Digital signal processing ybu edu (3)

YILDIRIM BEYAZIT UNIVERSITY EE307 Digital Signal Processing Student no: Name:

EE307 Digital Signal Processing (Homework 2) Date: (30 December 2016)

QUESTIONS

j

1. Let X(e ) be the Fourier transform of x(n), where x* denotes the complex conjugate of x. a. Find the FT of x*(n). b. Find the FT of x*(-n).

2. Let us given an ideal low-pass filter, with the frequency response H(ej) having a cutoff frequency c. If the output is y(n)={...,0,0,1n=0,1,1,1,1,1,1,0,0,...}, find c.

3. Find H(ej) of a 5-point moving averager, given by the input-output relation: 4

y (n)  15  x(n  k ) k 0

4. Find the z-transforms of the following sequences. a. (0.5)nu(n) b. -(0.5)nu(-n-1) c. (0.5)nu(-n) d. (n-1) e. (n+1) f. (0.5)n{u(n) -u(-n-10)} g. (0.5)|n| 5. Find the transfer function H(z) of the LTI system, whose step response is given by y(n)=(0.5)n-1u(n+1).

6. The pole and zeros of a transfer function of a causal LTI system is given as zp=-0.2, 0.2±0.5j, ±0.3±0.3j (i.e. seven poles in total!) and zz=0, 2±0.4j (i.e. three zeros!) Is this system stable? Also list all possible RoC for the given pole-zero configuration.

7. Let the sequence x(n)=cos(n/4) be obtained by sampling xc(t)=cos(0t) at a rate of 1000 samples/s. Find at least two possible values of 0 that gives x(n).

8. A band limited signal xc(t), whose Fourier transform Xc(j)=0 for ||≥2x104, is sampled with the period T to produce x(n)= xc(nT). What is maximum value of T to avoid aliasing (and thus xc(t) can be recovered!)?

9. Let x(n) be applied to a digital filter with an impulse response h(n). The filter output against x (n) is y (n)  T

n

 x(k ) . What is h(n)?

k 

10. Consider a signal x(t)=10 cos(20tn/4)-5 cos(50t) is sampled. a. What condition should the sampling frequency fs satisfy for y(t)= x(t)? b. How should fs be selected such that y(t)=A+10 cos(20tn/4)? c. Find the value of A.

Prof. Dr. Hüseyin Canbolat

YILDIRIM BEYAZIT UNIVERSITY EE307 Digital Signal Processing Student no: Name: 11. Consider the following transfer function H ( z )  system (that is, |H(ej)|=1).

z 1  z0* . Show that this is an all-pass 1  z0 z 1

12. Previous function has complex coefficients. To get a function with real coefficients

z 2  2 Re( z0 ) z 1  z0 z 1  z0* z 1  z0 consider H ( z )  which corresponds to a  1  z0 z 1 1  z*0 z 1 1  2 Re( z0 ) z 1  z0 2 z 2 difference equation y(k)-2Re(z0)y(k-1)+|z0|2y(k-2)=x(k-2)-2Re(z0)x(k-1)+|z0|2x(k). Show that this is also an all-pass system.

2

13. (For this question, you need to read some text on inverse, MP and LP systems!) In control theory and signal processing, a LTI system is said to be minimum-phase if the system and its inverse are causal and stable. For the system whose transfer 1  0.5 z 1 1  4 z 2   function is given as H ( z )  .

1  0.64 z 2 a. find a minimum phase (MP) transfer function, H1(z), and an all-pass transfer function, Hap(z), such that H(z)=H1(z)Hap(z). b. find another minimum phase, H2(z), and a linear phase (LP) FIR filter transfer function Hlin(z), such that H(z)=H2(z)Hlin(z).

14. Consider given pole-zero configuration of a LTI system: If the corresponding H is causal and stable, is it possible to have a causal and stable inverse system Hi (i.e.

H(z)Hi (z)=1)? Note that, the system has a zero at infinity!

15. Suppose a discrete-time ideal low-pass filter with the impulse h(n), given by 1   0.25  .

H (e j )    0 0.25     Let a system with h1(n)=h(2n) be given. Express H1(ej).

Prof. Dr. Hüseyin Canbolat