parti.m

DSP and fourier transforms

clear all %Clear all variables in memory

Trep = 1e-6;                  % Define the time interval that will make signals look continuous
t1 = [0:Trep:.01];            % Define a vector representing time
f_tone1=1000;                 % Set frequency to 1000Hz

Tsmp = 1e-4;                  % Sample interval?

%Part I a)defines a vector representing the input continuous time signal

xt = cos(2*pi*f_tone1*t1);

%Part I b)Plots the input continuous time signal

subplot(6,1,1), plot(xt), title('Continuous time Signal: x(t)');

%Part I c)Calculates and plots the Fourier Transform spectrum of x(t)

xt_FT = fft(xt);
xt_FT_shift = fftshift(xt_FT);
f_axis=linspace(-1/Trep/2,1/Trep/2,length(xt_FT));
subplot(6,1,2), plot(f_axis,abs(xt_FT_shift)), axis([-1500, 1500,0,5000]), title('Frequency Spectrum - Fourier Transform  of X(JW)');


%Part I d)Defines a second vector representing the impulse train p(t)

samplingInterval = Tsmp/Trep;
p = zeros(1,length(t1));
p(1:samplingInterval:end) = 1;

%Part I e)Ideally samples the input signal x(t) to get xs(t)

xS = xt.*p;
subplot(6,1,3), plot(t1,xS), title('Xs(t): sampled x(t)');

%Part I f)Calculates and plots Xs(jw), the Fourier Transform spectrum of
%the sampled version of x(t)

xS_FT = fft(xS);
xS_FT_Shift = fftshift(xS_FT);
xS_FT_Shift(find(abs(xS_FT_Shift)<1.5))=0;
subplot(6,1,4), plot(f_axis,xS_FT_Shift), axis([-20000, 20000,0,60]), title('FFT Spectrum: Xs(JW)');

%Part I g)Ideally filters the sampled signal Xs(t) with an ideal low-pass
%          filter defined

xsFilter = xS_FT_Shift;
for n=1:length(f_axis)
    xsFilter(n) = lowPassFilter( xS_FT_Shift(n), Tsmp, f_axis(n) * 2 * pi );
end

%Part I h)Plots the resulting spectrum of the reconstructed signal Xr(jw)

xsFilter_FT = fft(xsFilter);
xsFilter_FT_Shift = fftshift(xsFilter_FT);
subplot(6,1,5), plot(f_axis,abs(xsFilter)), axis([-1500, 1500,0,.01]), title('Reconstruction of Signal : Xr(JW)');

%Part I i)Calculate the reconstructed signal Xr(t) by computing the
%          inverse Fourier Transform of Xr(jw)

xs_IFFT = invFT(xsFilter);
subplot(6,1,6),
plot(t1,xs_IFFT), title('Comparison of Signals: x(t) AND Xr(jw)');
hold on;
Scaled_out=max(xs_IFFT)/max(xt)*xt;
plot(t1,Scaled_out);
hold off;