Stochastic Processes (II)

Solutions to exercises

Source: Gray (2004) . Sec. 5.10, p. 349.

Exercise 1:

For the two-sided process, the mean is

\mathbb{E}\{W_n\} = (1+r)m

and the autocorrelation is

R_W(k, j) = m^2 (1+r)^2 + (1+r^2)\sigma^2 \delta[k-j] + r \sigma^2\delta[k-j-1] + r \sigma^2 \delta[k-j+1]

Wn is strictly stationary.

For the one-sided process, the mean is

\mathbb{E}\{W_n\} = \left[ \begin{array}{ll} (1+r)m & n>0 \\ m & n=0 \end{array}, \right.

and the autocorrelation is

R_w(k, j) = \left[ \begin{array}{ll} m^2 (1+r)^2 + (1+r^2)\sigma^2 \delta[k-j] + r \sigma^2\delta[k-j-1] + r \sigma^2 \delta[k-j+1], & k>0, \quad j>0 \\ m^2 + \sigma^2 + r \sigma^2 \left(\delta[k+j-1]\right) & k=0 \text{ or } j = 0 \\ \end{array}, \right. .

For n > 0, \mathbb{E}\{W_n\} and R(n, n+k) do not depend on n.

Exercise 2:

SOLUTION:

\mathbb{E}\{U_0\} = \frac{5}{2} \\ R_U[k] = \frac{25}{4} + \frac{9}{4} \delta[k] + \delta[k-1] + \delta[k+1]

RESOLUTION:

The mean is

\mathbb{E}\{U_0\} = \mathbb{E}\{X_0\} + \mathbb{E}\{X_{-1}\} + \mathbb{E}\{Y_{0}\} = 1 + 1 + \frac{1}{2} = \frac{5}{2}

and the autocorrelation is

R_U[k] = \mathbb{E}\{U_{n+k} U_n\} \\ \qquad = \mathbb{E}\{(X_{n+k}+ X_{n+k-1} + Y_{n+k})(X_n + X_{n-1} + Y_n)\} \\ \qquad = \mathbb{E}\{(X_{n+k} + X_{n+k-1})(X_n + X_{n-1})\} + \mathbb{E}\{(X_{n+k} + X_{n+k-1}) Y_n\} + \mathbb{E}\{Y_{n+k}(X_n + X_{n-1})\} + \mathbb{E}\{Y_{n+k} Y_n\}

Since X_n and Y_n are independent, we have

R_U[k] = \mathbb{E}\{(X_{n+k} + X_{n+k-1})(X_n + X_{n-1})\} + \mathbb{E}\{X_{n+k} + X_{n+k-1}\} \mathbb{E}\{Y_n\} + \mathbb{E}\{Y_{n+k}\} \mathbb{E}\{X_n + X_{n-1}\} + \mathbb{E}\{Y_{n+k} Y_n\} \\ \qquad = 2R_X[k] + R_X[k-1] + R_X[k+1] + 4 \mathbb{E}\{X_0\}\mathbb{E}\{Y_0\} + R_Y[k]

And, since both X_n and Y_n are iid processes, we can write

R_X[k] = \mathbb{E}\{X_0\}^2 + \text{var}\{X_0\} \delta[k] = 1 + \delta[k] \\ R_Y[k] = \mathbb{E}\{Y_0\}^2 + \text{var}\{Y_0\} \delta[k] = \frac{1}{4} + \frac{1}{4} \delta[k]

therefore

R_U[k] = 2(1 + \delta[k]) + (1 + \delta[k-1]) + (1 + \delta[k+1]) + 2 + (\frac{1}{4} + \frac{1}{4} \delta[k]) \\ \qquad = (2 + 1 + 1 + 2 + \frac{1}{4}) + (2 + \frac14) \delta[k] + \delta[k-1] + \delta[k+1] \\ \qquad = \frac{25}{4} + \frac{9}{4} \delta[k] + \delta[k-1] + \delta[k+1]


Exercise 4:

R_W(k) = m^2 (1+r)^2 + (1+r^2)\sigma^2 \delta[k] + r \sigma^2\delta[k-1] + r \sigma^2 \delta[k+1]

therefore

S_W(\omega) = (1+r^2)\sigma^2 + 2 \pi m^2 (1+r)^2 \delta(\omega) + 2 r \sigma^2 \cos(\omega), \qquad \qquad -\pi \le \omega \le \pi

Also

S_U(\omega) = \frac{9}{4} + \frac{17\pi}{2} \delta(\omega) + 2 \cos(\omega) \qquad \qquad -\pi \le \omega \le \pi

(both S_W(\omega) and S_U(\omega) are periodic with period 2\pi)

Exercise 5:

(a)

