Alternative Smoothing: transformed to matrix method, minor bug fixes, merged test scripts
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@ -5,13 +5,10 @@ Author: C. Jansen, J.A. de Jong - ASCEE V.O.F.
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Smooth data in the frequency domain.
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TODO: This function is rather slow as it
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used Python for loops. The implementations should be speed up in the near
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future.
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TODO: check if the smoothing is correct: depending on whether the data points
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are spaced lin of log in frequency, they should be given different weights.
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TODO: accept data that is not equally spaced in frequency
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TODO: output data that is log spaced in frequency
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TODO: This function is rather slow as it uses [for loops] in Python. Speed up.
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NOTE: function requires lin frequency spaced input data
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TODO: accept input data that is not lin spaced in frequency
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TODO: it makes more sense to output data that is log spaced in frequency
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Cutoff frequencies of window taken from
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http://www.huennebeck-online.de/software/download/src/index.html 15-10-2021
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@ -30,14 +27,13 @@ where b = 3 for 1/3rd octave, f = frequency, fm = mid-band frequency, g = gain.
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Gain is related to magnitude; power is related to gain^2
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"""
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__all__ = ['SmoothingType', 'smoothSpectralData', 'SmoothingWidth', 'smoothSpectralDataAlt', 'intHann']
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__all__ = ['SmoothingType', 'smoothSpectralData', 'SmoothingWidth']
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from enum import Enum, unique
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import bisect
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import copy
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import numpy as np
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from math import ceil, floor, sin
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from numpy import arange, log2, log10, pi
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@unique
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class SmoothingWidth(Enum):
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@ -95,14 +91,17 @@ def smoothCalcMatrix(freq, sw: SmoothingWidth):
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tr = 2 # truncate window after 2x std; shorter is faster and less accurate
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Noct = sw.value[0]
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assert Noct > 0, "'Noct' must be absolute positive"
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assert np.isclose(freq[-1]-freq[-2], freq[1]-freq[0]), "Input data must "\
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"have a linear frequency spacing"
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if Noct < 1:
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raise Warning('Check if \'Noct\' is entered correctly')
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# Initialize
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L = len(freq)
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Q = np.zeros(shape=(L, L), dtype=np.float16) # float16: keep size small
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Q[0, 0] = 1 # in case first point is skipped
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x0 = 1 if freq[0] == 0 else 0 # Skip first data point if zero frequency
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# Loop over indices of raw frequency vector
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for x in range(x0, L):
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# Find indices of data points to calculate current (smoothed) magnitude
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@ -129,9 +128,9 @@ def smoothCalcMatrix(freq, sw: SmoothingWidth):
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# Find indices corresponding to frequencies
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xl = bisect.bisect_left(freq, fl) # index corresponding to fl
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xu = bisect.bisect_left(freq, fu)
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xu = bisect.bisect_right(freq, fu)
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xl = xu-1 if xu-xl <= 0 else xl # Guarantee window length of at least 1
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xl = xu-1 if xu-xl <= 0 else xl # Guarantee window length >= 1
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# Calculate window
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xg = np.arange(xl, xu) # indices
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@ -140,6 +139,10 @@ def smoothCalcMatrix(freq, sw: SmoothingWidth):
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gs /= np.sum(gs) # normalize: integral=1
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Q[x, xl:xu] = gs # add to matrix
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# Normalize to conserve input power
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Qpower = np.sum(Q, axis=0)
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Q = Q / Qpower[np.newaxis, :]
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return Q
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@ -154,7 +157,7 @@ def smoothSpectralData(freq, M, sw: SmoothingWidth,
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fractional octave smoothing is related to log spaced data. In this
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implementation, the window extends with a fixed frequency step to either
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side. The deviation is largest when Noct is small (e.g. coarse smoothing).
