Update the analysis

matlab/approximate_psd_ifft.m

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+%% 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]);

matlab/approximate_psd_tf.m

@@ -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]);

matlab/compute_psd_levels.m

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+%% 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

matlab/spectral_analysis_basics.m

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+%% 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]);

ref.bib

@@ -17,4 +17,12 @@
   doi =          {10.1007/978-94-017-2840-9},
   pages =        {nil},
   series =       {Solid Mechanics and Its Applications},
+}
+
+@book{stoica05_spect,
+  author =       {Stoica, Petre and Moses, Randolph L and others},
+  title =        {Spectral analysis of signals},
+  year =         2005,
+  publisher =    {Pearson Prentice Hall Upper Saddle River, NJ},
+  keywords =     {},
 }