• Home
  • Textbooks
  • Probability, Statistics, and Random Processes For Electrical Engineering
  • Analysis and Processing of Random Signals

Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia

Chapter 10

Analysis and Processing of Random Signals - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $g(x)$ denote the triangular function shown in Fig. P10.1.
(a) Find the power spectral density corresponding to $R_X(\tau)=g(\tau / T)$.
(b) Find the autocorrelation corresponding to the power spectral density $S_X(f)=g(f / W)$.

Check back soon!
03:05

Problem 2

Let $p(x)$ be the rectangular function shown in Fig. P10.2. Is $R_X(\tau)=p(\tau / T)$ a valid autocorrelation function?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 3

(a) Find the power spectral density $S_Y(f)$ of a random process with autocorrelation function $R_X(\tau) \cos \left(2 \pi f_0 \tau\right)$, where $R_X(\tau)$ is itself an autocorrelation function.
(b) Plot $S_Y(f)$ if $R_X(\tau)$ is as in Problem 10.1a.

Check back soon!

Problem 4

(a) Find the autocorrelation function corresponding to the power spectral density shown in Fig. P10.3.
(b) Find the total average power.
(c) Plot the power in the range $|f|>f_0$ as a function of $f_0>0$.

Check back soon!
01:16

Problem 5

A random process $X(t)$ has autocorrelation given by $R_X(\tau)=\sigma_X^2 e^{-\tau^2 / 2 \alpha^2}, \alpha>0$.
(a) Find the corresponding power spectral density.
(b) Find the amount of power contained in the frequencies $|f|>k \alpha$, where $k=1,2,3$.

James Kiss
James Kiss
Numerade Educator
03:20

Problem 6

Let $Z(t)=X(t)+Y(t)$. Under what conditions does $S_Z(f)=S_X(f)+S_Y(f)$ ?

Uma Kumari
Uma Kumari
Numerade Educator
02:17

Problem 7

Show that
(a) $\quad R_{X, Y}(\tau)=R_{Y, X}(-\tau)$.
(b) $S_{X, Y}(f)=S_{Y, X}^*(f)$.

Gennady Notowidigdo
Gennady Notowidigdo
Numerade Educator

Problem 8

Let $Y(t)=X(t)-X(t-d)$.
(a) Find $R_{X . Y}(\tau)$ and $S_{X, Y}(f)$.
(b) Find $R_Y(\tau)$ and $S_Y(f)$.

Check back soon!

Problem 9

Do Problem 10.8 if $X(t)$ has the triangular autocorrelation function $g(\tau / T)$ in Problem 10.1 and Fig. P 10.1.

Check back soon!

Problem 10

Let $X(t)$ and $Y(t)$ be independent wide-sense stationary random processes, and define $Z(t)=X(t) Y(t)$.
(a) Show that $Z(t)$ is wide-sense stationary.
(b) Find $R_Z(\tau)$ and $S_Z(f)$.

Check back soon!

Problem 11

In Problem 10.10 , let $X(t)=a \cos \left(2 \pi f_0 t+\Theta\right)$ where $\Theta$ is a uniform random variable in $(0,2 \pi)$. Find $R_Z(\tau)$ and $S_Z(f)$.

Check back soon!
03:01

Problem 12

Let $R_X(k)=4 \alpha^{|k|},|\alpha|<1$.
(a) Find $S_X(f)$.
(b) Plot $S_X(f)$ for $\alpha=0.25$ and $\alpha=0.75$, and comment on the effect of the value of $\alpha$.

Matthew Allcock
Matthew Allcock
Numerade Educator
03:01

Problem 13

Let $R_X(k)=4(\alpha)^{|k|}+16(\beta)^{|k|}, \alpha<1, \beta<1$.
(a) Find $S_X(f)$.
(b) Plot $S_X(f)$ for $\alpha=\beta=0.5$ and $\alpha=0.75=3 \beta$ and comment on the effect of value of $\alpha / \beta$.

Matthew Allcock
Matthew Allcock
Numerade Educator

Problem 14

Let $R_X(k)=9(1-|k| / N)$, for $|k|<N$ and 0 elsewhere. Find and plot $S_X(f)$.

Check back soon!
02:19

Problem 15

. Let $X_n=\cos \left(2 \pi f_0 n+\Theta\right)$, where $\Theta$ is a uniformly distributed random variable in the interval $(0,2 \pi)$. Find and plot $S_X(f)$ for $f_0=0.5,1,1.75, \pi$.

