H = np.r_[np.real(Hc), np.imag(Hc)] # best estimate in real, imag
UH = np.cov(np.c_[np.real(HMC), np.imag(HMC)],
rowvar=0) # covariance of real, imag
UH = make_semiposdef(UH, verbose=True) # correct for numerical errors
bF, UbF = model_est.invLSFIR_unc(
H, UH, N, tau, f,
Fs) # Calculation of FIR deconvolution filter and its assoc. unc.
CbF = UbF / (np.tile(np.sqrt(np.diag(UbF))[:, np.newaxis], (1, N + 1)) *
np.tile(np.sqrt(np.diag(UbF))[:, np.newaxis].T,
(N + 1, 1))) # correlation of filter coefficients
# Deconvolution Step1: lowpass filter for noise attenuation
fcut = f0 + 20e3
low_order = 100 # cut-off frequency and filter order
blow, lshift = kaiser_lowpass(low_order, fcut,
Fs) # FIR low pass filter coefficients
shift = tau + lshift # delay of low-pass plus that of the FIR deconv filter
# Deconvolution Step2: Application of deconvolution filter
xhat, Uxhat = FIRuncFilter(yn, noise, bF, UbF, shift,
blow) # apply low-pass and FIR deconv filter
# Plot of results
fplot = np.linspace(0, 80e3, 1000)
Hc = dsp.freqz(b, a, 2 * np.pi * fplot / Fs)[1]
Hif = dsp.freqz(bF, 1.0, 2 * np.pi * fplot / Fs)[1]
Hl = dsp.freqz(blow, 1.0, 2 * np.pi * fplot / Fs)[1]
plt.figure(1)
plt.clf()
plt.plot(fplot * 1e-3, db(Hc), fplot * 1e-3, db(Hif * Hl), fplot * 1e-3,
db(Hc * Hif * Hl))
import numpy as np
from PyDynamic.misc.testsignals import rect
from PyDynamic.uncertainty.propagate_filter import FIRuncFilter
from PyDynamic.misc.tools import col_hstack, make_semiposdef
from PyDynamic.misc.filterstuff import kaiser_lowpass
import PyDynamic.uncertainty.propagate_MonteCarlo as MC
# parameters of simulated measurement
Fs = 100e3
Ts = 1 / Fs
# nominal system parameters
fcut = 20e3
L = 100
b = kaiser_lowpass(L, fcut, Fs)[0]
# uncertain knowledge: cutoff between 19.5kHz and 20.5kHz
runs = 1000
FC = fcut + (2 * np.random.rand(runs) - 1) * 0.5e3
B = np.zeros((runs, L + 1))
for k in range(runs):
B[k, :] = kaiser_lowpass(L, FC[k], Fs)[0]
Ub = make_semiposdef(np.cov(B, rowvar=0))
# simulate input and output signals
time = np.arange(0, 499 * Ts, Ts)
noise = 1e-3
x = rect(time, 100 * Ts, 250 * Ts, 1.0, noise=noise)
import matplotlib.pyplot as plt
##### some definitions for all tests
# parameters of simulated measurement
Fs = 100e3 # sampling frequency (in Hz)
Ts = 1 / Fs # sampling interval length (in s)
# nominal system parameters
fcut = 20e3 # low-pass filter cut-off frequency (6 dB)
L = 100 # filter order
b1 = kaiser_lowpass(L, fcut,Fs)[0]
b2 = kaiser_lowpass(L-20,fcut,Fs)[0]
# uncertain knowledge: cutoff between 19.5kHz and 20.5kHz
runs = 20
FC = fcut + (2*np.random.rand(runs)-1)*0.5e3
B = np.zeros((runs,L+1))
for k in range(runs): # Monte Carlo for filter coefficients of low-pass filter
B[k,:] = kaiser_lowpass(L,FC[k],Fs)[0]
Ub = make_semiposdef(np.cov(B,rowvar=0)) # covariance matrix of MC result
# simulate input and output signals
nTime = 500
time = np.arange(nTime)*Ts # time values
