Ma1254 Random Processes :: Unit 5 :: Correlation & Spectral Densities

CORRELATION FUNCTION AND POWER SPECTRAL DENSITY

Part-A 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

What is meant by spectral analysis? Define the power spectral density function of a stationary process? State any 2 properties of the PSD of a stationary process. Define cross power spectral density and state any two of its properties. The power spectral density of a random process X(t) is given by Sxx(w) = {∏ ; Iwl≤1 Sxx(w) = 0 otherwise Find its auto correlation function. Write down the Wiener - Khinchine relations. What is the use of Wiener - Khinchine theorem? Find the auto correlation function of a stationary process whose PSD is given Sxx(w)= {w2 ; Iwl≤1 Sxx(w) = 0 otherwise DescribelilHneaf'llystem Define a system. When is it called a linear system? When a system is said it be time - invariant? What is unit impulse response of a system? Give:the. Relation between cross• power density spectrum and crosscorrelation function. Define the power of a random process. Write the propertie of a linear system.

Part - B 1. A WSS process has an auto correlation function R(r) = p.e-3\t\, where p is constant. Find its PSD of the process. 2. Find the power spectral density of a stationary random process for which the auto correlation function is Rxx(r) = A2e-αlr│ 3. The auto correlation of an aperiodic power signal is Rxx(r) = e-r2 (α2 /2.) Find the power spectral density of the signal 4. Find the autocorrelation function ofthe random process X(t) for which the power spectral density is given by S( w) =2α / α2+w2 5. Find the power spectral density of the random process whose autocorrelation function is R(t) = e-α │r│cos{βr).

6. For the process {x(t)} where x(t) = acos(bt + y), where y is uniformly distributed over (-pi, ∏). Find the auto correlation function and the spectral density. 7. The impulse response of a low pass filter is ae-atu(t) where a = ie• If a zero mean white Gaussian process {N(t)} is an input into this filter, find the auto correlation function and mean square value of the output process. . 8. Find the auto correlation function corresponding to the PSD Sxx(w) = 4/, 1+.(w 2│4) -α< W < α 9.Given the power spectral density S x x (w) = 4+1w2' find the average power of the process. 10. A WSS random process X(t) with auto correlation Rxx(r) = Ae-alTI where 'A' and 'a' are real positive constants, is applied to the input of an linear . invariant system with impulse response h(t) = e-btu(t) where b is a real positive constant. Find the auto correlation of the output Y (t) of the system. 11. X(t) is the input and Y(t) is the output of a system. Also {X(t); t €T} is a stationary random process with μx = 0 and Rxx(r) = e-α1T1• Find μ y, Syy(w) and Ryy (t), if the power transfer function is H (w) = R/( R+iL( w) ). 12. A system has a transfer function as1/( l+j(f/l00) if the input to the system is a zero-mean stationary random process with power density spectrum as 10-9. Find the power of the output. 13. X(t), a stationary random process with zero mean is given as input to a system with transfer function H(f) = R/(R+i(2∏)L'The auto correlation of the input is e-βItI. Find the mean and power of output process. Check whether output process is stationary. 14.

Given the power spectral of a continuous process as Sxx(w) =1/ (w4+5w+4 ) find the mean square of the process.