ECGR4124 Digital Final Spring 2014

Name: ______

LAST 4 NUMBERS of Student Number: _____

Do NOT begin until told to do so Make sure that you have all pages before starting NO TEXTBOOK, NO CALCULATOR, NO CELL PHONES/WIRELESS DEVICES Open handouts, 2 sheet front/back notes, NO problem handouts, NO exams, NO quizzes DO ALL WORK IN THE SPACE GIVEN Do NOT use the back of the pages, do NOT turn in extra sheets of work/paper Multiple-choice answers should be within 5% of correct value Show ALL work, even for multiple choice

ACADEMIC INTEGRITY: Students have the responsibility to know and observe the requirements of The UNCC Code of Student Academic Integrity. This code forbids cheating, fabrication or falsification of information, multiple submission of academic work, plagiarism, abuse of academic materials, and complicity in academic dishonesty.

Unless otherwise noted: F{} denotes Discrete time Fourier transform {DTFT, DFT, or Continuous, as implied in problem} F-1{} denotes inverse Fourier transform ω denotes frequency in rad/sample, Ω denotes frequency in rad/second ∗ denotes linear convolution, N denotes circular convolution x*(t) denotes the conjugate of x(t)

Useful constants, etc: e ≈ 2.72 π ≈ 3.14 e2 ≈ 7.39 e4 ≈ 54.6 e-0.5 ≈ 0.607 e-0.25 ≈ 0.779 1/e ≈ 0.37 √2 ≈ 1.41 e-2 ≈ 0.135 √3 ≈ 1.73 e-4 ≈ 0.0183 √5 ≈ 2.22 √7 ≈ 2.64 √10 ≈ 3.16 ln( 2 ) ≈ 0.69 ln( 4 ) ≈ 1.38 log10( 2 ) ≈ 0.30 log10( 3 ) ≈ 0.48 log10( 10 ) ≈ 1.0 log10( 0.1 ) ≈ -1 1/π ≈ 0.318 cos(π / 4) ≈ 0.71 cos( A ) cos ( B ) = 0.5 cos(A - B) + 0.5 cos(A + B) ejθ = cos(θ) + j sin(θ)

1/13 5 Points Each, Circle the Best Answer

1. A discrete-time sinusoid with ω = π rad/sample (sampled at 1000 samples/second) has a corresponding continuous-time frequency of

a) 125 Hz b) 250 Hz c) 500 Hz d) none above

2. If x[n] = 2δ[n-2] - u[n-2] then, x[2] equals

a) 0 b) 1 c) 2 d) none above

2 3. The first two points of x[n] (at n=0,1) in sampling x(t)= t u(t) at 2 samples/second are:

a) {0, 0.25} b) {0, 0.25} c) {1, 0.5} d) none above

4. ((-1))4 =

a) 0 b) 1 c) 2 d) none above

5. The signal x[n] = cos[ n/π ] has a discrete-time frequency in rad/sample of ω =

2 a) 1/π b) 1 c) π d) none above

2/13 5 Points Each, Circle the Best Answer

6. Circle the causal BIBO stable impulse response below.

-n n a) h[n]=(0.9) u[-n] b) h[n]= 2 u[n-1] c) h[n]= n u[n] - n u[n-2] d) none above

7. If a causal filter with output y[n] and input x[n] has difference equation y[n] = x[n] - x[n-2], then the filter is

a) IIR b) FIR c) none above

8. The circular convolution of two 4-point sequences x[n] = {1,0,0,1} and y[n] = {1,2,3,0}

a) {3,5,3,1} b) {1,2,3,1} c) {1,3,4,5} d) none above

9. The difference equation for a LTI (linear time-invariant) system with output y[n] is y[n] = 2x[n] + y[n-1]. The impulse response of the system is h[n] =

