TP2 delivered.

TP2/DiogoEliseu_TP2_1.m

@@ -1,11 +1,24 @@
+%% Inicialização do ambiente
+clear ; close all; clc
+
 %% Exercício 1
 
+figure(1)
 p = [1.8, 0.5, -0.3];
+d = [1, 0.3, -0.2];
+w = -4*pi:(8*pi/999):4*pi;
+freqz(p, d, w)
+
+figure(2)
+p = [1];
 d = [1, 0.5];
 w = -4*pi:(8*pi/999):4*pi;
-X = freqz(p, d, w);
+freqz(p, d, w)
 
-subplot(2,1,1),plot(w, abs(X))
+figure(3)
-xlabel('w'),title('Magnitude')
+p = [1 2 3 4 1 3];
-subplot(2,1,2),plot(w, angle(X))
+w = -4*pi:(8*pi/999):4*pi;
-xlabel('w'),title('Phase')
+%[cena, coisa] = freqz(p, 1, w);
+%plot(unwrap(angle(cena)));
+%figure(4)
+freqz(p, 1, w);

TP2/DiogoEliseu_TP2_2.m

@@ -13,12 +13,14 @@ DTFT2 = fft(x2, 1024);
 
 subplot(2,1,1)
 stem(x2)
+xlabel('n'), ylabel('Amplitude'), title('Sinal x2')
+
 subplot(2,1,2)
 hold on
-stem(X(16),abs(DFT2))
+stem(X(16), abs(DFT2), "LineStyle", "none")
 plot(X(1024), abs(DTFT2))
+xlabel('\omega/2\pi'), ylabel('Modulo'), title('DFT / DTFT de x2')
 hold off
-xlabel('cenas'), ylabel('Modulo'), title('DFT / DTFT de x2')
 
 function f = X(N)
 f = 0:(1/N):(1 - 1/N);

TP2/DiogoEliseu_TP2_3.m

@@ -1,24 +1,27 @@
-% Exercício 3
+%% Inicialização do ambiente
+clear ; close all; clc
+
+%% Exercício 3
 % signals
 N = 256;
 n = (0:N-1);
-X1 = 2*N*0.97.^n;
+X1 = 2.*n.*0.97.^n;
-X2 = cos(6*pi*n/N);
+X2 = cos(6*pi.*n./N);
-X3 = sin(12*pi*n/N);
+X3 = sin(12*pi.*n./N);
-
-% freq conv
-X1_fft = fft(X1);
-X2_fft = fft(X2);
-X3_fft = fft(X3);
-X1_X2_f = ifft(X1_fft.*X2_fft);
-X1_X3_f = ifft(X1_fft.*X3_fft);
-X2_X3_f = ifft(X2_fft.*X3_fft);
 
 % time conv
 X1_X2_t = conv(X1, X2);
 X1_X3_t = conv(X1, X3);
 X2_X3_t = conv(X2, X3);
 
+% freq conv
+X1_fft = fft([X1 zeros(1,N-1)]);
+X2_fft = fft([X2 zeros(1,N-1)]);
+X3_fft = fft([X3 zeros(1,N-1)]);
+X1_X2_f = ifft(X1_fft.*X2_fft);
+X1_X3_f = ifft(X1_fft.*X3_fft);
+X2_X3_f = ifft(X2_fft.*X3_fft);
+
 %--- plots
 
 % signals
@@ -38,36 +41,38 @@ plot(X3);
 xlabel("n")
 ylabel("X3")
 
+x_axis = -(N-1):(N-1);
+
 % time
 subplot(3,3,4)
-plot(X1_X2_t);
+plot(x_axis, X1_X2_t);
 %title("X1 * X2 no domínio do tempo")
 xlabel("n")
 ylabel("X1(t) * X2(t)")
 subplot(3,3,5)
-plot(X1_X3_t);
+plot(x_axis, X1_X3_t);
 title("*(t)")
 xlabel("n")
 ylabel("X1(t) * X3(t)")
 subplot(3,3,6)
-plot(X2_X3_t);
+plot(x_axis, X2_X3_t);
 %title("X2 * X3 no domínio do tempo")
 xlabel("n")
 ylabel("X2(t) * X3(t)")
 
