A causal ltiv discrete-time system is described by the

Question

1. Determine the Z-transform and the corresponding ROC of x[n] = αⁿu[n + 1] + βⁿu[n + 2] where |β| > |α| > 0. Show its pole-zero plots and indicate clearly the ROC in the plot.

2. Determine the Z-transform and the corresponding ROC of x[n] = αⁿu[n- 2] + βⁿu[-n- 1] where |β| > |α| > 0. Show its pole-zero plots and indicate clearly the ROC in the plot.

3. Determine the Z-transform and the corresponding ROC of y[n] = n²αⁿu[n].

4. Determine the Inverse Z-transform of X(Z) = 7+3.6Z^-1/1+0.9Z^-1+0.18Z^-2. Assume that the sequence x[n] is causal (right-sided).

5. The Z-transform of a right-sided sequence h[n] is: H(Z) = Z+1.7/(Z+0.3)(Z-0.5). Find its Inverse Z-transform h[n].

6. Consider the digital filter structure shown below, where H₁(Z) = 2.1 + 3.3Z^-1 + 0.7Z^-2, H₂(Z) = 1.4 - 5.2Z^-1 + 0.8Z^-2, and H₃(Z) = 3.2 + 4.5Z^-1 + 0.9Z^-2. Determine the overall system transfer function H(Z).

7. A causal LTIV discrete-time system is described by the difference equation: y[n] = 0.4y[n - 1] + 0.05y[n - 2] + 3x[n], where x[n] and y[n] are, respectively, the input and the output sequences of the system.

(a) Determine the transfer function H(Z) of the system.

(b) Determine the impulse response h[n] of the system.

Answer 1

These few questions are on Z-transform 1. Find out the Z-transform and the equivalent ROC of x[n] = anu[n + 1] + ßnu[n + 2] where |ß| > |a| > 0. Illustrate its pole-zero plots and indicate clearly the ROC in the plot. 2. Find out the Z-transform and the analogous ROC of x[n] = anu[n- 2] + ßnu[-n- 1] where |ß| > |a| > 0. Show its pole-zero plots and indicate clearly the ROC in the plot. 3. Find out the Z-transform and the resultant ROC of y[n] = n2anu[n]. 4. Find out the Inverse Z-transform of X(Z) = 7+3.6Z-1/1+0.9Z-1+0.18Z-2. Assume that the sequence x[n] is causal (right-sided). 5. The Z-transform of a right-sided sequence h[n] is: H(Z) = Z+1.7/(Z+0.3)(Z-0.5). Discover its Inverse Z-transform h[n]. 6. Consider the digital filter structure exposed below, where H1(Z) = 2.1 + 3.3Z-1 + 0.7Z-2, H2(Z) = 1.4 - 5.2Z-1 + 0.8Z-2, and H3(Z) = 3.2 + 4.5Z-1 + 0.9Z-2. Verify the overall system transfer function H(Z).