## Step 1: Analytical Fourier Transform ### **Signal Description** The signal \( x(t) \) is a trapezoidal pulse: - \( x(t) = t + 2 \) for \( -2 \leq t < -1 \) - \( x(t) = 1 \) for \( -1 \leq t \leq 1 \) - \( x(t) = -t + 2 \) for \( 1 < t \leq 2 \) - \( x(t) = \) otherwise ### **Decomposition** We can express \( x(t) \) as a combination of two shifted ramp functions and a rectangular pulse: \[ x(t) = \text{rect}\left(\frac{t}{2}\right) + r(t+2) - 2r(t+1) - r(t-2) + 2r(t-1) \] But for simplicity, we can treat it as the sum of three parts: 1. Left ramp (\(-2 \leq t < -1\)): \( x_1(t) = t+2 \) 2. Rectangle (\(-1 \leq t \leq 1\)): \( x_2(t) = 1 \) 3. Right ramp (\(1 < t \leq 2\)): \( x_3(t) = -t+2 \) ### **Fourier Transform Calculation** #### **1. Rectangle Pulse (\( x_2(t) \)):** \[ x_2(t) = 1, \quad -1 \leq t \leq 1 \] Its Fourier transform: \[ X_2(f) = \int_{-1}^{1} 1 \cdot e^{-j2\pi ft} dt = \frac{\sin(2\pi f)}{\pi f} \] #### **2. Left Ramp (\( x_1(t) = t+2 \), \( -2 \leq t < -1 \)):** \[ X_1(f) = \int_{-2}^{-1} (t+2)e^{-j2\pi f t} dt \] Let \( u = t+2 \rightarrow t = u-2 \), \( du = dt \), limits: \( t = -2 \rightarrow u = \), \( t = -1 \rightarrow u = 1 \) \[ X_1(f) = \int_{}^{1} u e^{-j2\pi f (u-2)} du = e^{j4\pi f} \int_{}^{1} u e^{-j2\pi f u} du \] This integral can be solved using integration by parts. #### **3. Right Ramp (\( x_3(t) = -t+2 \), \( 1 < t \leq 2 \)):** Similarly, \[ X_3(f) = \int_{1}^{2} (-t+2)e^{-j2\pi ft} dt \] Let \( v = -t+2 \rightarrow t = 2-v \), \( dt = -dv \), limits: \( t=1 \rightarrow v=1 \), \( t=2 \rightarrow v= \) \[ X_3(f) = \int_{1}^{} v e^{-j2\pi f (2-v)} (-dv) = \int_{}^{1} v e^{-j2\pi f (2-v)} dv \] \[ = e^{-j4\pi f} \int_{}^{1} v e^{j2\pi f v} dv \] #### **Combine All Parts** The total Fourier transform \( X(f) = X_1(f) + X_2(f) + X_3(f) \). **Explanation:** We broke the signal into its three sections, used standard Fourier transform properties and change of variables, and set up the integrals for each. --- ## Step 2: MATLAB FFT and Plot ### **Numerical FFT (MATLAB code):** ```matlab % Define time vector t = linspace(-4, 4, 100); x = zeros(size(t)); % Build x(t) according to the piecewise definition x((t>=-2) & (t<-1)) = t((t>=-2) & (t<-1)) + 2; x((t>=-1) & (t<=1)) = 1; x((t>1) & (t<=2)) = -t((t>1) & (t<=2)) + 2; % Compute FFT N = length(t); X_fft = fftshift(fft(x, N)); f = linspace(-.5, .5, N) / (t(2) - t(1)); % Frequency vector % Magnitude and phase figure; subplot(2,1,1); plot(f, abs(X_fft)); title('Magnitude Spectrum'); xlabel('Frequency (Hz)'); ylabel('|X(f)|'); subplot(2,1,2); plot(f, angle(X_fft)); title('Phase Spectrum'); xlabel('Frequency (Hz)'); ylabel('Phase (radians)'); ``` **Explanation:** We implemented the signal in MATLAB, used `fft` to compute its spectrum, and plotted both magnitude and phase for direct comparison to the analytical result. --- ## **Final Answer** - **Step 1:** The analytical Fourier transform is the sum of the transforms of the three segments, which can be explicitly computed as shown above. - **Step 2:** The MATLAB FFT implementation numerically computes the magnitude and phase spectrum, which should match the analytical prediction. **Conclusion:** By comparing both spectra, you can verify the correctness of your analytical Fourier transform for the given trapezoidal signal.