tdehaeze/spectral-analysis
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update the analysis

Browse Source
This commit is contained in:
Thomas Dehaeze
2019-12-02 15:47:40 +01:00
parent aaaf94f7c3
commit 6b798bc821
25 changed files with 2655 additions and 1359 deletions
Download Patch File Download Diff File Expand all files Collapse all files

107
matlab/approximate_psd_ifft.m Normal file
View File

@@ -0,0 +1,107 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Signal's PSD
% We load the PSD of the signal we wish to replicate.
load('./mat/dist_psd.mat', 'dist_f');
% We remove the first value with very high PSD.
dist_f.f = dist_f.f(3:end);
dist_f.psd_gm = dist_f.psd_gm(3:end);
% The PSD of the signal is shown on figure ref:fig:psd_original.
figure;
hold on;
plot(dist_f.f, dist_f.psd_gm)
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([0.1, 500]);
% Algorithm
% We define some parameters that will be used in the algorithm.
Fs = 2*dist_f.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
N = 2*length(dist_f.f); % Number of Samples match the one of the wanted PSD
T0 = N/Fs; % Signal Duration [s]
df = 1/T0; % Frequency resolution of the DFT [Hz]
% Also equal to (dist_f.f(2)-dist_f.f(1))
% We then specify the wanted PSD.
phi = dist_f.psd_gm;
% We create amplitudes corresponding to wanted PSD.
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(phi(i)*df);
end
% Finally, we add some random phase to =C=.
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
% Obtained Time Domain Signal
% The time domain data is generated by an inverse FFT.
% The =ifft= Matlab does not take into account the sampling frequency, thus we need to normalize the signal.
u = N/sqrt(2)*ifft(Cx); % Normalisation of the IFFT
t = linspace(0, T0, N+1); % Time Vector [s]
figure;
plot(t, u)
xlabel('Time [s]');
ylabel('Amplitude');
xlim([t(1), t(end)]);
% PSD Comparison
% We duplicate the time domain signal to have a longer signal and thus a more precise PSD result.
u_rep = repmat(u, 10, 1);
% We compute the PSD of the obtained signal with the following commands.
nx = length(u_rep);
na = 16;
win = hanning(floor(nx/na));
[pxx, f] = pwelch(u_rep, win, 0, [], Fs);
% Finally, we compare the PSD of the original signal and the obtained signal on figure ref:fig:psd_comparison.
figure;
hold on;
plot(dist_f.f, dist_f.psd_gm, 'DisplayName', 'Original PSD')
plot(f, pxx, 'DisplayName', 'Computed')
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
legend('location', 'northeast');
xlim([0.1, 500]);