import numpy as np from PyDynamic.misc.testsignals import rect from PyDynamic.uncertainty.propagate_filter import FIRuncFilter from PyDynamic.misc.tools import col_hstack, make_semiposdef from PyDynamic.misc.filterstuff import kaiser_lowpass import PyDynamic.uncertainty.propagate_MonteCarlo as MC # parameters of simulated measurement Fs = 100e3 # sampling frequency (in Hz) Ts = 1 / Fs # sampling interval length (in s) # nominal system parameters fcut = 20e3 # low-pass filter cut-off frequency (6 dB) L = 100 # filter order b = kaiser_lowpass(L,fcut,Fs)[0] # uncertain knowledge: cutoff between 19.5kHz and 20.5kHz runs = 1000 FC = fcut + (2*np.random.rand(runs)-1)*0.5e3 B = np.zeros((runs,L+1)) for k in range(runs): # Monte Carlo for filter coefficients of low-pass filter B[k,:] = kaiser_lowpass(L,FC[k],Fs)[0] Ub = make_semiposdef(np.cov(B,rowvar=0)) # covariance matrix of MC result # simulate input and output signals time = np.arange(0,499*Ts,Ts) # time values noise = 1e-5 # std of white noise x = rect(time,100*Ts,250*Ts,1.0,noise=noise) # input signal
Hc = np.mean(HMC, dtype=complex, axis=0)
H = np.r_[np.real(Hc), np.imag(Hc)]
UH = np.cov(np.c_[np.real(HMC), np.imag(HMC)], rowvar=0)
UH = make_semiposdef(UH)
# Calculation of FIR deconvolution filter and its assoc. unc.
bF, UbF = deconv.LSFIR_unc(H, UH, N, tau, f, Fs)
# correlation of filter coefficients
CbF = UbF / (np.tile(np.sqrt(np.diag(UbF))[:, np.newaxis], (1, N + 1)) *
np.tile(np.sqrt(np.diag(UbF))[:, np.newaxis].T, (N + 1, 1)))
# Deconvolution Step1: lowpass filter for noise attenuation
fcut = f0 + 20e3
low_order = 100
blow, lshift = kaiser_lowpass(low_order, fcut, Fs)
shift = -tau - lshift
# Deconvolution Step2: Application of deconvolution filter
xhat, Uxhat = FIRuncFilter(yn, noise, bF, UbF, shift, blow)
# Plot of results
fplot = np.linspace(0, 80e3, 1000)
Hc = dsp.freqz(b, a, 2 * np.pi * fplot / Fs)[1]
Hif = dsp.freqz(bF, 1.0, 2 * np.pi * fplot / Fs)[1]
Hl = dsp.freqz(blow, 1.0, 2 * np.pi * fplot / Fs)[1]
plt.figure(1)
plt.clf()
plt.plot(fplot * 1e-3, db(Hc), fplot * 1e-3, db(Hif * Hl), fplot * 1e-3,
db(Hc * Hif * Hl))
plt.legend(
def conduct_validation_of_FIRuncFilter():
# parameters of simulated measurement
Fs = 100e3 # sampling frequency (in Hz)
Ts = 1 / Fs # sampling interval length (in s)
# nominal system parameters
fcut = 20e3 # low-pass filter cut-off frequency (6 dB)
L = 100 # filter order
b1 = kaiser_lowpass(L, fcut, Fs)[0]
b2 = kaiser_lowpass(L - 20, fcut, Fs)[0]
# uncertain knowledge: cutoff between 19.5kHz and 20.5kHz
runs = 1000
FC = fcut + (2 * np.random.rand(runs) - 1) * 0.5e3
B = np.zeros((runs, L + 1))
for k in range(
runs): # Monte Carlo for filter coefficients of low-pass filter
B[k, :] = kaiser_lowpass(L, FC[k], Fs)[0]
Ub = make_semiposdef(np.cov(
B, rowvar=False)) # covariance matrix of MC result
# simulate input and output signals
nTime = 500
time = np.arange(nTime) * Ts # time values
# different cases
sigma_noise = 1e-2 # 1e-5
for kind in ["float", "corr", "diag"]:
for blow in [None, b2]:
print(kind, type(blow))
if kind == "float":
# input signal + run methods
x = rect(time, 100 * Ts, 250 * Ts, 1.0, noise=sigma_noise)
Uy, UyMC, y, yMC = _conduct_FIRuncFilter_and_MonteCarlo(
Ub, b1, blow, kind, runs, sigma_noise, x)
