Obisidian vault auto-backup: 05-01-2026 14:28:22 on . 7 files edited

ISEN/Traitement du signal/CIPA4/TP/TP2/TP2_Experience1.m

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+%% Experiment 1 : FFT of sinusoidal mixtures (magnitude spectra only)
+
+clc;
+clear;
+close all;
+
+%% ========================== PART i ==========================
+%% i) Ts = 0.2 ms, t in [0, 4 s]
+
+Ts1   = 0.2e-3;               % Sampling period (s)
+Fs1   = 1 / Ts1;              % Sampling frequency (Hz)
+t1    = 0 : Ts1 : 4;          % Time vector
+N1    = length(t1);           % Number of samples
+
+fi1   = Fs1 / N1;             % Frequency resolution (Hz)
+k1    = 0 : (N1-1);           % FFT index vector
+f_full_1 = k1 * fi1;          % Full frequency axis
+
+% Signal: x(t) = sin(2000 t + 56) * sin(56 - 6000 t) + 2 cos(4000 t)
+x1 = sin(2000*t1 + 56) .* sin(56 - 6000*t1) + 2*cos(4000*t1);
+
+%% i-a) Magnitude spectrum of x(t)  (single-sided)
+
+X1 = fft(x1);                     % N-point FFT
+half_N1   = floor(N1/2) + 1;      % Length of single-sided spectrum
+X1_half   = X1(1:half_N1);        % Positive frequencies only
+f1_half   = f_full_1(1:half_N1);  % Frequency axis (Hz)
+magX1     = abs(X1_half);         % Magnitude of complex spectrum
+
+figure('Name','Part i-a: |X(f)| of x(t)');
+plot(f1_half, magX1, 'r-');
+title('Part i-a: magnitude spectrum |X(f)| of x(t)');
+xlabel('Frequency (Hz)');
+ylabel('|X(f)|');
+grid on;
+
+%% i-b) Magnitude spectrum of x^2(t)  (single-sided)
+
+x2 = x1.^2;                   % x^2(t)
+X2 = fft(x2);                 % FFT of x^2(t)
+
+X2_half   = X2(1:half_N1);
+magX2     = abs(X2_half);
+% Same frequency axis f1_half
+
+figure('Name','Part i-b: |X_2(f)| of x^2(t)');
+plot(f1_half, magX2, 'm-');
+title('Part i-b: magnitude spectrum |X_2(f)| of x^2(t)');
+xlabel('Frequency (Hz)');
+ylabel('|X_2(f)|');
+grid on;
+
+%% ========================== PART ii ==========================
+%% ii) Fs = 5 kHz, t in [0, 2 s]
+% x(t) = cos(2000 t^2) + 0.01 sin(2000 t) + 0.01 sin(10000 t)
+
+Fs2   = 5000;                % Sampling frequency (Hz)
+Ts2   = 1 / Fs2;             % Sampling period (s)
+t2    = 0 : Ts2 : 2;         % Time vector
+N2    = length(t2);          % Number of samples
+
+fi2   = Fs2 / N2;            % Frequency resolution
+k2    = 0 : (N2-1);
+f_full_2 = k2 * fi2;
+
+x3 = cos(2000*t2.^2) + 0.01*sin(2000*t2) + 0.01*sin(10000*t2);
+
+X3 = fft(x3);                % FFT
+
+half_N2   = floor(N2/2) + 1;
+X3_half   = X3(1:half_N2);
+f2_half   = f_full_2(1:half_N2);
+magX3     = abs(X3_half);
+
+figure('Name','Part ii: |X(f)| of chirp signal');
+plot(f2_half, magX3, 'b-');
+title('Part ii: magnitude spectrum |X(f)| of x(t)');
+xlabel('Frequency (Hz)');
+ylabel('|X(f)|');
+grid on;

ISEN/Traitement du signal/CIPA4/TP/TP2/TP2_Experience2.m

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+%% Experiment 2 : Recreation of a signal
+
+clc;
+clear;
+close all;
+
+Fs = 4000;                 % Sampling frequency (Hz)
+N  = 2048;                 % Number of samples
+t  = (0:N-1)/Fs;           % Time vector (s)
+
+df = Fs/N;                 % Frequency resolution (Hz)
+
+% --- 1) Desired peak frequencies (Hz) ---
+f_target = [300 400 700 1300 1600 1700];
+
+% Align each frequency with the closest FFT bin to avoid spectral leakage.
+k = round(f_target/df);    % Integer FFT bin indices
+f = k * df;                % Actual frequencies used (bin-aligned)
+
+% --- 2) Desired peak heights in |FFT| ---
+H = [1000 2000 3250 3250 2000 1000];
+
+% --- 3) Amplitudes of the cosines for those peak heights ---
+% For a cosine perfectly aligned to a bin, the peak in |FFT| is about N*A/2,
+% so we choose A = 2*H/N to obtain the target height H. 
+A = 2*H / N;
+
+% --- 4) Build the time-domain signal y(t) as a sum of cosines ---
+% Random phases are used because the magnitude spectrum does not depend on phase.
+phi = 2*pi*rand(size(f));  % Random phase for each component
+y   = zeros(size(t));      % Initialize signal
+
+for i = 1:length(f)
+    y = y + A(i) * cos(2*pi*f(i)*t + phi(i));
+end
+
+% --- 5) Compute FFT and plot the magnitude spectrum ---
+Y  = fft(y);               % Complex FFT of y(t)
+Py = abs(Y);               % Magnitude of the FFT (two-sided, unnormalized) 
+
+f_axis = (0:N-1) * df;     % Frequency axis (Hz)
+
+figure;
+stem(f_axis, Py, 'g');     % Discrete spectrum as vertical lines
+xlim([0 Fs/2]);            % Show only positive frequencies up to Nyquist
+xlabel('Frequency (Hz)');
+ylabel('|Y(f)|');
+title('Magnitude spectrum with target peak heights');
+grid on;