\mathbb{E}\{Y_n\} = 0 \\ S_Y(\omega) = \dfrac{\sin^2\left(\frac{K\omega}{2}\right)} {\sin^2\left(\frac{\omega}{2}\right)}

(b)

Y_n is a zero-mean Gaussian random variable with variance \frac{1}{K}
f_{Y_n}(y) = \sqrt{\frac{K}{2\pi}} = \exp\left(-\frac{K Y_n^2}{2}\right)
The characteristic function is M_{Y_n}(ju) = \exp\left(-\frac{u^2}{2K}\right) [You can skip questions about the characteristic function]

(c)

\delta_n = \frac{1}{K}\left(\delta_n - \delta_{n-K}\right)

(d)

C_Y(n) = \mathbb{E}\{W_n^2\} = \frac{1}{K^2}\left(2\delta_n - \delta_{n-K} - \delta_{n+K}\right)

(e) [You can skip questions about convergence]

Exercise 6:

(a) \mathbb{E}\{X_n\} = 0

(b) \mathbb{E}\{X_n^2\} = \dfrac{1}{12n^2}

(c) K_X(i, j) = \dfrac{1}{12n^2} \delta_{i-k}

(d) \mathbb{E}\{S_n\} = 0

(e) [You can skip questions about convergence]

(f) \mathbb{E}\{Y_n\} = 0

(g)

K(i, j)= \left[ \begin{array}{ll} \dfrac{1}{24} \min(i, j) (-1)^{\frac{i+j}{2}}, & i\ge 2, \quad, j\ge 2, \quad i, j \text{ even} \\ - \dfrac{1}{24} \min(i, j) (-1)^{\frac{i+j}{2}}, & i\ge 2, \quad, j\ge 2, \quad i, j \text{ odd} \\ 0, & \text{otherwise} \end{array} \right.

(h)

Z = Y_6 + Y_4 = 6 X_6 \quad \Rightarrow F_Z(z) = \left[\begin{array}{ll} z + \frac{1}{2}, & -\frac{1}{2} \le z \le \frac{1}{2} \\ 0, & \text{otherwise} \end{array}\right.

Exercise 8:

Z_n = X_n U_n
Y_n = X_n + U_n
W_n = X_n + U_0
\mathbb{E}\{Z_n\} = \mathbb{E}\{Y_n\} = \mathbb{E}\{W_n\} = 0
\text{cov}(Z_n) = \mathbb{E}\{X_n^2\} \mathbb{E}\{U_n^2\} = \sigma^2
\text{cov}(Y_n) = \mathbb{E}\{X_n^2\} + \mathbb{E}\{U_n^2\} = 1 + \sigma^2
\text{cov}(W_n) = \mathbb{E}\{X_n^2\} + \mathbb{E}\{U_0^2\} = 1 + \sigma^2

R_{Z}(n) = \sigma^2 \delta_n \Rightarrow S_Z(\omega) = \sigma^2
R_{Y}(n) = (1 + \sigma^2) \delta_n \Rightarrow S_Y(\omega) = 1 + \sigma^2
R_{W}(n) = 1 + \sigma^2 \delta_n \Rightarrow S_W(\omega) = 2\pi \delta(w) + \sigma^2, \quad -\pi \le \omega \le \pi

K_{Z, Y}(n, m) = \mathbb{E}\{Z_n Y_m\} = \mathbb{E}\{X_n U_n (X_m + U_m)\} = 0
K_{Z, W}(n, m) = \mathbb{E}\{Z_n W_m\} = \mathbb{E}\{X_n U_n (X_m + U_0)\} = 0
K_{Y, W}(n, m) = \mathbb{E}\{Y_n W_m\} = \mathbb{E}\{(X_n + U_n)(X_m + U_0\} = \sigma^2 \delta_{n-m} + \delta_n

Exercise 13:

p_{W_n}(w) = \left[\begin{array}{ll} p^2 + (1-p)^2, & w = 0 \\ 2p(1-p), & w = 1 \end{array}\right.
R_W(k, j) = 4q^2 (1 - \delta_{k-j} - \delta_{k-j-1} - \delta_{k-j+1}) + 2 q\delta_{k-j} + q (\delta_{k-j-1} + \delta_{k-j+1})
where q = p(1-p)
S_W(\omega) = 2 q \left(4 q \pi \delta(\omega) + (1 - 2q) + (1 - 4q) \cos(\omega) \right)

Source:  Beichelt, F. (2016). Chapter 6: Basics of Stochastic processes: exercises (p.252)

Exercise 6.12

SOLUTION:

C_Y[k] = R_Y[k] = \frac{0.8^{|k|}}{0.36}

RESOLUTION:

