Digital signal processing ybu edu (14)

YILDIRIM BEYAZIT UNIVERSITY EE208 Signals and Systems Student no: Name:

EE208 Signals and Systems (Homework) Date: (2 May 2016)

QUESTIONS 1. For the systems given below, investigate their linearity, time-invariance, stability and causality properties. Explain your answers. i. y(t )  e x (t / 2) ii. y(n)  x(n)  x(n 1) iii. y(t )  sin(t ) x(t )  x 2 (2t ) iv.

n4



y(n)  x(0) 

x( k )

k n2

2. The output of a linear system is sin(kt) against the input signals of the form tk (k is a constant). Answer the following: i. Find the output for x(t )    6t 2  47t 5 t 6 .

1  t 27 ii. Find the output for x(t )  .

1 t9 iii. Is this system a LTI system? Justify your answer!

3. Two discrete-time LTI system is cascade connected as in the figure (|a|<1). x(n)

w(n)

h1(n)=u(n)

h2(n)=anu(n)

y(n)

a. Find the overall impulse response of the cascade system. b. Find the response y(n) to the input x(n)=(n)-a(n-1).

4. The step response of a LTI system is given by y(t )  e2t u(t )  0.5u(1  t ) . Find the response to the input signal given in the figure (below). x(t)

0

1

2

3

t

5. Consider the LTI systems in Question 3. Now suppose that the systems are connected in parallel. a. Find the overall impulse response of the cascade system. b. Find the response y(n) to the input x(n)=(n)(n1)+(n2).

6. Determine and sketch the even and odd parts of the signals y(t) and x(t) in Question 4 and x(n) in Question 5.

7. Determine the periodicity of the following signals. If it is periodic find the period! a. {sin(5t-2)}2 b. sin(5n-/4) c. (-1)ncos(n/7) d. (-2) -ncos(n/7)

Assoc. Prof. Dr. Hüseyin Canbolat

YILDIRIM BEYAZIT UNIVERSITY ECE101 Introduction to Electronics and Communication Engineering Student no: Name: 8. Consider an LTI system. Its response to the input x1(t) is y1(t). The graphs of x1(t) and y1(t) are given below. Find and sketch the output of this system against x2(t) given below.

9. Consider the signal given below. Sketch the following signals, x(1-t/3) and x(t-2)[ u(t)u(t-2)].

10. Consider we discretize the signal in Question 9 with a period of 0.2 seconds. Sketch the corresponding discrete time signal x1(n).

11. Now sample the signal in Question 9 with period 0.5 seconds. Sketch the corresponding discrete time signal x2(n).

12. Sketch x2(2-n) (Question 11) and x1(3n+1) (Question 10).

Assoc. Prof. Dr. HĂźseyin Canbolat