ISEN/Traitement du signal/CIPA4/TP/TP2/TP2_Experience3.m

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+%% Experiment 3 : Power spectrum of "holiday_offer"
+% Plot the power spectrum with the same style as the TP figure. [web:81][web:155]
+
+clc;
+clear;
+close all;
+
+% 1) Load signal from MAT-file
+data = load('holiday_offer.mat');
+varNames   = fieldnames(data);
+signalName = varNames{1};      
+x = data.(signalName);         % time-domain signal vector
+
+Fs = 11025;                    % Sampling frequency (Hz)
+
+% 2) FFT-based power spectrum (periodogram)
+N    = length(x);
+Xdft = fft(x);
+Pxx  = (1/N) * abs(Xdft).^2;   % power spectrum estimate [web:154]
+
+% We keep the FULL spectrum (both sides) as in the given figure
+f = (0:N-1) * (Fs/N);          % frequency axis from 0 to Fs (≈11025 Hz)
+
+% 3) Plot
+figure;                       
+plot(f, Pxx, 'g');            
+xlabel('Frequency (Hz)');
+ylabel('Power');
+title('Power spectrum of holiday\_offer signal');
+grid on;
+xlim([0 Fs]);                  

ISEN/Traitement du signal/CIPA4/TP/TP2/TP2_Experience4.m

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+%% Experiment 4 : Magnitude spectra of composed signals
+
+clc;
+clear;
+close all;
+
+%% ========================== PART i ==========================
+% i) Set the sampling period to 0.5 ms and time interval [0, 2 s]
+%    x(t) = sin(4000 t) + 2 cos(4000 t - 32) + cos^2(2000 t)
+
+Ts1 = 0.5e-3;               % Sampling period (s)
+Fs1 = 1 / Ts1;              % Sampling frequency (Hz) -> 2000 Hz
+t1  = 0 : Ts1 : 2;          % Time vector (s)
+N1  = length(t1);           % Number of samples
+
+% Define the signal x1(t)
+x1 = sin(4000*t1) + 2*cos(4000*t1 - 32) + cos(2000*t1).^2;
+
+% FFT and magnitude spectrum (single-sided)
+X1    = fft(x1);                       % Complex FFT of x1
+magX1 = abs(X1);                       % Magnitude spectrum
+halfN1 = floor(N1/2) + 1;              % First half (real signal) [web:171]
+magX1_half = magX1(1:halfN1);
+f1 = (0:halfN1-1) * (Fs1/N1);          % Frequency axis in Hz
+
+figure('Name','Experiment 4 - Part i');
+plot(f1, magX1_half, 'g');             % Green magnitude spectrum
+xlabel('Frequency (Hz)');
+ylabel('|X_1(f)|');
+title('Experiment 4 - Part i : Magnitude spectrum of x(t)');
+grid on;
+
+%% ========================== PART ii ==========================
+% ii) Set the sampling frequency to 5 kHz and time interval [0, 2 s]
+%     x(t) = 0.1 sin(2000 * sqrt(t)) + 0.01 sin(4000 t) + 0.01 cos(7000 t)
+% Note: Interpret "2000 t1/2" as 2000 * sqrt(t), i.e., t^(1/2).
+
+Fs2 = 5000;                 % Sampling frequency (Hz)
+Ts2 = 1 / Fs2;              % Sampling period (s)
+t2  = 0 : Ts2 : 2;          % Time vector (s)
+N2  = length(t2);           % Number of samples
+
+% Avoid sqrt(0) issues in instantaneous frequency by allowing t = 0 (OK in MATLAB)
+x2 = 0.1 * sin(2000 * sqrt(t2)) + 0.01 * sin(4000*t2) + 0.01 * cos(7000*t2);
+
+% FFT and magnitude spectrum (single-sided)
+X2    = fft(x2);
+magX2 = abs(X2);
+halfN2 = floor(N2/2) + 1;
+magX2_half = magX2(1:halfN2);
+f2 = (0:halfN2-1) * (Fs2/N2);          % Frequency axis in Hz
+
+figure('Name','Experiment 4 - Part ii');
+plot(f2, magX2_half, 'g');             % Green magnitude spectrum
+xlabel('Frequency (Hz)');
+ylabel('|X_2(f)|');
+title('Experiment 4 - Part ii : Magnitude spectrum of x(t)');
+grid on;