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) + x²(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) = sin²t
(iii) x[n] = cos[n/2]
(iv) x[n] = cos²[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) = H₁(z) H₂(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)