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Consider a random process where rectangular pulses of width 1 are separated in time by intervals...
nice handwriting please. Question 2 (30 Points) Consider a random process where rectangular pulses of width...
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Question 2 (30 Points) Consider a random process where rectangular pulses of width T, are separated in time by intervals of T seconds. The amplitude of each pulse is determined independently and with equal probability to be either 1, 0, or -1. Pulses begin at periodic time instants to tnt where to is a random variable that is uniformly distributed over the range 0 to T. A sample function is shown below. X(t). T; to-T to +...
5.4.8 Electrical pulses with independent and identically distributed random ampl tudes ξ1,$2, arr...
5.4.8 Electrical pulses with independent and identically distributed random ampl tudes ξ1,$2, arrive at a detector at random times W1, W2 according to a Poisson process of rate λ. The detector output 6k(t) for the kth pulse at time t is for t Wk That is, the amplitude impressed on the detector when the pulse arrives is ξk, and its effect thereafter decays exponentially at rate α. Assume that the detector is additive, so that if N(t) pulses arrive during...
A random process is generated as follows: X(t) = e−A|t|, where A is a random variable...
A random process is generated as follows: X(t) = e−A|t|, where A is a random variable with pdf fA(a) = u(a) − u(a − 1) (1/seconds). a) Sketch several members of the ensemble. b) For a specific time, t, over what values of amplitude does the random variable X(t) range? c) For a specific time, t, find the mean and mean-squared value of X(t). d) For a specific time, t, determine the pdf of X(t).
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events...
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events (think of incoming phone calls, customers entering a bank, car accidents, etc.) that occur up to and including time t where the occurrence times of these events satisfy the following three conditions Events only occur after time 0, i.e., X(t)0 for t0 If N (1, 2], where 0< t t2, denotes the number of events that occur in the time interval (t1,...
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where...
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where A and B are two statistically independent Gaussian random variables, each with zero mean and variance o?. a) Plot two sample functions of X(t). b) Find E{X(0)} c) Find the autocorrelation function Rx(t,t +T). d) Find the pdf of the random variable Y = X(1). e) Is X(t) a Gaussian process? Prove your result.
9. Random binary signal x(t) transmit one digit every Tb seconds. A binary bit '1, is transmitted...
9. Random binary signal x(t) transmit one digit every Tb seconds. A binary bit '1, is transmitted by a rectangular pulse of width T&/2 and amplitude of 1. A binary bit 'O' is transmitted by no signal. The digits1 and '0' are equal likely and occur randomly. Determine the autocorrelation function and the power spectrum density. (20 points) 3
9. Random binary signal x(t) transmit one digit every Tb seconds. A binary bit '1, is transmitted by a rectangular pulse...
2. Consider the random process x(t) defined by x(t) a cos(wt + 6).where w and a...
2. Consider the random process x(t) defined by x(t) a cos(wt + 6).where w and a are constants, and 0 is a random variable uniformly distributed in the range (-T, ) Sketch the ensemble (sample functions) representing x(t). (2.5 points). a. b. Find the mean and variance of the random variable 0. (2.5 points). Find the mean of x(t), m (t) E(x(t)). (2.5 points). c. d. Find the autocorrelation of x(t), R (t,, t) = E(x, (t)x2 (t)). (5 points)....
7 (10pt) Signal s(t) is created by multiplying a rectangular pulse with a sinusoidal signal: s(t) A cos(2mfet) rect where rect(t) is a rectangular pulse with width 1 and amplitude 1 which occupies -0...
7 (10pt) Signal s(t) is created by multiplying a rectangular pulse with a sinusoidal signal: s(t) A cos(2mfet) rect where rect(t) is a rectangular pulse with width 1 and amplitude 1 which occupies -0.5 to 0.5 in time domain. Please find out s(t)'s null-to-null bandwidth.
7 (10pt) Signal s(t) is created by multiplying a rectangular pulse with a sinusoidal signal: s(t) A cos(2mfet) rect where rect(t) is a rectangular pulse with width 1 and amplitude 1 which occupies -0.5 to...
