Questions tagged [autocorrelation]

Autocorrelation is the cross-correlation of a signal with itself.

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doubt on demostration autocorrelation theorem

I’m stuck on this demonstration. It is written " If $x'$ is held constant, $dx=dy$ ". But how $x’$ can be constant? shouldn't $x'$ be a variable?
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Extremes of the integral of the Fourier transform of $\rho(\tau)$

The definition of correlation function of a velocity component, in turbulence (and, more in general, in all the signal), is on a limited domain $\rho(\tau)=\frac {1}{T}\int_{t_{0}}^{t_ {0}+T}v(t)^{.}...
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Power spectrum is phase blind

I came accros the phrase " It is well known that the autocorrelation and power spectrum are phase blind". Is there an analytical way to prove this, so I can understand why this is correct.
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Isolate g(t) starting from the definition of power spectrum

Starting from the following definition of power spectrum of a signal $g(t)$ in time: $$S(f)=\int_{-\infty}^{+\infty}\rho(\tau)e^{-i2\pi f \tau} d\tau$$ where $$\rho(\tau)=\int_{-\infty}^{+\infty}g(t)^{...
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47 views

Expected Value of a sequence with two random variables

If I have a signal of the form $x\left(n\right)=Acos\left(nω+ϕ\right)$ where $\omega \in \left[\omega -\lambda ,\omega \:+\lambda \:\right]$ is a uniform random variable and $\phi $ is also a uniform ...
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Calculate ACF in C++?

I would like to manually reproduce the method that authors of an article used in their research (DOI: 10.1038/s41598-017-02750-9 (Page 8. top)). It is mentioned as "ACF", so I wrote ...
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27 views

Signal “seasonality” detection

I am wanting to do a seasonality detection on a signal, but am struggling big time to find a solution. I have tried several different things, including statsmodels STL/seasonal_decompose. It seems ...
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1answer
41 views

autocorrelation of multiple signals

Problem: I am looking at an adaptive filtering application where the eigenvaluespread of the autocorrelation matrix $R$ is important for the convergence of the algorithm. For a single channel system ...
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63 views

Ringing/Oscillation in the reconstructed Periodogram

I have an original periodogram that I need to model with autoregressive process. However the model isn't right as it is not fitted well to the original periodogram. I am suspecting I am doing ...
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72 views

Processes/Transforms involved to get brainwave data from raw EEG? (Autocorrelation confusion)

Not clear on what the autocorrelation function of raw EEG means physically why can't you take the FT of a the EEG itself and get frequency data? With BCI & basic electrode setups you can ...
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87 views

Find the autocorrelation function of signal $x(t) = u(t) - u(t-1)$

I have used the energy-type signal autocorrelation function: $$\mathcal{R}_{xx}(\tau)=\int_{-\infty}^{\infty}x(t)x^*(t+\tau)dt$$ I have rewritten the equation as: $$\begin{align} \int_{-\infty}^{\...
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How to find the output mean and autocorrelation of a non-linear system

I need help with this question. I am sure this is the right StackExchange forum for this type of question. Consider a nonlinear device such that the output is $Y(t) = aX^2(t)$, where the input X(t) ...
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Cross-correlation estimation in MATLAB using a method similar to pwelch for the PSD

Is there a function in MATLAB to estimate the cross correlation function of two wide-sense stationary signals (or the autocorrelation of the signal with itself) in a way similar to the estimation of ...
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27 views

Generating data with given Auto and Cross Correlations

I have two discrete vectors $\mathbf{x}_1$ and $\mathbf{x}_2$, and I'm trying to generate more data $\mathbf{f}_1$ and $\mathbf{f}_2$ that has some basics properties of $\mathbf{x}_1$ and $\mathbf{x}...
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What are some really accurate ways to get the value of a peak (local maximum) given some points around it? (To be used for autocorrelation peaks.)

