Determining stochastic process to give a specified autocorrelation function
$begingroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
647
$endgroup$
add a comment |
$begingroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
647
$endgroup$
add a comment |
$begingroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
647
$endgroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
647
asked Jan 12 at 20:17
AvedisAvedis
647
edited Jan 12 at 22:26
Avedis
asked Jan 12 at 20:17
AvedisAvedis
647
asked Jan 12 at 20:17
AvedisAvedis
647
asked Jan 12 at 20:17
AvedisAvedis
647
647
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3071337%2fdetermining-stochastic-process-to-give-a-specified-autocorrelation-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3071337%2fdetermining-stochastic-process-to-give-a-specified-autocorrelation-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