James Kiss
James Kiss
Numerade Educator

Problem 16

Let $D_n=X_n-X_{n-d}$, where $d$ is an integer constant and $X_n$ is a zero-mean, WSS random process.
(a) Find $R_D(k)$ and $S_D(f)$ in terms of $R_X(k)$ and $S_X(f)$. What is the impact of $d$ ?
(b) Find $E\left[D_n^2\right]$.

Check back soon!

Problem 17

. Find $R_D(k)$ and $S_D(f)$ in Problem 10.16 if $X_n$ is the moving average process of Example 10.7 with $\alpha=1$.

Check back soon!

Problem 18

Let $X_n$ be a zero-mean, bandlimited white noise random process with $S_X(f)=1$ for $|f|<f_c$ and 0 elsewhere, where $f_c<1 / 2$.
(a) Show that $R_X(k)=\sin \left(2 \pi f_c k\right) /(\pi k)$.
(b) Find $R_X(k)$ when $f_c=1 / 4$.

Check back soon!

Problem 19

Let $W_n$ be a zero-mean white noise sequence, and let $X_n$ be independent of $W_n$.
(a) Show that $Y_n=W_n X_n$ is a white sequence, and find $\sigma_Y^2$.
(b) Suppose $X_n$ is a Gaussian random process with autocorrelation $R_X(k)=(1 / 2)^{\text {k }}$. Specify the joint pmf's for $Y_n$.

Check back soon!

Problem 20

Evaluate the periodogram estimate for the random process $X(t)=a \cos \left(2 \pi f_0 t+\Theta\right)$, where $\Theta$ is a uniformly distributed random variable in the interval $(0,2 \pi)$. What happens as $T \rightarrow \infty$ ?

Check back soon!

Problem 21

(a) Show how to use the FFT to calculate the periodogram estimate in Eq. (10.32).
(b) Generate four realizations of an iid zero-mean Gaussian sequence of length 128 . Calculate the periodogram.
(c) Calculate 50 periodograms as in part $b$ and show the average of the periodograms after every 10 additional realizations.

Check back soon!

Problem 22

Let $X(t)$ be a differentiable WSS random process, and define

$$
Y(t)=\frac{d}{d t} X(t)
$$

Find an expression for $S_Y(f)$ and $R_Y(\tau)$. Hint: For this system, $H(f)=j 2 \pi f$.

Check back soon!

Problem 23

Let $Y(t)$ be the derivative of $X(t)$, a bandlimited white noise process as in Example 10.3.
(a) Find $S_Y(f)$ and $R_Y(\tau)$.
(b) What is the average power of the output?

Check back soon!

Problem 24

Repeat Problem 10.23 if $X(t)$ has $S_X(f)=\beta^2 e^{-\pi f^2}$.

Check back soon!

Problem 25

Let $Y(t)$ be a short-term integration of $X(t)$ :

$$
Y(t)=\frac{1}{T} \int_{t \sim T}^t X\left(t^{\prime}\right) d t^{\prime}
$$

(a) Find the impulse response $h(t)$ and the transfer function $H(f)$.
(b) Find $S_Y(f)$ in terms of $S_X(f)$.

Check back soon!

Problem 26

In Problem 10.25 , let $R_X(\tau)=(1-|\tau| / T)$ for $|\tau|<T$ and zero elsewhere,
(a) Find $S_Y(f)$.
(b) Find $R_Y(\tau)$.
(c) Find $E\left[Y^2(t)\right]$.

Check back soon!

Problem 27

The input into a filter is zero-mean white noise with noise power density $N_0 / 2$. The filter has transfer function

$$
H(f)=\frac{1}{1+j 2 \pi f}
$$

(a) Find $S_{Y, X}(f)$ and $R_{Y, X}(\tau)$.
(b) Find $S_Y(f)$ and $R_Y(\tau)$.
(c) What is the average power of the output?

Check back soon!

Problem 28

A bandlimited white noise process $X(t)$ is input into a filter with transfer function $H(f)=1+j 2 \pi f$.
(a) Find $S_{Y, X}(f)$ and $R_{Y, X}(\tau)$ in terms of $R_X(\tau)$ and $S_X(f)$.
(b) Find $S_Y(f)$ and $R_Y(\tau)$ in terms of $R_X(\tau)$ and $S_X(f)$.
(c) What is the average power of the output?