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Casper Jansen, 07-05-2021
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07-05-2021
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Update 16-01-2023: speed up algorithm
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- Smoothing is performed using matrix multiplication
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@ -227,63 +230,10 @@ def smoothSpectralData(freq, M, sw: SmoothingWidth,
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return Psm
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# %% Alternative algorithm
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from numpy import arange, log2, log10, pi, ceil, floor, sin
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def smoothSpectralDataAlt(freq, Mdb, sw: SmoothingWidth):
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Noct = 1/sw.value[0]
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# M = Mdb
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M = 10**(Mdb/20)
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f0 = 0
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if freq[0] == 0:
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f0 += 1
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Nfreq = len(freq)
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test_smoothed = np.array(M)
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ifreq = freq/(freq[1]-freq[0])
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ifreq = np.array(ifreq.astype(int))
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ifreqMin = ifreq[f0]
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ifreqMax = ifreq[Nfreq-1]
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sfact = 2**(Noct/2)
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maxNkp = ifreqMax - floor((ifreqMax-1)/sfact**2)+1
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W = np.zeros(maxNkp)
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kpmin = np.floor(ifreq/sfact).astype(int)
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kpmax = np.ceil(ifreq*sfact).astype(int)
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for ff in range(f0, len(M)):
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if kpmin[ff] < ifreqMin:
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kpmin[ff] = ifreqMin
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kpmax[ff] = ceil(ifreq[ff]**2/ifreqMin)
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NoctAct = log2(kpmax[ff]/kpmin[ff])
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elif kpmax[ff] > ifreqMax:
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kpmin[ff] = floor(ifreq[ff]**2/ifreqMax)
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kpmax[ff] = ifreqMax
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NoctAct = log2(kpmax[ff]/kpmin[ff])
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else:
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NoctAct = Noct
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kp = arange(kpmin[ff], kpmax[ff]+1)
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if NoctAct:
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Phi1 = log2((kp - 0.5)/ifreq[ff])/NoctAct
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Phi2 = log2((kp + 0.5)/ifreq[ff])/NoctAct
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for ii in range(len(kp)):
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W[ii] = intHann(Phi1[ii], Phi2[ii])
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test_smoothed[ff] = np.dot(M[kpmin[ff]-ifreq[0]:kpmax[ff]-ifreq[0]+1], W[:ii+1])
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test_smoothed = 20*log10(test_smoothed)
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return test_smoothed
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# %% Integrated Hann window
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# Integrated Hann window
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def intHann(x1, x2):
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if (x2 <= -1/2) or (x1 >= 1/2):
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return 0
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@ -299,6 +249,242 @@ def intHann(x1, x2):
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return (sin(2*pi*x2) - sin(2*pi*x1))/(2*pi) + (x2-x1)
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def smoothSpectralDataAlt(freq, MdB, sw: SmoothingWidth,
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st: SmoothingType = SmoothingType.levels):
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"""
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According to Tylka_JAES_SmoothingWeights.pdf
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"A Generalized Method for Fractional-Octave Smoothing of Transfer Functions
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that Preserves Log-Frequency Symmetry"
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https://doi.org/10.17743/jaes.2016.0053
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par 1.3
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eq. 16
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"""
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Noct = 1/sw.value[0]
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# M = MdB
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M = 10**(MdB/20)
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f0 = 0
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if freq[0] == 0:
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f0 += 1
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Nfreq = len(freq) # Number of frequenties
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test_smoothed = np.array(M) # Input [Power]
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ifreq = freq/(freq[1]-freq[0]) # Frequency, normalized to step=1
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ifreq = np.array(ifreq.astype(int))
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ifreqMin = ifreq[f0] # Min. freq, normalized to step=1
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ifreqMax = ifreq[Nfreq-1] # Max. freq, normalized to step=1
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sfact = 2**(Noct/2) # bounds are this factor from the center freq
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maxNkp = ifreqMax - floor((ifreqMax-1)/sfact**2)+1
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# W = np.zeros(int(np.round(maxNkp)))
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kpmin = np.floor(ifreq/sfact).astype(int) # min freq of window
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kpmax = np.ceil(ifreq*sfact).astype(int) # max freq of window
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for ff in range(f0, len(M)): # loop over input freq
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if kpmin[ff] < ifreqMin:
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kpmin[ff] = ifreqMin
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kpmax[ff] = ceil(ifreq[ff]**2/ifreqMin) # achieved Noct
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if np.isclose(kpmin[ff], kpmax[ff]):
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kpmax[ff] += 1
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NoctAct = log2(kpmax[ff]/kpmin[ff])
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elif kpmax[ff] > ifreqMax:
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kpmin[ff] = floor(ifreq[ff]**2/ifreqMax) # achieved Noct
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kpmax[ff] = ifreqMax
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if np.isclose(kpmin[ff], kpmax[ff]):
