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] = j^{n} h[n]. The value of g[8] equals ________

(give the answer up to three decimal places)

- 0.090
- 0.097
- 0.80
- 0.087

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

- 0.29
- 0.36
- 0.15
- 0.24

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

- 2z + 2 – 8/z + 7/z
^{2}– 3/z^{3} - -2z + 2 – 6/z + 1/z
^{2}– 3/z^{3} - -2z – 2 + 8/z – 7/z
^{2}+ 3/z^{3} - 4z – 2 – 8/z – 1/z
^{2}+ 3/z^{3}

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

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

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

- 0 and 0
- 0 and 1
- 1 and 0
- 1 and 1

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

- 1/3 < |Z| < 3
- 1/3 < |Z| < 1/2
- 1/2 < |Z| < 3
- 1/3 < |Z| < 2

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

- a
^{n-1} - a
^{n} - na
^{n} - na
^{n-1}

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

- one pole and one zero
- one pole and two zeros
- two poles and one zero
- two poles and two zeros

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

- also has a pole at 0.5∠30°
- has a constant phase response over the z-plane: arg|H(z)| = constant
- is stable only if it anticausal
- has a constant phase response over the unit circle: arg|H(e
^{jΩ})| = constant

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)
^{k}δ(k) - δ(k) – (-1)
^{k}u(k) - (-1)
^{k}u(k) - u(k) – (-1)
^{k}δ(k)