Check back soon!
10:23

Problem 29

(a) A WSS process $X(t)$ is applied to a linear system at $t=0$. Find the mean and autocorrelation function of the output process. Show that the output process becomes WSS as $t \rightarrow \infty$.

Ruirui Liu
Ruirui Liu
Numerade Educator

Problem 30

Let $Y(t)$ be the output of a linear system with impulse response $h(t)$ and input $X(t)$. Find $R_{Y, X}(\tau)$ when the input is white noise. Explain how this result can be used to estimate the impulse response of a linear system.

Check back soon!

Problem 31

. (a) A WSS Gaussian random process $X(t)$ is applied to two linear systems as shown in Fig. P10.4. Find an expression for the joint pdf of $Y\left(t_1\right)$ and $W\left(t_2\right)$.
(b) Evaluate part a if $X(t)$ is white Gaussian noise.

Check back soon!

Problem 32

2. Repeat Problem 10.31 b if $h_1(t)$ and $h_2(t)$ are ideal bandpass filters as in Example 10.11 . Show that $Y(t)$ and $W(t)$ are independent random processes if the filters have nonoverlapping bands.

Check back soon!
05:21

Problem 33

Let $Y(t)=h(t) * X(t)$ and $Z(t)=X(t)-Y(t)$ as shown in Fig. P10.5.
(a) Find $S_Z(f)$ in terms of $S_X(f)$.
(b) Find $E\left[Z^2(t)\right]$.

Robin Corrigan
Robin Corrigan
Numerade Educator

Problem 34

Let $Y(t)$ be the output of a linear system with impulse response $h(t)$ and input $X(t)+N(t)$. Let $Z(t)=X(t)-Y(t)$.
(a) Find $R_{X, Y}(\tau)$ and $R_Z(\tau)$.
(b) Find $S_Z(f)$.
(c) Find $S_Z(f)$ if $X(t)$ and $N(t)$ are independent random processes.

Check back soon!

Problem 35

A random telegraph signal is passed through an ideal lowpass filter with cutoff frequency $W$. Find the power spectral density of the difference between the input and output of the filter. Find the average power of the difference signal.

Check back soon!

Problem 36

Let $Y(t)=a \cos \left(2 \pi f_c t+\Theta\right)+N(t)$ be applied to an ideal bandpass filter that passes the frequencies $\left|f-f_c\right|<W / 2$. Assume that $\Theta$ is uniformly distributed in $(0,2 \pi)$. Find the ratio of signal power to noise power at the output of the filter.

Check back soon!
05:20

Problem 37

Let $Y_n=\left(X_{n+1}+X_n+X_{n-1}\right) / 3$ be a "smoothed" version of $X_n$. Find $R_Y(k), S_Y(f)$, and $E\left[Y^2{ }_n\right]$.

SA
Safina Aman
Numerade Educator

Problem 38

. Suppose $X_n$ is a white Gaussian noise process in Problem 10.37. Find the joint pmf for $\left(Y_n, Y_{n+1}, Y_{n+2}\right)$.

Check back soon!

Problem 39

Let $Y_n=X_n+\beta X_{n-1}$, where $X_n$ is a zero-mean, first-order autoregressive process with autocorrelation $R_X(k)=\sigma^2 \alpha^k,|\alpha|<1$.
(a) Find $R_{Y, X}(k)$ and $S_{Y, X}(f)$.
(b) Find $S_Y(f), R_Y(k)$, and $E\left[Y_n^2\right]$.
(c) For what value of $\beta$ is $Y_n$ a white noise process?

Check back soon!

Problem 40

A zero-mean white noise sequence is input into a cascade of two systems (see Fig. P10.6). System 1 has impulse response $h_n=(1 / 2)^n u(n)$ and system 2 has impulse response $g_n=(1 / 4)^n u(n)$ where $u(n)=1$ for $n \geqq 0$ and 0 elsewhere.
(a) Find $S_Y(f)$ and $S_Z(f)$.
(b) Find $R_{W, Y}(k)$ and $R_{W, Z}(k)$; find $S_{W, Y}(f)$ and $S_{W, Z}(f)$. Hint: Use a partial fraction expansion of $S_{W, Z}(f)$ prior to finding $R_{W, Z}(k)$.
(c) Find $E\left[Z_n^2\right]$.

Check back soon!

