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I don't really understand what do you mean by multiply them in the time domain and multiply them with window function. I think that you are trying to implement the Welch's PSD calculation. If so, steps should be: Split your data into possibly overlapping segments of length $N$ Preferably remove the mean of each segment. Window each of the segments by ...


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Let $$\hat{\phi}_{yu}(\omega) = \sum_{k = -M}^{M} w(k) \ \hat{r}_{yu}(k) e^{-i\omega k}\tag{1}$$ apply a change of variable $$k'=k+M+1\Rightarrow k=k'-M-1$$ and $(1)$ becomes $$\begin{align}\hat{\phi}_{yu}(\omega) &= \sum_{k' = 1}^{N} w(k'-M-1) \ \hat{r}_{yu}(k'-M-1) e^{-i\omega (k'-M-1)}\\[10pt] &=e^{i\omega (M+1)}\sum_{k' = 1}^{N} w(k'-M-1) \ \...


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The computational requirements for various non-parametric PSD estimation methods is discussed in [*]. The complexity depends on the length $N$ of your data record. Roughly speaking: The most straightforward periodogram method which calculates the magnitude squared of the DFT of $N$ samples has a complexity of $N \log N$. Bartlett and Welch methods use $M$ ...


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If your nonlinearity can be expressed as a polynomial (i.e., in terms of addition and multiplication), you can make use of: The linearity of the Fourier transform, i.e., if $f$ and $g$ are (benign) functions, $a$ and $b$ are numbers and $ℱ$ denotes the Fourier transform, then: $$ℱ(a·f+b·g) = a·ℱ(f) + b·ℱ(g)$$ The convolution theorem, which states that ...


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TLDR; The mistake is here: If I calculate the expected value, which is according to definition an average, this way $E\left\{\widehat{P_x}(e^{jω})\right\} = \frac{1}{2π}\int_{<2π>} > \widehat{P_x}(e^{jω})dω$ The mistake is in your interpretation of expectation. It is not an average over all frequencies. For each fixed $\omega$, the ...


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You seem to understand the issues at hand. In your situation, if the trial runs would be identical, except for the noise, averaging the signals (or the complex values) will definitely work, and I recommend that it how you do it. The notion of averaging magnitudes (or magnitudes squared) across DFTs comes from when you are chunking a signal into multiple ...


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The simplest way to approximate an AR-2 process in Matlab / Octave is the following: N = 1024; % number of process samples. a = [1, -0.9, 0.2]; % denominator coefficients, p = 2. b = [1.0]; % numerator coefficient. x = filter(b,a, randn(1,N)); % generate N sample of AR-2 x[n]. Note: an AR process requires a ...


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First of, if you can read German: I was author on a thing, back in the day. There is a step in which the signal vector is multiplied by its hermitian transpose and the correlation matrix of the product is computed. What do both of these steps accomplish from a high level? So, first of all, MUSIC can be used for a couple of things. Originally, it was used ...


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Work in progress: wait till I am done before reading (and throwing brickbats!) This question is difficult to answer without getting into a lot of details about basic signal analysis and Fourier transform theory. Because of the way my brain works, I will discuss only real-valued continuous-time deterministic signals, and will get into the stochastic and ...


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Your FFT method of finding the phase of a single tone is only valid if your 's' signal contains an exact integer number of cycles. I.E., no spectral leakage. And that is NOT the case for your 's' signal.


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For simplicity I will show an approach Id' use on 1D signal (A row of real world image). You will be able to extend it and I will add few remarks on how you can even gain from having 2D data. The general idea is as sketched in Estimate the Discrete Fourier Series of a Signal with Missing Samples. The trick here is to exploit prior information. In our case ...


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The standard Periodogram uses all the data and computes its power spectrum spectrum estimatation at once; therefore it provides more spectral resolution but less estimation reliability (larger estimation variance). The Welch's modification to the standard Periodogram is the concept of dividing the signal into shorter blocks and averaging the computed per ...