 % freq
 subplot(3,3,7)
-plot(X1_X2_f);
+plot(x_axis, X1_X2_f);
 %title("X1 * X2 no domínio da freq")
 xlabel("n")
 ylabel("X1(\omega) * X2(\omega)")
 subplot(3,3,8)
-plot(X1_X3_f);
+plot(x_axis, X1_X3_f);
 title("*(\omega)")
 xlabel("n")
 ylabel("X1(\omega) * X3(\omega)")
 subplot(3,3,9)
-plot(X2_X3_f);
+plot(x_axis, X2_X3_f);
 %title("X2 * X3 no domínio da freq")
 xlabel("n")
 ylabel("X2(\omega) * X3(\omega)")

TP2/DiogoEliseu_TP2_4.m

@@ -0,0 +1,121 @@
+%% Inicialização do ambiente
+clear ; close all; clc
+
+% Definimos uma seed estatica para o gerador de números aleatórios de forma
+% a podermos repetir os experimentos com resultados determinísticos
+rng(42,'twister');
+
+
+%% Exercício 4
+%a)
+N = 255;
+n = 0:N;
+X0 = 0.25.*n.*exp(-0.03.*n);
+Ra = (cos(0.82*pi.*n) + sin(0.85*pi.*n) + sin(0.91*pi.*n) + cos(0.95*pi.*n))/4;
+Rb = (0.6-(-0.6)).*rand(1,N+1) + (-0.6);
+
+Xa = X0+Ra;
+Xb = X0+Rb;
+Xc = X0 + Ra + Rb .* Xa;
+
+figure(1)
+subplot(2,3,1)
+plot(X0);
+title("X0");
+subplot(2,3,2)
+plot(Ra);
+title("Ra");
+subplot(2,3,3)
+plot(Rb);
+title("Rb");
+subplot(2,3,4)
+plot(Xa);
+title("Xa");
+subplot(2,3,5)
+plot(Xb);
+title("Xb");
+subplot(2,3,6)
+plot(Xc);
+title("Xc");
+
+%b)
+DFT_range1 = fft(Xa, N+1); % entre 0 e 2pi
+DFT_range2 = [ DFT_range1((N+1)/2+1 : (N+1)) , DFT_range1(1 : (N+1)/2) ];
+DFT_range3 = DFT_range1(1 : (N+1)/2);
+
+figure(2)
+subplot(2,3,1)
+plot(X(N+1, 0, 2),abs(DFT_range1))
+xlabel('\omega/2\pi'), ylabel('|X(\omega)|'), title('[0,2\pi]')
+
+subplot(2,3,2)
+plot(X(N+1, -1, 1),abs(DFT_range2))
+xlabel('\omega/2\pi'), ylabel('|X(\omega)|'), title('[-\pi,\pi]')
+
+subplot(2,3,3)
+plot(X((N+1)/2, 0, 1),abs(DFT_range3))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('[0,\pi]')
+
+subplot(2,3,4)
+plot(X(N+1, 0, 2),abs(DFT_range1))
+xlabel('\omega/2\pi'), ylabel('\theta(\omega)'), title('[0,2\pi]')
+
+subplot(2,3,5)
+plot(X(N+1, 0, 2),abs(DFT_range1))
+xlabel('\omega/2\pi'), ylabel('\theta(\omega)'), title('[-\pi,\pi]')
+
+subplot(2,3,6)
+plot(X(N+1, 0, 2),abs(DFT_range1))
+xlabel('\omega/\pi'), ylabel('\theta(\omega)'), title('[0,\pi]')
+
+%c)
+DFT_X0 = fft(X0, N+1);
+DFT_X0 = DFT_X0(1 : (N+1)/2);
+
+DFT_Ra = fft(Ra, N+1);
+DFT_Ra = DFT_Ra(1 : (N+1)/2);
+
+DFT_Rb = fft(Rb, N+1);
+DFT_Rb = DFT_Rb(1 : (N+1)/2);
+
+DFT_Xa = fft(Xa, N+1);
+DFT_Xa = DFT_Xa(1 : (N+1)/2);
+
+DFT_Xb = fft(Xb, N+1);
+DFT_Xb = DFT_Xb(1 : (N+1)/2);
+
+DFT_Xc = fft(Xc, N+1);
+DFT_Xc = DFT_Xc(1 : (N+1)/2);
+
+figure(3)
+subplot(2,3,1)
+plot(X((N+1)/2, 0, 1),abs(DFT_X0))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('X0')
+
+subplot(2,3,2)
+plot(X((N+1)/2, 0, 1),abs(DFT_Ra))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('Ra')
+
+subplot(2,3,3)
+plot(X((N+1)/2, 0, 1),abs(DFT_Rb))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('Rb')
+
+subplot(2,3,4)
+plot(X((N+1)/2, 0, 1),abs(DFT_Xa))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('Xa')
+
+subplot(2,3,5)
+plot(X((N+1)/2, 0, 1),abs(DFT_Xb))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('Xb')
+
+subplot(2,3,6)
+plot(X((N+1)/2, 0, 1),abs(DFT_Xc))
+xlabel('\omega/\pi'), ylabel('|X(\omega)|'), title('Xc')
+
+
+
+
+function f = X(N, inicio, fim)
+f = (0:(1/N):(1 - 1/N))*(fim-inicio)+inicio;
+end
+

