Question

NB! This task is required to be solved in matlab. this task also requires the use of the function displayDualSpectrum();
which i have pasted in the bottom.  the tasks that i need help with are A), B) and C). this is a multi-part question.

Task - Frequency mixing
We use a basic signal that can be described mathematically as follows:  
x(t) = A, + Acos (21 fi:t) + A2.cos(21.f2.t)


with this We shall then make an amplitude modulated signal:    where fc is the carrier frequency.  the code below specifies the parameters :

clear;
close all;
%-------------------------------------------------------
% Parameters
%-------------------------------------------------------
Fs = 240; % sampling frequency
F1 = 8;
F2 = 12;
A0 = 0.48;
A1 = 0.31;
A2 = 0.19;
Fc = 50; % Carrier Frequency
T = 20; % signal duration in seconds

A) Complete the code in the field below so that we have two signals in the form of row vectors xt and yt  as described by the formula above. We shall then get a plot of the frequency spectrum into one baseband signal and one frequency modulated signal.

code given by task:

%-------------------------------------------------------
% Signal generation
%-------------------------------------------------------
t = 0:1/Fs:(T-1/Fs); % Gives exactly Fs * T amount of samples.

%-------------------------------------------------------
% Plot of signal spectrum. ( i do not think this one needs to be changed)
%-------------------------------------------------------
subplot(2,1,1);
displayDualSpectrum(xt,Fs);
title('BaseBand-signal x(t)');
subplot(2,1,2);
displayDualSpectrum(yt,Fs);
title('FrequencyMixed signal y(t)');

B)  

Try adjusting the different signal parameters and answer the following control questions:

- Why do we have different amplitudes for the signals in subplot 1 and subplot 2.

- What determines the bandwidth of the signal y(t) ? 

- Try to set the carrier frequency fc to 113 Hz or higher. what happens ?

C) 

Adjust the code and use loops so that the signal x(t) can be given by a variable number of sine components depending on the following formula:

Where the parameters of Ai and fi are given by elements in the row vectors A and F, both with M amounts of elements.

displayDualSpectrum function below:

function displayDualSpectrum(xn,Fs)
L = length(xn);
Xk = fft(xn);
Fk = -Fs/2:Fs/L:(Fs/2-Fs/L);
Xk = [Xk(L/2+1:L) Xk(1:L/2)];

%figure();
%set(gcf, 'Visible', 'on');
  
%plot(Fk, abs(Xk)./(Fs*T), 'LineWidth', 2); axis([max(Fc +F_min,-0.5) (Fc+F_max) 0 (A0+0.02)]);
plot(Fk, abs(Xk)./(L), 'LineWidth', 2);
xlim([-Fs/2 Fs/2]);
xlabel('Frequency (Hz)');
ylabel('Amplitude');
grid on;

end

Answer 1

Matlab code for Q9 function Q9 clear; close all; %---------- % Parameters 96---- Fs = 240; % sampling frequency f1 = 8; f2 = 12; A0 = 0.48; Al = 0.31; A2 = 0.19; fc = 50; % Carrier Frequency T = 20; % signal duration in seconds t = 0:1/Fs:(T-1/Fs); % Gives exactly Fs * T amount of samples. % -- - ---- % Signal generation %-- xt=AO+Al*cos(2*pi*fl*t)+A2* cos(2*pi*f2*t); yt=xt.* (cos(2*pi*fc*t)); % ------------- % Plot of signal spectrum. ( i do not think this one needs to be changed) %-- subplot(2,1,1); displayDualSpectrum(xt, Fs); title('BaseBand-signal x()'); subplot(2,1,2); displayDualSpectrum(yt, Fs); title('FrequencyMixed signal y(t) for fc=113'); Matlab code for display DualSpectrum; function displayDualSpectrum( xn,Fs) L = length(xn); Xk = fft(xn); Fk = -Fs/2:Fs/L:(Fs/2-Fs/L); Xk = [Xk(L/2+1:L) Xk(1:L/2)); %figure(); %set(gcf, 'Visible', 'on'); %plot(Fk, abs(Xk)./(Fs*T), 'Line Width', 2); axis([max(Fc +F_min,-0.5) (Fc+F_max) 0 (A0+0.02)]); plot(Fk, abs(Xk)./(L), 'LineWidth', 2); xlim([-Fs/2 Fs/2]); xlabel('Frequency (Hz)'); ylabel('Amplitude'); grid on; end

BaseBand-signal X(t) Amplitude F- 1 - 100 -100 -50 0 50 100 Frequency (Hz) Frequency Mixed signal y(t) Amplitude -100 -50 50 100 Frequency (Hz)

Part b solution figures

part 1

BaseBand-signal X(t) Amplitude F- -100 - 500 Frequency (Hz) 50 100 Frequency Mixed signal y(t) for carrier 2cos(2pifct) Amplitude -100 -50 50 100 Frequency (Hz)

part b

2)

BaseBand-signal X(t) Amplitude . -100 - 500 Frequency (Hz) 50 100 Frequency Mixed signal y(t) for 12=20 Amplitude -100 -50 50 100 Frequency (Hz)

part b

3)

BaseBand-signal X(t) Amplitude FE -100 50 100 - 500 Frequency (Hz) FrequencyMixed signal y(t) for fc=113 Amplitude 0.1 . .. ............................... -100 -50 50 100 Frequency (Hz)

% part b % 1) since signal x(t) multiplied by cos(2*pi*fc*t) will translate the spectrum % of x(1) to hihg frequency range say 0 to 50 but reduce the amplitude y % let multiply x(t) by 2*cos(2*pi*fc*t), the amplitude remain same % 2) the bandwidth of y(t) depend upon f2 as f2 > f1. If f2 is high the bandwidth % of y(t) signal is also high %plot for f2=20, the spectrum is broader in comparison to f2=12 #3) the higher carrir frequency fc=113 will shift the spectrum of x(t) to higher % frequency say the 0 frequency will shift to 113 hz % for fc= 113 the plot will be at far distance in comaprison to fc =50hz %part c, for give range of vector Ai and fi, sum of signal % Matlab code for sum signal function sum=sumsignal(A,f) t=0:0.01:1; sum=zeros(1,length(t)); for i=1:length(A) sum sum+A(i)*cos(2*pi*f(i)*t); end plot(t, sum); xlabel(' time t'); ylabel(' signal x(t);"); title('plot of summed signal'); end command line instruction for part c >> A=1:1:3; >> f=1:1:3; >> sumsumsignal(A,f);

plot of summed signal signal x(t); 0.2 0.4 0.6 0.8 timet