Problem 41

A moving average process $X_n$ is produced as follows:

$$
X_{i 4}=W_n+\alpha_1 W_{n-1}+\cdots+\alpha_p W_{n-p}
$$

where $W_n$ is a zero-mean white noise process.
(a) Show that $R_X(k)=0$ for $|k|>p$.
(b) Find $R_X(k)$ by computing $E\left[X_{n+k} X_n\right]$, then find $S_X(f)=\mathscr{F}\left\{R_X(k)\right\}$.
(c) Find the impulse response $h_n$ of the linear system that defines the moving average process. Find the corresponding transfer function $H(f)$, and then $S_X(f)$. Compare your answer to part b.

Check back soon!

Problem 42

Consider the second-order autoregressive process defined by

$$
Y_n=\frac{3}{4} Y_{n-1}-\frac{1}{8} Y_{n-2}+W_n
$$

where the input $W_n$ is a zero-mean white noise process.
(a) Verify that the unit-sample response is $h_n=2(1 / 2)^n-(1 / 4)^n$ for $n \geq 0$, and 0 otherwise.
(b) Find the transfer function.
(c) Find $S_Y(f)$ and $R_Y(k)=\mathscr{w}^{-1}\left\{S_Y(f)\right\}$.

Check back soon!

Problem 43

Suppose the autoregressive process defined in Problem 10.42 is the input to the following moving average system:

$$
Z_n=Y_n-1 / 4 Y_{n-1}
$$

(a) Find $S_Z(f)$ and $R_Z(k)$.
(b) Explain why $Z_n$ is a first-order autoregressive process.
(c) Find a moving average system that will produce a white noise sequence when $Z_n$ is the input.

Check back soon!

Problem 44

An autoregressive process $Y_n$ is produced as follows:

$$
Y_n=\alpha_1 Y_{n-1}+\cdots+\alpha_q Y_{n-q}+W_n
$$

where $W_n$ is a zero-mean white noise process.
(a) Show that the autocorrelation of $Y_n$ satisfies the following set of equations:

$$
\begin{aligned}
& R_Y(0)=\sum_{i=1}^q \alpha_i R_Y(i)+R_W(0) \\
& R_Y(k)=\sum_{i=1}^q \alpha_i R_Y(k-i)
\end{aligned}
$$

(b) Use these recursive equations to compute the autocorrelation of the process in Example 10.22.

Check back soon!

Problem 45

(a) Show that the signal $x(t)$ is recovered in Figure 10.10 (b) as long as the sampling rate is above the Nyquist rate.
(b) Suppose that a deterministic signal is sampled at a rate below the Nyquist rate. Use Fig. 10.10 (b) to show that the recovered signal contains additional signal components from the adjacent bands. The error introduced by these components is called aliasing.
(c) Find an expression for the power spectral density of the sampled bandlimited random process $X(t)$.
(d) Find an expression for the power in the aliasing error components.
(e) Evaluate the power in the error signal in part c if $S_X(f)$ is as in Problem 10.1b.

Check back soon!

Problem 46

An ideal discrete-time lowpass filter has transfer function:

$$
H(f)=\left\{\begin{array}{lll}
1 & \text { for } & |f|<f_c<1 / 2 \\
0 & \text { for } & f_c<|f|<1 / 2
\end{array}\right.
$$

(a) Show that $H(f)$ has impulse response $h_n=\sin \left(2 \pi f_c n\right) / \pi n$.
(b) Find the power spectral density of $\mathrm{Y}(k T)$ that results when the signal in Problem 10.1 b is sampled at the Nyquist rate and processed by the filter in part a.
(c) Let $Y(t)$ be the continuous-time signal that results when the output of the filter in part b is fed to an interpolator operating at the Nyquist rate. Find $S_Y(f)$.

Check back soon!

Problem 47

. In order to design a differentiator for bandlimited processes, the filter in Fig. 10.10(c) is designed to have transfer function:

$$
H(f)=j 2 \pi f / T \text { for }|f|<1 / 2
$$
(a) Show that the corresponding impulse response is:

$$
h_0=0, h_n=\frac{\pi n \cos \pi n-\sin \pi n}{\pi n^2 T}=\frac{(-1)^n}{n T} n \neq 0
$$

(b) Suppose that $X(t)=a \cos \left(2 \pi f_0 t+\Theta\right)$ is sampled at a rate $1 / T=4 f_0$ and then input into the above digital filter. Find the output $Y(t)$ of the interpolator.

Check back soon!