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kpmin[ff] -= 1
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NoctAct = log2(kpmax[ff]/kpmin[ff])
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else:
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NoctAct = Noct # Noct = smoothing width (Noct=6 --> 1/6th octave)
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kp = arange(kpmin[ff], kpmax[ff]+1) # freqs of window
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Phi1 = log2((kp - 0.5)/ifreq[ff])/NoctAct # integration bounds for hann window
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Phi2 = log2((kp + 0.5)/ifreq[ff])/NoctAct
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W = np.zeros(len(kp))
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for ii in range(len(kp)):
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W[ii] = intHann(Phi1[ii], Phi2[ii]) # weight = integration of hann window between Phi1 and Phi2
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test_smoothed[ff] = np.dot( M[kpmin[ff]-ifreq[0]:kpmax[ff]-ifreq[0]+1], W[:ii+1] ) # eq 16
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test_smoothed = 20*log10(test_smoothed)
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return test_smoothed
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def smoothCalcMatrixAlt(freq, sw: SmoothingWidth):
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"""
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Args:
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freq: array of frequencies of data points [Hz] - equally spaced
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sw: SmoothingWidth
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Returns:
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freq: array frequencies of data points [Hz]
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Q: matrix to smooth power: {fsm} = [Q] * {fraw}
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Warning: this method does not work on levels (dB)
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According to Tylka_JAES_SmoothingWeights.pdf
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"A Generalized Method for Fractional-Octave Smoothing of Transfer Functions
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that Preserves Log-Frequency Symmetry"
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https://doi.org/10.17743/jaes.2016.0053
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par 1.3
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eq. 16
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"""
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# Settings
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tr = 2 # truncate window after 2x std; shorter is faster and less accurate
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Noct = sw.value[0]
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assert Noct > 0, "'Noct' must be absolute positive"
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assert np.isclose(freq[-1]-freq[-2], freq[1]-freq[0]), "Input data must "\
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"have a linear frequency spacing"
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if Noct < 1:
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raise Warning('Check if \'Noct\' is entered correctly')
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# Initialize
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L = len(freq)
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Q = np.zeros(shape=(L, L), dtype=np.float16) # float16: keep size small
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Q[0, 0] = 1 # in case first point is skipped
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x0 = 1 if freq[0] == 0 else 0 # Skip first data point if zero frequency
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Noct /= 1.5 # empirical correction factor: window @ -6 dB at Noct bounds
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ifreq = freq/(freq[1]-freq[0]) # frequency, normalized to step=1
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ifreq = np.array(ifreq.astype(int))
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ifreqMin = ifreq[x0] # min. freq, normalized to step=1
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ifreqMax = ifreq[L-1] # max. freq, normalized to step=1
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sfact = 2**((1/Noct)/2) # bounds are this factor from the center freq
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kpmin = np.floor(ifreq/sfact).astype(int) # min freq of window
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kpmax = np.ceil(ifreq*sfact).astype(int) # max freq of window
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for ff in range(x0, len(M)): # loop over input freq
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if kpmin[ff] < ifreqMin:
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kpmin[ff] = ifreqMin
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kpmax[ff] = ceil(ifreq[ff]**2/ifreqMin) # decrease smooth. width
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if np.isclose(kpmin[ff], kpmax[ff]):
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kpmax[ff] += 1
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NoctAct = 1/log2(kpmax[ff]/kpmin[ff])
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elif kpmax[ff] > ifreqMax:
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kpmin[ff] = floor(ifreq[ff]**2/ifreqMax) # decrease smooth. width
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kpmax[ff] = ifreqMax
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if np.isclose(kpmin[ff], kpmax[ff]):
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kpmin[ff] -= 1
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NoctAct = 1/log2(kpmax[ff]/kpmin[ff])
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else:
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NoctAct = Noct
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kp = arange(kpmin[ff], kpmax[ff]+1) # freqs of window
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# Integration bounds for Hann window
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Phi1 = log2((kp - 0.5)/ifreq[ff]) * NoctAct
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Phi2 = log2((kp + 0.5)/ifreq[ff]) * NoctAct
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# Weights within window = integration of hann window between Phi1, Phi2
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W = np.zeros(len(kp))
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for ii in range(len(kp)):
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W[ii] = intHann(Phi1[ii], Phi2[ii])
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# Insert W at input freq ii, starting at index 'kpmin[ff]-ifreq[0]'
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Q[ff, kpmin[ff]-ifreq[0]:kpmax[ff]-ifreq[0]+1] = W
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# Normalize to conserve input power
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Qpower = np.sum(Q, axis=0)
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Q = Q / Qpower[np.newaxis, :]
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return Q
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def smoothSpectralDataAltMatrix(freq, M, sw: SmoothingWidth,
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st: SmoothingType = SmoothingType.levels):
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"""
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Apply fractional octave smoothing to magnitude data in frequency domain.
