Newest questions tagged parseval - Signal Processing Stack Exchange most recent 30 from dsp.stackexchange.com 2019-07-22T21:50:23Z https://dsp.stackexchange.com/feeds/tag?tagnames=parseval&sort=newest http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://dsp.stackexchange.com/q/59530 1 reference parseval theorem limited signal Luca Mirtanini https://dsp.stackexchange.com/users/44065 2019-07-15T20:57:19Z 2019-07-16T11:13:03Z <p>some days ago I asked here <a href="https://dsp.stackexchange.com/questions/59451/parseval-for-a-continuos-but-limited-signal">parseval for a continuos but limited signal</a> if the Parseval can be applied for limited signal.</p> <p>Can you recommend me a book or a paper that I can use as reference for this?</p> <p>Can someone show me the demonstration considering a limited domain</p> <p>Thank you </p> https://dsp.stackexchange.com/q/59458 1 demonstration using parseval Arkadiusz duka https://dsp.stackexchange.com/users/44130 2019-07-12T17:18:20Z 2019-07-12T20:03:41Z <p>I have to do a demonstration. If we do the Parseval identity of the signals <span class="math-container">$x(t)$</span>, <span class="math-container">$y(t)$</span> and <span class="math-container">$z(t)$</span> that go from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> and that are real, we have:</p> <p><span class="math-container">$\int_{0}^{T} x(t)^2dt=\int_{-\infty}^{\infty}|X(f)|^2df$</span></p> <p><span class="math-container">$\int_{0}^{T} y(t)^2dt=\int_{-\infty}^{\infty}|Y(f)|^2df$</span></p> <p><span class="math-container">$\int_{0}^{T} z(t)^2dt=\int_{-\infty}^{\infty}|Z(f)|^2df$</span> (1)</p> <p>If we sum up the these 3 equations, and we divide both sides by the interval <span class="math-container">$T$</span>, we obtain:</p> <p><span class="math-container">$\frac{1}{T}\int_{0}^{T} (x(t)^2+y(t)^2+z(t)^2)dt=\frac{1}{T}\int_{-\infty}^{\infty}(|X(f)|^2+|Y(f)|^2+|Z(f)|^2)df$</span> (2)</p> <p>Can you confirm me that everything is ok in this demonstration? I am trying to demonstrate this because my advisor says that since the PSD of the signals <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span>, which are <span class="math-container">$S_{X}$</span>, <span class="math-container">$S_{Y}$</span>, <span class="math-container">$S_{Z}$</span>, respect this relation:</p> <p><span class="math-container">$\int_{-\infty}^{\infty}(S_{X}(f)+S_{Y}(f)+S_{Z}(f))df=\frac{1}{T}\int_{0}^{T} (x(t)^2+y(t)^2+z(t)^2)dt$</span> (3)</p> <p>then (and this is the point that is totally wrong for me) he says that <span class="math-container">$S_{X}+ S_{Y}+S_{Z}$</span> is the Fourier transform of ( <span class="math-container">$x(t)^2 +y(t)^2+z(t)^2$</span>):</p> <p><span class="math-container">$S_{X}+ S_{Y}+S_{Z} = \int_{-\infty}^{\infty}( x(t)^2 +y(t)^2+z(t)^2)\exp(-itf2\pi)dt$</span> (4)</p> <p>this cannot be possible in my opinion. I already tried to show him the procedure that leads to the PSD of a signal etc. but he still believe (maybe because I am young, not expert and just arrived) that is true what he said. So I am trying to do the above mentioned demonstration, saying that since <span class="math-container">$|X(f)|^2$</span> is different from the absolute value of the Fourier transform of <span class="math-container">$x(t)^2$</span>, it cannot be contemporary true what it is stated in the eq. (4) and the application of the Parseval identity (2).</p> <p>I beg you to tell me if my procedure is good. I want to convince him with something that is unassailable. (I am almost desperate, I have been trying to convince him that is wrong since 3 month ago)</p> https://dsp.stackexchange.com/q/59451 2 parseval for a continuos but limited signal Luca Mirtanini https://dsp.stackexchange.com/users/44065 2019-07-12T15:05:36Z 2019-07-12T15:44:03Z <p>I have a question about the parseval relation written here <a href="https://en.wikipedia.org/wiki/Parseval%27s_theorem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Parseval%27s_theorem</a> (In the chapter Notation used in physics).