Consider a random process X(t) defined by X(t) - Ycoset, 0st where o is a constant 1. and Y is a uniform random variabl...
Consider a random process X(t) defined by X(t) - Ycoset, 0st where o is a constant 1. and Y is a uniform random variable over (0,1) (a) Classify X(t) (b) Sketch a few (at least three) typical sample function of X(t) (c) Determine the pdfs of X(t) at t 0, /4o, /2, o. (d) EX() (e) Find the autocorrelation function Rx(t,s) of X(t) (f) Find the autocovariance function Rx(t,s) of X(t)
Consider a random process X(t) defined by X(t) -...
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?
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Question 2 (30 Points) Consider a random process where rectangular pulses of width T, are separated in time by intervals of T seconds. The amplitude of each pulse is determined independently and with equal probability to be either 1, 0, or -1. Pulses begin at periodic time instants to tnt where to is a random variable that is uniformly distributed over the range 0 to T. A sample function is shown below. X(t). T; to-T to +...
5.4.8 Electrical pulses with independent and identically distributed random ampl tudes ξ1,$2, arrive at a detector at random times W1, W2 according to a Poisson process of rate λ. The detector output 6k(t) for the kth pulse at time t is for t Wk That is, the amplitude impressed on the detector when the pulse arrives is ξk, and its effect thereafter decays exponentially at rate α. Assume that the detector is additive, so that if N(t) pulses arrive during...
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events (think of incoming phone calls, customers entering a bank, car accidents, etc.) that occur up to and including time t where the occurrence times of these events satisfy the following three conditions Events only occur after time 0, i.e., X(t)0 for t0 If N (1, 2], where 0< t t2, denotes the number of events that occur in the time interval (t1,...
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where A and B are two statistically independent Gaussian random variables, each with zero mean and variance o?. a) Plot two sample functions of X(t). b) Find E{X(0)} c) Find the autocorrelation function Rx(t,t +T). d) Find the pdf of the random variable Y = X(1). e) Is X(t) a Gaussian process? Prove your result.
9. Random binary signal x(t) transmit one digit every Tb seconds. A binary bit '1, is transmitted by a rectangular pulse of width T&/2 and amplitude of 1. A binary bit 'O' is transmitted by no signal. The digits1 and '0' are equal likely and occur randomly. Determine the autocorrelation function and the power spectrum density. (20 points) 3
9. Random binary signal x(t) transmit one digit every Tb seconds. A binary bit '1, is transmitted by a rectangular pulse...
2. Consider the random process x(t) defined by x(t) a cos(wt + 6).where w and a are constants, and 0 is a random variable uniformly distributed in the range (-T, ) Sketch the ensemble (sample functions) representing x(t). (2.5 points). a. b. Find the mean and variance of the random variable 0. (2.5 points). Find the mean of x(t), m (t) E(x(t)). (2.5 points). c. d. Find the autocorrelation of x(t), R (t,, t) = E(x, (t)x2 (t)). (5 points)....
7 (10pt) Signal s(t) is created by multiplying a rectangular pulse with a sinusoidal signal: s(t) A cos(2mfet) rect where rect(t) is a rectangular pulse with width 1 and amplitude 1 which occupies -0.5 to 0.5 in time domain. Please find out s(t)'s null-to-null bandwidth.
7 (10pt) Signal s(t) is created by multiplying a rectangular pulse with a sinusoidal signal: s(t) A cos(2mfet) rect where rect(t) is a rectangular pulse with width 1 and amplitude 1 which occupies -0.5 to...
Consider a random process X(t) defined by X(t) - Ycoset, 0st where o is a constant 1. and Y is a uniform random variable over (0,1) (a) Classify X(t) (b) Sketch a few (at least three) typical sample function of X(t) (c) Determine the pdfs of X(t) at t 0, /4o, /2, o. (d) EX() (e) Find the autocorrelation function Rx(t,s) of X(t) (f) Find the autocovariance function Rx(t,s) of X(t)
Consider a random process X(t) defined by X(t) -...
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