Is there an equivalent of Parseval's theorem for wavelets?Wavelets and cryptographyData fusion using 2d...
How exactly is a normal force exerted, at the molecular level?
How did installing this RPM create a file?
What exactly did Ant-Man see that made him say that their plan worked?
Who are these Discworld wizards from this picture?
How would an order of Monks that renounce their names communicate effectively?
Does a return economy-class seat between London and San Francisco release 5.28 t of CO2e?
Do launching rockets produce a sonic boom?
Handling a player (unintentionally) stealing the spotlight
How can I write a panicked scene without it feeling like it was written in haste?
My colleague is constantly blaming me for his errors
How to answer "write something on the board"?
Find first and last non-zero column in each row of a pandas dataframe
How did researchers find articles before the Internet and the computer era?
Does any Greek word have a geminate consonant after a long vowel?
Why is Japan trying to have a better relationship with Iran?
Can two or more lightbeams (from a laser for example) have visible interference when they cross in mid-air*?
Is there a legal way for US presidents to extend their terms beyond four years?
How hard is it to sell a home which is currently mortgaged?
Do the 26 richest billionaires own as much wealth as the poorest 3.8 billion people?
Single level file directory
How can a valley surrounded by mountains be fertile and rainy?
Golf the smallest circle!
How to securely dispose of a smartphone?
What does the phrase "building hopping chop" mean here?
Is there an equivalent of Parseval's theorem for wavelets?
Wavelets and cryptographyData fusion using 2d discrete wavelet transform (DWT)Nonlinear wavelets transform?Wavelets in time series predictionsDoes windowing affect Parseval's theorem?Parseval's Theorm and Effective BandwidthBand energy and Parseval theoremChecking Parseval's Theorem for Gaussian Signal by Using ScipyWhat are the constraints in design of discrete orthogonal wavelets?Is the convolution between a decomposition and reconstruction filter a scalar?
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0;
}
$begingroup$
Parseval's theorem can be interpreted as:
... the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For the case of a signal $x(t)$ and its Fourier transform $X(omega)$, the theorem says:
$$
int{|x(t)|^2 ; dt} = int{|X(omega)|^2 ; domega}
$$
For the case of discrete wavelet transform (DWT), or wavelet packet decomposition (WPD), we get a 2D array of coefficients along the time and frequency (or scale) axis:
|
| c{1,f}
| ...
freq | c{1,2}
| c{1,1} c{2, 1} ... c{t, 1}
|______________________________
time
Can a sum of this series somehow be understood as a signal's energy? Is there an equivalent rule to Parseval's theorem?
wavelet signal-energy dwt parseval
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
$endgroup$
add a comment |
$begingroup$
Parseval's theorem can be interpreted as:
... the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For the case of a signal $x(t)$ and its Fourier transform $X(omega)$, the theorem says:
$$
int{|x(t)|^2 ; dt} = int{|X(omega)|^2 ; domega}
$$
For the case of discrete wavelet transform (DWT), or wavelet packet decomposition (WPD), we get a 2D array of coefficients along the time and frequency (or scale) axis:
|
| c{1,f}
| ...
freq | c{1,2}
| c{1,1} c{2, 1} ... c{t, 1}
|______________________________
time
Can a sum of this series somehow be understood as a signal's energy? Is there an equivalent rule to Parseval's theorem?
wavelet signal-energy dwt parseval
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
$endgroup$
add a comment |
$begingroup$
Parseval's theorem can be interpreted as:
... the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For the case of a signal $x(t)$ and its Fourier transform $X(omega)$, the theorem says:
$$
int{|x(t)|^2 ; dt} = int{|X(omega)|^2 ; domega}
$$
For the case of discrete wavelet transform (DWT), or wavelet packet decomposition (WPD), we get a 2D array of coefficients along the time and frequency (or scale) axis:
|
| c{1,f}
| ...
freq | c{1,2}
| c{1,1} c{2, 1} ... c{t, 1}
|______________________________
time
Can a sum of this series somehow be understood as a signal's energy? Is there an equivalent rule to Parseval's theorem?