elif kind == "corr":
# get an instance of noise, the covariance and the covariance-matrix
# with
# the specified color
color = "red"
noise = power_law_noise(N=nTime,
color_value=color,
std=sigma_noise)
Ux = power_law_acf(nTime, color_value=color, std=sigma_noise)
# input signal
x = rect(time, 100 * Ts, 250 * Ts, 1.0, noise=noise)
# build Ux_matrix from autocorrelation Ux
Ux_matrix = toeplitz(trimOrPad(Ux, nTime))
# run methods
y, Uy = FIRuncFilter(
x, Ux, b1, Ub, blow=blow,
kind=kind) # apply uncertain FIR filter (GUM formula)
yMC, UyMC = MC(
x, Ux_matrix, b1, np.ones(1), Ub, runs=runs,
blow=blow) # apply uncertain FIR filter (Monte Carlo)
elif kind == "diag":
sigma_diag = sigma_noise * (
1 +
np.heaviside(np.arange(len(time)) - len(time) // 2.5, 0)
) # std doubles after half of the time
noise = sigma_diag * white_gaussian(len(time))
# input signal + run methods
x = rect(time, 100 * Ts, 250 * Ts, 1.0, noise=noise)
Uy, UyMC, y, yMC = _conduct_FIRuncFilter_and_MonteCarlo(
Ub, b1, blow, kind, runs, sigma_diag, x)
# compare FIR and MC results
plt.figure(1)
plt.cla()
plt.plot(time, x, label="input")
plt.plot(time, y, label="output FIR direct")
plt.plot(time, yMC, label="output FIR MC")
plt.xlabel("time [s]")
plt.ylabel("signal amplitude [1]")
plt.legend()
plt.figure(2)
plt.cla()
plt.plot(time, Uy, label="FIR formula")
plt.plot(time, np.sqrt(np.diag(UyMC)), label="Monte Carlo")
plt.xlabel("time [s]")
plt.ylabel("signal uncertainty [1]")
plt.legend()
plt.show()
b_,a_ = dsp.bilinear(bc_,ac_,Fs) # transform to digital filter coefficients
HMC[k,:] = dsp.freqz(b_,a_,2*np.pi*f/Fs)[1] # calculate DFT frequency response
Hc = np.mean(HMC,dtype=complex,axis=0) # best estimate (complex-valued)
H = np.r_[np.real(Hc), np.imag(Hc)] # best estimate in real, imag
UH= np.cov(np.c_[np.real(HMC),np.imag(HMC)],rowvar=0) # covariance of real, imag
UH= make_semiposdef(UH, verbose=True) # correct for numerical errors
bF, UbF = deconv.LSFIR_unc(H,UH,N,tau,f,Fs) # Calculation of FIR deconvolution filter and its assoc. unc.
CbF = UbF/(np.tile(np.sqrt(np.diag(UbF))[:,np.newaxis],(1,N+1))*
np.tile(np.sqrt(np.diag(UbF))[:,np.newaxis].T,(N+1,1))) # correlation of filter coefficients
# Deconvolution Step1: lowpass filter for noise attenuation
fcut = f0+20e3; low_order = 100 # cut-off frequency and filter order
blow, lshift = kaiser_lowpass(low_order, fcut, Fs) # FIR low pass filter coefficients
shift = tau + lshift # delay of low-pass plus that of the FIR deconv filter
# Deconvolution Step2: Application of deconvolution filter
xhat,Uxhat = FIRuncFilter(yn,noise,bF,UbF,shift,blow) # apply low-pass and FIR deconv filter
# Plot of results
fplot = np.linspace(0, 80e3, 1000)
Hc = dsp.freqz(b,a, 2*np.pi*fplot/Fs)[1]
Hif = dsp.freqz(bF, 1.0, 2 * np.pi * fplot / Fs)[1]
Hl = dsp.freqz(blow, 1.0, 2 * np.pi * fplot / Fs)[1]
plt.figure(1); plt.clf()
plt.plot(fplot*1e-3, db(Hc), fplot*1e-3, db(Hif*Hl), fplot*1e-3, db(Hc*Hif*Hl))
plt.legend(('System freq. resp.', 'compensation filter','compensation result'))
# plt.title('Amplitude of frequency responses')