Equality between two equations
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
asked 2 days ago
Matthias LauberMatthias Lauber
283
add a comment |
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
asked 2 days ago
Matthias LauberMatthias Lauber
283
1
good catch. interesting too.
– Nick
2 days ago
add a comment |
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
asked 2 days ago
Matthias LauberMatthias Lauber
283
at the moment I am reading the following paper
Benno, Steven A., and José MF Moura. "On translation invariant
subspaces and critically sampled wavelet transforms." Multidimensional
Systems and Signal Processing 8.1-2 (1997): 89-110.
The step between the equations (10) and (11) I can't comprehend. The step is as follows
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega)k}Big)domega =
$$
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k delta(f-omega+k)Big) domega.
$$
I know that $e^{-i2pi k}$ for $kinmathbb{Z}$ is an orthnormal basis, but not over $mathbb{R}$ and I have no idea, why $k$ comes into the dirac function with an "+". At most I woud expect something like $delta(f-omega)$ since it is in a product with k.
Has someone an idea about this?
Thanks Matthias
exponential-function dirac-delta
exponential-function dirac-delta
asked 2 days ago
Matthias LauberMatthias Lauber
283
asked 2 days ago
Matthias LauberMatthias Lauber
283
asked 2 days ago
Matthias LauberMatthias Lauber
283
asked 2 days ago
Matthias LauberMatthias Lauber
283
asked 2 days ago
Matthias LauberMatthias Lauber
283
283
1
good catch. interesting too.
– Nick
2 days ago
add a comment |
1
good catch. interesting too.
– Nick
2 days ago
1
1
good catch. interesting too.
– Nick
2 days ago
good catch. interesting too.
– Nick
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
add a comment |
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%2f3062961%2fequality-between-two-equations%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
add a comment |
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
add a comment |
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
Clearly there is a typo in equation (10), the correct expression is
$$int_R G(omega) overline{tilde{G}(omega)} e^{-j2piomegatau} Big(sum_k e^{-j2pi(f-omega+k)}Big)domega,
$$
which can be seen by substituting the definition of $a_k$ exactly as the authors describe.
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
answered 2 days ago
pre-kidneypre-kidney
12.8k1748
12.8k1748
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
add a comment |
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct.
– Matthias Lauber
yesterday
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3062961%2fequality-between-two-equations%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
1
good catch. interesting too.
– Nick
2 days ago