Question

Please answer in MATLAB, thank you!

2. Calculate the energy of time domain signal x (t) and z (t) for the range of 0SIS2.5 Also calculate the energy of these signals in frequency domain using Parseval's theorem. Plot Energy (X) and Energy (Z) as a function of frequency f in a 2xl subplot (Energy vs frequency plot is know as energy spectrum of a signal).

Answer 1

Signal energy using Parseval's theorem is given by: N-1 N-1 where X[k] is the DFT of x[n], both of length N.

Matlab Code is below:

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close all,
clear all,
clc,

Fs = 1000;
F1 = 5;
F2 = 10;
Amp = 1;

t = 0:(1/Fs):2.5;

Xt = Amp * sin(2*pi*F1*t);
Zt = Amp * sin(2*pi*F2*t);

subplot(2,3,1); plot(Xt); str = strcat('Signal X(t) in Time Domain, F = ',num2str(F1),' Hz'); title(str);
subplot(2,3,4); plot(Zt); str = strcat('Signal Z(t) in Time Domain, F = ',num2str(F2),' Hz'); title(str);

SignalEnergy_TimeDomain_X = num2str(sumsqr(Xt))
SignalEnergy_TimeDomain_Z = num2str(sumsqr(Zt))

L=length(Xt);
NFFT = 2^nextpow2(L); % Next power of 2 from length of y
Y = fft(Xt,NFFT)/L;
EnergySpectrum_X = abs(Y).^2/L;
subplot(2,3,3); plot(EnergySpectrum_X); title('Energy Spectrum - X(t)');

f = Fs/2*linspace(0,1,NFFT/2);
SignalEnergy_FreqDomain_X = num2str(sumsqr(abs(Y))/L)

% Plot single-sided amplitude spectrum.
subplot(2,3,2); plot(2*abs(Y(1:NFFT/2)));
str = strcat('X(t) -> FFT at N = ',num2str(NFFT),' Points'); title(str);
xlabel('Frequency (Hz)'); ylabel('|Y(f)|');

L=length(Zt);
NFFT = 2^nextpow2(L); % Next power of 2 from length of y
Y = fft(Zt,NFFT)/L;
EnergySpectrum_Z = abs(Y).^2/L;
f = Fs/2*linspace(0,1,NFFT/2);
SignalEnergy_FreqDomain_Z = num2str(sumsqr(abs(Y))/L)
subplot(2,3,6); plot(EnergySpectrum_Z); title('Energy Spectrum - Z(t)');

% Plot single-sided amplitude spectrum.
subplot(2,3,5); plot(f,2*abs(Y(1:NFFT/2)));
str = strcat('Z(t) -> FFT at N = ',num2str(NFFT),' Points'); title(str);
xlabel('Frequency (Hz)'); ylabel('|Y(f)|');

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Signal Xt) in Time Domain, F-5 Hz XIt)-> FFT at N-4095 Points Energy Spectrum X) 0.8 0.6 0.4 0.5 0.2 0 500 1000 1500 2000 2500 3000 500 1000 500 2000 2500 000 2000 3000 4000 5000 Frequency (Hz) Signal Zt) in Time Domain, F 10 Hz Zt)-> FFT at N406 Points Energy Spectrum-Zi) 0.8 0.8 0.6 0.6 0.4 0.5 0.2 0.2 200 0 500 1000 1500 2000 2500 3000 100 300 400 500 000 2000 3000 4000 5000 Frequency (Hz)