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Problem 1 [15 points] Consider two discrete-time LTI systems…

Problem 1 [15 points] Consider two discrete-time LTI systems with input-output relationships $$S_1: y_1[n] = x[n – 2] \quad \text{and} \quad S_2: y_2[n] = \sum_{k = -\infty}^{n} x[k]$$We place them in cascade to form a composite system:$$x[n] \rightarrow \boxed{S_1} \rightarrow \boxed{S_2} \rightarrow y[n]$$We input a periodic signal \(x[n]\) to this cascade, which has Fourier series coefficients \(\{a_k\}\). Letting \(\{b_k\}\) be the Fourier series coefficients for the output \(y[n]\), give an expression for \(b_k\) in terms of \(a_k\). (You do not need to consider \(k = 0\).) Problem 2 [15 points] An LTI system has a frequency response$$H(j\omega) = \frac{3 + j\omega}{jw(2 + j\omega)}.$$If the output of the system is$$y(t) = e^{-2t}u(t),$$determine the input \(x(t)\) being applied to the system. Problem 3 [35 points] Consider a periodic signal $$x(t) = 2 + \sin\left(\frac{2\omega_0t}{3}+\frac{\pi}{2}\right) + e^{j\left(\omega_0t+\frac{\pi}{4}\right)}$$ (a) [15 pts] Determine the period. And hence find the Fourier Series coefficients \(\{a_k\}\) of \(x(t)\).    (Hint: Use Euler’s equations) (b) [10 pts] Determine the Fourier Transform of \(x(t)\). (c) [10 pts] Use Parseval’s relation (of CTFS) to determine the overall average power \(P_{\infty}\) of \(x(t)\). Problem 4 [35 points] Consider an LTI system. Given an input signal,$$x(t) = \frac{\sin(\pi t)}{\pi t},$$the corresponding output signal is,$$y(t) = \left(\frac{\sin(3t)}{\pi t}\right)^2$$ (a) [15 pts] Determine and plot the spectrum of both \(X(j\omega)\) and \(Y(j\omega)\). Make sure you mark all the key points (both frequency and amplitude). (b) [15 pts] If \(x_1(t) = \cos(\pi t) + \sin(2 \pi t)\) is fed to the same system, determine the corresponding output signal \(y_1(t)\)? (c) [5 pts] If we think of this system as a low-pass filter, what is the cut-off frequency? Congratulations, you are almost done with Midterm Exam 2.  DO NOT end the Honorlock session until you have submitted your work to Gradescope.  When you have answered all questions:  Use your smartphone to scan your answer sheets and notes pages and save the scan as a PDF. Make sure your scan is clear and legible.  Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.).  Click this link to go to Gradescope to submit your work: Midterm Exam 2 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam.  End the Honorlock session.