2. Suppose $x(t) = \left| \text{sinc} \left( \frac{t}{2} \right) \cos(10t) + 3 \cos(10t) \right|$ is input to a filter whose frequency response is $H(\omega) = \text{rect}(\omega)$. \newline Find the output $y(t)$. Show all your work. Hint: $\left| \cos(10t) \right| = \frac{2}{\pi} + \sum_{n=1}^{\infty} \frac{1}{\pi} \frac{(-1)^n}{n^2 - \frac{1}{4}} \cos(20nt)$.
Added by Jaime C.
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First, let's find the Fourier transform of the input signal x(t). The Fourier transform of sinc(t) is Rect(w/2π), and the Fourier transform of cos(10t) is π[δ(w-10) + δ(w+10)]. Therefore, the Fourier transform of x(t) is: X(w) = Rect(w/2π) * π[δ(w-10) + δ(w+10)] Show more…
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