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Transliterations of Ukrainian names have different avatars in English (and in others languages as well). You can find Kravchuk polynomials, and other papers like On Krawtchouk Transforms or Krawtchouk polynomials and Krawtchouk matrices. You can find as well Kravchuk orthogonal polynomials. As they form an orthogonal basis of polynomials (as well as many ...


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By projecting a vector x using PCA (on the PCs), you maximize the variance in the reduced space. Initially, the space is not optimal in terms of maximizing the variance. So: PCA projects vector 𝑥 to a space of 𝑝 dimensions where the difference between the initial vector and the projection has maximum energy. (initially the is no maximum variance ...


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You can't design a filter that creates a phase shift that's constant with frequency for real valued input (if that's what you are trying to do). A Hilbert transformer appears to be doing this. However, the problem is, you can't implement a perfect Hilbert transformer since it's non causal with an infinite length impulse response. The tricky part is that ...


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Note that the antiderivative of a function is only defined up to a constant. Furthermore, note that if you integrate a periodic function, the result is not necessarily periodic. Let $$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t}\tag{1}$$ Now we integrate $(1)$ with an arbitrary lower integration limit $t_0$. I'll say more about that later. $$\begin{...


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As with Fourier, as with IIR filtering, discrete wavelet forward implementation, and how they can be inversed, depends of lot on how you treat the samples that you don't know, for standard finite support data. In other words: as you convolve, what do you do on the left side and the right side of the signal. There is a huge literature on how to perform a non-...


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Q1: You should ask yourself if $$\frac{z}{3z^2 - 4z + 1}\stackrel{?}{=}\frac{\frac 3 2}{z-1} - \frac{\frac 1 2 }{z-\frac 1 3}$$ really holds. You'll find out that you forgot to scale correctly. Q2: There is no reason to consider $X(z)/z$ instead of $X(z)$ in this case. And concerning your question about the anti-causal sequence, a multiplication by $z^{-...


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$$(-1)^n = \cos(n\pi)=(e^{j\pi})^n=(e^{-j\pi})^n=e^{j\pi n} = e^{-j\pi n}$$ All of these would work. However, applying a DFT is bit tricky since it's directly at the Nyquist frequency and it technically violates the sampling theorem. You can certainly do the math but interpreting the result is not straightforward. For example if you phase shift $\cos(n\pi)$ ...


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Oh, hell... Let's say you have two periodic functions, $x(t)$ and $y(t)$ having exactly the same period (and fundamental frequency): $$ x(t) \triangleq \sum\limits_{k=-\infty}^{\infty} a_k \, e^{j k \omega_0 t} $$ $$ y(t) \triangleq \sum\limits_{k=-\infty}^{\infty} b_k \, e^{j k \omega_0 t} $$ where the period common to both is $\frac{2 \pi}{\omega_0}$....


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It appears to me that the "initial value" issue would affect only the DC value, $a_0$, of the Fourier series. In my opinion, the author should have left the integrals as indefinite integrals (which are the same as the "anti-derivative") or have expressed this relationship only in terms of differentiation, and not integration. Actually, now that I think of ...


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[Beginning of the story] Remember the discrete wavelet curse: in 1D, with 2-scale or dyadic wavelets, you cannot have finite support, realness, orthogonality and linear phase (symmetry/antisymmetry) at the same time, except for the Haar wavelet, which lacks of regularity and overlap. You have to lift one constraint to have the other fulfilled. For instance : ...


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Note that the DFT transforms a periodic sequence into a periodic sequence. Hence, the result of your IFFT is intrinsically periodic and you should view it as such. The parts you are seeing at the end of it are thus just as well behind the start as they are before it. Hence, this part may as well be some form of pre-ringing, stemming from a nonzero group ...


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In typical (or all) applications, the Laplace Transform is used for continuous time systems and the z-Transform for discrete systems. However, the Laplace Transform for Discrete Time Systems certainly exists but would be more complicated than necessary to solve. The z transform exists as a mathematical simplification of the Laplace Transform that can be ...


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According to the properties of Z-transform, the ROC of sum of two signals is the intersection of their individual ROCs. If you look at the signal $$ x[n] = (\frac{1}{3})^n u[n] + (2)^n u[-n-1] $$ you see that it consists of two signals; one right sided $(\frac{1}{3})^n u[n]$ with a pole at $z = 1/3$ and whose Z-transform is $X(z) = 1/(1 - (1/3)z^{-1})$ ...


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It's not formally defined, however I've seen $\int_{-\infty}^\infty \omega |\Psi(\omega)|^2 d\omega$ be used in the literature (i.e. the centre of mass of $|\Psi(\omega)|^2$), where $\Psi$ is the Fourier transform of the wavelet $\psi$.


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I did use Differential Evolution to calculate the coefficients. But you can re-design the filter pair easily using the HIIR library by Laurent de Soras (its source code will automatically unzip to a subdirectory hiir). You can use this C++ HilbertDesign.cpp source and compile with g++ using the compile-command quoted on the first line: // -*- compile-...


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