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6.1 For the random process $X(t)$ of Example 6.1, find (a) $f_{X(\pi/4)}(x)$, $f_{X(\pi/2)}(x)$, and $f_X(x)$ (b) $f_{X(\pi/4), X(\pi/2)}(x_1, x_2)$ and $f_{X(\pi/4), X(\pi)}(x_1, x_2)$ (c) $\bar{X}(\pi/4)$, $\bar{X}(\pi)$, and $\bar{X}(t)$ in terms of $t$ (d) $R_X(\pi/2, \pi)$ and $C_X(t + \tau, t)$ for every $t$ and $\tau$. Example 6.1: A Sinusoidal Signal with Binary Random Phase Consider a sinusoidal signal $X(t) = \cos(t + \phi)$, where $\phi$ is a RV determined by tossing a coin: $\phi = \begin{Bmatrix} 0 & \text{head shows} \\ \pi & \text{tail shows} \end{Bmatrix}$. Note that • $X(t)$ is random since its values cannot be determined surely beforehand. • $X(t)$ is not a RV because a time function rather than a number is assigned to it given an outcome of the random experiment. • Sample space = \{$x_1(t), x_2(t)$\} = \{$\cos t$, $-\cos t$\} since $\cos(t + \pi) = -\cos t$. • For any given time $t_1$, $X(t_1)$ is a RV with at most two possible values: $\cos t_1$ and $-\cos t_1$.

          6.1 For the random process $X(t)$ of Example 6.1, find
(a) $f_{X(\pi/4)}(x)$, $f_{X(\pi/2)}(x)$, and $f_X(x)$
(b) $f_{X(\pi/4), X(\pi/2)}(x_1, x_2)$ and $f_{X(\pi/4), X(\pi)}(x_1, x_2)$
(c) $\bar{X}(\pi/4)$, $\bar{X}(\pi)$, and $\bar{X}(t)$ in terms of $t$
(d) $R_X(\pi/2, \pi)$ and $C_X(t + \tau, t)$ for every $t$ and $\tau$.
Example 6.1: A Sinusoidal Signal with Binary Random Phase
Consider a sinusoidal signal $X(t) = \cos(t + \phi)$, where $\phi$ is a RV determined by
tossing a coin: $\phi = \begin{Bmatrix} 0 & \text{head shows} \\ \pi & \text{tail shows} \end{Bmatrix}$. Note that
• $X(t)$ is random since its values cannot be determined surely beforehand.
• $X(t)$ is not a RV because a time function rather than a number is assigned
to it given an outcome of the random experiment.
• Sample space = \{$x_1(t), x_2(t)$\} = \{$\cos t$, $-\cos t$\} since $\cos(t + \pi) = -\cos t$.
• For any given time $t_1$, $X(t_1)$ is a RV with at most two possible values: $\cos t_1$
and $-\cos t_1$.

6.1 For the random process X(t) of Example 6.1, find
(a) fX(π/4)(x), fX(π/2)(x), and fX(x)
(b) fX(π/4), X(π/2)(x1, x2) and fX(π/4), X(π)(x1, x2)
(c) X̅(π/4), X̅(π), and X̅(t) in terms of t
(d) RX(π/2, π) and CX(t + τ, t) for every t and τ.
Example 6.1: A Sinusoidal Signal with Binary Random Phase
Consider a sinusoidal signal X(t) = cos(t + ϕ), where ϕ is a RV determined by
tossing a coin: ϕ = 0 head shows
π tail shows. Note that
• X(t) is random since its values cannot be determined surely beforehand.
• X(t) is not a RV because a time function rather than a number is assigned
to it given an outcome of the random experiment.
• Sample space = {x1(t), x2(t)} = {cos t, -cos t} since cos(t + π) = -cos t.
• For any given time t1, X(t1) is a RV with at most two possible values: cos t1
and -cos t1.

Added by Fernando T.

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