University of Illinois Spring 2016ECE210 - Homework 11Due Apr 13th, 2016 18:001. For each part of this problem, find and sketch y (t) = f(t) ∗ h(t).(a) f(t) = u(t),h(t) = u(t)(b) f(t) = tu(t),h(t) = u(t)(c) f(t) = δ(t) − 5δ(t − 2) + δ(t − 4),h(t) = rect(t/2)(d) f(t) = u(t),h(t) = e−tsin(t)u(t)2. Sketch the convolution of these two signals. You do not need to specify the functional form of y(t) (you don’tneed to give it in the form of an equation), but your sketch should specify the values of y(t) at each integer t,and should show roughly what the shape looks like in between these points.f(t) =1 − |t + 5|, −6 ≤ t ≤ −41 0 ≤ t ≤ 10 otherwiseh(t) =(1 0 ≤ t ≤ 10 otherwise3. Consider a system whose impulse response ish(t) = rect(t)Suppose that there are two signals, f(t) and g(t), such thatf(t) ∗ h(t) = t2rect(t2)g(t) ∗ h(t) = max(0, 1 − (t − 5)2)Express g(t) in terms of f(t) and δ(t)4. Use row 13 of table 7.1, together with the Fourier transform and convolution properties of the impulse, to proverow 8 of table 7.1.5. Any real-valued signal x(t) can be written as the sum of a real-valued even part, xe(t), and a real-valued o ddpart, xo(t), such thatxe(−t) = xe(t) (1)xo(−t) = xo(t) (2)x(t) = xe(−t) + xo(t) (3)(a) Prove that Xe(), the Fourier transform of xe(t), is pure real. Hint: use the following equation, which is truewhenever its comp onent integrals are well defined:Z∞−∞f(t)dt =Z∞−∞(f(t) + f (−t))dt(b) Prove that Xo(), the Fourier transform of xo(t), is pure imaginary.(c) The energy spectrum of a signal is defined asWx(ω) = |X(ω)|2Use Eq. 3 and the results of parts (a) and (b) to express Wx() in terms of X2e(ω) and X2o(ω). Note thatthese are the squared Fourier transforms, not the squared magnitude Fourier transforms, therefore one ofthem is positive and one of them is negative (which one? why?).Page 1 of 2(d) The auto correlation of a signal is defined as the inverse FT of its energy spectrum, thusrx(t) =12πZ∞−∞Wx(ω)ejωtdωUse the results of parts (a) through (c) to express rx(t) in terms of xe(t) and xo(t).(e) Consider the following signalx(t) =3, −1 ≤ t ≤ 01, 0 ≤ t ≤ 10, otherwisei. Find xe(t) and xo(t).ii. Find rx(t) . Specify the values rx(−2) ,rx(−1) ,rx(0) ,rx(1) ,rx(2) .Page 2 of