How to plot the graph with different amplitudes at different frequencies?

Question

TEOH CHEE JIN 2022-12-28

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1885582-how-to-plot-the-graph-with-different-amplitudes-at-different-frequencies

回答： Sayan 2023-9-15

For the amplitude of D(w), I have used the command "simplify" and obtained that there are two different amplitudes at different frequency. At w=40pi, the absolute amplitude is 50pi, whereas at w=4960pi and w=5040pi, the amplitudes are 25pi, what should I do to plot the graph showing different amplitudes at different frequency?

%create symbolic functions x, a, b, c, d, e, f_c with independent variable t

syms x(t) a(t) h(t) b(t) c(t) d(t) e f_c(t) f_c1(t) f_c2(t) t tau

%create symbolic functions A, B, C, D, and E with independent variable w

syms A(w) B(w) C(w) D(w) E(w) w

x(t) = cos(100*pi*t);

a(t) = x(0.4*t);

h(t) = dirac(t-0.02);

b(t) = int(a(tau)*h(t-tau), 'tau', -inf, inf);

f_c(t) = 10*cos(2500*pi*t);

f_c1(t) = f_c(t);

f_c2(t) = f_c(t);

c(t) = b(t)*f_c1(t);

d(t) = c(t)*f_c2(t);

% f_c1(t)

% f_c2(t)

% figure

%

% subplot (3,1,1)

% fplot(a(t))

% xlim([-0.05 0.05]),ylim([-1.5 1.5])

% title ('Time domain of signal a(t)')

% xlabel('Time, t')

% ylabel('Amplitude, a(t)')

% grid on

%

% A(w) = fourier(a(t), w);

% w = -60*pi:0.1*pi:60*pi;

% subsA = A(w);

% % Replace Inf value with suitable value

% idx = subsA == Inf;

% subsA(idx) = pi; % choose suitable value from the expression of fourier transform.

% subplot (3,1,2)

% plot(w,real(subsA));

% ylim([0 4])

% title ('Frequency domain of signal A(\omega)')

% ylabel("\Re(A(\omega))");

% xlabel("\omega");

% xticks(-40*pi:20*pi:40*pi);

% xticklabels({'-40\pi', '-20\pi', '0', '20\pi', '40\pi'})

%

% subplot (3,1,3)

% plot(w, angle(A(w)))

% title ('Phase spectrum of signal A(\omega)')

% ylabel("\angle(A(\omega))");

% xlabel("\omega");

%

% figure

%

% subplot(3,1,1)

% fplot(b(t))

% xlim([-0.05 0.05]),ylim([-1.5 1.5])

% title ('Time domain of signal b(t)')

% xlabel('Time, t')

% ylabel('Amplitude, b(t)')

% grid on

%

% syms B w

% B(w) = fourier(b(t), w);

% w = -60*pi:0.1*pi:60*pi;

% subsB = B(w);

% % Replace Inf value with suitable value

% idx = abs(subsB) == Inf;

% subsB(idx) = pi; % choose suitable value from the expression of fourier transform.

% subplot (3,1,2)

% plot(w,real(subsB));

% ylim([0 4])

% title ('Frequency domain of signal B(\omega)')

% ylabel("\Re(B(\omega))");

% xlabel("\omega");

% xticks(-60*pi:20*pi:60*pi);

% xticklabels({'-40\pi', '-20\pi', '0', '20\pi', '40\pi'})

%

% subplot (3,1,3)

% plot(w, angle(B(w)))

% ylim([-4 4])

% title ('Phase spectrum of signal B(\omega)')

% ylabel("\angle(B(\omega))");

% xlabel("\omega");

% figure

%

% subplot(3,1,1)

% fplot(c(t))

% xlim([-0.05 0.05]),ylim([-12 12])

% title ('Time domain of signal c(t)')

% xlabel('Time, t')

% ylabel('Amplitude, c(t)')

% grid on

%

% syms C w

% C(w) = fourier(c(t), w);

% simplify(C(w))

% w = -2800*pi:20*pi:2800*pi;

% subsC = C(w);

% % Replace Inf value with suitable value

% idx = abs(subsC) == Inf;

% subsC(idx) = 5*pi; % choose suitable value from the expression of fourier transform.

% subplot (3,1,2)

% plot(w,real(subsC));

%

% ylim([0 20])

% title ('Frequency domain of signal C(\omega)')

% ylabel("\Re(C(\omega))");

% xlabel("\omega");

% xticks([-2540*pi -2460*pi 0 2460*pi 2540*pi]);

% xticklabels({'-2540\pi', '-2460\pi', '0', '2460\pi', '2540\pi'})

%

% subplot (3,1,3)

% plot(w, angle(C(w)))

% ylim([-4 4])

% title ('Phase spectrum of signal C(\omega)')

% ylabel("\angle(C(\omega))");

% xlabel("\omega");

figure

subplot(3,1,1)

fplot(d(t))

xlim([-0.1 0.1]),ylim([-110 110])

title ('Time domain of signal d(t)')

xlabel('Time, t')

ylabel('Amplitude, d(t)')

grid on

syms D w

D(w) = fourier(d(t), w);

w = -5040*pi:10*pi:5040*pi;

simplify(D(w))

ans =

subsD = D(w);

% Replace Inf value with suitable value

idx = abs(subsD) == Inf;

subsD(idx) = 25*pi; % choose suitable value from the expression of fourier transform.

subplot (3,1,2)

plot(w,real(subsD));

ylim([0 160])

title ('Frequency domain of signal D(\omega)')

ylabel("\Re(D(\omega))");

xlabel("\omega");

subplot (3,1,3)

plot(w, angle(D(w)))

ylim([-4 4])

title ('Phase spectrum of signal D(\omega)')

ylabel("\angle(D(\omega))");

xlabel("\omega");

This is the calculation that I have done to verify the answer.

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Answer 1

Sayan 2023-9-15

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1885582-how-to-plot-the-graph-with-different-amplitudes-at-different-frequencies#answer_1310941

在 MATLAB Online 中打开

Hi @TEOH CHEE JIN,

I understand from your query that it is required to plot the co-efficient of the "dirac" function of the fourier transform. However, in the code, all the coefficients of the "dirac" function are assigned to 25*pi. This can be found in the below code snippet.

subsD = D(w);
% Replace Inf value with suitable value
idx = abs(subsD) == Inf;
subsD(idx) = 25*pi;%%here the "idx" contains the indices of dirac functions and all the values of "subsD" corresponding to "idx" is assigned to 25*pi

The possible fix is to find the indices of the correlating angular frequency and use them to assign the correct value. This is shown in the below code snippet.

%create a function "ap" to return the indices of the corresponding 
%frequency 
function n=ap(f) 
n=int32(((5040*pi+f)/(10*pi))+1);%as the value of "w" are in arithmetic progrssion indices are obtained using the formula 
end 
subsD = D(w);
%manually assign the values to the "dirac" of the corresponding frequency 
subsD(ap(-5040*pi))=25*pi; 
subsD(ap(5040*pi))=25*pi; 
subsD(ap(-4960*pi))=25*pi; 
subsD(ap(4960*pi))=25*pi; 
subsD(ap(-40*pi))=50*pi; 
subsD(ap(40*pi))=50*pi; 
%now the amplitude of subsD can be plotted 
plot(w,subsD); 

Further information on "dirac" function can be found in the following MATLAB documentation:

https://www.mathworks.com/help/releases/R2022b/symbolic/sym.dirac.html#buerxvk-2

Hope this helps in resolving the query.

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How to plot the graph with different amplitudes at different frequencies?

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How to plot the graph with different amplitudes at different frequencies?

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