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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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You have mistyped the formula, replace this line sum = sum + y(i).*(cos((pi.*(2.*y(i)+1).*u(j))/(2*N))); with the one below, and it works fine. sum = sum + y(i).*(cos((pi.*(2.*u(i)+1).*u(j))/(2*N)));

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This question is hard to answer because it depends on your definition of "transform". The Hilbert transform can be written as a convolution with the kernel $$h(t)=\frac{1}{\pi t}\tag{1}$$ and, consequently, it can be represented by a linear time-invariant (LTI) system with an impulse response given by $(1)$. So the application of any LTI system to a signal ...

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Your confusion comes from the fact that you use $X(\cdot)$ for denoting both functions, the function of $\omega$ and the function of $f$, but they are really two different functions, because $$X(\omega)=\hat{X}(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt,\quad\omega=2\pi f\tag{1}$$ That's why in all the correspondences you mentioned you can just replace $... 3 The properties that you cite are for actually changing the frequency components, by stretching/compressing or shifting, not for just changing the units on the axis. When you do the conversion,$\omega=2\pi f$, you are doing a mapping from radians per second to Hertz but the frequencies in the signal are not changing. When you do the frequency stretching/... 3 For clarity, I would write this DCT as: $$F(u) = \alpha(u)\sum_{i=0}^{N-1}f(i)\cos\left(\frac{\pi u}{2N}(2i+1)\right)$$ We note that, with this 1-indexing of Matlab: $$y[i+1] = f(i)\,.$$ Then I would modify the inner limit (from$0$to$N-1$instead of$1$to$N$): for i = 0:N-1 sum = sum + y(i+1)*(cos((pi*(2*i+1)*u(j))/(2*N))); end and you can remove the ... 2 To have all the samples in the interpolated waveform land on the original samples implies using an integer interpolation rate (otherwise time domain distortion would need to be introduced, and I can't think of an application where that would make sense to do). For integer interpolation, a very simple approach is to use the DFT with zero padding since the ... 2 Parseval's identity and Plancherel's theorem finally boil down to orthogonality. When one decomposes a data (with samples), via a scalar product, onto an orthogonal sequence (yielding coefficients), there exists a certain preservation (equality, up to a proportionality factor) of energy between samples and coefficients. There are some technical conditions, ... 1 Please to refer this video link. In my mind, the widths argument in cwt indicates the scale in wavelet equation. 1 Hint: Question says$y[n]$has length$10$, but the alternate DFT coefficients of$y[n]$i.e.$Y(e^{j\omega})|_{\omega = 2\pi \frac{k}{5}}$, matches with$X(e^{j\omega})$evaluated at those$\omega$exactly. This should draw your attention towards upsampling of 5-point DFT$X(e^{j\omega})|_{\omega = 2\pi \frac{k}{5}}$or equivalently periodization of a ... 1 The Karhunen–Loeve Transform is the equivalent of PCA analysis for continuous signals, you could seek more informations on this type of Feature extraction. 1/The idea is to compute the covariance matrix on known signals (i don't know, maybe the ECG of a person suffering from a particular heart disease).$C = (x-\bar{x})(x-\bar{x})^T$where X is your ... 1 Consider that you're calculating the dB value of the magnitude of the signal, which is the absolutely value of the amplitude. By taking the magnitude, you've discarded the sign, as you're interesting only in the size, not direction. So, you can't recover the signs. That means this will work only for positively-offset signals. But also, we are not usually ... 1 You have the abs(X) in the formula, which is not a reversible operation. All the negative values in the original audio sample has been converted to positive. 1 Probably the value of the DCT coefficient and its frequency (i.e., a histogram plot). However, it's impossible to say that with certainty given the information you have provided. If you found that plot in the linked paper, then the definitions of the x- and y-axes are in all likelihood given therein. The paper is behind a paywall, so it's not helpful to ... 1 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-... 1 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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