Question

2. Consider the following random sequence $x(n)$: $x(n) = \cos(\omega_0 n + \theta)$ where $\omega_0$ is a non-random constant and $\theta$ is a random variable uniformly distributed between 0 and $2\pi$ such that: $p_\theta(\theta) = \begin{cases} \frac{1}{2\pi}, & 0 \le \theta \le 2\pi \\ 0, & \text{otherwise} \end{cases}$ a) Compute the mean of the sequence $x(n)$, i.e. $E[x(n)]$. b) Compute the auto-correlation of the sequence $x(n)$, i.e. $r_{xx}(n)$. c) Is the sequence $x(n)$ a wide-sense stationary sequence? Explain.

          2.
Consider the following random sequence $x(n)$:

$x(n) = \cos(\omega_0 n + \theta)$

where $\omega_0$ is a non-random constant and $\theta$ is a random variable
uniformly distributed between 0 and $2\pi$ such that:

$p_\theta(\theta) = \begin{cases} \frac{1}{2\pi}, & 0 \le \theta \le 2\pi \\ 0, & \text{otherwise} \end{cases}$

a) Compute the mean of the sequence $x(n)$, i.e. $E[x(n)]$.
b) Compute the auto-correlation of the sequence $x(n)$, i.e. $r_{xx}(n)$.
c) Is the sequence $x(n)$ a wide-sense stationary sequence? Explain.

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2.
Consider the following random sequence x(n):

x(n) = cos(ω0 n + θ)

where ω0 is a non-random constant and θ is a random variable
uniformly distributed between 0 and 2π such that:

pθ(θ) = (1)/(2π), 0 ≤θ≤ 2π
0, otherwise

a) Compute the mean of the sequence x(n), i.e. E[x(n)].
b) Compute the auto-correlation of the sequence x(n), i.e. rxx(n).
c) Is the sequence x(n) a wide-sense stationary sequence? Explain.

Added by Victor P.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Manish Jain

Step 1

Since x(n) = cos(Î¸n + 0), we can rewrite it as x(n) = cos(Î¸n). The expected value of cos(Î¸n) can be found by integrating over the range of Î·, which is uniformly distributed between 0 and 2. E[x(n)] = ∫[0,2] cos(Î¸n) dÎ· Since Î· is uniformly distributed Show more…

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Consider the following random sequence x(n): x(n) = cos(Î¸n + 0) where Î¸ is a non-random constant and Î· is a random variable uniformly distributed between 0 and 2, such that: P(0 < Î· < 2) = 1 otherwise, P(Î·) = 0 a) Compute the mean of the sequence x(n), i.e. E[x(n)]. b) Compute the auto-correlation of the sequence x(n), i.e. r(Î·,n). c) Is the sequence x(n) a wide-sense stationary sequence? Explain.

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