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5. 20 points You have an RC-circuit whose impulse response for a and output voltage y(t)...
In each step to follow, the signals h(t), a(t), and y(t) denote respectively the impulse response, input, and output of a continuous-time LTI system. Accordingly, H(w), X(w) and Y(w) denote their Fourier transforms. Hint: Carefully consider for each step whether to work in the time domain or frequency domain. (b) (25 points) On the axes below, provide a clearly labeled sketch of y(t) for all t given Σ H(w)-( ) sine? (w/8) j2Tt r(t)-e δ(t-n/2) and with sinc(t) = sin(t)/t...
We may design a dc power supply by cascading a full-wave rectifier and an RC circuit. The full wave rectifier output x(t) = |cos(120t)I is the input to the RC circuit whose frequency response is 4. a) Find the Fourier transform for the input signal x(t) b) Find the Fourier transform for the output signal y(t) c) Find the range for the RC such that magnitude of the first harmonic of the ripple in y(t) is less than 1% of...
3-(10 points) Consider a C-T. LTI system given below X(t) - h(t) y(t) The impulse response is h(t)=sinc(200t). We apply an input signal x(t)=sinc(100t) to produce the output y(t). Find and plot Y(m). Find y(t).
(e) Consider an LTI system with impulse response h(t) = π8ǐnc(2(t-1). i. (5 pts) Find the frequency response H(jw). Hint: Use the FT properties and pairs tables. ii. (5 pts) Find the output y(t) when the input is (tsin(t) by using the Fourier Transform method. 3. Fourier Transforms: LTI Systems Described by LCCDE (35 pts) (a) Consider a causal (meaning zero initial conditions) LTI system represented by its input-output relationship in the form of a differential equation:-p +3讐+ 2y(t)--r(t). i....
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1, (20 points) Suppose we have a İTİ system with impulse response(h(t) described as following h(t) 6u(t) where u(t) is unit step function. The output(Y (s)) is expressed as the product of input (R(s)) and transfer function Y(s) = R(s)H(s) The Laplace transform is defined as LTI system R(H) Y (s) Figure 1: LTI system in s-plane (a) (5 points) Find the tranisfer function(H(s)) of the LITI system. (b) (5 points) Find the Laplace transform of the input(r(t)....
3. (20 points) A system has an impulse response given by h (t) sin (2t) rt (a) Find the frequency response function of this system H (w). (b) Find the frequency domain output Y (w) if the input to the system is z (t) cos (3t).
For a continuous time linear time-invariant system, the
input-output relation is the following (x(t) the input, y(t)
the
output):
, where h(t) is the impulse response function of the
system.
Please explain why a signal like e/“* is always an eigenvector
of
this linear map for any w. Also, if ¥(w),X(w),and H(w) are
the
Fourier transforms of y(t),x(t),and h(t), respectively.
Please
derive in detail the relation between Y(w),X(w),and H(w),
which means to reproduce the proof of the basic convolution
property...
Consider the LTI system with input ??(??) = ?? ?????(??) and the
impulse response ?(??) = ?? ?2????(??). A. (3 points) Determine
??(??) and ??(??) and the ROCs B. (3 points) Using the
convolutional property of the Laplace transform, determine ??(??),
the Laplace transform of the output, ??(??) C. (3 points) From the
answer of part B, find ??(??)
9 points) Consider the LTI system with input x(t)eu(t) and the impulse response h(t)-e-2u(t) A. 3 points) Determine X(s) and H(s)...
(a) Show that for any B 〉 0 and any c E R. 3, sinc is a Fourier transform pair. You may assume the Fourier transform pair Pr(t) ←→ τ sine ( (b) An ideal bandpass filter has frequency response w) 0, otherwise 2(t-1 Find the output response y(t) when the input is (t)-sinc 2
(a) Show that for any B 〉 0 and any c E R. 3, sinc is a Fourier transform pair. You may assume the Fourier...
5. Fourier Transform and System Response (12 pts) A signal æ(t) = (e-t-e-3t)u(t) is input to an LTI system T with impulse response h(t) and the output has frequency content Y(jw) = 3;w – 4w2 - jw3 (a) (10 pts) Find the Fourier transform H(jw) = F{h(t)}, i.e., the frequency response of the system. (b) (2 pts) What operation does the system T perform on the input signal x(t)?