150
matlab/approximate_psd_tf.m Normal file
View File

@@ -0,0 +1,150 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Signal's PSD
% We load the PSD of the signal we wish to replicate.
load('./mat/dist_psd.mat', 'dist_f');
% We remove the first value with very high PSD.
dist_f.f = dist_f.f(3:end);
dist_f.psd_gm = dist_f.psd_gm(3:end);
% The PSD of the signal is shown on figure ref:fig:psd_ground_motion.
figure;
hold on;
plot(dist_f.f, dist_f.psd_gm)
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([0.1, 500]);
% Transfer Function that approximate the ASD
% Using =sisotool= or any other tool, we create a transfer function $G$ such that its magnitude is close to the Amplitude Spectral Density $\Gamma_x = \sqrt{S_x}$:
% \[ |G(j\omega)| \approx \Gamma_x(\omega) \]
G_gm = 0.002*(s^2 + 3.169*s + 27.74)/(s*(s+32.73)*(s+8.829)*(s+7.983)^2);
% We compare the ASD $\Gamma_x(\omega)$ and the magnitude of the generated transfer function $|G(j\omega)|$ in figure [[]].
figure;
hold on;
plot(dist_f.f, sqrt(dist_f.psd_gm))
plot(dist_f.f, abs(squeeze(freqresp(G_gm, dist_f.f, 'Hz'))))
hold off;
xlabel('Frequency [Hz]');
ylabel('Amplitude Spectral Density');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlim([0.1, 500]);
% Generated Time domain signal
% We know that a signal $u$ going through a LTI system $G$ (figure [[fig:velocity_to_voltage_psd_bis]]) will have its ASD modified according to the following equation:
% \begin{equation}
% \Gamma_{yy}(\omega) = \left|G(j\omega)\right| \Gamma_{uu}(\omega)
% \end{equation}
% #+NAME: fig:velocity_to_voltage_psd_bis
% #+CAPTION: Schematic of the instrumentation used for the measurement
% [[file:figs/velocity_to_voltage_psd.png]]
% Thus, if we create a random signal with an ASD equal to one at all frequency and we pass this signal through the previously defined LTI transfer function =G_gm=, we should obtain a signal with an ASD that approximate the original ASD.
% To obtain a random signal with an ASD equal to one, we use the following code.
Fs = 2*dist_f.f(end); % Sampling Frequency [Hz]
Ts = 1/Fs; % Sampling Time [s]
t = 0:Ts:500; % Time Vector [s]
u = sqrt(Fs/2)*randn(length(t), 1); % Signal with an ASD equal to one
% We then use =lsim= to compute $y$ as shown in figure [[fig:velocity_to_voltage_psd_bis]].
y = lsim(G_gm, u, t);
% The obtained time domain signal is shown in figure [[fig:time_domain_u]].
figure;
plot(t, y);
xlabel('Time [s]');
ylabel('Amplitude');
% Comparison of the Power Spectral Densities
% We now compute the Power Spectral Density of the computed time domain signal $y$.
nx = length(y);
na = 16;
win = hanning(floor(nx/na));
[pxx, f] = pwelch(y, win, 0, [], Fs);
% Finally, we compare the PSD of the original signal and the obtained signal on figure ref:fig:psd_comparison.
figure;
hold on;
plot(dist_f.f, dist_f.psd_gm, 'DisplayName', 'Original PSD')
plot(f, pxx, 'DisplayName', 'Computed')
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
legend('location', 'northeast');
xlim([0.1, 500]);
% Simulink
% One advantage of this technique is that it can be easily integrated into simulink.
% The corresponding schematic is shown in figure [[fig:simulink_psd_generate]] where the block =Band-Limited White Noise= is used to generate a random signal with a PSD equal to one (parameter =Noise Power= is set to 1).
% Then, the signal generated pass through the transfer function representing the wanted ASD.
% #+name: fig:simulink_psd_generate
% #+caption: Simulink Schematic
% [[file:figs/simulink_psd_generate.png]]
% We simulate the system shown in figure [[fig:simulink_psd_generate]].
out = sim('matlab/generate_signal_psd.slx');
% And we compute the PSD of the generated signal.
nx = length(out.u_gm.Data);
na = 8;
win = hanning(floor(nx/na));
[pxx, f] = pwelch(out.u_gm.Data, win, 0, [], 1e3);
% Finally, we compare the PSD of the generated signal with the original PSD in figure [[fig:compare_psd_original_simulink]].
figure;
hold on;
plot(dist_f.f, dist_f.psd_gm, 'DisplayName', 'Original PSD')
plot(f, pxx, 'DisplayName', 'Computed from Simulink')
hold off;
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density');
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
legend('location', 'northeast');
xlim([0.1, 500]);

93
matlab/compute_psd_levels.m Normal file
View File

@@ -0,0 +1,93 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Time Domain Signal
% Let's first define the number of sample and the sampling time.
N = 10000; % Number of Sample
dt = 0.001; % Sampling Time [s]
t = dt*(0:1:N-1)'; % Time vector [s]
% We generate of signal that consist of:
% - a white noise with an RMS value equal to =anoi=
% - two sinusoidal signals
% The parameters are defined below.
asig = 0.1; % Amplitude of the signal [V]
fsig = 10; % Frequency of the signal [Hz]
ahar = 0.5; % Amplitude of the harmonic [V]
fhar = 50; % Frequency of the harmonic [Hz]
anoi = 1e-3; % RMS value of the noise
% The signal $x$ is generated with the following code and is shown in figure [[fig:time_domain_x_zoom]].
x = anoi*randn(N, 1) + asig*sin((2*pi*fsig)*t) + ahar*sin((2*pi*fhar)*t);
figure;
plot(t, x);
xlabel('Time [s]');
ylabel('Amplitude');
xlim([0, 1]);
% Estimation of the magnitude of a deterministic signal
% Let's compute the PSD of the signal using the =blackmanharris= window.
nx = length(x);
na = 8;
win = blackmanharris(floor(nx/na));
[pxx, f] = pwelch(x, win, 0, [], 1/dt);
% Normalization of the PSD.
CG = sum(win)/(nx/na);
NG = sum(win.^2)/(nx/na);
fbin = f(2) - f(1);
pxx_norm = pxx*(NG*fbin/CG^2);
% We determine the frequency bins corresponding to the frequency of the signals.
isig = round(fsig/fbin)+1;
ihar = round(fhar/fbin)+1;
% The theoretical RMS value of the signal is:
srmt = asig/sqrt(2); % Theoretical value of signal magnitude
% And we estimate the RMS value of the signal by either integrating the PSD around the frequency of the signal or by just taking the maximum value.
srms = sqrt(sum(pxx(isig-5:isig+5)*fbin)); % Signal spectrum integrated
srmsp = sqrt(pxx_norm(isig) * NG*fbin/CG^2); % Maximum read off spectrum
% Estimation of the noise level
% The noise level can also be computed using the integration method.
% The theoretical RMS noise value is.
nth = anoi/sqrt(max(f)) % Theoretical value [V/sqrt(Hz)]
% We can estimate this RMS value by integrating the PSD at frequencies where the power of the noise signal is above the power of the other signals.
navg = sqrt(mean(pxx_norm([ihar+10:end]))) % pwelch output averaged

