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NOTE: For all the problems below, please include your derivation process which you used to get your solution. (Derivation process is 50% of your score, except simple z-transformation process.)

1. Consider the linear discrete-time shift-invariant system with unit sample response h[n] = au[n], where a < 1. Determine the steady-state response, i.e., the response for large n, to the excitation x[n] = -ju[n], where j = -1.

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01:02

Linear time-invariant (LTI) system with impulse response h[n] = y[n] = 2 u[n] when the input is x[n]:u[n] yields the output.a) Determine an expression for the frequency response of the system.b) Determine an expression for X(e^jÏ‰) i.e the Discrete Time Fourier Transform of x[n].c) Determine an expression for x[n].

05:20

Consider the system y(n) = 0.4y(n-1) + %@tro-4. Find the frequency response H(e^e) of this system. Find the steady-state response Yss(n) when x(n) = Scos(n*t/3 – 1/4).

05:04

Evaluate the following integral:1.25 + cos(Ï‰)

If the input to an LTI system is x(n) and the system’s time impulse response is h(n), find the output y(n) if:x[n] = âˆš(-1) * j * 0.5 * sin(Ï‰ * n)H(e^jÏ‰) = 2 + jÏ‰

The unit impulse response for a Linear Time Invariant Discrete Time System is h[n] and its output is the discrete time unit impulse function Î´(n). Find its input x[n] if the DTFT of h[n] is defined as:1 + 0.7e^jÏ‰

Let I(n) be a real discrete signal and let X(e^jÏ‰) be its DTFT. Find y(n) if:Y(e^jÏ‰) = X(e^j(Ï€/3))

05:49

Also, use z-transform to verify your answer. x(k) = k^3 – u(k). Use a convolution sum to find the output y(k) if the input is h(k) = 0.2u(k). A certain discrete-time LTI system has the following impulse response:

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