Problem 48

Complete the proof of the sampling theorem by showing that the mean square error is zero. Hint: First show that $E[(X(t)-(\hat{X}((t) X(k T)]=0$, all $k$.

Check back soon!
03:04

Problem 49

Plot the power spectral density of the amplitude modulated signal $Y(t)$ in Example 10.18, assuming $f_c>W ; f_c<W$. Assume that $A(t)$ is the signal in Problem 10.1b.

James Kiss
James Kiss
Numerade Educator

Problem 50

Suppose that a random telegraph signal with transition rate $\alpha$ is the input signal in an amplitude modulation system. Plot the power spectral density of the modulated signal assuming $f_c=\alpha / \pi$ and $f_c=10 \alpha / \pi$.

Check back soon!

Problem 51

. Let the input to an amplitude modulation system be $2 \cos \left(2 \pi f_1+\Phi\right)$, where $\Phi$ is uniformly distributed in $(-\pi, \pi)$. Find the power spectral density of the modulated signal assuming $f_c>f_1$.

Check back soon!

Problem 52

Find the signal-to-noise ratio in the recovered signal in Example 10.18 if $S_N(f)=\alpha f^2$ for $\left|f \pm f_c\right|<W$ and zero elsewhere.

Check back soon!

Problem 53

The input signals to a QAM system are independent random processes with power spectral densities shown in Fig. P10.7. Sketch the power spectral density of the QAM signal.

Check back soon!

Problem 54

Under what conditions does the receiver shown in Fig. P10.8 recover the input signals to a QAM signal?

Check back soon!
05:22

Problem 55

Show that Eq. (10.67b) implies that $S_{B, A}(f)$ is a purely imaginary, odd function of $f$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 56

Let $X_\alpha=Z_\alpha+N_\alpha$ as in Example 10.22 , where $Z_\alpha$ is a first-order process with $R_Z(k)=4(3 / 4)^{|k|}$ and $N_\alpha$ is white noise with $\sigma_N^2=1$.
(a) Find the optimum $p=1$ filter for estimating $Z_\alpha$.
(b) Find the mean square error of the resulting filter.

Check back soon!

Problem 57

Let $X_\alpha=Z_\alpha+N_\alpha$ as in Example 10.21 , where $Z_\alpha$ has $R_Z(k)=\sigma_Z^2\left(r_1\right)^{|k|}$ and $N_\alpha$ has $R_N(k)=\sigma_N^2 r_2^k$, where $r_1$ and $r_2$ are less than one in magnitude.
(a) Find the equation for the optimum filter for estimating $Z_\alpha$.
(b) Write the matrix equation for the filter coefficients.
(c) Solve the $p=2$ case, if $\sigma_Z^2=9, r_1=2 / 3, \sigma_N^2=1$, and $r_2=1 / 3$.
(d) Find the mean square error for the optimum filter in part c .
(e) Use the matrix function of Octave to solve parts c and d for $p=3,4,5$.

Check back soon!

Problem 58

Let $X_\alpha=Z_\alpha+N_\alpha$ as in Example 10.21 , where $Z_\alpha$ is the first-order moving average process of Example 10.7, and $N_\alpha$ is white noise.
(a) Find the equation for the optimum filter for estimating $Z_o$.
(b) For the $p=1$ and $p=2$ cases, write and solve the matrix equation for the filter coefficients.
(c) Find the mean square error for the optimum filter in part b.

Check back soon!

Problem 59

Let $X_\alpha=Z_\alpha+N_\alpha$ as in Example 10.19, and suppose that an estimator for $Z_\alpha$ uses observations from the following time instants: $I=\{n-p, \ldots, n, \ldots, n+p\}$.
(a) Solve the $p=1$ case if $Z_\alpha$ and $N_\alpha$ are as in Problem 10.56.
(b) Find the mean square error in part a.
(c) Find the equation for the optimum filter.
(d) Write the matrix equation for the $2 p+1$ filter coefficients.
(e) Use the matrix function of Octave to solve parts a and $b$ for $p=2,3$.

Check back soon!

Problem 60

Consider the predictor in Eq. (10.86b).
(a) Find the optimum predictor coefficients in the $p=2$ case when $R_Z(k)=9(1 / 3)^{k \mid}$.
(b) Find the mean square error in part a.
(c) Use the matrix function of Octave to solve parts a and b for $p=3,4,5$.

Check back soon!