TP2/DiogoEliseu_TP2_5.m

@@ -0,0 +1,108 @@
+%% Inicialização do ambiente
+clear ; close all; clc
+
+%% Exercício 5
+file_id = fopen("INS_10k.txt", "r");
+A = fscanf(file_id, "%f %f %f %f %f %f %f", [7 Inf]);
+A=A';
+
+X  = A(:,1); % ang_rate ( angles/sec )
+Y  = A(:,2);
+Z  = A(:,3);
+DX = A(:,4);
+DY = A(:,5);
+DZ = A(:,6);
+t  = A(:,7);
+
+fclose(file_id);
+
+L = length(X); % Length of signal
+total_time = (t(L)-t(1))/1000; % Time the sampling took in seconds
+T = 0.02;  % Sampling period in seconds
+SF = 1/T;  % Sampling frequency (cycles per sec)
+
+tv = 0:1/SF:total_time; % secs (duh.)
+F = SF*(1:(L))/L; % cycles per sec (Hertz)
+
+% -- Normalizations -- %
+F_rad = (2*pi) .* F ./ SF; % rads per sample
+X_rad = X .* (pi/180); % ( radians / sec)
+X_hertz = X_rad .* (1/(2*pi));
+
+DFT_0_2pi = fft(X_rad, L); % entre 0 e 2pi
+DFT_0_pi = DFT_0_2pi(1 : (L)/2);
+
+%1)
+figure(1)
+subplot(2,2,1)
+plot(tv,X) % X(t)
+title("X(t)"); ylabel("Amplitude (angles)"); xlabel("Time (s)"); 
+subplot(2,2,2)
+plot(F_rad, abs(DFT_0_2pi)); % dft in 0:2pi
+title("FFT of X in [0:2\pi]"); ylabel("Magnitude"); xlabel("\omega/\pi");
+subplot(2,2,3)
+plot(F_rad(1 : (L)/2), abs(DFT_0_pi)); %dft in 0:pi
+title("FFT of X in [0:\pi]"); ylabel("Magnitude"); xlabel("\omega/\pi");
+subplot(2,2,4)
+plot(F,X_hertz); % X(f)
+title("X in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+
+%2)
+Y_rad = Y .* (pi/180);
+Y_hertz = Y_rad .* (1/(2*pi));
+Z_rad = Z .* (pi/180);
+Z_hertz = Z_rad .* (1/(2*pi));
+
+figure(2)
+subplot(2,3,1)
+plot(F,X_hertz); % X(f)
+title("X in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+subplot(2,3,2)
+plot(F,Y_hertz); % X(f)
+title("Y in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+subplot(2,3,3)
+plot(F,Z_hertz); % X(f)
+title("Z in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+
+subplot(2,3,4)
+plot(F,abs(fft(X_hertz)));
+title("DFT of X in function of the frequency (Hz)"); ylabel("Magnitude"); xlabel("Frequency (Hertz)");
+subplot(2,3,5)
+plot(F,abs(fft(Y_hertz)));
+title("DFT of Y in function of the frequency (Hz)"); ylabel("Magnitude"); xlabel("Frequency (Hertz)");
+subplot(2,3,6)
+plot(F,abs(fft(Z_hertz)));
+title("DFT of Z in function of the frequency (Hz)"); ylabel("Magnitude"); xlabel("Frequency (Hertz)");
+
+%3)
+DX_rad = DX .* (pi/180);
+DX_hertz = DX_rad .* (1/(2*pi));
+DX_hertz = DX_hertz(1:601);
+DY_rad = DY .* (pi/180);
+DY_hertz = DY_rad .* (1/(2*pi));
+DY_hertz = DY_hertz(1:601);
+DZ_rad = DZ .* (pi/180);
+DZ_hertz = DZ_rad .* (1/(2*pi));
+DZ_hertz = DZ_hertz(1:601);
+F3 = F(1:601);
+
+figure(3)
+subplot(2,3,1)
+plot(F3,DX_hertz); % X(f)
+title("X' in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+subplot(2,3,2)
+plot(F3,DY_hertz); % X(f)
+title("Y' in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+subplot(2,3,3)
+plot(F3,DZ_hertz); % X(f)
+title("Z' in function of the frequency (Hz)"); ylabel("Amplitude (Hertz)"); xlabel("Frequency (Hertz)");
+
+subplot(2,3,4)
+plot(F3,abs(fft(DX_hertz)));
+title("DFT of X in function of the frequency (Hz)"); ylabel("Magnitude"); xlabel("Frequency (Hertz)");
+subplot(2,3,5)
+plot(F3,abs(fft(DY_hertz)));
+title("DFT of Y in function of the frequency (Hz)"); ylabel("Magnitude"); xlabel("Frequency (Hertz)");
+subplot(2,3,6)
+plot(F3,abs(fft(DZ_hertz)));
+title("DFT of Z in function of the frequency (Hz)"); ylabel("Magnitude"); xlabel("Frequency (Hertz)");