I have looked everywhere on the internet for this and, surprisingly, haven't found much useful information. Given 3 or more points closest to a peak (local maximum) what are some of the most accurate ...
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196 views

Autocorrelation function of a triangular wave

I am interested to find the analytical expression for the autocorrelation function of a signal that comprises triangular pulses. I have followed the derivation in "Statistical Theory of ...
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78 views

Calculate periodicity of audio signal

I have to calculate the periodicity of a audio signal like this: You can see by eye, that the volume rises the highest every second "blob". The spectogram (that is visualized ...
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Auto-correlation of absolute squared stochastic process

Consider the stochastic process $a(t) \in \mathbb{C}$. Its autocorrelation function is given as $$ \phi_{aa}(\tau)=\left(a(t)\star a(t)\right)(\tau)=\int_{-\infty}^{\infty}a^*(t)\cdot a(t+\tau) \...
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71 views

Prove that a specific auto-correlation function is even

Background (from digital signal processing): The $\delta$ function is defined as below: $$\delta[n]=\begin{cases} 1 \quad \mbox{if} \hspace{.4em} n=0\\ 0 \quad \mbox{o.w.} \end{cases}$$ In an LTI (...
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How to compute autocorrelation with exponential IIR filter efficiently

I have a discrete dataset (called V). I want to compute the autocorrelation of V at multiple time horizons. Normally I know that I can use the IFFT of the PSD. But in my case I need it with ...
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How to apply the window function when finding autocorrelation?

I want to get frequencies from amplitudes and I'm following Chap 5.1.1 from Speech Signal Processing by Praat As the in pictures, I will multiply the signal $s(t)$ with the Hanning window $w(t)$, ...
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PSD of linearly modulated signal using autocorrelation?

Consider a signal $v(t)$ given by $$v(t)=\sum_{n=-\infty}^{\infty} b[n]p(t-nT).$$ Assume that $b[n]$ is uncorrelated with zero mean, i.e. $\mathbb{E}[b[n]b^*[m]]=\mathbb{E}[|b[n]|^2]\delta[n-m]$ and $\...
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Calculation of eigendecomposition of a signal in its Fourier domain?

I want to find the eigendecomposition of a 3-dimensional discretely sampled signal $X$, where each sample $X_{i,j,k}$ is treated as a vector $\langle i, j, k\rangle$ (with the origin at the middle of ...
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253 views

Autocorrelation for periodic signals

Autocorrelation for power signals is defined by $$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tx(t)x^*(t-\tau)dt\tag{1}$$ Is it true that for periodic signals $(1)$ can be computed by $$R_x(\tau)=...
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60 views

How is time used in this autocorrelation expression?

There are two papers I'm working through that explain applications of autocorrelation. The first - by Monti - is quite clear to me: It runs a sum over the time axis. The second one by Brown makes no ...
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finding power spectral density from a vector

I have been given a vector: \begin{equation} v= \:\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix} \end{equation} my job is to find the power spectral density from this vector \...
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Estimating Signal-To-Noise ratio from discrete time series

A hidden variable $x(t)$ (in my case, neuronal signal averaged over a brain area) exhibits dynamics on multiple timescales from milliseconds to seconds. An observable $y(t)$ is produced by convolving $...
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Computing Autocorrelation via FFT

I have seen several methods to calculate Autocorrelations using FFTs, and am confused about why they differ. Zero-Pad it to double its original length.Take the FFT. Then replace all the coefficients ...
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156 views

Proving the upper bound of cross correlation

I am reading about cross-correlation from this document and equation (5) states that The maximum value of the crosscorrelation is not always when the shift equals zero; however, we can prove the ...
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Self-correlation vs Digital frequency: figure from paper

In the paper, is discussed a MIMO-OFDM system over the frequency self-correlation of i.i.d. Rayleigh channels and subcarrier grouping. in this paper is shown the following figure. it shows the self-...
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60 views

What's the relation between frequency band of $X(j\omega)$ and $\Phi_{xx}(j\omega)$?

in which: $x_{c}(t)$ is a continuous-time signal $X(j\Omega)$ is the Fourier Transform of $x_{c}(t)$ $\Phi_{xx}(j\Omega)$ is the Power Spectrum Density of $x_{c}(t)$ which defined as Fourier ...
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65 views

auto-correlation of differentiator

How to do this entirely in the time-domain, without using frequency domain? Let's say I have a continuous-time system that is a differentiator, and X(t) is WSS: $$x(t) \rightarrow \boxed{\frac{d}{dt}} ...
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325 views

Power Spectral Density from Probability Density Function

The samples of a signal $x[n]$ are i.i.d. and follow a triangular pdf with $a = 0,\ b = 2,\ c = 1$: The DC-power of the signal is $$\mu_x^2 = \big(\mathbb{E}(X)\big)^2 = \left(\int_{-\infty}^{\infty} ...
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correlation matrix vs. correlation function?

