The signal x(t)=4+cos(4 πt)-sin(8 πt) forms the input to a filter whose impulse response is h(t)

First, letu0027s find the Fourier transform of the input signal $x(t)$.u000AThe Fourier transform of $4$

Question

The signal $x(t)=4+\cos (4 \pi t)-\sin (8 \pi t)$ forms the input to a filter whose impulse response is $h(t)$, as shown. Find the response $y(t)$. $$ x(t) \longrightarrow \text { filter } h(t) \rightarrow y(t) $$ (a) $h(t)=\sin (5 t)$ (b) $h(t)=\operatorname{sinc}(5 t-2)$ (c) $h(t)=\operatorname{sinc}^2(5 t-2)$ (d) $h(t)=e^{-t} u(t)$ (e) $h(t)=\delta(t)-e^{-t} u(t)$ (f) $h(t)=\operatorname{sinc}(t) \cos (8 \pi t)$ (g) $h(t)=\operatorname{sinc}^2(t) \cos (5 \pi t)$ (h) $h(t)=\operatorname{sinc}^2(t) \cos (16 \pi t)$

   The signal $x(t)=4+\cos (4 \pi t)-\sin (8 \pi t)$ forms the input to a filter whose impulse response is $h(t)$, as shown. Find the response $y(t)$.
$$
x(t) \longrightarrow \text { filter } h(t) \rightarrow y(t)
$$
(a) $h(t)=\sin (5 t)$
(b) $h(t)=\operatorname{sinc}(5 t-2)$
(c) $h(t)=\operatorname{sinc}^2(5 t-2)$
(d) $h(t)=e^{-t} u(t)$
(e) $h(t)=\delta(t)-e^{-t} u(t)$
(f) $h(t)=\operatorname{sinc}(t) \cos (8 \pi t)$
(g) $h(t)=\operatorname{sinc}^2(t) \cos (5 \pi t)$
(h) $h(t)=\operatorname{sinc}^2(t) \cos (16 \pi t)$

Analog and digital signal processing

Ambardar, Ashok 2nd Edition

Chapter 9, Problem 40

Instant Answer

Step 1

The Fourier transform of $4$ is $\delta(\omega)$ (a Dirac delta function). The Fourier transform of $\cos(4\pi t)$ is $\frac{1}{2}[\delta(\omega-4\pi)+\delta(\omega+4\pi)]$ (using the Fourier transform of $\cos(at)$). The Fourier transform of $\sin(8\pi t)$ is Show more…

Show all steps

Thanks for your feedback!

The signal $x(t)=4+\cos (4 \pi t)-\sin (8 \pi t)$ forms the input to a filter whose impulse response is $h(t)$, as shown. Find the response $y(t)$. $$ x(t) \longrightarrow \text { filter } h(t) \rightarrow y(t) $$ (a) $h(t)=\sin (5 t)$ (b) $h(t)=\operatorname{sinc}(5 t-2)$ (c) $h(t)=\operatorname{sinc}^2(5 t-2)$ (d) $h(t)=e^{-t} u(t)$ (e) $h(t)=\delta(t)-e^{-t} u(t)$ (f) $h(t)=\operatorname{sinc}(t) \cos (8 \pi t)$ (g) $h(t)=\operatorname{sinc}^2(t) \cos (5 \pi t)$ (h) $h(t)=\operatorname{sinc}^2(t) \cos (16 \pi t)$

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback