Obisidian vault auto-backup: 05-01-2026 14:58:35 on . 2 files edited

ISEN/Traitement du signal/CIPA4/TP/TP2/TP2_Experience7.m

@@ -1,41 +1,52 @@
 %% Experiment 7 : Spectral Analysis of the raised cosine waveform
+% Raised cosine pulse:
+%   h(t) = sinc(t/T) * cos(pi*beta*t/T) / (1 - (4*beta^2*t.^2)/T^2)
+% where sinc(x) = sin(pi*x)/(pi*x).
+%
+% For this experiment:
+%   T   = 2 ms
+%   beta = 0.5 (then 0.25)
+%   t in [-10T, 10T]
+%   Ts  = 0.13*T  (sampling period)
+%
+% We generate the pulse h(t), plot it in time (ms) and frequency (Hz),
+% and compute the theoretical bandwidth B = (1/(2T))*(1+beta). [web:222][web:228]
 
 clc;
 clear;
 close all;
 
-%% Common parameters
+%% 1) Common parameters
+
 T   = 2e-3;                 % Symbol period (s)
 Ts  = 0.13 * T;             % Sampling period (s)
 Fs  = 1 / Ts;               % Sampling frequency (Hz)
 t   = -10*T : Ts : 10*T;    % Time vector from -10T to 10T
 N   = length(t);            % Number of samples
 
-%% Helper function: raised cosine pulse (inline)
+% Frequency axis for FFT (centered around 0)
-% We handle the singularities at t = 0 and t = ±T/(2*beta) using limits. 
+f_axis = (-N/2 : N/2-1) * (Fs / N);   % Hz
-rc_pulse = @(beta) ...
+
-    arrayfun(@(tt) ...
+% Inline handle to generate the pulse for any beta value
-        raisedCosineSample(tt, T, beta), t);
+rc_pulse = @(beta) arrayfun(@(tt) raisedCosineSample(tt, T, beta), t);
+
+%% ===================== beta = 0.5 =====================
 
-%% ===== a) & b) for beta = 0.5 =====
 beta1 = 0.5;
 
-h1 = rc_pulse(beta1);       % Time-domain raised cosine pulse
+% Time-domain pulse
+h1 = rc_pulse(beta1);
 
-% --- Time-domain plot (x-axis in ms) ---
 figure('Name','Experiment 7 - Time domain, beta = 0.5');
-plot(t*1e3, h1, 'g');       % t in ms
+plot(t*1e3, h1, 'g');           % t in ms
 xlabel('Time (ms)');
 ylabel('h(t)');
 title('Raised cosine pulse h(t), \beta = 0.5');
 grid on;
 
-% --- Frequency-domain magnitude spectrum ---
+% Frequency-domain magnitude spectrum
-H1 = fft(h1);               % FFT of the pulse
+H1    = fft(h1);                % FFT of the pulse [web:8]
-magH1 = abs(fftshift(H1));  % Center zero frequency; take magnitude
+magH1 = abs(fftshift(H1));      % Magnitude, zero frequency centered [web:202]
-
-% Frequency axis in Hz, centered around 0
-f_axis = (-N/2:N/2-1) * (Fs/N);
 
 figure('Name','Experiment 7 - Frequency domain, beta = 0.5');
 plot(f_axis, magH1, 'g');
@@ -44,19 +55,18 @@ ylabel('|H(f)|');
 title('Magnitude spectrum of raised cosine pulse, \beta = 0.5');
 grid on;
 
-% --- c) Measure bandwidth (approximate)
+% Theoretical bandwidth
-% For a raised cosine with symbol rate Rs = 1/T, ideal theoretical
+Rs       = 1 / T;                      % Symbol rate (Hz)
-% baseband bandwidth is B = (Rs/2)*(1+beta). 
+BW_theo1 = (Rs/2) * (1 + beta1);       % B = (Rs/2)*(1+beta) [web:222][web:228]
-Rs      = 1/T;                    % Symbol rate (Hz)
+fprintf('Theoretical bandwidth for beta = 0.5 : %.2f Hz\n', BW_theo1);
-BW_theo1 = (Rs/2) * (1 + beta1);  % Theoretical bandwidth (Hz)
+
-fprintf('Theoretical bandwidth for beta=0.5: %.2f Hz\n', BW_theo1);
+%% ===================== beta = 0.25 =====================
 
-%% ===== d) Repeat for beta = 0.25 =====
 beta2 = 0.25;
 
+% Time-domain pulse
 h2 = rc_pulse(beta2);
 
-% Time-domain plot
 figure('Name','Experiment 7 - Time domain, beta = 0.25');
 plot(t*1e3, h2, 'g');
 xlabel('Time (ms)');
@@ -65,8 +75,9 @@ title('Raised cosine pulse h(t), \beta = 0.25');
 grid on;
 
 % Frequency-domain magnitude spectrum
-H2 = fft(h2);
+H2    = fft(h2);
 magH2 = abs(fftshift(H2));
+
 figure('Name','Experiment 7 - Frequency domain, beta = 0.25');
 plot(f_axis, magH2, 'g');
 xlabel('Frequency (Hz)');
@@ -74,23 +85,34 @@ ylabel('|H(f)|');
 title('Magnitude spectrum of raised cosine pulse, \beta = 0.25');
 grid on;
 
+% Theoretical bandwidth
 BW_theo2 = (Rs/2) * (1 + beta2);
-fprintf('Theoretical bandwidth for beta=0.25: %.2f Hz\n', BW_theo2);
+fprintf('Theoretical bandwidth for beta = 0.25 : %.2f Hz\n', BW_theo2);
 
 %% ===== Local function for one sample of raised cosine =====
 function h = raisedCosineSample(t, T, beta)
     % Raised cosine pulse sample at time t (scalar).
     % h(t) = sinc(t/T) * cos(pi*beta*t/T) / (1 - (4*beta^2*t^2)/T^2)
-    % Special cases at t = 0 and t = ±T/(2*beta) use L'Hospital limits. 
+    % with sinc(x) = sin(pi*x)/(pi*x).
+    % Special cases at t = 0 and t = ±T/(2*beta) use limit values. [web:223]
 
+    x = t / T;
+
+    % Manual definition of sinc(x) = sin(pi*x)/(pi*x)
+    if abs(x) < 1e-12
+        sx = 1;                       % limit at x -> 0
+    else
+        sx = sin(pi*x) / (pi*x);
+    end
+
+    % Handle critical points of the raised cosine denominator
     if abs(t) < 1e-12
-        % Limit at t -> 0
+        % At t -> 0, h(0) = 1 (sinc(0)=1 and cos(0)/(1-0)=1)
-        h = 1;  % sinc(0) = 1 and cos(0)/(1-0) = 1
+        h = 1;
     elseif beta ~= 0 && abs(abs(t) - T/(2*beta)) < 1e-12
-        % Singularities at t = ±T/(2*beta) -> use known limit value 
+        % Limit at t = ±T/(2*beta) [web:223]
         h = (beta/pi) * sin(pi/(2*beta));
     else
-        x = t / T;
+        h = sx * cos(pi*beta*x) / (1 - (4*beta^2*x^2));
-        h = sinc(x) * cos(pi*beta*x) / (1 - (4*beta^2*x^2));
     end
 end