What is Autocorrelation: Definition and 63 Discussions

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.
Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance.
Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.

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  1. Salmone

    How is an autocorrelation function computed? (Dynamic Light Scattering)

    In an experiment of Dynamic Light Scattering, how is an autocorrelation like the one in the image computed? Mathematically a correlation function can be written as ##G(\tau)=\langle I(t)I(t+\tau) \rangle##, in an experiment like the one I mentioned the scattered intensity light is collected by...
  2. yklee

    Ising model autocorrelation function calculation

    Dear Mr. and Ms., I am trying to measure the autocorrelation functions of 2D ising model based on the equation given by where A(t) denote a measure. I calculate a c(t) of magnetization. I calculated in this way data_path = f"../../trajectory/data.txt" data = np.loadtxt(data_path)...
  3. I

    Interferometric autocorrelation of sech^2 pulse?

    I know that from intensity autocorrelation, I simply need to divide the FWHM by 1.53 for sech2 pulses. But I can't seem to be able to find any reference on how to get pulse width from interferometric autocorrelation signal. Can someone help me?
  4. T

    Understanding Waveform & Autocorrelation in Pulse Lasers

    So, I was working on autocorrelation for my pulse laser system and I started to wonder what is the difference between single wave with 400nm of wavelength and two 800nm waves overlapped. In the knowledge, I know of, is that wavelength is the length between two picks. And when it is pulse laser...
  5. mertcan

    I Derivation of the Variance of Autocorrelation

    Hi everyone in this link (https://stats.stackexchange.com/questions/226334/ljung-box-finite-sample-adjustments) I see the variance of autocorrelation related to specific lag is demonstrated in the following: $$ Var(r_k) = \frac {\sum_{t=k+1}^n a_t*a_{t-k}} {\sum_{t=1}^n a_t^2}$$ where ##r_k## is...
  6. iVenky

    Power of noise after passing through a system h(t)

    **Reposting this again, as I was asked to post this on a homework forum** 1. Homework Statement Hi, I am trying to solve this math equation (that I found on a paper) on finding the variance of a noise after passing through an LTI system whose impulse response is h(t) X(t) is the input noise...
  7. SchroedingersLion

    Equivalence of msd and velocity autocorrelation

    Greetings, I am searching for an explanation that connects the mean squared displacement in an ensemble of particles to the velocity autocorrelation functions. I know that they are connected since both of them can be used to calculate the diffusion constant. But I do not find a mathematical...
  8. CivilSigma

    Autocorrelation and Spectral Density

    Homework Statement For a constant power signal x(t) = c, determine the auto correlation function and the spectral density function. Homework Equations The auto correlation function is: $$R_x (\tau) = \int_{-\infty}^{\infty} E(x(t) \cdot x(t+\tau)) d\tau$$ To my understanding, here to find...
  9. CivilSigma

    Auto-correlation Integral

    Homework Statement I am computing the auto correlation and spectral density functions of the following signal: $$f(t)=Ae^{-ct}sin(\omega t)$$ $$AutoCorrelation = R_x(\tau) = \int_{-\infty}^{\infty} f(x)f(x+\tau) \cdot \frac{1}{T} dx$$ $$SpectralDensity = S_x(\omega) = \frac{1}{2\pi}...
  10. tworitdash

    Autocorrelation of a Bernoulli Coin Flipping Experiment

    I am confused at one point. The coin flipping Bernoulli Process has a probability of p of getting HEADS and a probability of 1-p of getting TAILS. Let's define a random variable x[n], which takes the value +1 when it is a HEADS, and -1 when it is a TAILS. The mean or estimation of x[n] becomes...
  11. P

    I Autocorrelation Error Interpretation

    Hello, I have a question on Autocorrelation of OLS model. So I encountered a autocorrelation error, and use Cochrane - Orcutt Two-step Procedure(CORC) to fix it. And my OLS became GLS. And I have no idea how to interpret the coefficient on my initial X and Y with the new model. I mean something...
  12. M

    MATLAB Understanding Autocorrelation in Matlab: Exploring Functions for Different Lags

    Hello everyone. Iam a little bit confused about the autocorrelation and using Matlab so I hope someone can help me out. As far as understand Matlab computes the sampled autocorrelation where the lag between the samples is given by x(n)* x(n-l). Buy what if the autocorrelation depends on the...
  13. baby_1

    Normalized Autocorrelation of two signals

    Homework Statement I obtain Autocorrelation function of two signal: as Homework Equations [/B]The Attempt at a Solution [/B] I want to know what is Normalized Autocorrelation of above equation? Thanks
  14. baby_1

    Autocorrelation function of the output of the nonlinear device

    Hello for an input signal with a noise we have and for obtain Power spectural density we use autocorrelation function where hkm is but I need to know what is autocorrelation function for different inputs with different frequencies? such as Any help will appericate
  15. F

    Estimating Atmospheric Params w/ Autocorrelation & Spectral Density Funcs.

    Homework Statement An experiment probes a coherent process and the receiver is coherent. The autocorrelation function and the spectral density function are estimated and transformaed into atmospheric parameters. See Table 1 below 1) Calculate the maximum velocity (in m/s) that may be...
  16. J

    Autocorrelation function of a Wiener process & Poisson process

    Homework Statement 3. The Attempt at a Solution [/B] ***************************************** Can anyone possibly explain step 3 and 4 in this solution?
  17. A

    What is more efficient, autocorrelation or SSA?