wavelet signal-energy dwt parseval
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
$endgroup$
Parseval's theorem can be interpreted as:
... the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For the case of a signal $x(t)$ and its Fourier transform $X(omega)$, the theorem says:
$$
int{|x(t)|^2 ; dt} = int{|X(omega)|^2 ; domega}
$$
For the case of discrete wavelet transform (DWT), or wavelet packet decomposition (WPD), we get a 2D array of coefficients along the time and frequency (or scale) axis:
|
| c{1,f}
| ...
freq | c{1,2}
| c{1,1} c{2, 1} ... c{t, 1}
|______________________________
time
Can a sum of this series somehow be understood as a signal's energy? Is there an equivalent rule to Parseval's theorem?
wavelet signal-energy dwt parseval
wavelet signal-energy dwt parseval
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
asked 8 hours ago
hazrmardhazrmard
1287 bronze badges
1287 bronze badges
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
- the decomposition is not "expansive", i.e. the number of samples and of coefficients is the same.
- wavelet filter coefficients are not re-scaled, as happens in some applications (like lifting wavelets, to keep integer computations).
- wavelets are orthogonal (this is not the case in JPEG2000 compression).
You can verify this indirectly, looking at approximation coefficients. At each level, their number of samples is halved, and their amplitudes have around a $1.4$ scale factor, which is just $sqrt{2}$. This feature is used for instance to estimate the Gaussian noise power from wavelet coefficients:
$$ hat{sigma} = textrm{median} (w_i)/0.6745$$
A little further, there is a notion that generalizes (orthonormal) bases: frames. A set of functions $(phi_i)_{iin mathcal{I}}$ ($mathcal{I}$ is a finite or infinite index set) is a frame if for all vectors $x$:
$$ C_flat|x|^2 le sum_{iin mathcal{I}} |<x,phi_i>|^2le C_sharp|x|^2$$
with $0<C_flat,C_sharp < infty$. This is a more general Parseval-Plancherel-like result used for general wavelets.
In other words, it "approximately preserves energy" by projection (inner product). If the constants $ C_flat$ and $C_sharp $ are equal, the frame is said to be tight. Orthonormal bases are non-redundant sets of vectors with $ C_flat=C_sharp = 1 $.
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "295"
};
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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%2fdsp.stackexchange.com%2fquestions%2f59113%2fis-there-an-equivalent-of-parsevals-theorem-for-wavelets%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
$begingroup$
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
- the decomposition is not "expansive", i.e. the number of samples and of coefficients is the same.
- wavelet filter coefficients are not re-scaled, as happens in some applications (like lifting wavelets, to keep integer computations).
- wavelets are orthogonal (this is not the case in JPEG2000 compression).
You can verify this indirectly, looking at approximation coefficients. At each level, their number of samples is halved, and their amplitudes have around a $1.4$ scale factor, which is just $sqrt{2}$. This feature is used for instance to estimate the Gaussian noise power from wavelet coefficients:
$$ hat{sigma} = textrm{median} (w_i)/0.6745$$
A little further, there is a notion that generalizes (orthonormal) bases: frames. A set of functions $(phi_i)_{iin mathcal{I}}$ ($mathcal{I}$ is a finite or infinite index set) is a frame if for all vectors $x$:
$$ C_flat|x|^2 le sum_{iin mathcal{I}} |<x,phi_i>|^2le C_sharp|x|^2$$
with $0<C_flat,C_sharp < infty$. This is a more general Parseval-Plancherel-like result used for general wavelets.
In other words, it "approximately preserves energy" by projection (inner product). If the constants $ C_flat$ and $C_sharp $ are equal, the frame is said to be tight. Orthonormal bases are non-redundant sets of vectors with $ C_flat=C_sharp = 1 $.
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
$endgroup$
add a comment |
$begingroup$
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
- the decomposition is not "expansive", i.e. the number of samples and of coefficients is the same.
- wavelet filter coefficients are not re-scaled, as happens in some applications (like lifting wavelets, to keep integer computations).
- wavelets are orthogonal (this is not the case in JPEG2000 compression).
