HW1 - Hw 1 PDF

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University of Illinois

Spring 2021

ECE 310: Problem Set 1 Due: 5pm, Friday, February 5, 2021 1. Sketch (by hand) the following discrete-time signals, for all values of n at which the signal is non-zero. Here, u[n] is the unit step function, and δ[n] is the unit pulse (Kronecker delta). (a) x[n] = u[n − 2] − u[n + 3]

(b) x[n] = n2 u[−n + 2]u[n]   (c) x[n] = cos nπ (n − 2) (u[n] − u[n − 5]) 3 √ (d) x[n] = n δ[2n − 4] 2. Determine whether each of the following signals is periodic. If it is, determine its (smallest) period. (a) x[n] = sin( π7 n) (b) x[n] = cos( π4 n)u[n] (c) x[n] = sin(π + n) (d) x[n] = e−j7πn 3. Consider the sequence {x[n]}2n=−3 = {3, −2, 0, 1, 2, 1}. (a) Sketch the sequence y[n] defined by y[n] = x[−n − 1]. (b) Sketch the sequence w[n] defined by w[n] = 2x[2n − 1].

(c) Express x[n] in terms of the unit pulse function (the Kronecker delta) δ[n].

4. Evaluate each of the following expressions and represent your final answer in both cartesian and polar forms. Simplify your answer as much as possible. √ (a) 26 45◦ + 26 (−30◦ ). (1 + j)3 (b) . (1 − j)2   2 + j4 −1 + j 3 2 (c) − 1 + j2 1+j 5. Determine all roots of the equation z 4 − 9 = 0, and sketch them on the complex plane. 6. Consider the systems specified by the following input-output relations, where x[n] is the input and y[n] the output: (a) y[n] = x[n + 1] − 2x[n − 2]

(b) y[n] = 5x[−n] + 1 (c) y[n] = x[3] · x[n] (d) y[n] = ejπn x[n]

For each system, determine if it is: (i) linear or non-linear, (ii) time-invariant or time-varying, (iii) causal or non-causal. Justify your answers....