Problem 61

Let $X(t)$ be a WSS, continuous-time process.
(a) Use the orthogonality principle to find the best estimator for $X(t)$ of the form

$$
\hat{X}(t)=a X\left(t_1\right)+b X\left(t_2\right)
$$

where $t_1$ and $t_2$ are given time instants.
(b) Find the mean square error of the optimum estimator.
(c) Check your work by evaluating the answer in part b for $t=t_1$ and $t=t_2$. Is the answer what you would expect?

Check back soon!

Problem 62

Find the optimum filter and its mean square error in Problem 10.61 if $i_1 \neq t-d$ and $t_2=t+d$.

Check back soon!

Problem 63

Find the optimum filter and its mean square error in Problem 10.61 if $t_1=t-d$ and $t_2=\boldsymbol{t}$ $-2 d$, and $R_X(\tau)=e^{-\alpha|\tau|}$ Compare the performance of this filter to the performance of the optimum filter of the form $\hat{X}(t)=a X(t-d)$.

Check back soon!

Problem 64

Modify the system in Problem 10.33 to obtain a model for the estimation error in the optimum infinite-smoothing filter in Example 10.24. Use the model to find an expression for the power spectral density of the error $e(t)=Z(t)-Y(t)$, and then show that the mean square error is given by:

$$
E\left[e^2(t)\right]=\int_{-\infty}^{\infty} \frac{S_Z(f) S_N(f)}{S_Z(f)+S_N(f)} d f
$$

Hint: $E\left[e^2(t)\right]=R_c(0)$.

Check back soon!

Problem 65

Solve the infinite-smoothing problem in Example 10.24 if $Z(t)$ is the random telegraph signal with $\alpha=1 / 2$ and $N(t)$ is white noise. What is the resulting mean square error?

Check back soon!

Problem 66

. Solve the infinite-smoothing problem in Example 10.24 if $Z(t)$ is bandlimited white noise of density $N_1 / 2$ and $N(t)$ is (infinite-bandwidth) white noise of noise density $N_0 / 2$. What is the resulting mean square error?

Check back soon!

Problem 67

Solve the infinite-smoothing problem in Example 10.24 if $Z(t)$ and $N(t)$ are as given in Example 10.25. Find the resulting mean square error.

Check back soon!

Problem 68

Let $X_n=Z_n+N_n$, where $Z_n$ and $N_n$ are independent, zero-mean random processes.
(a) Find the smoothing filter given by Eq. (10.89) when $Z_n$ is a first-order autoregressive process with $\sigma_X^2=9$ and $\alpha=1 / 2$ and $N_n$ is white noise with $\sigma_N^2=4$.
(b) Use the approach in Problem 10.64 to find the power spectral density of the error $S_e(f)$.
(c) Find $R_e(k)$ as follows: Let $Z=e^{j 2 \pi f}$, factor the denominator $S_e(f)$, and take the inverse transform to show that:

$$
R_e(k)=\frac{\sigma_X^2 z_1}{\alpha\left(1-z_1^2\right)} z_1^{|k|} \text { where } 0<z_1<1
$$

(d) Find an expression for the resulting mean square error.

Check back soon!

Problem 69

Find the Wiener filter in Example 10.25 if $N(t)$ is white noise of noise density $N_{\mathrm{t}} / 2=1 / 3$ and $Z(t)$ has power spectral density

$$
S_z(f)=\frac{4}{4+4 \pi^2 f^2}
$$

Check back soon!

Problem 70

. Find the mean square error for the Wiener filter found in Example 10.25. Compare this with the mean square error of the infinite-smoothing filter found in Problem 10.67.

Check back soon!

Problem 71

Suppose we wish to estimate (predict) $X(t+d)$ by

$$
\hat{X}(t+d)=\int_0^{\infty} h(\tau) X(t-\tau) d \tau
$$

(a) Show that the optimum filter must satisfy

$$
R_X(\tau+d)=\int_0^{\infty} h(x) R_X(\tau-x) d x \quad \tau \geq 0
$$

(b) Use the Wiener-Hopf method to find the optimum filter when $R_X(\tau)=e^{-2 \mid \tau}$.

Check back soon!

