Cc

Bogazici University

Institute of Biomedical Engineering BM 504–Systems Theory for Biomedical Science

Homework Assignment 1 Due to: Feb 22, 2019, 17:00 February 16, 2019

Spring Term

Instructor: Ahmet Ademoglu, PhD

1. Using the Euler identity 1 1 cos(θ) = ejθ + e−jθ 2 2 Show that (a) cos(2θ) = 2 cos2 (θ) − 1 (b) cos(4θ) = cos4 (θ) − 6 cos2 (θ) sin2 (θ) + sin4 (θ) 2. Express the numbers below in polar form and plot them on the complex plane. √ (a) (1 − j 3)3 (b)

ejπ/3√ −1 1+j 3

3. Show whether the following systems are i) linear, ii) time-invariant; (a) cos(3t)x(t) ( 0 (b) y(t) = x(t) + x(t − 2)

x(t) < 0, x(t) ≥ 0

2 (c) y(t) = ( dx dt )

4. Evaluate the following integrals (a)

R∞

− τ )dτ , b)

R∞

Rt

for the signal below

−∞ x(τ )δ(t

5. Find and sketch

−∞ x(t)dt

3 −∞ (t

+ 4)δ(1 − t)dt, c)

R∞

−∞ e

(x−1) cos(π/2(x

− 5))δ(x − 3)dx

6. Consider the signal x(t) show below and (a) sketch v(t) = 3x(−1/2(t + 1)) (b) determine the energy and power of v(t) (c) sketch the even part of v(t)

7. Consider the signal −1/2x(−3t + 2) shown below (a) Sketch the original signal x(t) (b) Sketch the even part of x(t) (c) Sketch the odd part of x(t)

8. For each of the following pairs of waveforms, use the convolution integral to find the response y(t) of the LTI system with impulse response h(t) to the input x(t). Sketch your results. (a) x(t) = e−αt u(t) (b) h(t) = e−βt u(t) Do this both when α 6= β and when α = β (c) x(t) = u(t) − 2u(t − 2) + u(t − 5) and h(t) = e2t u(l − t) (d) x(t) and h(t) given as in Figure a), b) and c) below;

(e)

i. Consider an LTI system with input and output related through the equation Z∞

y(t) =

e−(t−τ ) x(τ − 2)dτ

−∞

What is the impulse response of the system?

ii. Determine the response of the system when the input x(t) is as shown in Figure below;

9. Determine the equivalent impulse response h(t) of the following cascade system where h1 (t) = h2 (−t) = e−t u(t)

10. Show that if the the response of the system to the input x(t) is y(t), then its response to dx(t)/dt is dy(t)/dt. 11. Consider the linear time invariant system S and a signal x(t) = 2e−3t u(t − 1). If x(t) −→ y(t) and dx(t)/dt −→ −3y(t) + e−2t u(t) determine the impulse response h(t) of S.