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The purpose of this assignment is to practice concepts related to the wide-sense stationary processes, filtering,...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
2. (30 points) Let X(t) be a wide-sense stationary (WSS) random signal with power spectral density...
2. (30 points) Let X(t) be a wide-sense stationary (WSS) random signal with power spectral density S(f) = 1011(f/200), and let y(t) be a random process defined by Y(t) = 10 cos(2000nt + 1) where is a uniformly distributed random variable in the interval [ 027]. Assume that X(t) and Y(t) are independent. (a) Derive the mean and autocorrelation function of Y(t). Is Y(t) a WSS process? Why? (b) Define a random signal Z(t) = X(t)Y(t). Determine and sketch the...
Let X(t) be a wide-sense stationary random process with the autocorrelation function :
Let X(t) be a wide-sense stationary random process with the autocorrelation function : Rxx(τ)=e-a|τ| where a> 0 is a constant. Assume that X(t) amplitude modulates a carrier cos(2πf0t+θ), Y(t) = X(t) cos(2πf0t+θ) where θ is random variable on (-π,π) and is statistically independent of X(t). a. Determine the autocorrelation function Ryy(τ) of Y(t), and also give a sketch of it. b. Is y(t) wide-sense stationary as well?
t + τ Proof From Definition 10.17, RİT (r) yields Rn(t) = Elx()r(t + τ)]. Making the substitution u Since X(0) and Y(O) are jointly wide sense stationary, Ryr(u, -t for random sequences Rx-r). The...
t + τ Proof From Definition 10.17, RİT (r) yields Rn(t) = Elx()r(t + τ)]. Making the substitution u Since X(0) and Y(O) are jointly wide sense stationary, Ryr(u, -t for random sequences Rx-r). The proof is similar i: 10.11X(t) is a wide sense stationary stochastic process with autocorrelation function Rx(r). (2) Express the autocorrelation function of Y(C) in terms of Rx(r) Is r) wide sense (2) Express the cross-correlation function of x(t) and Y (t) in terms of Rx(t)...
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with...
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with common power spec- tral density given by SX (f) = SX (f) = 112B(f) watt/Hz. Define x(t) = Xe(t) cos(2tht)--Xs(t) sin(2tht) where fo 》 (a) Is X(t) a Gaussian process? (b) Find the mean E(X (t), autocorrelation function Rx (t,t + T), and power spectral density Sx(f) of the process X(t) (c) Find the pdf of X(O) (d) The process X(t) is passed through...
5. Arandom prices of X(t) is known to be wide-sense stationary with E[X (t)] 11. Give one or more...
5. Arandom prices of X(t) is known to be wide-sense stationary with E[X (t)] 11. Give one or more reasons why each of the following expressions cannot by the autocorrelation function of the process: a. Rult, t + r) = cos(8t)exp(-(t+r)2) b. R (tt)sin(2)+2) R,x(t, t + τ) = 1 1 sin(5(T-2))/(5(r-2)) Rxx(t,t+r)=-11e" d.
5. Arandom prices of X(t) is known to be wide-sense stationary with E[X (t)] 11. Give one or more reasons why each of the following expressions...
Show work for every step please Consider a wide-sense stationary process zlnl] with power spectrum Pole'")...
Show work for every step please
Consider a wide-sense stationary process zlnl] with power spectrum Pole'") = 1 13 12 cos w 10+6 cosw 1. Find the autocorrelation sequence ra[k] of zIn). 2. Find a stable and causal whitening filter H(z) that produces unit variance white noise when the input is rn]. Determine the impulse response hin] of this whitening filter
A. For each of the following randomn processes, state whether it is wide-sense stationary (WSS) and why in 1-3 Sente...
A. For each of the following randomn processes, state whether it is wide-sense stationary (WSS) and why in 1-3 Sentences (a) A Poisson random process N(t) with mean function mN () =M and autocovariance function CN(t,t2) = Ati. (b) A Gaussian random process W (t) with mean function mw (t) = 3t and autocovariance function Cw (l,t,) = 9e 2t2 0 and antocorrelation function (c) An exponential random process Z(t) with mean function mz(1) RZ(t1,t2) = e 42 Ll
A....
