Let X(t) be a complex-valued random process defined as X(t) = Aej(@t+), Uniform(0, 2), and A is a random variable independent of with EA = μ and where Var(A) = 6². ~ a. Find the mean function of X(t), µx(t). b. Find the autocorrelation function of X(t), Rx(t₁, t2).
Let X(t) be a complex-valued random process defined as X(t) = Aej(@t+), Uniform(0, 2), and A is a random variable independent of with EA = μ and where Var(A) = 6². ~ a. Find the mean function of X(t), µx(t). b. Find the autocorrelation function of X(t), Rx(t₁, t2).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![Problem 10
(Complex Random Processes) In some applications, we need to work with complex-valued random
processes. More specifically, a complex random process X(t) can be written as
X(t) = X(t) + jX; (t),
where X, (t) and X;(t) are two real-valued random processes and j = √-1. We define the mean
function and the autocorrelation function as
µx(t) = E[X(t)]
= E[X, (t)] + jE[X;(t)]
= μx,(t) + jux;(t);
Rx(t1, 1₂) = E[X(t₁)X* (t₂)]
E [(X,(t₁) + jX;(t₁)) (X, (t2) − jX;(t2))].
Let X(t) be a complex-valued random process defined as
X(t) = Ae(@t+0)
where Φ ~ Uniform(0, 2л), and A is a random variable independent of with EA = μ and
Var(A) = 6².
a. Find the mean function of X(t), µx(t).
b. Find the autocorrelation function of X(t), Rx(t1, t2).
9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47f6c1da-c6df-426c-81ec-6020162a0815%2Ff6ee3be1-3c02-4dd7-9f9b-60dba36c27bb%2F2vgdsin_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 10
(Complex Random Processes) In some applications, we need to work with complex-valued random
processes. More specifically, a complex random process X(t) can be written as
X(t) = X(t) + jX; (t),
where X, (t) and X;(t) are two real-valued random processes and j = √-1. We define the mean
function and the autocorrelation function as
µx(t) = E[X(t)]
= E[X, (t)] + jE[X;(t)]
= μx,(t) + jux;(t);
Rx(t1, 1₂) = E[X(t₁)X* (t₂)]
E [(X,(t₁) + jX;(t₁)) (X, (t2) − jX;(t2))].
Let X(t) be a complex-valued random process defined as
X(t) = Ae(@t+0)
where Φ ~ Uniform(0, 2л), and A is a random variable independent of with EA = μ and
Var(A) = 6².
a. Find the mean function of X(t), µx(t).
b. Find the autocorrelation function of X(t), Rx(t1, t2).
9
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