3. [28 points] A causal discrete-time LTI system's input-output relationship is defined by the following difference equa

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3 28 Points A Causal Discrete Time Lti System S Input Output Relationship Is Defined By The Following Difference Equa 1 (61.31 KiB) Viewed 21 times

3. [28 points] A causal discrete-time LTI system's input-output relationship is defined by the following difference equation: y[n] -0.4y[n 1] = x[n], y[−1] = 0. a. [2 pts] Is this system defined recursively or not? b. [4 pts] Show that this system's impulse response is h[n] = (0.4)¹u[n]. c. [4 pts] Find the z-transform H(z). d. [6 pts] Find the output of this system to an input x₁ [n] = cos (n+30°). e. [4 pts] Now, we consider another input x₂ [n] = n(2)"u[n]. Find the z-transform of this input X₂(z). f. [8 pts] The z-transform of the output to input x₂ [n] can be found by Y(z) = H(z)X₂ (Z). Find Y(z). Find the output y[n] by taking the inverse z-transform of Y(z).