279
matlab/spectral_analysis_basics.m Normal file
View File

@@ -0,0 +1,279 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% #+RESULTS:
% [[file:figs/velocity_to_voltage.png]]
% #+NAME: fig:velocity_to_voltage
% #+CAPTION: Schematic of the instrumentation used for the measurement
% #+RESULTS: fig:velocity_to_voltage
% [[file:figs/velocity_to_voltage.png]]
% Let's say, we know that the sensitivity of the geophone used is
% \[ G_g(s) = G_0 \frac{\frac{s}{2\pi f_0}}{1 + \frac{s}{2\pi f_0}} \quad \left[\frac{V}{m/s}\right] \]
% The parameters are defined below.
G0 = 88; % Sensitivity [V/(m/s)]
f0 = 2; % Cut-off frequency [Hz]
Gg = G0*(s/2/pi/f0)/(1+s/2/pi/f0);
% And the dynamics of the amplifier in the bandwidth of interest is just a gain: $G_a(s) = 1000$.
Ga = 1000;
% #+NAME: fig:voltage_to_velocity
% #+CAPTION: Schematic of the instrumentation used for the measurement
% #+RESULTS: fig:voltage_to_velocity
% [[file:figs/voltage_to_velocity.png]]
% Let's load the measured $x$.
data = load('mat/data_028.mat', 'data'); data = data.data;
t = data(:, 3); % Time vector [s]
x = data(:, 1)-mean(data(:, 1)); % The offset if removed (coming from the voltage amplifier) [v]
dt = t(2)-t(1); % Sampling Time [s]
Fs = 1/dt; % Sampling Frequency [Hz]
% We simulate this system with matlab using the =lsim= command.
v = lsim(inv(Gg*Ga), x, t);
% And we plot the obtained velocity
figure;
plot(t, v);
xlabel("Time [s]");
ylabel("Velocity [m/s]");
% Power Spectral Density and Amplitude Spectral Density
% From the Matlab documentation:
% #+begin_quote
% The goal of spectral estimation is to describe the distribution (over frequency) of the power contained in a signal, based on a finite set of data.
% #+end_quote
% We now have the velocity $v$ in the time domain:
% \[ v(t)\ [m/s] \]
% The Power Spectral Density (PSD) $S_v(f)$ of the time domain $v(t)$ can be computed using the following equation:
% \[ S_v(f) = \frac{1}{f_s} \sum_{m=-\infty}^{\infty} R_{xx}(m) e^{-j 2 \pi m f / f_s} \ \left[\frac{(m/s)^2}{Hz}\right] \]
% where
% - $f_s$ is the sampling frequency in Hz
% - $R_{xx}$ is the autocorrelation
% The PSD represents the distribution of the (average) signal power over frequency.
% $S_v(f)df$ is the infinitesimal power in the band $(f-df/2, f+df/2)$ and the total power in the signal is obtained by integrating these infinitesimal contributions.
% To compute the Power Spectral Density with matlab, we use the =pwelch= function ([[https://fr.mathworks.com/help/signal/ref/pwelch.html?s_tid=doc_ta][documentation]]).
% The use of the =pwelch= function is:
% =[pxx,w] = pwelch(x,window,noverlap,nfft, fs)=
% With:
% - =x= is the discrete time signal
% - =window= is a window that is used to smooth the obtained PSD
% - =overlap= can be used to have some overlap from section to section
% - =nfft= specifies the number of FFT points for the PSD
% - =fs= is the sampling frequency of the data =x= in Hertz
% As explained in cite:schmid12_how_to_use_fft_matlab, it is recommended to use the =pwelch= function the following way:
% First, define a window (preferably the =hanning= one) by specifying the averaging factor =na=.
nx = length(v);
na = 8;
win = hanning(floor(nx/na));
% Then, compute the power spectral density $S_v$ and the associated frequency vector $f$ with no overlap.
[Sv, f] = pwelch(v, win, 0, [], Fs);
% The obtained PSD is shown in figure [[fig:psd_velocity]].
figure;
plot(f, Sv);
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density $\left[\frac{(m/s)^2}{Hz}\right]$')
xlim([0.1, 500]);
% #+NAME: fig:psd_velocity
% #+CAPTION: Power Spectral Density of the measured velocity
% #+RESULTS: fig:psd_velocity
% [[file:figs/psd_velocity.png]]
% The Amplitude Spectral Density (ASD) is defined as the square root of the Power Spectral Density and is shown in figure [[fig:asd_velocity]].
% \begin{equation}
% \Gamma_{vv}(f) = \sqrt{S_{vv}(f)} \quad \left[ \frac{m/s}{\sqrt{Hz}} \right]
% \end{equation}
figure;
plot(f, sqrt(Sv));
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
ylabel('Amplitude Spectral Density $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
xlim([0.1, 500]);
% #+NAME: fig:velocity_to_voltage_psd
% #+CAPTION: Schematic of the instrumentation used for the measurement