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I don't think that FM has anything to do with this, unless you are using the term "spectrum inversion" in an unusual context. The original signal is presumably a real signal centered at 1200 MHz with a bandwidth of 100 MHz. If you sketch its spectrum conceptually, it will have some content from 1150 to 1250 MHz, and a conjugate-symmetric version of the ...


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One way of doing power spectrum estimation of random processes actually requires finding a set of ARMA model parameters that best approximates the given observed random process $x[n]$. In your case, you selected an all-pole signal model with $p$ poles as the roots of the denominator polynomial $A_p(z)$ with coefficients $a_p[k]$, and a $b(0)$ related with ...


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As you have taken 1024 point FFT, multiply the x-axis with a factor of 2*pi/1024. For now your peak is at sample number 200. 200*2*pi/1024 approximately equal to 0.4*pi


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I think I know the answer now. If I made an error, please correct me. Assume there are $m$ sensors and $d$ sources. As the eigenvectors in $S$ and $G$ of the Hermitian matrix $R$ is selected to be orthogonal and normalized, $$UU^*=\begin{bmatrix}S&G\end{bmatrix}\begin{bmatrix}S^*\\G^*\end{bmatrix}=SS^*+GG^*=UU^{-1}=I_m;$$ $$S^*S=I_d;$$ $$\mathrm{span}\{...


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I've found this: Implementing Welch's method for Power Spectral Density That seems to answer the question. However, the author encountered a problem, which I believe may be addressed by modifying the code that was identified as problematic in the thread by: dataA(istart:iend,j) = data1(istart:iend,j).*window; dataB(istart:iend,j) = data2(istart:iend,j)....


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Apart from the stochastic literature, there are a set of deterministic algorithms that claim exact(?) recovery of a signal form its partial Fourier description, collectively referred to as: signal reconstruction from DTFT phase or magnitude alone They are generally iterative in nature and exact convergence is hard to achieve in practice. Monson Hayes [et ...


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Since the data records $x_i$ are uncorrelated realizations of the same random process, $\hat P^{per}_{(i)}(e^{j\omega})$ are all uncorrelated random variables with identical means and variances (given by the mean and variance of the Bartlett Method's PSD estimator). So, for any fixed $\omega$, $\mathbf E[\hat P^{per}_{(i)}(e^{j\omega})]$ are equal $\forall i$...


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Noise analysis in Spice (Berkeley Spice) is done by summing up the power spectral density from every noise source in the circuit. There are a couple caveats. The circuit is assumed to be linear. In other words the circuit is first solved for a specific DC operating point then each of the components' equivalent linear noise sources are modeled as thermal ...


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A first thing to try is to reduce the unwanted variables. What happens if you set $\ell=s=0$? $$ U_M(r,0,0)= \left\{\begin{matrix} 0 & r<-M\\ 1-\frac{r}{M}& -M\leq r \leq 0 \\ 1-\frac{0}{M}& 0 < r \leq 0\\ 1-\frac{r}{M}& 0<r \leq M\\ 0 & r>M \end{matrix}\right.$$ Luckily, this definition seems to fit with $V_M$, since ...


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If the data is cyclic by its nature the best thing would work using its spectrum. You can easily build a system which checks sub set of data to verify periodic and the once you establish your groups checking the affinity of new data is easy - add it to each series. It should belong to the one creates less spread in the frequency (Namely, it follows the ...


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It looks like you are trying to recreate the functionality in spectrogram. Spectrogram will give you a short-time Fourier transform using whatever blocksize you specify. Much of your code looks like it is re-inventing this.


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Is there a method to identify location of non-zero Fourier coefficients of a signal (just locations, not values) with minimum computational cost (less that computations of FFT)? So, first of all, computation of FFT: Remember, the $10N$-point FFT, which you need for the resolution you demand: Suppose my signal is a vector of length $N$ associated with ...


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