</p> <p>If I have a signal continuous but limited (so it does not go from <span class="math-container">$\infty$</span> to <span class="math-container">$\infty$</span> but from 0 to T, can The Parseval theorem be applied?</p> https://dsp.stackexchange.com/q/59113 4 Is there an equivalent of Parseval's theorem for wavelets? hazrmard https://dsp.stackexchange.com/users/19559 2019-06-25T17:33:52Z 2019-06-25T20:47:40Z <p><a href="https://en.wikipedia.org/wiki/Parseval%27s_theorem" rel="nofollow noreferrer">Parseval's theorem</a> can be interpreted as:</p> <blockquote> <p>... the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.</p> </blockquote> <p>For the case of a signal <span class="math-container">$x(t)$</span> and its Fourier transform <span class="math-container">$X(\omega)$</span>, the theorem says:</p> <p><span class="math-container">$$\int{|x(t)|^2 \; dt} = \int{|X(\omega)|^2 \; d\omega}$$</span></p> <p>For the case of <a href="https://en.wikipedia.org/wiki/Discrete_wavelet_transform" rel="nofollow noreferrer">discrete wavelet transform (DWT)</a>, or <a href="https://en.wikipedia.org/wiki/Wavelet_packet_decomposition" rel="nofollow noreferrer">wavelet packet decomposition (WPD)</a>, we get a 2D array of coefficients along the time and frequency (or scale) axis:</p> <pre><code> | | c{1,f} | ... freq | c{1,2} | c{1,1} c{2, 1} ... c{t, 1} |______________________________ time </code></pre> <p>Can a sum of this series somehow be understood as a signal's energy? Is there an equivalent rule to Parseval's theorem?</p> https://dsp.stackexchange.com/q/55471 1 Checking Parseval's Theorem for Gaussian Signal by Using Scipy user10942748 https://dsp.stackexchange.com/users/40575 2019-02-14T02:40:43Z 2019-04-15T09:02:00Z <p>I'm trying to check Parseval's theorm for Gaussian signal. It's well known that fourier transform of <span class="math-container">$\exp(-t^2)$</span> is <span class="math-container">$\sqrt{\pi}\exp(-\pi^2 k^2)$</span>. So I implement it by using quad and simps. I think that the one is continuous integral, the other is integration which uses sample.</p> <p>Code</p> <pre><code>from scipy import integrate import numpy as np def f(x): return (np.exp(-x**2))**2 def F(k): return ((np.pi**0.5)*np.exp((-np.pi**2)*(k**2)))**2 a=integrate.quad(f,-np.inf,np.inf) print(a) #&gt;&gt;&gt;(1.2533141373155017, 4.4674503165883495e-09) b=integrate.quad(F,-np.inf,np.inf) print(b) #&gt;&gt;&gt;(1.2533141373155066, 1.592743224014847e-08) </code></pre> <p>a=b so above code follows Parseval's Theorm. The problem is on following code.</p> <pre><code>import scipy.fftpack as fft N=50000 t = np.linspace(-1000,1000, N) h=np.exp(-t**2) H=2*np.abs(fft.fftshift(fft.fft(h)/N)) freq=fft.fftshift(fft.fftfreq(H.shape,t-t)) S_h=integrate.simps(h**2,t) print(S_h) &gt;&gt;&gt;1.25331413732 S_H=integrate.simps(H**2,freq) print(S_H) &gt;&gt;&gt;1.25326400525e-06 </code></pre> <p><code>S_h</code> is not equals to <code>S_H</code>. What is the problem? How can I fix it??</p> https://dsp.stackexchange.com/q/54833 0 Autocorrelation sequence in terms of Fourier transform of the underlying signal Oliver https://dsp.stackexchange.com/users/8475 2019-01-15T07:20:41Z 2019-01-15T16:15:52Z <p>Let <span class="math-container">$x(n)$</span> be a sequence of length <span class="math-container">$N$</span>, which is zero outside the interval <span class="math-container">$(0,N-1)$</span>. Let <span class="math-container">$X(k), k=0,1,\cdots,N-1$</span> be the FFT coefficients of <span class="math-container">$x(n)$</span>, that is, <span class="math-container">$X(k)=\sum_{n=0}^{N-1}x(n) \exp\left( -\frac{j2\pi k n}{N}\right)$</span>. How to relate the autocorrelation sequence <span class="math-container">$y(l)=\sum_{n=0}^{N-1-l}x(n)x^{*}(n+l)$</span> in terms of <span class="math-container">$X(k)$</span>? </p> <p>My try:</p> <p><span class="math-container">\begin{equation} \begin{split} y(l)&amp; =\sum_{n=0}^{N-1-l}x(n)x^{*}(n+l)\\ &amp; =\sum_{n=0}^{N-1-l}x(n) \left[ \sum_{k=0}^{N-1}X^*(k) \exp\left( -\frac{j2\pi (k+l) n}{N}\right) \right]\\ &amp; = \sum_{k=0}^{N-1}X^*(k) \left[\sum_{n=0}^{N-1-l}x(n) \exp\left( -\frac{j2\pi (k+l) n}{N}\right) \right]\\ \end{split} \end{equation}</span> </p> <p>How to proceed from here?</p>