Can someone help me understand the difference between a 1-dim autocorrelation function and a 2-dim autocorrelation matrix of a random process aka time series? My Leon-Garcia textbook defines CX(τ) and ...
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Are ACF plots sufficient to detect white noise assuming probable existence of non linear associations?

ACF plots use autocorrelations tests to determine associations in time series. Autocorrelations, however, only measure linear associations. To my understanding, certain non-linear associations can ...
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aperiodic auto correlation properties

I'm reading about PSC (primary synchronization code) in UMTS. It is a so-called Golay sequence. It is said it has a good aperiodic auto correlation properties. What does it mean (in terms of aperiodic)...
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56 views

Welch's method, no windowing function

I'm trying to pick a method to increase the bin-width on several discrete Fourier transforms I'm incorporating into some scientific analysis code. I have several questions about how this could work ...
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51 views

Autocorrelation not as expected

I am trying to find the autocorrelation function on the intensity of a pixel over time stored in an array "Intensity". Usually one would expect the highest point to be at t = 0 and then decay. I am ...
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Discrete-time sampling of filtered white noise

I am trying to understand how I can relate a discrete-time random process to a continuous-time random process sampled at discrete times. Suppose I have a noise source $N_\tau(t)$ which is derived ...
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Error in measuring wavelength/frequency using autocorrelation

I am currently using these two fonctions to compute to autocorrelation of a signal in order to measure the wavelength: ...
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594 views

Autocorrelation function and correlation integral

I am confused by the definition of autocorrelation function. It is originally defined as the expected value $$R_{XX}(\tau) = E[(X(t)X(t+\tau)] = \langle X(t)X(t+\tau)\rangle\tag{1}$$ where $\langle\...
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62 views

Evaluating discrete spectral density at only a few frequencies

I'm trying to obtain the spectral density at three particular frequencies for a computational chemistry problem that I'm working on (if you are curious, it has to do with the estimation of Nuclear ...
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54 views

argmax of the cross correlation

I want to find the shift between 2 signals $\sin a$ and $\sin b$ where (with Matlab code) $$ a = \{0, 1,2,3,4\} $$ $$ b = a +1 $$ $$ \implies b = \{1,2,3,4,5\} $$ then $$ \arg\max \sin a = 2 $$ and ...
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Calculation of channel autocorrelation for LMMSE

I am currently trying to implement an LMMSE estimator/equalizer for a python based ofdm receiver. The actual reception is done with a SDR and iq-samples are stored in a file for the "ofdm-receiver" ...
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101 views

Blackman-Tukey Autopower equation

I am looking over Blackman-Tukey method for Autopower. It uses the DFT after a window is applied on a autocorrelation. $$ Power Spectrum = \frac1{2\pi} \ \sum_{(k=-(N-1))}^{N-1} w[k] R[k] e^{-i\...
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What does $x^*$ mean in autocorrelation

Suppose $x[n]$ is a sequence of length $N$. Following formual defines circular autocorrelation sequence of $x[n]$: $$r_{xx}[m] = \sum_0^{N-1}x[n]x^*[n-m \mod N]$$ What does $x^*$ mean in this equation?...
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118 views

Calculate autocorrelation of a sinus

I am a beginer in signal processing. I am currently studying a signal $s(t) = \cos(h(t))$, and I have to calculate its autocorellation. Since $h(t)$ is periodic of period $T_s$, I calculated the value ...
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2answers
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Contradictory results from power spectral densities and autocorrelation in shot noise

I have a white noise generator circuit which involves a Zener diode meant to undergo breakdown across a resistor $R1$, and am trying to compare power spectral densities and autocorrelation of the ...
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151 views

Does a signal and its autocorrelation ever have the same Fourier spectra?

My understanding is that Fourier power spectra give the frequencies and corresponding intensities/amplitudes contained within a signal. In certain techniques, such as molecular dynamics, it is common ...
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145 views

Questions about autocorrelation, correlation, and randomness

I'm learning about autocorrelation functions and trying to calculate some from a time series. Conceptually, what is the difference between the following two autocorrelation functions? $$g_{1}(\tau)=\...

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