5. A stationary random process X[n] is input to a discrete time LTI system with frequency...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency response j“)-10 zero mean given as A(e nmay be expressed as where Wnlis a zero mea a-HS1 unit variancei.i.d. (independent identically distributed) Gaussian sequence and c, d are constants. Let Yl be the output random a)Determine the mean function for the output random sequence Yn in terms ofa, c and d b) Determine S7 (e), the power spectral density ofthe output random sequence Yn]...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
2. (30 points) Let X(t) be a wide-sense stationary (WSS) random signal with power spectral density S(f) = 1011(f/200), and let y(t) be a random process defined by Y(t) = 10 cos(2000nt + 1) where is a uniformly distributed random variable in the interval [ 027]. Assume that X(t) and Y(t) are independent. (a) Derive the mean and autocorrelation function of Y(t). Is Y(t) a WSS process? Why? (b) Define a random signal Z(t) = X(t)Y(t). Determine and sketch the...
t + τ Proof From Definition 10.17, RİT (r) yields Rn(t) = Elx()r(t + τ)]. Making the substitution u Since X(0) and Y(O) are jointly wide sense stationary, Ryr(u, -t for random sequences Rx-r). The proof is similar i: 10.11X(t) is a wide sense stationary stochastic process with autocorrelation function Rx(r). (2) Express the autocorrelation function of Y(C) in terms of Rx(r) Is r) wide sense (2) Express the cross-correlation function of x(t) and Y (t) in terms of Rx(t)...
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with common power spec- tral density given by SX (f) = SX (f) = 112B(f) watt/Hz. Define x(t) = Xe(t) cos(2tht)--Xs(t) sin(2tht) where fo 》 (a) Is X(t) a Gaussian process? (b) Find the mean E(X (t), autocorrelation function Rx (t,t + T), and power spectral density Sx(f) of the process X(t) (c) Find the pdf of X(O) (d) The process X(t) is passed through...
5. Arandom prices of X(t) is known to be wide-sense stationary with E[X (t)] 11. Give one or more reasons why each of the following expressions cannot by the autocorrelation function of the process: a. Rult, t + r) = cos(8t)exp(-(t+r)2) b. R (tt)sin(2)+2) R,x(t, t + τ) = 1 1 sin(5(T-2))/(5(r-2)) Rxx(t,t+r)=-11e" d.
5. Arandom prices of X(t) is known to be wide-sense stationary with E[X (t)] 11. Give one or more reasons why each of the following expressions...
Show work for every step please
Consider a wide-sense stationary process zlnl] with power spectrum Pole'") = 1 13 12 cos w 10+6 cosw 1. Find the autocorrelation sequence ra[k] of zIn). 2. Find a stable and causal whitening filter H(z) that produces unit variance white noise when the input is rn]. Determine the impulse response hin] of this whitening filter
A. For each of the following randomn processes, state whether it is wide-sense stationary (WSS) and why in 1-3 Sentences (a) A Poisson random process N(t) with mean function mN () =M and autocovariance function CN(t,t2) = Ati. (b) A Gaussian random process W (t) with mean function mw (t) = 3t and autocovariance function Cw (l,t,) = 9e 2t2 0 and antocorrelation function (c) An exponential random process Z(t) with mean function mz(1) RZ(t1,t2) = e 42 Ll
A....
5. A stationary random process X[n] is input to a discrete time LTI system with frequency response j“)-10 zero mean given as A(e nmay be expressed as where Wnlis a zero mea a-HS1 unit variancei.i.d. (independent identically distributed) Gaussian sequence and c, d are constants. Let Yl be the output random a)Determine the mean function for the output random sequence Yn in terms ofa, c and d b) Determine S7 (e), the power spectral density ofthe output random sequence Yn]...
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