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Smoothing is performed to power, using a sliding Gaussian window with
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variable length. The window is truncated after 2x std at either side.
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The implementation is not exact, because f is linearly spaced and
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fractional octave smoothing is related to log spaced data. In this
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implementation, the window extends with a fixed frequency step to either
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side. The deviation is largest when Noct is small (e.g. coarse smoothing).
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07-05-2021
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Update 16-01-2023: speed up algorithm
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- Smoothing is performed using matrix multiplication
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- The smoothing matrix is not calculated if it already exists
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Args:
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freq: array of frequencies of data points [Hz] - equally spaced
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M: array of either power, transfer functin or dB points. Depending on
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the smoothing type `st`, the smoothing is applied.
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Returns:
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freq : array frequencies of data points [Hz]
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Msm : float smoothed magnitude of data points
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"""
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# TODO: Make this function multi-dimensional array aware.
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# Safety
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MM = copy.deepcopy(M)
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Noct = sw.value[0]
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assert len(M) > 0, "Smoothing function: input array is empty" # not sure if this works
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assert Noct > 0, "'Noct' must be absolute positive"
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if Noct < 1:
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raise Warning('Check if \'Noct\' is entered correctly')
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assert len(freq) == len(M), 'f and M should have equal length'
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# if st == SmoothingType.ps:
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# assert np.min(M) >= 0, 'absolute magnitude M cannot be negative'
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if st == SmoothingType.levels and isinstance(M.dtype, complex):
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raise RuntimeError('Decibel input should be real-valued')
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# Initialize
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L = M.shape[0] # number of data points
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if st == SmoothingType.levels:
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P = 10**(MM/10) # magnitude [dB] --> power
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else:
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P = MM # data already given as power
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# TODO: This does not work due to complex numbers. Should be split up in
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# magnitude and phase.
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# elif st == SmoothingType.tf:
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# P = P**2
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# P is power while smoothing. x are indices of P.
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Psm = np.zeros_like(P) # Smoothed power - to be calculated
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if freq[0] == 0:
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Psm[0] = P[0] # Reuse old value in case first data..
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# ..point is skipped. Not plotted any way.
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# # Re-use smoothing matrix Q if available. Otherwise, calculate.
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# # Store in dict 'Qdict'
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# nfft = int(2*(len(freq)-1))
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# key = f"nfft{nfft}_Noct{Noct}" # matrix name
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# if 'Qdict' not in globals(): # Guarantee Qdict exists
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# global Qdict
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# Qdict = {}
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# if key in Qdict:
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# Q = Qdict[key]
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# else:
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# Q = smoothCalcMatrixAlt(freq, sw)
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# Qdict[key] = Q
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Q = smoothCalcMatrixAlt(freq, sw)
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# Apply smoothing
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Psm = np.matmul(Q, P)
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if st == SmoothingType.levels:
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Psm = 10*np.log10(Psm)
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return Psm
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# %% Test
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if __name__ == "__main__":
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""" Test function for evaluation and debugging
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@ -307,53 +493,76 @@ if __name__ == "__main__":
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points. They should be treated and weighted differently.