Problem 72

Let $X_n=Z_n+N_n$, where $Z_{n d}$ and $N_n$ are independent random processes, $N_n$ is a white noise process with $\sigma_N^2=1$, and $Z_n$ is a first-order autoregressive process with $R_Z(k)=$ $4(1 / 2)^k$. We are interested in the optimum filter for estimating $Z_n$ from $X_n, X_{n-1}, \ldots$
(a) Find $S_X(f)$ and express it in the form:

$$
S_X(f)=\frac{\frac{1}{2 z_1}\left(1-\frac{1}{z_1} e^{-j 2 \pi f}\right)\left(1-z_1 e^{j 2 \pi f}\right)}{\left(1-\frac{1}{2} e^{-j 2 \pi f}\right)\left(1-\frac{1}{2} e^{j 2 \pi f}\right)}
$$

(b) Find the whitening causal filter.
(c) Find the optimal causal filter.

Check back soon!

Problem 73

. If $W_n$ and $N_n$ are Gaussian random processes in Eq. (10.102), are $Z_n$ and $X_n$ Markov processes?

Check back soon!

Problem 74

Derive Eq. (10.120) for the mean square prediction error.

Check back soon!

Problem 75

Repeat Example 10.26 with $a=0.5$ and $a=2$.

Check back soon!

Problem 76

Find the Kalman algorithm for the case where the observations are given by

$$
X_n=b_n Z_n+N_n
$$

where $b_n$ is a sequence of known constants.

Check back soon!

Problem 77

Verify Eqs. (10.125) and (10.126) for the periodogram and the autocorrelation function estimate.

Check back soon!
View

Problem 78

Generate a sequence $X_n$ of iid random variables that are uniformly distributed in ( 0,1 ).
(a) Compute several 128 -point periodograms and verify the random behavior of the periodogram as a function of $f$. Does the periodogram vary about the true power spectral density?
(b) Compute the smoothed periodogram based on 10,20, and 50 independent periodograms. Compare the smoothed periodograms to the true power spectral density.

Shu Naito
Shu Naito
Numerade Educator

Problem 79

Repeat Problem 10.78 with $X_n$ a first-order autoregressive process with autocorrelation function: $R_X(k)=(.9)^{|k|} ; R_X(k)=(1 / 2)^{|k|} ; R_X(k)=(.1)^{|k|}$.

Check back soon!

Problem 80

Consider the following estimator for the autocorrelation function

$$
\hat{r}_k^{\prime}(m)=\frac{1}{k-|m|} \sum_{n=0}^{k-|m|-1} X_n X_{n+m}
$$

Show that if we estimate the power spectrum of $X_n$ by the Fourier transform of $\hat{r}_k^{\prime}(m)$, the resulting estimator has mean

$$
E\left[\widetilde{p}_k(f)\right]=\sum_{m^{\prime}=-(k-1)}^{k-1} R_X\left(m^{\prime}\right) e^{-j 2 \pi f m^{\prime}}
$$

Why is the estimator biased?

Check back soon!

Problem 81

Let $X(t)$ have power spectral density given by $S_X(f)=\beta^2 e^{-f^2 / 2 W_0^2} / \sqrt{2 \pi}$.
(a) Before performing an FFT of $S_X(f)$, you are asked to calculate the power in the aliasing error if the signal is treated as if it were bandlimited with bandwidth $k W_0$.
What value of $W$ should be used for the FFT if the power in the aliasing error is to be less than $1 \%$ of the total power? Assume $W_0=1000$ and $\beta=1$.
(b) Suppose you are to perform $N=2 M$ point FFT of $S_X(f)$. Explore how $W, T$, and $t_0$ vary as a function of $f_0$. Discuss what leeway is afforded by increasing $N$.
(c) For the value of $W$ in part a, identify the values of the parameters $f_0, T$, and $t_0$ for $N=128,256,512,1024$.
(d) Find the autocorrelation $\left\{R_X\left(k t_0\right)\right\}$ by applying the FFT to $S_X(f)$. Try the options identified in part c and comment on the accuracy of the results by comparing them to the exact value of $R_X(\tau)$.

Check back soon!

Problem 82

Use the FFT to calculate and plot $S_X(f)$ for the following discrete-time processes:
(a) $R_X(k)=4 \alpha^{[k]}$, for $\alpha=0.25$ and $\alpha=0.75$.
(b) $R_X(k)=4(1 / 2)^{|k|}+16(1 / 4)^{|k|}$.
(c) $X_n=\cos \left(2 \pi f_0 n+\Theta\right)$, where $\Theta$ is a uniformly distributed in $(0,2 \pi]$ and $f_0=1000$.

Check back soon!

