**Z-transform Analysis of Sampled-Data Control Systems**

**1. The DFT of a signal x(n) of length N is X(k). When X(k) is given and x(n) is computed from it, the length of x(n)**

a) Is increased to infinity

b) Remains N

c) Becomes 2N – 1

d) Becomes N2

**Answer:** a

**Explanation:** When X (k) is given and x (n) is computed from it, the length of x (n) is increased to infinity.

**2. The system having input x (n) related to output y(n) as y (n) = log |x (n)| is:**

a) Nonlinear, causal, stable

b) Linear, noncausal, stable

c) Nonlinear, causal, not stable

d) Linear, noncausal, not stable

**Answer:** a

**Explanation:** As y (n) is the function of x (n) hence it is nonlinear but it is bounded and also depends upon past and present values therefore it is stable and causal respectively.

**3. Zero-order hold used in practical reconstruction of continuous-time signals is mathematically represented as a weighted-sum of rectangular pulses shifted by:**

a) Any multiples of the sampling interval

b) Integer multiples of the sampling interval

c) One sampling interval

d) 1 second intervals

**Answer:** b

**Explanation:** Zero-order hold is used to reconstruct the continuous-time signal which is represented as a weighted sum of rectangular pulses shifted by integer multiples of the sampling interval.

**4. When two honest coins are simultaneously tossed, the probability of two heads on any given trial is:**

a) 1

b) 3/4

c) 1/2

d) ¼

**Answer:** d

**Explanation:** Total outcomes =4

Favorable outcomes =1

Hence probability =1/4.

**5. A continuous-time periodic signal x(t), having a period T, is convolved with itself. The resulting signal is**

a) Not periodic

b) Periodic having a period T

c) Periodic having a period 2T

d) Periodic having a period T/2

**Answer:** b

**Explanation:** Periodic having a period T Convolution of a periodic signal (period T) with itself will give the same period T.

**6. If the Fourier series coefficients of a signal are periodic then the signal must be**

a) Continuous-time, periodic

b) Discrete-time, periodic

c) Continuous-time, non-periodic

d) Discrete-time, non-periodic

**Answer:** b

**Explanation:** Discrete-time, periodic these are the properties of the discrete-time periodic signal.

**7. The region of convergence of a causal finite duration discrete-time signal is**

a) The entire z-plane except z = 0

b) The entire z-plane except z = ∞

c) The entire z-plane

d) A strip in z-plane enclosing jω–axis

**Answer:** a

**Explanation:** For discrete time signal for any causal system the region of convergence is always entire z-plane but except z =0.

**8. The probability cumulative distribution function must be monotone and**

a) Increasing

b) Decreasing

c) Non-increasing

d) Non-decreasing

**Answer:** d

**Explanation:** The cumulative distribution function that is monotone and non-decreasing or can be increasing.

**9. Convolution is used to find:**

a) The impulse response of an LTI System

b) Frequency response of a System

c) The time response of a LTI system

d) The phase response of a LTI system

**Answer:** c

**Explanation:** Convolution is the combination of addition and multiplication of the signals and used to find the impulse response of the LTI system.

**10. The Fourier Transform of a rectangular pulse is**

a) Another rectangular pulse

b) Triangular pulse

c) Sinc function

d) Impulse

**Answer:** c

**Explanation:** The Fourier Transform of a rectangular pulse that is the sinc function.