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"""
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import matplotlib.pyplot as plt
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import time
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plt.close('all')
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# Initialize
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Noct = 2 # Noct = 6 for 1/6 oct. smoothing
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Noct = 3 # Noct = 6 for 1/6 oct. smoothing
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# Create dummy data set 1: noise
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fmin = 1e3 # [Hz] min freq
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fmax = 24e3 # [Hz] max freq
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Ndata = 200 # number of data points
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freq = np.linspace(fmin, fmax, Ndata) # frequency points
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M = abs(0.4*np.random.normal(size=(Ndata,)))+0.01 #
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M = 20*np.log10(M)
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# # Create dummy data set 2: dirac delta
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# fmin = 3e3 # [Hz] min freq
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# # Create dummy data set 1: noise
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# fmin = 1e3 # [Hz] min freq
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# fmax = 24e3 # [Hz] max freq
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# Ndata = 200 # number of data points
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# freq = np.linspace(fmin, fmax, Ndata) # frequency points
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# M = 0 * abs(1+0.4*np.random.normal(size=(Ndata,))) + 0.01 #
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# M[int(100)] = 1
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# # freq = np.hstack((0, freq))
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# M = abs(0.4*np.random.normal(size=(len(freq),)))+0.01 #
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# M = 20*np.log10(M)
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# Apply function
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# Create dummy data set 2: single tone
|
||||
fmin = 0 # [Hz] min freq
|
||||
fmax = 5e3 # [Hz] max freq
|
||||
Ndata = 2501 # number of data points
|
||||
freq = np.linspace(fmin, fmax, Ndata) # frequency points
|
||||
M = 1e-4*np.random.normal(size=(Ndata,))
|
||||
M[500] = 1
|
||||
MdB = 20*np.log10(abs(M))
|
||||
|
||||
class sw:
|
||||
value = [Noct]
|
||||
st = SmoothingType.levels # so data is given in dB
|
||||
# st = SmoothingType.ps # so data is given in power
|
||||
|
||||
# Smooth
|
||||
Msm = smoothSpectralData(freq, M, sw, st)
|
||||
if 'Qdict' in globals():
|
||||
del Qdict
|
||||
|
||||
t0 = time.time()
|
||||
Msm = smoothSpectralData(freq, MdB, sw, st) # current algorithm
|
||||
t1 = time.time()
|
||||
MsmAlt = smoothSpectralDataAlt(freq, MdB, sw, st) # alternative algorithm
|
||||
t2 = time.time()
|
||||
MsmAltMatrix = smoothSpectralDataAltMatrix(freq, MdB, sw, st) # alternative algorithm, matrix method
|
||||
t3 = time.time()
|
||||
fsm = freq
|
||||
|
||||
print(f"Smoothing time: {t1-t0} s")
|
||||
print(f"Smoothing time: {t2-t1} s (Alt)")
|
||||
print(f"Smoothing time: {t3-t2} s (Alt Matrix)")
|
||||
|
||||
# Plot - lin frequency
|
||||
plt.figure()
|
||||
plt.plot(freq, M, '.b')
|
||||
plt.plot(freq, MdB, '.b')
|
||||
plt.plot(fsm, Msm, 'r')
|
||||
plt.plot(fsm, MsmAlt, 'g')
|
||||
plt.plot(fsm, MsmAltMatrix, '--k')
|
||||
plt.xlabel('f (Hz)')
|
||||
plt.ylabel('magnitude')
|
||||
plt.xlim([100, fmax])
|
||||
plt.xlim((0, fmax))
|
||||
plt.ylim((-90, 1))
|
||||
plt.grid('both')
|
||||
plt.title('lin frequency')
|
||||
plt.legend(['Raw', 'Smooth'])
|
||||
plt.legend(['Raw', 'Smooth', 'SmoothAlt', 'SmoothAltMatrix'])
|
||||
|
||||
# Plot - log frequency
|
||||
plt.figure()
|
||||
plt.semilogx(freq, M, '.b')
|
||||
plt.semilogx(freq, MdB, '.b')
|
||||
plt.semilogx(fsm, Msm, 'r')
|
||||
plt.semilogx(fsm, MsmAlt, 'g')
|
||||
plt.semilogx(fsm, MsmAltMatrix, '--k')
|
||||
plt.xlabel('f (Hz)')
|
||||
plt.ylabel('magnitude')
|
||||
plt.xlim([100, fmax])
|
||||
plt.xlim((100, fmax))
|
||||
plt.ylim((-90, 1))
|
||||
plt.grid('both')
|
||||
plt.title('log frequency')
|
||||
plt.legend(['Raw', 'Smooth 1'])
|
||||
plt.legend(['Raw', 'Smooth', 'SmoothAlt', 'SmoothAltMatrix'])
|
||||
|
||||
Loading…
Reference in New Issue
Block a user