Problem 83

Use the FFT to calculate and plot $R_X(k)$ for the following discrete-time processes:
(a) $S_X(f)=1$ for $|f|<f_c$ and 0 elsewhere, where $f_c=1 / 8,1 / 4,3 / 8$.
(b) $S_X(f)=1 / 2+1 / 2 \cos 2 \pi f$ for $|f|<1 / 2$.

Check back soon!

Problem 84

. Use the FFT to find the output power spectral density in the following systems:
(a) Input $X_n$ with $R_X(k)=4 \alpha^{|k|}$, for $\alpha=0.25, H(f)=1$ for $|f|<1 / 4$.
(b) Input $X_n=\cos \left(2 \pi f_0 n+\Theta\right)$, where $\Theta$ is a uniformly distributed random variable and $H(f)=j 2 \pi f$ for $|f|<1 / 2$.
(c) Input $X_n$ with $R_X(k)$ as in Problem 10.14 with $N=3$ and $H(f)=1$ for $|f|<1 / 2$.

Check back soon!
01:51

Problem 85

(a) Show that

$$
R_X(\tau)=2 \operatorname{Re}\left\{\int_0^{\infty} S_X(f) e^{-\rho 2 \pi f \tau} d f\right\}
$$

(b) Use approximations to express the above as a DFT relating $N$ points in the time domain to $N$ points in the frequency domain.
(c) Suppose we meet the $t_0 f_0=1 / N$ requirement by letting $t_0=f_0=1 / \sqrt{N}$. Compare this to the approach leading to Eq. (10.142).

James Kiss
James Kiss
Numerade Educator

Problem 86

(a) Generate a sequence of 1024 zero-mean unit-variance Gaussian random variables and pass it through a system with impulse response $h_n=e^{-2 n}$ for $n \geq 0$.
(b) Estimate the autocovariance of the output process of the digital filter and compare it to the theoretical autocovariance.
(c) What is the pdf of the continuous-time process that results if the output of the digital filter is fed into an interpolator?

Check back soon!

Problem 87

(a) Use the covariance matrix factorization approach to generate a sequence of 1024 Gaussian samples with autocovariance $h(t)=e^{-2 i t!}$.
(b) Estimate the autocovariance of the observed sequence and compare to the theoretical result.

Check back soon!

Problem 88

Does the pulse amplitude modulation signal in Example 9.38 have a power spectral density? Explain why or why not. If the answer is yes, find the power spectral density.

Check back soon!

Problem 89

Compare the operation and performance of the Wiener and Kalman filters for the signals discussed in Example 10.26.

Check back soon!
01:35

Problem 90

(a) Find the power spectral density of the ARMA process in Example 10.15 by finding the transfer function of the associated linear system.
(b) For the ARMA process find the cross-power spectral density from $E\left[Y_n X_m\right]$, and then the power spectral density from $E\left[Y_n Y_m\right]$.

AP
Andreas Papavassiliou
Numerade Educator

Problem 91

Let $X_1(t)$ and $X_2(t)$ be jointly WSS and jointly Gaussian random processes that are input into two linear time-invariant systems as shown below:

$$
\begin{aligned}
& X_1(t) \rightarrow h_1(t) \rightarrow Y_1(t) \\
& X_2(t) \rightarrow h_2(t) \rightarrow Y_2(t)
\end{aligned}
$$

(a) Find the cross-correlation function of $Y_1(t)$ and $Y_2(t)$. Find the corresponding crosspower spectral density.
(b) Show that $Y_1(t)$ and $Y_2(t)$ are jointly WSS and jointly Gaussian random processes.
(c) Suppose that the transfer functions of the above systems are nonoverlapping, that is, $\left|H_1(f)\right|\left|H_2(f)\right|=0$. Show that $Y_1(t)$ and $Y_2(t)$ are independent random processes.
(d) Now suppose that $X_1(t)$ and $X_2(t)$ are nonstationary jointly Gaussian random processes. Which of the above results still hold?

Check back soon!

Problem 92

Consider the communication system in Example 9.38 where the transmitted signal $X(t)$ consists of a sequence of pulses that convey binary information. Suppose that the pulses $p(t)$ are given by the impulse response of the ideal lowpass filter in Figure 10.6. The signal that arrives at the receiver is $Y(t)=X(t)+N(t)$ which is to be sampled and processed digitally.
(a) At what rate should $Y(t)$ be sampled?
(b) How should the bit carried by each pulse be recovered based on the samples $Y(n T)$ ?
(c) What is the probability of error in this system?

Check back soon!