MATLAB PROBLEM: We wish to transmit the following 20 bits of data, using various
ID: 3794837 • Letter: M
Question
MATLAB PROBLEM:
We wish to transmit the following 20 bits of data, using various modulation schemes: [1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1]
1. Using a data rate of 1 bit per second, a carrier of 5 Hz., and a sampling frequency of 50 samples per second, create a MATLAB plot of the OOK modulated signal, along with a plot of the PSD of the signal. Use the functions "plot_fft_spectrum" and "fft_spectrum" as a starting point, and modify as necessary.
2. Repeat for BPSK
3. Now create an FSK modulation of the same signal, using 4 Hz. and 6 Hz. to represent binary zeros and ones respectively.
------------------------------------------------------------------------------------------
% "plot_fft_spectrum"
function [f,X] = fft_spectrum(t,x)
%this function uses the built-in fft to compute the frequency spectrum of a
%signal x. It also generates the appropriate frequency scale, f, and
%applies the "fftshift" so that X and f vectors are available directly for
%plotting.
%the inputs x and t should be vectors of the same length.
%the outputs, X and f will also be vectors of that length
%this program is very simple, and will only work if the time axis either starts with 0, or is
%balanced positive and negative, with 0 in the center.
%extract the parameters of t vector
N=length(x);
dt=t(2)-t(1);
%create frequency vector
df=1/(N*dt);
%note the difference between odd length and even length vectors
if rem(N,2)==0 %N even
f= ( (-N/2):(N/2-1) )*df;
else %N odd
f= ( (-(N-1)/2):((N-1)/2) )*df;
end
%find the index of the t=0 element
N0=find(t==0);
%do the fft to create the spectrum samples
%use proper shifting depending on the time scale and scale amplitude by dt
if N0==1 %time scale is nonnegative
%X=fftshift(fft(fftshift(x)))*dt;
X=fftshift(fft((x)))*dt;
elseif rem(N,2)==0 && N0==N/2+1 %time scale is balanced, N is even
X=fftshift(fft(x))*dt;
elseif rem(N,2)==1 && N0==(N+1)/2 %time scale is balanced, N is odd
X=fftshift(fft(x))*dt;
else %time scale is unbalanced, so this function won't handle it
disp('Unbalanced time scale.')
end
-----------------------------------------
% "fft_spectrum"
function [f,X] = plot_fft_spectrum(t,x,fignum)
%this function calls fft_spectrum to compute the fft of an arbitrary
%signal, and plot the magnitude and phase spectrum
%It calls the function fft_spectrum to do the computation
%INPUTS
%t is the vector of time samples on which x is defined
%x is the vector of samples of the function x(t)
%fignum is the figure number you wish MATLAB to plot in
%compute the spectrum and the frequency axis values
if t(1)==0 %positive time scale
[f,X]=fft_spectrum(t,x);
else %balanced time scale, so put the t=0 value in the first element
[f,X]=fft_spectrum(t,fftshift(x));
end
figure(1), clf
subplot(3,1,1), plot(t,x,'b','LineWidth',2)
title('Time Signal')
ylabel('x(t)')
xlabel('Time (seconds)')
grid on
subplot(3,1,2), plot(f,abs(X),'b','LineWidth',2)
title('Magnitude Spectrum')
ylabel('|X(f)|')
xlabel('Frequency (Hz.)')
axis([min(f) max(f) 0 1.2*max(abs(X))])
grid on
subplot(3,1,3), plot(f,angle(X),'r','LineWidth',2)
title('Phase Spectrum')
ylabel('Phi(X(f))')
xlabel('Frequency (Hz.)')
grid on
% subplot(5,1,4), plot(f,real(X),'b')
% grid on
% subplot(5,1,5), plot(f,imag(X),'r')
% grid on
Explanation / Answer
% "plot_fft_spectrum"
function [f,X] = fft_spectrum(t,x)
%this function uses the built-in fft to compute the frequency spectrum of a
%signal x. It also generates the appropriate frequency scale, f, and
%applies the "fftshift" so that X and f vectors are available directly for
%plotting.
%the inputs x and t should be vectors of the same length.
%the outputs, X and f will also be vectors of that length
%this program is very simple, and will only work if the time axis either starts with 0, or is
%balanced positive and negative, with 0 in the center.
%extract the parameters of t vector
N=length(x);
dt=t(2)-t(1);
%create frequency vector
df=1/(N*dt);
%note the difference between odd length and even length vectors
if rem(N,2)==0 %N even
f= ( (-N/2):(N/2-1) )*df;
else %N odd
f= ( (-(N-1)/2):((N-1)/2) )*df;
end
%find the index of the t=0 element
N0=find(t==0);
%do the fft to create the spectrum samples
%use proper shifting depending on the time scale and scale amplitude by dt
if N0==1 %time scale is nonnegative
%X=fftshift(fft(fftshift(x)))*dt;
X=fftshift(fft((x)))*dt;
elseif rem(N,2)==0 && N0==N/2+1 %time scale is balanced, N is even
X=fftshift(fft(x))*dt;
elseif rem(N,2)==1 && N0==(N+1)/2 %time scale is balanced, N is odd
X=fftshift(fft(x))*dt;
else %time scale is unbalanced, so this function won't handle it
disp('Unbalanced time scale.')
end
-----------------------------------------
% "fft_spectrum"
function [f,X] = plot_fft_spectrum(t,x,fignum)
%this function calls fft_spectrum to compute the fft of an arbitrary
%signal, and plot the magnitude and phase spectrum
%It calls the function fft_spectrum to do the computation
%INPUTS
%t is the vector of time samples on which x is defined
%x is the vector of samples of the function x(t)
%fignum is the figure number you wish MATLAB to plot in
%compute the spectrum and the frequency axis values
if t(1)==0 %positive time scale
[f,X]=fft_spectrum(t,x);
else %balanced time scale, so put the t=0 value in the first element
[f,X]=fft_spectrum(t,fftshift(x));
end
figure(1), clf
subplot(3,1,1), plot(t,x,'b','LineWidth',2)
title('Time Signal')
ylabel('x(t)')
xlabel('Time (seconds)')
grid on
subplot(3,1,2), plot(f,abs(X),'b','LineWidth',2)
title('Magnitude Spectrum')
ylabel('|X(f)|')
xlabel('Frequency (Hz.)')
axis([min(f) max(f) 0 1.2*max(abs(X))])
grid on
subplot(3,1,3), plot(f,angle(X),'r','LineWidth',2)
title('Phase Spectrum')
ylabel('Phi(X(f))')
xlabel('Frequency (Hz.)')
grid on
% subplot(5,1,4), plot(f,real(X),'b')
% grid on
% subplot(5,1,5), plot(f,imag(X),'r')
% grid on
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