% #+RESULTS: fig:velocity_to_voltage_psd
% [[file:figs/velocity_to_voltage_psd.png]]
% Similarly, the ASD of $y$ is:
% \begin{equation}
% \Gamma_{yy}(\omega) = \left|G(j\omega)\right| \Gamma_{uu}(\omega)
% \end{equation}
% Thus, we could have computed the PSD of $x$ and then obtain the PSD of the velocity with:
% \[ S_{v}(\omega) = |G_a(j\omega) G_g(j\omega)|^{-1} S_{x}(\omega) \]
% The PSD of $x$ is computed below.
nx = length(x);
na = 8;
win = hanning(floor(nx/na));
[Sx, f] = pwelch(x, win, 0, [], Fs);
% And the PSD of $v$ is obtained with the below code.
Svv = Sx.*squeeze(abs(freqresp(inv(Gg*Ga), f, 'Hz'))).^2;
% The result is compare with the PSD computed from the $v$ signal obtained with the =lsim= command in figure [[fig:psd_velocity_lti_method]].
figure;
hold on;
plot(f, Sv, 'DisplayName', 'lsim technique');
plot(f, Svv, 'DisplayName', 'LTI technique');
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density $\left[\frac{(m/s)^2}{Hz}\right]$');
xlim([0.1, 500]);
legend('location', 'southwest');
% #+NAME: fig:velocity_to_displacement_psd
% #+CAPTION: Schematic of the instrumentation used for the measurement
% #+RESULTS: fig:velocity_to_displacement_psd
% [[file:figs/velocity_to_displacement_psd.png]]
% We then have the relation between the PSD of $d$ and the PSD of $v$:
% \begin{equation}
% S_{dd}(\omega) = \left|\frac{1}{j \omega}\right|^2 S_{vv}(\omega)
% \end{equation}
% Using a frequency variable $f$ in Hz:
% \begin{equation}
% S_{dd}(f) = \left| \frac{1}{j 2\pi f} \right|^2 S_{vv}(f)
% \end{equation}
% For the Amplitude Spectral Density:
% \begin{equation}
% \Gamma_{dd}(f) = \frac{1}{2\pi f} \Gamma_{vv}(f)
% \end{equation}
% Note here that the PSD (and ASD) of one variable and its derivatives/integrals are equal at one particular frequency $f = 1\ rad/s \approx 0.16\ Hz$:
% \begin{equation}
% S_{xx}(\omega = 1) = S_{vv}(\omega = 1)
% \end{equation}
% With Matlab, the PSD of the displacement can be computed from the PSD of the velocity with the following code.
Sd = Sv.*(1./(2*pi*f)).^2;
% The obtained PSD of the displacement can be seen in figure [[fig:psd_velocity_displacement]].
figure;
hold on;
plot(f, Sd, 'DisplayName', '$S_d$ in $\left[\frac{m^2}{Hz}\right]$');
plot(f, Sv, 'DisplayName', '$S_v$ in $\left[\frac{(m/s)^2}{Hz}\right]$');
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density');
xlim([0.1, 500]);
legend('location', 'southwest');
% Cumulative Power/Amplitude Spectrum
% The Cumulative Power Spectrum is the cumulative integral of the Power Spectral Density:
% #+NAME: eq:cps
% \begin{equation}
% CPS_v(f) = \int_0^f PSD_v(\nu) d\nu \quad [(m/s)^2]
% \end{equation}
% It is also possible to integrate from high frequency to low frequency:
% #+NAME: eq:cps_inv
% \begin{equation}
% CPS_v(f) = \int_f^\infty PSD_v(\nu) d\nu \quad [(m/s)^2]
% \end{equation}
% The Cumulative Power Spectrum taken at frequency $f$ thus represent the power in the signal in the frequency band $0$ to $f$ or $f$ to $\infty$ depending on the above definition taken.
% The choice of the integral direction depends on the shape of the PSD.
% If the power is mostly present at low frequencies, it is preferable to use equation [[eq:cps_inv]].
% The Cumulative Amplitude Spectrum is defined as the square root of the Cumulative Power Spectrum:
% \[ CAS_v(f) = \sqrt{CPS_v(f)} = \sqrt{\int_f^\infty PSD_v(\nu) d\nu} \quad [m/s] \]
% The Root Mean Square value of the velocity corresponds to the Cumulative Amplitude Spectrum when integrated at all frequencies:
% \[ v_{\text{rms}} = \sqrt{\int_0^\infty PSD_v(\nu) d\nu} = CAS_v(0) \quad [m/s \ \text{rms}] \]
% With Matlab, the Cumulative Power Spectrum can be computed with the below formulas and the results are shown in figure [[fig:cps_integral_comp]].
CPS_v = cumtrapz(f, Sv); % Cumulative Power Spectrum from low to high frequencies
CPS_vv = flip(-cumtrapz(flip(f), flip(Sv))); % Cumulative Power Spectrum from high to low frequencies
figure;
ax1 = subplot(1, 2, 1);
hold on;
plot(f, CPS_v);
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum [$(m/s)^2$]')
ax2 = subplot(1, 2, 2);
hold on;
plot(f, CPS_vv);
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
linkaxes([ax1,ax2],'xy');
xlim([0.1, 500]); ylim([1e-15, 1e-11]);