TP2/DiogoEliseu_TP2_6.m

@@ -0,0 +1,101 @@
+%% Análise de Resultados
+%[+] Para n = 256
+% [>] conv demorou 0.000481
+% [>] ifft demorou 0.001027
+%
+%[+] Para n = 1337
+% [>] conv demorou 0.000601
+% [>] ifft demorou 0.003004
+%
+%[+] Para n = 31337
+% [>] conv demorou 0.456320
+% [>] ifft demorou 0.038267
+%
+%[+] Para n = 424242
+% [>] conv demorou 5.849966
+% [>] ifft demorou 0.885959
+%
+%
+% Verifica-se que para n pequenos temos conv mais eficiente do que ifft,
+% Para n maiores temos ifft mais eficiente do que conv.
+
+%% Inicialização do ambiente
+clear ; close all; clc
+
+%% Exercício 6
+% signals
+N = 256;
+n = (0:N-1);
+X1 = 2.*n.*0.97.^n;
+X2 = cos(6*pi.*n./N);
+
+% time conv
+tic
+X1_X2_t = conv(X1, X2);
+tocked1 = toc;
+
+% freq conv
+tic
+X1_fft = fft([X1 zeros(1,N-1)]);
+X2_fft = fft([X2 zeros(1,N-1)]);
+X1_X2_f = ifft(X1_fft.*X2_fft);
+tocked2 = toc;
+
+fprintf("[+] Para n = %d\n [>] conv demorou %f\n [>] ifft demorou %f\n\n",N, tocked1, tocked2);
+
+N = 1337;
+n = (0:N-1);
+X1 = 2.*n.*0.97.^n;
+X2 = cos(6*pi.*n./N);
+
+% time conv
+tic
+X1_X2_t = conv(X1, X2);
+tocked1 = toc;
+
+% freq conv
+tic
+X1_fft = fft([X1 zeros(1,N-1)]);
+X2_fft = fft([X2 zeros(1,N-1)]);
+X1_X2_f = ifft(X1_fft.*X2_fft);
+tocked2 = toc;
+
+fprintf("[+] Para n = %d\n [>] conv demorou %f\n [>] ifft demorou %f\n\n",N, tocked1, tocked2);
+
+N = 31337;
+n = (0:N-1);
+X1 = 2.*n.*0.97.^n;
+X2 = cos(6*pi.*n./N);
+
+% time conv
+tic
+X1_X2_t = conv(X1, X2);
+tocked1 = toc;
+
+% freq conv
+tic
+X1_fft = fft([X1 zeros(1,N-1)]);
+X2_fft = fft([X2 zeros(1,N-1)]);
+X1_X2_f = ifft(X1_fft.*X2_fft);
+tocked2 = toc;
+
+fprintf("[+] Para n = %d\n [>] conv demorou %f\n [>] ifft demorou %f\n\n",N, tocked1, tocked2);
+
+N = 424242;
+n = (0:N-1);
+X1 = 2.*n.*0.97.^n;
+X2 = cos(6*pi.*n./N);
+
+% time conv
+tic
+X1_X2_t = conv(X1, X2);
+tocked1 = toc;
+
+% freq conv
+tic
+X1_fft = fft([X1 zeros(1,N-1)]);
+X2_fft = fft([X2 zeros(1,N-1)]);
+X1_X2_f = ifft(X1_fft.*X2_fft);
+tocked2 = toc;
+
+fprintf("[+] Para n = %d\n [>] conv demorou %f\n [>] ifft demorou %f\n",N, tocked1, tocked2);