2. To represent a signal in frequency we need to use the Fourier transform. a. Derive in full the Fourier transform of the signal $M(f) = F\{m(t)\}$ and draw the result by hand. b. In Matlab, we are representing the signal in time using vectors with discrete values. Therefore, we need to apply a discrete version of the Fourier transform. i. Use the following command to generate a discrete Fourier transform of the signal $m$ for $\Delta t = 0.01$ sec and $N = 256$: $M = abs(fftshift(fft(m, N) * \Delta t));$ Now, generate a frequency vector: $f = [-\frac{N}{2}:\frac{N}{2} - 1]/(N * \Delta t)$ ii. Generate a figure presenting $M$ as a function of $f$, for range $-5 \leq f \leq 5$ Hz iii. Compare the curve in the figure to your hand drawn spectrum and provide an account for all discrepancies you observe. c. Read Matlab's documentation on the fft and fftshift functions at
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The DFT is a mathematical transformation that converts a sequence of complex numbers into another sequence of complex numbers, representing the frequency components of the original sequence. The fft function takes an input sequence and returns its DFT. Show more…
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4.21. Compute the Fourier transform of each of the following signals: (a) [e^{-αt} cos ω_0t]u(t), α > 0 (b) e^{-3|t|} sin 2t (c) x(t) = { 1 + cos πt, |t| ≤ 1; 0, |t| > 1 (d) ∑_{k=0}^{∞} α^k δ(t - kT), |α| < 1 (e) [te^{-2t} sin 4t]u(t) (f) [sin πt / πt][sin 2π(t-1) / π(t-1)] (g) x(t) as shown in Figure P4.21(a) (h) x(t) as shown in Figure P4.21(b) (i) x(t) = { 1 - t^2, 0 < t < 1; 0, otherwise (j) ∑_{n=-∞}^{+∞} e^{-|t-2n|} 4.22. Determine the continuous-time signal corresponding to each of the following transforms.
Adi S.
The discrete-time Fourier series (DTFS) is a frequency domain representation for periodic discrete-time sequences. For a signal x[n] with fundamental period N, the DTFS synthesis and analysis equations are given as: Synthesis equation: x[n] = ∑ Ck e^(jkw0n) Analysis equation: Ck = (1/N) ∑ x[n] e^(-jk(2̀́/N)n) In MATLAB environment, we can generate a rectangular signal using square function. The command A*square (2*pi*f*t,d) generates a square wave for the time vector t with frequency f and duty cycle d. You can read Matlab's Help Documentation about square function for more information. Using the parameters f = 2 and d = %50, generate a rectangular signal x(t) for a time vector of t = 0:0.1:1.
Sri K.
Use the following code as a reference to generate a signal and find the FFT. Fs = 1000; % Sampling frequency T = 1/Fs; % Sampling period L = 500; % Length of signal t = (0:L-1)*T; y0 = sin(2*pi*50*t); Y = fft(y0); P2 = abs(Y/L); % 2-sided spectrum P1 = P2(1:L/2+1); % single-sided spectrum f = Fs*(0:(L/2))/L; % Define the frequency vector to plot in the frequency domain figure(1) plot(f, P1) xlabel('Frequency (Hz)') ylabel('|P1(f)|') Q1. What is the sampling frequency of y0? Q2. What do you observe in the FFT plot?
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