You can verify this indirectly, looking at approximation coefficients. At each level, their number of samples is halved, and their amplitudes have around a $1.4$ scale factor, which is just $sqrt{2}$. This feature is used for instance to estimate the Gaussian noise power from wavelet coefficients:
$$ hat{sigma} = textrm{median} (w_i)/0.6745$$
A little further, there is a notion that generalizes (orthonormal) bases: frames. A set of functions $(phi_i)_{iin mathcal{I}}$ ($mathcal{I}$ is a finite or infinite index set) is a frame if for all vectors $x$:
$$ C_flat|x|^2 le sum_{iin mathcal{I}} |<x,phi_i>|^2le C_sharp|x|^2$$
with $0<C_flat,C_sharp < infty$. This is a more general Parseval-Plancherel-like result used for general wavelets.
In other words, it "approximately preserves energy" by projection (inner product). If the constants $ C_flat$ and $C_sharp $ are equal, the frame is said to be tight. Orthonormal bases are non-redundant sets of vectors with $ C_flat=C_sharp = 1 $.
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
$endgroup$
add a comment |
$begingroup$
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
- the decomposition is not "expansive", i.e. the number of samples and of coefficients is the same.
- wavelet filter coefficients are not re-scaled, as happens in some applications (like lifting wavelets, to keep integer computations).
- wavelets are orthogonal (this is not the case in JPEG2000 compression).
You can verify this indirectly, looking at approximation coefficients. At each level, their number of samples is halved, and their amplitudes have around a $1.4$ scale factor, which is just $sqrt{2}$. This feature is used for instance to estimate the Gaussian noise power from wavelet coefficients:
$$ hat{sigma} = textrm{median} (w_i)/0.6745$$
A little further, there is a notion that generalizes (orthonormal) bases: frames. A set of functions $(phi_i)_{iin mathcal{I}}$ ($mathcal{I}$ is a finite or infinite index set) is a frame if for all vectors $x$:
$$ C_flat|x|^2 le sum_{iin mathcal{I}} |<x,phi_i>|^2le C_sharp|x|^2$$
with $0<C_flat,C_sharp < infty$. This is a more general Parseval-Plancherel-like result used for general wavelets.
In other words, it "approximately preserves energy" by projection (inner product). If the constants $ C_flat$ and $C_sharp $ are equal, the frame is said to be tight. Orthonormal bases are non-redundant sets of vectors with $ C_flat=C_sharp = 1 $.
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
$endgroup$
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
- the decomposition is not "expansive", i.e. the number of samples and of coefficients is the same.
- wavelet filter coefficients are not re-scaled, as happens in some applications (like lifting wavelets, to keep integer computations).
- wavelets are orthogonal (this is not the case in JPEG2000 compression).
You can verify this indirectly, looking at approximation coefficients. At each level, their number of samples is halved, and their amplitudes have around a $1.4$ scale factor, which is just $sqrt{2}$. This feature is used for instance to estimate the Gaussian noise power from wavelet coefficients:
$$ hat{sigma} = textrm{median} (w_i)/0.6745$$
A little further, there is a notion that generalizes (orthonormal) bases: frames. A set of functions $(phi_i)_{iin mathcal{I}}$ ($mathcal{I}$ is a finite or infinite index set) is a frame if for all vectors $x$:
$$ C_flat|x|^2 le sum_{iin mathcal{I}} |<x,phi_i>|^2le C_sharp|x|^2$$
with $0<C_flat,C_sharp < infty$. This is a more general Parseval-Plancherel-like result used for general wavelets.
In other words, it "approximately preserves energy" by projection (inner product). If the constants $ C_flat$ and $C_sharp $ are equal, the frame is said to be tight. Orthonormal bases are non-redundant sets of vectors with $ C_flat=C_sharp = 1 $.
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
edited 5 hours ago
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
answered 6 hours ago
Laurent DuvalLaurent Duval
17.3k3 gold badges21 silver badges66 bronze badges
17.3k3 gold badges21 silver badges66 bronze badges
add a comment |
add a comment |
Thanks for contributing an answer to Signal Processing 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%2fdsp.stackexchange.com%2fquestions%2f59113%2fis-there-an-equivalent-of-parsevals-theorem-for-wavelets%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
