M.Tech DEGREE I SEMSTER EXAMINATION IN ELECTRONICS

(SIGNAL PROCESSING) – MODEL PAPER – 2012

SP101 – FUNDAMENTALS OF SPECTRAL ESTIMATION

Module - I

1). (A) Given the following spectral density functions , express them in minimum phase and all pass

system

(a) Ry(z)= 0.2−z−1

1−0.5z−1

(b) Ry(z)= 1+z−1−6z−2

1+1

4z−1−

1

8z−2

(5)

(B) By means of DFT and IDFT method find the circular convolution of the following sequences

X1(n)={2,1,2,1}

X2(n)={1,2,3,4} (5)

OR

2) . (A) find the inverse z transform of X(z)=log (1- αz-1) , |z|>|α| (3)

(B) Determine the Fourier transform of the signal

X(n)=

{ (3)

(C) Derive the DTFT of

(i) FT [u(n)] = 1

1−𝑒−𝑗𝜔+ π δ(w + 2πK)

∞

𝑘=∞ (4)

Module – II

3). The exponential density function is given by fx(x) = 1

𝑎 𝑒−𝑥

𝑎 𝑢(𝑥)

Where ‘a’ is a parameter and u(x) is a unit step function

A for –m ≤n ≤ m

0 else where

a) determine the mean , variance, skewness and kurtosis of the Rayleigh random variable with

a=1 .Comment on the significance of these moments in shape of the density function

b) determine the characteristic function of the exponential PDF (10)

OR

4). A) Let w(n) be a zero mean , uncorrelated Gaussian random sequence with variance 𝜎2(n)=1

a) characterise the random sequence w(n)

b) define x(n)=w(n)+w(n-1) , - ∞ < n <∞ , Determine the mean and autocorrelation of x(n). Also

characterize x(n) (5)

B) A WSS process with PSD Rx(𝑒𝑗𝜔 )=

1

1.64+1.6 cos 𝜔 is applied to a causal system described by the

following difference equation y(n)=0.6y(n-1)+x(n)+1.25x(n-1).

Compute a) the PSD of the output

b) the cross PSD Rxy(𝑒𝑗𝜔 ) between input and output (5)

OR

C) Find the eigen value and eigen vectors of the following matrix

A=[ ] Module-III

5. A) Consider a zero mean random sequence x(n) with PSD Rx(ej𝝎)=

5+3 cos 𝜔

17+8 cos 𝜔

a) Determine the innovations representation of the process x(n)

b) Find the autocorrelation sequence rx(l) (5)

B) Use the Yule-Walker equations to determine the autocorrelation and partial autocorrelation

coefficients of the following AR models, assuming that ω(n) ∼ WN(0,1)

a) x(n)=0.5x(n-1)+ ω(n)

b) x(n)=1.5x(n-1)-0.6x(n-2)+ ω(n)

What is the variance ςx2 of the resulting process (5)

OR

-2 1 1

-6 1 3

-12 -2 8

6. A) Find a minimum-phase model with autocorrelation ρ(0)=1, ρ(±1)=0.25 and ρ(l)=0 for lρl ≥2

(5)

B) Determine the spectral flatness measure of the following processes

a) x(n)=a1x(n-1)+a2x(n-2)+ ω(n) and

b) x(n)= ω(n)+b1 ω(n-1)+b2 ω(n-2), whose ω(n)is a white noise sequence (5)

Module-IV

7. A) Let xc(t), -α<t<α, be a continuous-time signal with Fourier transform Xc(F), -α<F<α and let x(n)

be obtained by sampling xc(t) every ‘T’ per sampling interval with its DTFT X(ejw).Show that the DTFT

X(ejw) is given by

X(ejw)=Fs Xc(f Fs− l Fs) ∝

𝑙=−∝ ω=2πF,Fs=1/T (5)

B) Determine analytically the DTFT of rectangular, Bartlett window over 0≤n≤N-1 (5)

OR

8. A) Explain the steps of power spectrum estimation using Blackmann-Tukey method (3)

B) Explain the periodogram method of PSD estimation and explain its limitation (7)

Module-V

9. A) Explain the criterion for selecting the order of an AR model (5)

B) Explain least squares modified Yule-Walker method (5)

OR

10. A) Starting from transfer function, derive an expression to determine the spectrum of first order

all pole models (5)

B) Explain the Durbin algorithm to find a moving average model (5)