
[0.015000000150000001, -1485.6313472729323],
[0.035000000350000005, -1485.6313472729153],
[0.054999999549999998, -1485.6313472729012],
[0.074999999750000004, -1485.6313472729255],
[0.094999999949999997, -1485.6313472729362]
]
"""
peak_pos = 1050.5*np.sqrt(2) #1485.6313472729364
peak_neg = -1050.5*np.sqrt(2) #1485.6313472729255
expected_max = [
(0.005, peak_pos),
(0.025, peak_pos),
(0.045, peak_pos),
(0.065, peak_pos),
(0.085, peak_pos)
]
expected_min = [
(0.015, peak_neg),
(0.035, peak_neg),
(0.055, peak_neg),
(0.075, peak_neg),
(0.095, peak_neg)
]
atol_time = 1e-5
tol_ampl = 2.5e-6
self._test_peak_template(analytic_wfm.ACV_A6,
expected_max, expected_min,
"ACV6",
atol_time, tol_ampl
)
class Test_peakdetect(_Test_peakdetect_template):
name = "peakdetect"
def __init__(self, *args, **kwargs):
super(Test_peakdetect, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect
class Test_peakdetect_fft(_Test_peakdetect_template):
name = "peakdetect_fft"
def __init__(self, *args, **kwargs):
super(Test_peakdetect_fft, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect_fft
class Test_peakdetect_parabola(_Test_peakdetect_template):
name = "peakdetect_parabola"
def __init__(self, *args, **kwargs):
super(Test_peakdetect_parabola, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect_parabola
class Test_peakdetect_sine(_Test_peakdetect_template):
name = "peakdetect_sine"
def __init__(self, *args, **kwargs):
super(Test_peakdetect_sine, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect_sine
class Test_peakdetect_sine_locked(_Test_peakdetect_template):
name = "peakdetect_sine_locked"
def __init__(self, *args, **kwargs):
super(Test_peakdetect_sine_locked, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect_sine_locked
class Test_peakdetect_spline(_Test_peakdetect_template):
name = "peakdetect_spline"
def __init__(self, *args, **kwargs):
super(Test_peakdetect_spline, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect_spline
class Test_peakdetect_zero_crossing(_Test_peakdetect_template):
name = "peakdetect_zero_crossing"
def __init__(self, *args, **kwargs):
super(Test_peakdetect_zero_crossing, self).__init__(*args, **kwargs)
self.func = peakdetect.peakdetect_zero_crossing
class Test_peakdetect_misc(unittest.TestCase):
def test__pad(self):
data = [1,2,3,4,5,6,5,4,3,2,1]
pad_len = 2
pad = lambda x, c: x[:len(x) // 2] + [0] * c + x[len(x) // 2:]
expected = pad(list(data), 2 **
peakdetect._n(len(data) * pad_len) - len(data))
received = peakdetect._pad(data, pad_len)
self.assertListEqual(received, expected)
def test__n(self):
self.assertEqual(2**peakdetect._n(1000), 1024)
def test_zero_crossings(self):
y = analytic_wfm.ACV_A1(linspace_peakdetect)
expected_indice = [1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000]
indice = peakdetect.zero_crossings(y, 50)
msg = "index:{0:d} should be within 1 of expected:{1:d}"
for rec, exp in zip(indice, expected_indice):
self.assertAlmostEqual(rec, exp, delta=1, msg=msg.format(rec, exp))
#class zero_crossings(unittest.TestCase):
if __name__ == "__main__":
tests_to_run = [
#Test_analytic_wfm,
Test_peakdetect,
Test_peakdetect_parabola,
Test_peakdetect_fft,
#Test_peakdetect_sine, #sine tests disabled pending rework
#Test_peakdetect_sine_locked,
Test_peakdetect_spline,
Test_peakdetect_zero_crossing,
Test_peakdetect_misc
]
suites_list = [unittest.TestLoader().loadTestsFromTestCase(test_class) for test_class in tests_to_run]
big_suite = unittest.TestSuite(suites_list)
unittest.TextTestRunner(verbosity=2).run(big_suite)
@simlan123

Thanks that is pretty useful!

@sixtenbe
Owner

Found an embarrassing error in the peakdetect_zero_crossing function, where it flipped the x and y values in the return. Also added some functions for fitting model functions to the waveform and a fft based interpolation, although at this time I only recommend the peakdetect and peakdetect_zero_crossing function. The reason for this is that I don't yet know if the other functions actually increases the accuracy of the peak detection. Being a metrologist I want to know the uncertainty of my peak detection funtion. For raw peak detection function this can easily be calculated as a function of the time resolution and signal frequency.

@simlan123

Oh the new fitting functions look good. I migth give those a try. I found a Savitzky-Golay filter pretty helpfull for my data. (Analytical chemistry)
Thanks for the update

@dojeda

Correct me if I'm wrong, but isn't the if index+lookahead >= length unnecessary in the peakdetect function, considering that the loop is from [0,length-lookahead) ?

@wuting

This is very useful! Thanks!

@alexlib

great piece of code

@chengdihan

great code, just a small issue correct me if I'm wrong, should
line 140 be mn = y
line 158 be mx = y

thanks

@madken

very useful, nice!

@thoo

Thanks for the code.

@aryabhatt

What is the license associated with this code?

@San08

I have a negative signal that has a peak, but this code lists all the values of x and y

@SebasStrife

Hi Sixtenbe! I implemented the peakdetect.py into the algorithm of my master thesis and it works like a charm:). Therefore I wanted to ask you what the license for this code is and how you want me to reference you? Thanks!

@Prash3r

great tool. i am using it for an acoustic experiment designed to behave like a quantum state. lots of data sets with lots of peaks; lots of manual work done by your script.
Thanks

highly appreciate your work.

@blink1073

Great work, @sixtenbe! Would you mind adding a license to the script? I would like to include it in a BSD-licensed library for metabolomics, https://github.com/metabolite-atlas/metatlas.

@daviddesancho

This works really well for identifying peaks in a saw-tooth pattern from atomic force microscopy. I would also want to ask for the license.

@har1eyk

Hi, @sixtenbe, thank you for coding this. Do you release this under the MIT license?

@Mazzafish

Killer code. Thanks a bunch for sharing!

@sixtenbe
Owner
sixtenbe commented Apr 19, 2016 edited

Just saw that I didn't actually get any kind of notifications for this code and lots of people are asking for the license.

The original piece of code that this was based on is in public domain see http://billauer.co.il/peakdet.html. The contributions I've made to the code can be considered licensed under http://www.wtfpl.net/, which means that you can do whatever you want with the code.

I might do some more with this code as I want to implement some automatic tests of it to verify that everything is returning sane values for a few analytic waveforms that will be based on an IEC standard for high voltage measurement systems.

@sixtenbe
Owner

Refactored a bit of the code and added some automatic testing of the peakdetection functions so that any new function can be verified to function to some standard of bare minimum.

I also changed the name of peakdetect_parabole to peakdetect_parabola, but added a legacy function so that the old name will still work.
I changed the name of the window variable in zero_crossings to window_len and added the variable window_f to enable specification of window type.

Future improvements that I really should do are to make all the function handle offsets without breaking and possibly add some further tests for triangle waveshapes ac defined in ACV_5 or ACV_6, but for this I would need to decide how I define what the maxima and minima are.

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