3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with...
3.34. Let (X.(t) and (x.(e)) denote two statistically independent zero mean stationary Gaussian random processes with common power spec- tral density given by Ste (f) = S, (f) = 112B(f) Watt/Ha Define X (t) X( t) cos(2 fo t) - Xs (t) sin(2r fot) ) - Xs(t) sin(2T fot where fo》 B (c) Find the pdf of X(0). (d) The process X(t) is passed through an ideal bandpass filter with transfer function given by otherwise. Let Y(t) denote the output...
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...
5.57 Let np(t) be a zero-mean white Gaussian noise with the power spectral density 20 let this noise be passed through an ideal bandpass filter with the bandwidth 2W centered at the frequency fe. Denote the output process by nt). 1. Assuming fo fe, find the power content of the in-phase and quadrature components of n(t). We were unable to transcribe this image 5.57 Let np(t) be a zero-mean white Gaussian noise with the power spectral density 20 let this...
Let x(t) = Acos(27/0t + ?) where fo is a given constant, A is a Rayleigh random variable with ? is a uniformly distributed random variable on [0, 2n, and A and ? are statistically independent. a) Find the mean E[X (t)h b) Find the autocorrelation function E(X(t)X(t+)). c) Is (X(t)) wide-sense stationary? d) Find the power spectral density Sx(f)
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...
help please For the following random process, where fe is called the carrier frequency in Hertz, and 6 is a random variab calculations le uniformly distributed over (-m..), answer the follwing questions showing your 1. Compute the meanx() 2. Compute the autocorrelation function Rx(t1.12) 3. Is x(t) stationary? Is it ergodic? Justify your answer. 4. Compute the power spectral density (PSD) Sx() 5. Compute the power of x), both in the time and the frequency domain.
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?
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. Show that if X(t) is a real random process with finite mean power and a mean power spectral density function Sx(S), then for all f. (a) Sx(f) > 0: (b) Sx(-f)=Sx(f). Hint: Recall that if x(t) is a real signal with spectrum X(f), then X (f) = X(-).