ECEA108 Finals 4Q2324 - Digital Signal Processing PDF

1. If x(n) = {-1, 7, 9, 2, -8, -1, -8}, determine: x(n+2) + x(n-3) Note: Bold and italic number is the origin A. {-1, 7, 8, 9, 1, 0, -9, 8, -6, -8, -1, -8} B. {1, -7, -8, -9, -1, 0, 9, -8, 6, 8, 1, 8} C. {-1, 7, 9, 1, -1, 7, 1, 1, 1, -16, -1, -8} D. {1, -7, -9, -1, 1, 7, -1, -1, -1, 1...

1. If x(n) = {-1, 7, 9, 2, -8, -1, -8}, determine: x(n+2) + x(n-3) Note: Bold and italic number is the origin A. {-1, 7, 8, 9, 1, 0, -9, 8, -6, -8, -1, -8} B. {1, -7, -8, -9, -1, 0, 9, -8, 6, 8, 1, 8} C. {-1, 7, 9, 1, -1, 7, 1, 1, 1, -16, -1, -8} D. {1, -7, -9, -1, 1, 7, -1, -1, -1, 16, 1, 8} Answer: A 2. If x(n) = {-4, 5, -7} and h(n) = {9, 2, 4, -2, 8}, determine y(n) = x(n) × h(n) Note: Bold and italic number is the origin A. {8, -20, -14} B. {9, -8, 20, 14, 8} C. {-8, 20, 14} D. {-8, 20, -14} Answer: C 3. If x(n) = {-3, -4, 7} and h(n) = {3, 5, 8, -8, 7}, determine the convolution of x(n) and h(n) Note: Bold and italic number is the origin A. {21, 4, -57, 103, -27, -23, -21} B. {9, 27, 23, -27, -67, 84, -49} C. {-21, -4, 57, -103, 27, 23, 21} D. {-9, -27, -23, 27, 67, -84, 49} Answer: D 4. If x(n) = {2, 1, 3} and h(n) = {-2, 5, -8, -9, -5}, determine the correlation (rxh(l)) of x(n) and h(n) Note: Bold and italic number is the origin A. {-10, -23, -40, -25, 23, 13, -6} B. {-10, -23, -40, -25, -23, 13, -6} C. {-4, 8, -17, -11, -43, -32, -15} D. {-4, -8, -17, -11, 43, -32, -15} Answer: B 5. If x(n) = {9, 5, 8, -2} and h(n) = {3, -1, 4, 6, -8, -7}, solve for x(n) − h(n) Note: Bold and italic number is the origin A. {3, -1, 4, 3, -3, 1, -2} B. {-3, 1, -4, 3, 13, 15, -2} C. {-3, 1, -4, 3, -3, 1, -2} D. {3, -1, 4, 3, 13, 15, -2} Answer: B 6. Determine the z-transform of the signal: x(n) = 4n u(n) 4𝑧 −1 A. 𝑋(𝑧) = (1−4𝑧−1 )2 𝑧 B. 𝑋(𝑧) = 4−𝑧 𝑧 C. 𝑋(𝑧) = 𝑧−4  𝑧 D. 𝑋(𝑧) = 𝑧+4 Answer: C 7. Determine the z-transform of the signal: x(n) = − 5n u(−n − 1) 𝑧 A. 𝑋(𝑧) = 5−𝑧 𝑧 B. 𝑋(𝑧) = 𝑧−5 5𝑧 −1 C. 𝑋(𝑧) = (1−5𝑧−1 )2 1 D. 𝑋(𝑧) = (1−5𝑧−1 )2 Answer: B 8. Determine the z-transform of the signal: x(n) = 6n u(−n − 1) 𝑧 A. 𝑋(𝑧) = 𝑧−6 𝑧 B. 𝑋(𝑧) = 6−𝑧 6𝑧 −1 C. 𝑋(𝑧) = (1−6𝑧−1 )2 1 D. 𝑋(𝑧) = 1+6𝑧−1 Answer: B 9. Determine the z-transform of the signal: x(n) = nu(n) 𝑧 A. 