16 MARKS:
UNIT I: DISCRETE RANDOM SIGNAL PROCESSING
(16 marks)
- i)Derive the power spectral density of the random process. (10)
ii) Determine the power spectrum of a WSS process having autocorrelation sequence rv(k)=(1/2)(A2coskω). (6)
- Explain Spectral factorization. What is regular process? State the properties of regular process. (16)
- Explain filtering random processes. (16)
- i) Obtain the filter to generate a random process with a power spectrum of Px(ejω)=
from white noise. (8)
ii) Explain Wide Sense Stationary process and its properties. (8)
- i) Explain Autocorrelation and Autocovariance matrices. Also explain its properties.(12)
ii) State and prove Wiener-Khintchine relation. (4)
- i) Calculate the mean and variance of the autocorrelation process. Also explain the cross correlation. (4)
ii) Find the autocorrelation sequence corresponding to the following power spectral densities
a) Px(Z)=
b) Px(ejω)=
(12)
(12)- i) Explain in detail about parameter estimation: Bias and Consistency. (6)
ii) Find the output of a first order LTI system, when it is excited by a white noise with 
and if the variance of the white noise is equal to 1. (10)
and if the variance of the white noise is equal to 1. (10)- (i) State and prove Parseval’s theorem. (6)
(ii) A discrete sequence is given by x(n)={0.3,0.1,0.15,0.23,0.29,0.28}.
Compute a) Mean, b) Variance, c) Autocorrelation. (10)
- i) Compute the autocorrelation for the given signal x(n)=(0.8)nu(n). (8)
ii) Obtain the mean and autocorrelation of the real valued harmonic process 
where A and
are constants and φ is a random variable uniformly distributed over the interval –π to π. (8)
where A and are constants and φ is a random variable uniformly distributed over the interval –π to π. (8)- i) The power spectral density of a wide sense stationary process x(n) is Px(ejω)=
.Find the whitening filter H(Z) that produces unit variance white noise when the input is x(n). (10)
ii) Compute the PSD of the given signal x(n)={1,3}. (6)
UNIT II: SPECTRUM ESTIMATION
(16 marks)
- Explain how the Periodogram is used to estimate the power spectrum? (16)
- Describe the performance measures of Periodogram based spectrum estimation technique. (16)
- The bias and variance of the Modified Periodogram is better than that of the ordinary Periodogram- Justify this statement analytically. (16)
- i) Explain power spectrum estimation using Barlett’s method. (8)
ii)
In the Welch method, calculate the variance of Welch power spectrum
estimate with the Bartlett window if there is 50% overlap between
successive sequences. (8)
- Derive the equation for power spectral density using Blackman-Tukey
method. (16)
- Compare the performance of periodogram, Bartlett, Welch and Blackman-Tukey methods of spectrum estimation. (16)
- Derive the expression for spectral estimation based on Auto Regressive
model. (16)
- Discuss MA and ARMA model based power spectrum estimations. (16)
- Elucidate the Levinson Durbin algorithm of solving the normal equations. (16)
- Explain how the power spectral estimate is calculated using parametric method of AR model. (16)
- Explain the power spectrum estimation using model based techniques. (16)
UNIT III: LINEAR ESTIMATION AND PREDICTION
(16 marks)
- Explain the steps in the design of FIR wiener filter that produces the minimum mean square estimate of the process. (16)
- Briefly explain forward linear prediction and backward linear prediction. (16)
- Derive the expression of Discrete Kalman filter. (16)
- Design a three step predictor for a random process having an autocorrelation sequence of the form rx(k)= δ(k)+(0.9) ׀k׀cos(πk/4). Determine the estimate of x(n+3) and also MMSE for the same. Compare the MMSE of multistep predictor with single step predictor. (16)
- Explain in detail about Causal and Noncausal IIR filter. (16)
- Discuss in detail how Levinson’s Durbin algorithm is used to solve the normal equations. (16)
- An AR(2) process has the autocorrelation sequence of rd(k)=αIkI with 0<α<1. Suppose d(n) is observed in the presence of uncorrelated white noise v(n), that has a variance of σv2.δ(k) and x(n)=d(n)+v(n). Design a second order FIR wiener filter to reduce the noise in x(n) with w(z)=w(0)+w(1)Z-1.Assume α=0.8. (16)
