Z-Transform MCQ

1. Consider a causal and stable LTI system with rational transfer function H(z), whose corresponding impulse response begins at n = 0. Furthermore, H(1) = 5/4. The poles of H(z) are

$p_k=\frac{1}{\sqrt{2}}exp\left ( j\frac{(2k-1)\pi }{4} \right ) for\; k = 1, 2, 3, 4$

The zeros of H(z) are all at z = 0. Let g[n] = jn h[n]. The value of g[8] equals ________

(give the answer up to three decimal places)

  1. 0.090
  2. 0.097
  3. 0.80
  4. 0.087
Answer
Answer. b

2. Let $S=\sum_{n=0}^{\infty }n\alpha ^{n}$ where |α| < 1. The value of α in the range 0 < α < 1, such that S = 2a is

  1. 0.29
  2. 0.36
  3. 0.15
  4. 0.24
Answer
Answer. a

3. The z-transform of a sequence x[n] is given as X[z] = 2z + 4 – 4/z + 3/z2. If y[n] is the first difference of x[n], then Y[z] is given by

  1. 2z + 2 – 8/z + 7/z2 – 3/z3
  2. -2z + 2 – 6/z + 1/z2 – 3/z3
  3. -2z – 2 + 8/z – 7/z2 + 3/z3
  4. 4z – 2 – 8/z – 1/z2 + 3/z3
Answer
Answer. a

4. Consider a discrete-time signal given by

x[n] = (-0.25)n u[n] + (0.5)n u[-n-1]

The region of convergence of its Z-transform would be

  1. the region inside the circle of radius 0.5 and centered at origin
  2. the region outside the circle of radius 0.25 and centered at origin
  3. the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.
  4. the entire z-plane
Answer
Answer. c

5. Let $X(z)=\frac{1}{1-z^{-3}}$ be the z-transform of a causal signal x[n]. Then, the values of x[2] and x[3] are

  1. 0 and 0
  2. 0 and 1
  3. 1 and 0
  4. 1 and 1
Answer
Answer. b

6. If $x[n]=(\frac{1}{3})^{\left | n \right |}-(\frac{1}{2})^{n}\; u[n]$ , then the region of convergence (ROC) of its z-transform in the z-plane will be

  1. 1/3 < |Z| < 3
  2. 1/3 < |Z| < 1/2
  3. 1/2 < |Z| < 3
  4. 1/3 < |Z| < 2
Answer
Answer. c

7. Given $X(z)=\frac{z}{(z-a)^{2}}$ with |Z| > a, the residue of X(z) Zn-1 at z = a for n ≥ 0 will be

  1. an-1
  2. an
  3. nan
  4. nan-1
Answer
Answer. d

8. H(z) is a transfer function of a real system when a signal x[n] = (1+j)n is the input to such a system, the output is zero. Further, the region of convergence (ROC) of $\left ( 1-\frac{1}{2}z^{-1} \right )H(z)$ is the entire z-plane (except z = 0). It can then be inferred that H(z) can have a minimum of

  1. one pole and one zero
  2. one pole and two zeros
  3. two poles and one zero
  4. two poles and two zeros
Answer
Answer. a

9. A discrete real all pass system has a pole at z = 2∠30°; it, therefore

  1. also has a pole at 0.5∠30°
  2. has a constant phase response over the z-plane: arg|H(z)| = constant
  3. is stable only if it anticausal
  4. has a constant phase response over the unit circle: arg|H(ejΩ)| = constant
Answer
Answer. c

10. If u(t) is the unit step and δ(t) is the unit impulse function, the inverse z-transform of F(z) = 1/z+1 for k > 0 is

  1. (-1)kδ(k)
  2. δ(k) – (-1)k u(k)
  3. (-1)k u(k)
  4. u(k) – (-1)k δ(k)
Answer
Answer. b