𝑋(𝑧) = 𝑧−1 𝑧 B. 𝑋(𝑧) = 1−𝑧 𝑧 −1 C. 𝑋(𝑧) = (1−𝑧−1 )2 1 D. 𝑋(𝑧) = 1+𝑧−1 Answer: C 10. Determine the z-transform of the signal: x(n) = 3n nu(−n − 1) −3𝑧 −1 A. 𝑋(𝑧) = 1−3𝑧−1 −3𝑧 −1 B. 𝑋(𝑧) = (1−3𝑧−1 )2 3𝑧 −1 C. 𝑋(𝑧) = 1−3𝑧−1 3𝑧 −1 D. 𝑋(𝑧) = (1−3𝑧−1 )2 Answer: B 1 11. Find the inverse z-transform of the signal: X(z) = 1−4z−1 ; |z| > |4| A. 𝑥(𝑛) = −4𝑛 𝑢(−𝑛 − 1) B. 𝑥(𝑛) = 4𝑛 𝑢(−𝑛 − 1) C. 𝑥(𝑛) = (−4)𝑛 𝑢(𝑛)  D. 𝑥(𝑛) = 4𝑛 𝑢(𝑛) Answer: D 1 12. Find the inverse z-transform of the signal: X(z) = 1−5z−1 ; |z| < |5| A. 𝑥(𝑛) = 5𝑛 𝑢(𝑛) B. 𝑥(𝑛) = −5𝑛 𝑢(−𝑛 − 1) C. 𝑥(𝑛) = (−5)𝑛 𝑢(𝑛) D. 𝑥(𝑛) = 5𝑛 𝑢(−𝑛 − 1) Answer: B z 13. Find the inverse z-transform of the signal: X(z) = 3−z ; |z| < |3| A. 𝑥(𝑛) = 3𝑛 𝑢(−𝑛 − 1) B. 𝑥(𝑛) = 3𝑛 𝑢(𝑛) C. 𝑥(𝑛) = (−3)𝑛 𝑢(𝑛) D. 𝑥(𝑛) = −3𝑛 𝑢(−𝑛 − 1) Answer: A z 14. Find the inverse z-transform of the signal: X(z) = z+4 ; |z| > |4| A. 𝑥(𝑛) = (−4)𝑛 𝑢(𝑛) B. 𝑥(𝑛) = 4𝑛 𝑢(−𝑛 − 1) C. 𝑥(𝑛) = 4𝑛 𝑢(𝑛) D. 𝑥(𝑛) = −4𝑛 𝑢(−𝑛 − 1) Answer: A 3z−1 15. Find the inverse z-transform of the signal: X(z) = (1−3z−1 )2 ; |z| > |3| 1 A. 𝑥(𝑛) = 𝑛3𝑛 𝑢(𝑛) 3 B. 𝑥(𝑛) = −𝑛3𝑛 𝑢(−𝑛 − 1) C. 𝑥(𝑛) = 𝑛3𝑛 𝑢(𝑛) D. 𝑥(𝑛) = 𝑛(−3)𝑛 𝑢(𝑛) Answer: C 16. Given the FIR Filter: 𝑦(𝑛) = 0.25𝑥(𝑛) + 0.2𝑥(𝑛 − 1) + 0.1𝑥(𝑛 − 2) + 0.3𝑥(𝑛 − 3), what is the filter length? A. 1 B. 2 C. 3 D. 4 Answer: C 17. An FIR band stop filter with a lower cut-off frequency of 1800 Hz, an upper cut-off frequency of 2200 Hz, and a sampling rate of 8000 Hz, determine the normalized upper cutoff frequency (Ωc). A. 0.55π radians  B. 0.45π radians C. 0.5π radians D. 1.7276 radians Answer: A 18. An FIR band stop filter with a lower cut-off frequency of 1800 Hz, an upper cut-off frequency of 2200 Hz, and a sampling rate of 8000 Hz, determine the normalized upper cutoff frequency (Ωc). A. 0.55π radians B. 0.45π radians C. 0.4π radians D. 1.4133 radians Answer: B 19. Consider the analog signal 3cos100πt, determine the minimum sampling rate required to avoid aliasing. A. 100 Hz B. 50 Hz C. 200 Hz D. 50π Hz Answer: A 20. Given the analog signal, 3cos50πt + 10sin300πt – cos100πt, what is the Nyquist frequency? A. 300 Hz B. 150 Hz C. 100 Hz D. 50 Hz Answer: A 21. What is the sampling period given the sampling frequency of 8000 Hz? A. 125 µs B. 125 s C. 4000 s D. 4000π s Answer: A