The covariance function is

C_Y[k] = \mathbb{E}\{(Y_{t+k}-m_Y)·(Y_t-m_Y)\}

where m_Y is the mean of the process. This mean can be computed through the recursive equation:

m_Y = \mathbb{E}\{Y_t\} = 0.8 \mathbb{E}{Y_{t-1}} + \mathbb{E}\{X_t\} = 0.8 \mathbb{E}\{Y_{t-1}\} = 0.8 m_Y

(were we have used that \mathbb{E}\{X_t\}=0). Therefore, m_Y=0 and the covariance function becomes equal to the autocorrelation function:

C_Y[k] = R_Y[k] = \mathbb{E}\{Y_{t+k}·Y_t\}

In order to compute the autocorrelation, note that the system driven by the input-output relation Y_t = 0.8 Y_{t-1} + X_t is linear and time invariant. Therefore, if h_t is the impulse response of the system, the autocorrelation function can be computed as

R_Y[k] = R_X[k]*h_k*h_{-k}

Therefore, we need to compute the impulse response. This can be done in the frequency domain. Computing the Fourier transform:

Y(\omega) = 0.8 e^{-j\omega} Y(\omega) + X(\omega)

Therefore

H(\omega) = \frac{Y(\omega)}{X(\omega)} = \frac{1}{1-0.8 e^{-j\omega}}

which is the Fourier transform of

h_t = 0.8^t u[t]

where u[t] is the step function.

Since X_t is a zero-mean iid process,

R_X[k] = \mathbb{E}\{X_t^2\}\delta_k = \delta_k

Therefore,

R_Y[k] = R_X[k]*h_k*h_{-k} = \delta_k * (0.8^ku[k])*(0.8^{-k}u[-k]) = \frac{0.8^{|k|}}{0.36}

Exercise 6.13

[This exercise can be solved by computing the impulse response through the frequency domain. The process is a bit tedious. Alternatively, you can apply eq. (6.50) in this book:]

C(n) = C(0) \dfrac{(1-y_1^2)y_2^{|n|+1} + (1-y_2^2)y_1^{|n|+1}} {(y_2-y_1)(1 + y_1 y_2)}

where

y_1 = 0.4 (4 + j)
y_2 = 0.4 (4 - j)
C(0) = 20.0

The output process is weakly stationary because |y_1|<1 and |y_2|<1 (otherwise, the system would be unstable).

Ex. 6.14

[It can be solved in the same way as Ex. 6.13]

C(n) = - 4.88 \left(0.19 \cdot (-0.1)^{|n|+1} + 0.99 \cdot 0.9^{|n|+1}\right)

Theodoridis, 2015. Chapter 2 (Secs. 2.4.1-2.4.3)

Ex. 2.9:

This problem is solved using the Cauchy-Schwartz inequality for random variables

|r(k)| = |\mathbb{E}\{U_{n+k} U_n \}| \le \sqrt{\mathbb{E}\{U_{n+k}^2\} \mathbb{E}\{U_n^2\}} = r(0)
|r_{uv}(k)| = |\mathbb{E}\{U_{n+k} V_n \}| \le \sqrt{\mathbb{E}\{U_{n+k}^2\} \mathbb{E}\{V_n^2\}} = \sqrt{r_u(0) r_v(0)}

Ex. 2.10.

Assuming that
D_n = U_n * w_n = \sum_{j=-\infty}^{\infty} U_j w_{n-j} \}
we can write
r_d(k) = \mathbb{E}\{D_{n+k} D_n \} = \\ \qquad = \mathbb{E}\{\sum_{i=-\infty}^{\infty} U_i w_{n+k-i} \sum_{j=-\infty}^{\infty} U_j w_{n-j} \} \\ \qquad = \sum_{i=-\infty}^{\infty} \sum_{j=-\infty}^{\infty} \mathbb{E}\{ U_i U_j \}w_{n+k-i} w_{n-j} \\ \qquad = \sum_{i=-\infty}^{\infty} \sum_{j=-\infty}^{\infty} r_u(i-j) w_{n+k-i} w_{n-j} \\
Applying the index change l=i-j,
r_d(k) = \sum_{l=-\infty}^{\infty} \sum_{j=-\infty}^{\infty} r_u(l) w_{n+k-l-j} w_{n-j} \\ \qquad = \sum_{l=-\infty}^{\infty} r_u(l) \sum_{j=-\infty}^{\infty} w_{n+k-l-j} w_{n-j} \\
and, applying the index change m=j-n,
r_d(k) = \sum_{l=-\infty}^{\infty} r_u(l) \sum_{m=-\infty}^{\infty} w_{-m} w_{(k-l)-m} \\ \qquad = \sum_{l=-\infty}^{\infty} r_u(l) \left[w_n ∗ w_{−n}\right]_{n=k-l} \qquad = r_u(k) ∗ w_k ∗ w_{−k}.

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