a) 2u[n] + u[n-1] b) 2δ[n] + u[n] c) 2u[n] d) none above

10. Implementation of convolution using the FFT results in circular convolution.

a) True b) False

3/13 5 Points Each, Circle the Best Answer

11. The impulse response of a system is h[n] = 2 δ[n-3], the is H(ω)=

-j3 -j2n -j2 a) 2e ω b) 2 e ω c) 3e ω d) none above

12. The difference equation of a causal system is y[n] = 2x[n] + 3y[n-1] . The z-transform of the system is H(z)=

2z 3z a) ;|z|>3 b) ;|z|>2 z − 3 z − 2

2z c) ;|z|>0.5 d) none above z − 0.5

n 13. The DTFT of x[n] = ( 0.2 ) u[n] is X(ω) =

-j -j -j5 n a) 1/( 1 – 5 e ω ) b) 1/( 1 – 0.2 e ω ) c) 5 e ω d) none above

14. The dc frequency response (H(ω) at ω=0) of a system with impulse response h[n] =u[n] – u[n-4] is

– j4 a) 1 - e ω b) 0 c) 4 d) none above

3z 15. If a filter has H(z) = and ROC |z|>0.5 then, the dc response of the filter is 0.5 + z

a) 3 b) 2 c) 1 d) none above

4/13 5 Points Each, Circle the Best Answer

z + 0.3 16. The system with z-transform H(z) = and ROC 1.2<|z|<2.2 is causal and (z + 2.2)(z −1.2) stable.

a) True b) False

-n 17. The z-transform of h[n]= 5 u[n] is H(z) =

0.2z 3z a) ;|z|>5 b) ;|z|>2 z − 5 z − 2

z c) ;|z|>0.2 d) none above z − 0.2

18. The DFT of the four point sequence x[n] ={1,0,-1,0} is X[k] =:

a) {0, 2, 0, 2} b) {1,0,-1,0} c) {0,1,-1,0} d) none above

!!! 19. The system with z-transform � � = ; |z|>0.01 would be best described as !!!.!"

a) lowpass b) bandpass c) bandstop d) highpass

20. For a system with output y[n] and difference equation y[n] = x[n] - 0.5 y[n-1], the magnitude of the frequency response of the system |H(ω)| is shown below.

a) True b) False

5/13 5 Points Each, Circle the Best Answer

21. An FIR filter Bartlett window has lower peak side-lobes than a rectangular window.

a) True b) False

-2n 22. The system with impulse response h[n] = 2 u[n-1] is an FIR filter.

a) True b) False

23. A 2 sample/second filter with impulse response h[n] is constructed using the impulse –t invariance method for h(t) = 2 u(t). Given this filter, h[n]=

–n -2n –n/2 a) (2 ) u[n] b) 2 u[n] c) 0.5 (2 ) u[n] d) none above

24. Pre-warped bilinear transform filter designs suffer from .

a) True b) False

25. The magnitude of the frequency spectrum, |X(Ω)|, for a continuous time signal x(t) is given below. To avoid aliasing, the signal x(t) must be sampled at a rate greater than

a) 15 Hz b) 20 Hz c) 25 Hz d) 30 Hz e) 40 Hz

1 |X(Ω)|

0 -80π 0 80π Ω

6/13 5 Points Each, Circle the Best Answer

26. In a system with a sample rate of 1000 samples/second, an ideal discrete-time lowpass filter with cutoff frequency of π/4 radians/sample would correspond to a continuous-time cutoff frequency of

a) 500 Hz b) 250 Hz c) 125 Hz d) none above

27. The 2-point Fourier matrix �! is

1 1 1 1 1 1 a) b) c) d) none above 1 −� 1 −1 1 �

28. The dc response of a system with difference equation y[n] = 3x[n] - 2x[n-1] - x[n-2] is:

a) 2 b) 1 c) 0 d) none above

z2 − 4 29. If a filter has H(z) = and ROC |z|>0 then, the first two points of the impulse 2z2 response are

a) {0,0.25} b) {0.5,-2} c) {0.5,0} d) none above

30. The frequency response of the system below is H(ω) =

1+ 2e− jω 1+ 5e− jω 1+ e5 jω a) b) c) d) none above 1− 5e− jω 1− 2e− jω 1+ e2 jω

x[n] Σ y[n]

Delay Delay 2 5

7/13 Assume the following system is initially at rest, and let h[n] be the impulse response.

