## Signals & Systems - Dec 2013

### Electronics & Telecomm. (Semester 4)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Determine whether the following signals are energy or power signals. Calculate their energy or power.

(i) x[n] = u[n]

(ii) x(t) = Asin(t) ; -∞ < t < ∞(4 marks)
**1 (b)** State and prove the following properties of Fourier Transform(i) Time Shifting.

(ii) Differentiation in time domain.(4 marks)
**1 (c) ** Check whether the following systems are linear or non-linear, time-invariant or time-variant, causal or non-causal, static or dynamic -

(i) y(t) = x(t) cos(100πt)

(ii) y(t) = x(t + 10) + x^{2}(t)(4 marks)
**1 (d)** Compare Discrete time Fourier Transform and Continuous Time Fourier Transform.(4 marks)
**1 (e)** State and discuss the properties of Region of Convergence for Z-Transform.(4 marks)
**2 (a)** Determine the exponential form of Fourier Series representation of the signal shown*(10 marks)
***2 (b)** Determine whether the following signals are periodic or non-periodic. If periodic, find the fundamental period.

(i) x(t) = cos(t)+ sin(√2 t)

(ii) x(t) = sin^{2}t

(iii) x[n] = cos[n/2]

(iv) x[n] = cos^{2}[πn/8](10 marks)
**3 (a)** The analog signal given below is sampled at 600 samples per second

x(t) = 2sin(480πt) + 3sin(720πt)

Calculate:

(i) Minimum Sampling rate to avoid aliasing.

(ii) If the signal is sampled at F_{s}=200Hz, what is the discrete time signal after sampling?

(iii) If the signal is sampled at F_{s}=75Hz, what is the discrete time signal after sampling?(10 marks)
**3 (b)** Determine the DT sequence associated with Z-Transform given below -*(10 marks)
***4 (a)** Draw Cascade and Parallel Realization. The transfer function of discrete time causal system is given by*(10 marks)
***4 (b)** Obtain the convolution of

x(t) = u(t) and h(t) = 1 for -1 ≤ t ≤ 1(10 marks)
**5 (a)** Obtain the inverse Laplace Transform of

*(10 marks)
***5 (b)** Find the Fourier Transform of:

(i) e^{-2(t-1)}u(t-1)

(ii) x(t)=|t|(10 marks)
**6 (a)** Determine the impulse response of DT-LTI system described by the difference equation (for n≥0)

y[n] - 0.5y[n-1] = x[n] + (1/3)x[n-1]

where all the initial conditions are zero.(10 marks)
**6 (b)** A causal LTI system has a Transfer Function H(z) = H_{1}(z) H_{2}(z) where

(i) If system is stable, give ROC condition.

(ii) Find the impulse response.

(iii) Find system response if X(z) = 1/(1-0.2z^{-1})

(iv) Draw pole-zero diagram.(10 marks)
**7 (a)** Using suitable method obtain the state transition matrix for the system matrix

(10 marks)
**7 (b)** The transfer function of the system is given as

Obtain the state variable model.(10 marks)