Consider the pulse function signal below

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Problem (8) fft, fftshift, abs functions in MATLAB Consider the pulse function signal below: x(t) = stp_fn(t+1) – stp_fn(t-1) Generate a time variable from -4 to 4 [sec] in 0.001 [sec] step size. Draw x(t) with respect to time using plot. Convert the signal into the “Frequency Domain” signal (amplitude only): – Use fft to convert x(t) into the frequency domain (find syntax using doc fft). – Use fftshift to correct the frequency axis. X(f) (you can also represent this in angular freq. X(@)) – Use abs to find the absolute value (amplitude) of the signals in frequency (find syntax using doc abs). X() (or in angular freq. |X()|) – Plot the amplitude with respect to frequency. |X(f)| vs f (2-D graph)

The Correct Answer and Explanation is:

To solve this problem, we will break it into steps and explain each part.

Problem Breakdown:

You are given the signal:x(t)=stp_fn(t+1)−stp_fn(t−1)x(t) = \text{stp\_fn}(t+1) – \text{stp\_fn}(t-1)x(t)=stp_fn(t+1)−stp_fn(t−1)

where stp_fn\text{stp\_fn}stp_fn is a step function. A step function is typically represented by:stp_fn(t)={0,t<01,t≥0\text{stp\_fn}(t) = \begin{cases} 0, & t < 0 \\ 1, & t \geq 0 \end{cases}stp_fn(t)={0,1,​t<0t≥0​

So, in this case:x(t)=step(t+1)−step(t−1)x(t) = \text{step}(t+1) – \text{step}(t-1)x(t)=step(t+1)−step(t−1)

This can be interpreted as a rectangular pulse that starts at t=−1t = -1t=−1 and ends at t=1t = 1t=1.

Step 1: Generating the Time Vector

We are asked to generate the time vector from -4 to 4 seconds with a step size of 0.001 seconds.

matlabCopyEditt = -4:0.001:4;  % Time vector from -4 to 4 with step size of 0.001 seconds

Step 2: Defining the Signal x(t)x(t)x(t)

Now, we can define x(t)x(t)x(t) based on the step function. The step_fn function can be created using the heaviside function in MATLAB.

matlabCopyEditx = heaviside(t + 1) - heaviside(t - 1);  % Pulse signal

Step 3: Plotting the Time Domain Signal

We plot the time-domain signal x(t)x(t)x(t) with respect to time.

matlabCopyEditfigure;
plot(t, x);
xlabel('Time (seconds)');
ylabel('Amplitude');
title('Time Domain Signal x(t)');
grid on;

Step 4: Convert the Signal to the Frequency Domain

To convert the time-domain signal into the frequency domain, we use the fft function. The fft function computes the Fast Fourier Transform of the signal.

matlabCopyEditX_f = fft(x);  % Apply FFT to the signal

Step 5: Correct the Frequency Axis Using fftshift

The fft function produces a frequency spectrum with negative frequencies on the left and positive frequencies on the right. We use fftshift to center the zero frequency in the middle of the plot.

matlabCopyEditX_f_shifted = fftshift(X_f);  % Shift the zero frequency to the center

Step 6: Find the Absolute Value of the Frequency Domain Signal

To get the amplitude of the frequency-domain signal, we take the absolute value of X(f)X(f)X(f) using the abs function.

matlabCopyEditX_f_abs = abs(X_f_shifted);  % Find the amplitude of the signal in the frequency domain

Step 7: Generate the Frequency Vector

The fft function assumes the time signal is periodic, so the frequency vector needs to correspond to the length of the signal. The frequency range is given by:f=−N/2Δt,−N/2+1Δt,…,N/2−1Δtf = \frac{-N/2}{\Delta t}, \frac{-N/2+1}{\Delta t}, …, \frac{N/2-1}{\Delta t}f=Δt−N/2​,Δt−N/2+1​,…,ΔtN/2−1​

where NNN is the number of points in XfX_fXf​ and Δt\Delta tΔt is the time step.

matlabCopyEditfs = 1 / 0.001;  % Sampling frequency
n = length(t);  % Number of points in the signal
f = (-n/2:n/2-1)*(fs/n);  % Frequency vector

Step 8: Plot the Amplitude Spectrum ∣X(f)∣|X(f)|∣X(f)∣

Finally, we plot the magnitude of the Fourier Transform ∣X(f)∣|X(f)|∣X(f)∣ with respect to frequency fff.

matlabCopyEditfigure;
plot(f, X_f_abs);
xlabel('Frequency (Hz)');
ylabel('Amplitude');
title('Frequency Domain Amplitude |X(f)|');
grid on;

Full MATLAB Code

matlabCopyEdit% Step 1: Generate time vector
t = -4:0.001:4;

% Step 2: Define the signal x(t)
x = heaviside(t + 1) - heaviside(t - 1);

% Step 3: Plot time domain signal
figure;
plot(t, x);
xlabel('Time (seconds)');
ylabel('Amplitude');
title('Time Domain Signal x(t)');
grid on;

% Step 4: Apply FFT to convert to frequency domain
X_f = fft(x);

% Step 5: Correct the frequency axis with fftshift
X_f_shifted = fftshift(X_f);

% Step 6: Compute the absolute value of X(f)
X_f_abs = abs(X_f_shifted);

% Step 7: Generate the frequency vector
fs = 1 / 0.001;  % Sampling frequency
n = length(t);   % Number of points
f = (-n/2:n/2-1)*(fs/n);  % Frequency vector

% Step 8: Plot the frequency domain amplitude
figure;
plot(f, X_f_abs);
xlabel('Frequency (Hz)');
ylabel('Amplitude');
title('Frequency Domain Amplitude |X(f)|');
grid on;

Explanation of the Process

  1. Time Domain Signal: The signal x(t)=step(t+1)−step(t−1)x(t) = \text{step}(t+1) – \text{step}(t-1)x(t)=step(t+1)−step(t−1) is a rectangular pulse between -1 and 1 seconds.
  2. FFT: The fft function computes the Fourier Transform of the signal, converting it from the time domain to the frequency domain.
  3. FFT Shift: The fftshift function is used to shift the zero frequency to the center of the frequency axis.
  4. Amplitude Spectrum: The abs function is used to compute the magnitude of the frequency components of the signal.
  5. Plotting: We plot both the time-domain and frequency-domain representations of the signal for visual inspection.

This process shows how to convert a time-domain signal to its frequency-domain representation using FFT in MATLAB.

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