x[n] Σ y[n] -1 z z-1 -1/4 1/4

z-1 2

5 Points Each

31. The second point of the impulse response is, h[1] =

a) 1/2 b) 1/4 c) 0 d)none above

32. h[2] =

a) 2 b) 1 c) 0 d)none above

33. The z-transform of the system is H(z)=

z2 − 0.25z + 2 z − 0.25 z − 0.5 a) ; z > 0.25 b) ; z > 2 c) ; z > 1 d)none above z2 − 0.25z z2 + 0.25z + 2 z2 + 2

34. The dc response of the system is

a) 1/2 b) 5/2 c) 9 d)none above

8/13 z2 −1 The following questions are for a filter with H(z) = and ROC |z|>0.5. 2(z2 + 0.25)

j F P Q T

A B C D E -1 1

U

R S G -j

5 Points Each, Circle the Best Answer

35. The proper locations of the poles of H(z) are at locations

a) B and D b) C and D c) T and U d) none above

36. The proper locations of the zeroes of H(z) are at locations

a) A and E b) F and G c) O and S d) none above

37. The impulse response h[n] is causal and BIBO stable.

a) True b) False

9/13 The following questions refer to the design of a filter with h[n]=δ[n]-δ [n-1], with a sampling rate of 8000 samples/second .

5 Points Each, Circle the Best Answer

38. For the discrete-time filter, H(z) =

6 1− z−1 a) −3 ; |z|>1 b) −3 ; |z|>0 1− z 1− z

c) 1 − �!!; |z|>0 d) none above

39. For the discrete-time filter, the dc gain is

a) 4 b) 1 c) 0 d) none above

40. Which of the following frequency responses |H(ω)| corresponds to the filter h[n]?

a) A b) B c) C d) D

A B

C D

10/13 In the following questions, a discrete-time filter is to be designed using the method. The sampling rate of the digital system is 10 samples/second. The causal continuous- time filter is given as:

1 H(s) = s + 5

5 Points Each, Circle the Best Answer

41. The z-transform of the discrete-time filter is H(z)=

0.1 -0.5 0.1 - 0.5 a) ;|z|> e b) ;|z|> e 1− e−5z−1 1− e−0.5z−1

0.1 -0.1 c) ; |z|>e d) none above 1− e−0.1z−1

42. For the discrete-time filter, the dc response is H(ω)|ω=0 is H(0)=

0.1 0.1 a) −10 b) c) 1 d) none above 1− e 1− e−0.5

43. The resulting discrete-time filter is BIBO stable.

a) True b) False

11/13 A discrete-time filter is to be designed using the bilinear transform method. The sampling rate of the digital system is 5 samples/second. The continuous-time filter is given as:

1 H(s) = s +10

5 Points Each, Circle the Best Answer

44. H(z)=

z +1 z +1 a) ;|z|>9/20 b) ;|z|>0.9 20z + 9 10z + 9

z +1 c) ; |z|>0 d) none above 20z

45. At analog frequency Ω=0 rad/second, the dc response of the analog filter |H(Ω)| equals

a) 0 b) 0.1 c) 0.5 d) none above

46. At frequency ω=0 rad/sample, the frequency response of the discrete-time filter |H(ω)| most nearly equals

a) 0.02 b) 0.1 c)0.25 d) 0.75

47. At frequency ω=π/2 rad/sample, the frequency response of the discrete-time filter |H(ω)| most nearly equals

a) 2 20 b) 0.25 c) 0.5 d) 1

12/13

5 Points Each (Circle the best answer) The following questions refer to the Java class below and the program main(). public class Green { private int a; private int b;

public Green(int aa, int bb) { a=aa; b=bb; } public void equals(Green c) { this.a= c.a; this.b=c.b; } public void fn(Green c) { this.a = 3*c.b – c.a; this.b = 2*c.a – this.b; } public void gg() { this.b = this.b -1; this.a = this.b – 1; }

public static void main(String[] args) { Green x = new Green (2,2); Green y = new Green (2,1); Green z = new Green (1,4); int xx=1,yy=2,zz=3; x.fn(y); z.gg(); } }

48. At the end of the main program, z.b=

a) 1 b) 2 c) 3 d) None above

49. At the end of the main program, x.a=

a) 1 b) 2 c) 3 d) None above

50. At the end of the main program, x.b=

a) 1 b) 2 c) 3 d) None above

13/13