- If the values of rx(k) for lags k-0 to 4 are rx(k)=[1.0, 0, 0.1, -0.2, -0.9]T .Solve the Wiener Holf equations and find the optimum three step predictor. (16)
- Describe how the kalman filter is used to estimate an unknown constant. (16)
- An AR(1) process has the autocorrelation sequence of rd(k)=αIkI with 0<α<1. Suppose d(n) is observed in the presence of uncorrelated white noise v(n), that has a variance of σv2.δ(k) and x(n)=d(n)+v(n). Design a first order FIR wiener filter to reduce the noise in x(n) with w(z)=w(0)+w(1)Z-1.Assume α=0.8. (16)
UNIT IV: ADAPTIVE FILTERS
(16 marks)
- Explain steepest decent adaptive filter. (16)
- Describe LMS algorithm for determining FIR adaptive filter coefficients and obtain its condition for convergence. (16)
- Briefly explain Adaptive Channel Equalization and Adaptive Echo Cancellation.(16)
- Explain in detail about exponentially weighted RLS. (16)
- Give complete discussion on how LMS algorithm is converging with necessary derivation of equation. (16)
- Derive the design equations for FIR adaptive wiener filter that would minimize the exponentially weighted least mean square error. (16)
- i) Describe sliding window RLS and derive its update equation. (8)
ii)
Explain how an Adaptive filter can be used as a noise canceller with a
block diagram. (8)
- Write a detailed note on any two applications of adaptive filters with neat
diagram. (16)
- i) State the difficulty in the design and implementation of LMS adaptive filter and describe how this problem is overcome with normalized LMS algorithm. (8)
ii) Derive the weight vector update equation for the LMS algorithm. (8)
- Explain adaptive linear prediction using LMS algorithm. (16)
UNIT V: MULTIRATE DIGITAL SIGNAL PROCESSING
(16 marks)
- Explain the mathematical description of change of sampling rate. (16)
- Define decimation and interpolation. Derive the equation of decimation and interpolation factor. (16)
- Discuss sampling rate conversion by a rational factor I/D. (16)
- Explain in detail the filter bank implementation of wavelet transforms. (16)
- Discuss how signal compression can be achieved using sub-band coding? (16)
- Discuss sampling rate conversion with time variant structures. (16)
- Briefly explain the polyphase structures for decimation and interpolation filters.(16)
- Describe the multistage implementation of multirate system. (16)
- Define wavelet transform. Describe filter bank implementation of wavelet expansion of signals. Discuss one application of wavelet transform. (16)
- Explain the application of multirate processing in sub band coding. (16)
- Describe the implementation of sampling rate conversion using direct form FIR structures. (16)
![\mathbf{R}_x = E[\mathbf{xx}^H] = \begin{bmatrix}
R_{xx}(0) & R^*_{xx}(1) & R^*_{xx}(2) & \cdots & R^*_{xx}(N-1) \\
R_{xx}(1) & R_{xx}(0) & R^*_{xx}(1) & \cdots & R^*_{xx}(N-2) \\
R_{xx}(2) & R_{xx}(1) & R_{xx}(0) & \cdots & R^*_{xx}(N-3) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
R_{xx}(N-1) & R_{xx}(N-2) & R_{xx}(N-3) & \cdots & R_{xx}(0) \\
\end{bmatrix}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vtTSKVOtJeKglwK1S3XkiRzVKud0BZM_S1V5g90hNKMCnA6CXY-NWO5Vquhd8QjE8NDY3QTvEtXQxXaIMEQv2noqUH4VLKr--YO1nwPlRFoHbf1_yI4mUIbOiFdcpDR69-bjPDG54WWjNqM2rMlg=s0-d)
![\Sigma
= \begin{bmatrix}
\mathrm{E}[(X_1 - \mu_1)(X_1 - \mu_1)] & \mathrm{E}[(X_1 - \mu_1)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_1 - \mu_1)(X_n - \mu_n)] \\ \\
\mathrm{E}[(X_2 - \mu_2)(X_1 - \mu_1)] & \mathrm{E}[(X_2 - \mu_2)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_2 - \mu_2)(X_n - \mu_n)] \\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\mathrm{E}[(X_n - \mu_n)(X_1 - \mu_1)] & \mathrm{E}[(X_n - \mu_n)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_n - \mu_n)(X_n - \mu_n)]
\end{bmatrix}.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u-Rg1sQzL_cG1tcaxLRbFuKS33BSZVEx7Lbd6ch88oixN_Jsumsr2-aQ6J3w6mUp4OoRjwUDL0wUnyra2l-DZElWQogZG_YMGRhzhJpM1_hAiMgMWjf4l93yTb-SkRKP5Ou0hm3iL__NKM6jM4vw=s0-d)