University of Illinois Spring 2016ECE210 - Homework 12Due (Wednesday, April 20 at 6:00 p.m.)1. Simplify the following expressions involving the impulse and/or shiftedimpulse and sketch the results:(a) g(t) = cos(πt)(dudt+ δ(t − 0.5)).(b) a(t) =´t−∞δ(τ − 1)dτ + rect(t3)δ(t − 2).(c) b(t) = δ(t − 4) ∗ u(t − 2).2. Consider an LTI system with impulse responseh(t) = e−4tu(t)Find the Fourier series represention of the output y(t) for the followinginputs.(a) f(t) = cos(2πt)(b) f(t) =P∞n=−∞δ(t − n)3. Determine the Fourier transform of the following signals — simplify the re-sults as much as you can. For parts (a), (b), and (c), sketch the magnitudeand phase of the result:(a) f(t) = 5 cos(5t) + 3 sin(15t).(b) x(t) = cos2(6t).(c) y(t) = e−tu(t) ∗ cos(2t).(d) z(t) = (1 + cos(3t))e−tu(t).4. The inverse of the sampling interval T –that is, T−1– is known as thesampling frequency and usually is specified in units of Hz. Determine theminimum sampling frequencies T−1needed to sample the following analogsignals without causing aliasing error.(a) Arbitrary signal f(t) with bandwidth 20 kHz.(b) f(t) = sinc(4000πt).(c) g(t) = sinc(4000πt) cos(8000πt).5. Determine whether of the following LTIC systems are BIBO stable andexplain why or why not:(a) h1(t) = 5δ(t) + 2e−2tu(t) + 3te−2tu(t)1(b) h2(t) = δ(t) + u(t)(c) h4(t) = −2δ(t − 3) − te−5tu(t)6. Consider the given zero-state input-output relations for a variety of sys-tems. In each case, determine whether the system is zero-state linear, timeinvariant, and causal(a) y(t) = f(t − 1) + f(t + 1)(b) y(t) = 5f(t) ∗ u(t)(c) y(t) = δ(t − 4) ∗ f(t) +´t−2−∞f2(τ)dτ(d) y(t) =´t+2−∞f(τ)dτ(e) y(t) =´t+2−∞f(τ2)dτ(f) y(t) = f3(t − 1)(g) y(t) = f((t −