Write dowm the difference equation for the infinite impulse response filter having z-transform H(z) = 1+0.9z−1 1−0.5z−12 = Y (z) X(z)

Problem 1

1. Derive the Fourier transform Xr(f) ∈C of a modulated rectangular pulse train xr(t)+ej2π1txr(t−4τ) where xr(t) is a rectangular pulse having height A and duration τ [15 pts] 2. Find the Fourier transform of x(t) = cos(2πf0t) + αxr(t)2 [15 pts].

Problem 2

1. If x(n) and h(n) are discrete sequences, write the mathematical definition of y(n) = x(n) ⋆ h(n) where ⋆ is convolution [5 pts]
2. Use the above definition to show that y(n) = [h1(n)+h2(n)]⋆x(n) is equal to h1(n)⋆x(n)+h2(n)⋆x(n) [5 pts]
3. A 3-port discrete-time LTI system has two inputs and one output, with impulse response from input k to output y(n) given by hk(n). If h1(n) = {1,0.5,−1,0,0,…} and h2(n) = {1,1,1,−1,−1,−1,0,0,…}, find the transform domain relationship Y (z) =Pk Hk(z)Xk(z) using the impulse responses. Assume inputs are denoted by z-transform pairs (xk(n),Xk(z)), where (n,z) ∈ Z,C,k = 1,2. Here n is time index, and z is the z-domain complex variable [10 pts]
4. Find the frequency domain relationship between the Fourier transforms of the two inputs and the output by evaluating the z-domain relation on the unit circle z = ejω [10 pts].
   Problem 3
5. A system produces y(t) when driven by x(t) via the transfer function y(t) = 100x(t−τ −τ2 + 10τ3). Here, τ is a constant time delay. Is this a linear or non-linear system? [10 pts] 2. A system produces Y (f) when driven by X(f) via the Fourier transfer function Y (f) = X(f)e−jγf2. Is this an LTI system? [10 pts] 3. A system produces y(t) when driven by x(t) via the transfer function y(t) =Rt λ=−∞ x(λ)dλ. What is the output for βx(t) where β ∈R. [10 pts] 4. Write dowm the difference equation for the infinite impulse response filter having z-transform H(z) = 1+0.9z−1 1−0.5z−12 = Y (z) X(z) [10 pts]
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