    What is more efficient in extracting the pattern in a time series analysis, autocorrelation or singular spectrum analysis?
  18. P

    Autocorrelation function MC simulation

    Dear Members, I've got a problem with following problem. I have build my first algorithm for Metropolis Monte Carlo. Now I am in a testing phase and one of my bosses ask me for autocorrelation function of my system. I find something in literature, but I am not sure exactly what the...
  19. G

    Autocorrelation Function of a Signal

    Homework Statement I'm posting this question here at this point. I am having difficulty understanding autocorrelation in terms of solving the problem below. I don't seem to understand the math behind this. A white noise process W(t) with unity (N_0/2 = 1) power spectral density is input to a...
  20. G

    Digital Communication Systems - Autocorrelation

    Homework Statement A white noise process W(t) with unity (N_0/2 = 1) power spectral density is input to a linear system. The output of the linear system is X(t), where X(t) = W(t) - W(t - 1) Determine the autocorrelation of X(t) and sketch it. Homework Equations Let τ denote a time...
  21. U

    MHB OLS standard error that corrects for autocorrelation but not heteroskedasticity

    Question: By mapping the OLS regression into the GMM framework, write the formula for the standard error of the OLS regression coefficients that corrects for autocorrelation but *not* heteroskedasticity. Furthermore, show that in this case, the conventional standard errors are OK if the $x$'s...
  22. N

    In matlab how do I write u(n) in order to get the autocorrelation?

    Homework Statement u(n) = .4s(n)+.7s(n) -.1s(n-2)+v(n) where v(n) is zero mean and has variance 0.003 and uncorrelated with s(n) I want the autocorrelation is E[u(n-k)u(n)] I can solve it easily by hand which is the follow so r(k) = 0.663 for k =0, 0.21 for k=1 , -0.04 for k=2 and the rest...
  23. K

    Nasty Autocorrelation Integral

    Hi guys, Long story short, I need to compute an autocorrelation integral. Here's the problem: There are two arbitrary gaussian pulses, one following the other by a fixed distance. By computing the autocorrelation over space(not time) and taking the derivative of the space-shift autocorrelation...
  24. ElijahRockers

    MATLAB - Autocorrelation Project

    I'm trying to design a program that can accurately detect the frequency of a real, periodic audio signal, such as several guitar notes in sequence. I have the skeleton of the program working. It does, at times, accurately detect the frequency. However, it is very noisy, and I'm not too sure how...
  25. stevendaryl

    Puzzle About Autocorrelation

    If f(x) is a square-integrable real-valued function on the reals, then we can define an auto-correlation function C(\theta) via: C(\theta) = \int dx f(x) f(x+\theta) I'm trying to get some insight on what C(\theta) is like, for small values of \theta in the case in which f(x) is highly peaked...
  26. iVenky

    Autocorrelation function from PDF?

    What is the exact prodecure for finding out the auto correlation function Rxx(τ) for a given pdf? Is it possible at all to find out the auto correlation function from the pdf? If not then what is given usually when you find out the auto correlation function Rxx(τ)? Thanks
  27. K

    Stationary vs. Autocorrelation.

    Dear All Salam I tried to make the derivation of the Auto correlation time difference dependency for the 2nd order stationary process, but i could not. Can anyone please help me. Regards, Salam
  28. J

    Physical meaning of autocorrelation

    Hi All, I was in a process of processing my vibration-test data. I now generated a plot of the autocorrelation function of the object acceleration. Please see the attachements (the second attachment is the close-up for small tau's). The x-axis in the plot is the time delay tau. You can see...
  29. A

    Autocorrelation of a wiener process

    Let \phi(t) be a Brownian Walk (Wiener Process), where \phi\in[0,2\pi). As such we work with the variable z(t)=e^{i\phi(t)}. I would like to calculate E(z(t)z(t+\tau)) This is equal to E(e^{i\phi(t)+i\phi(t+\tau)}) and I know that E(e^{i\phi(t)})=e^{-\frac{1}{2}\sigma^{2}(t)}, where the...
  30. W

    Finding the strongest beat in a sound wave using autocorrelation

    Hi everyone, I have a sound wave representing a piano piece played at a steady tempo, and would like to get a graph of the saliency of each beat (essentially, a probability distribution for how strong each possible tempo is). I understand that this is done by plotting the autocorrelation...
  31. V

    Calculating autocorrelation function

    Dear Sir/Madam, I am posting this question here because my field of research is biophysics and i am doing molecular dynamics (MD) simulation on bilayer system. Well, what I want is some explanations on calculating auto-correlation function using MD simulation data. I have 4000 frames out...
  32. D

    Autocorrelation function in Excel

    Autocorrelation function in Excel (SOLVED) Hi, I have tried reading through this forum looking for similar example but am unable to find one. Basically, I have a large set of data that I intend to do autocorrelation. =CORREL(A1:A10000,A2:A10001) =CORREL(A1:A10000,A3:A10002)...
  33. J

    Is the Fourier transf. of an autocorrelation functn always positive?

    I am trying to understand the IR spectra of liquid. I can get the autocorrelation function of atoms' velocity, <v_{i}(0)v_{i}(t)> make a Fourier Transformation, the vibrational density of state (VDOS) can be obtained. Does the VDOS always be positive? Or it can also take negative value...
  34. xaratustra

    Confused about autocorrelation and PSD

    I am confused a bit :confused:. I read in a paper that the following property holds, but can't find where it comes from. \mathsf{E}\left[X(\omega)X^*(\omega')\right]=2\pi\delta(\omega-\omega')S_{XX}(\omega) it says that the expected value of the Fourier transformed signal is proportional...
  35. R

    Autocorrelation Function of Markov Chain/Discrete Process

    Homework Statement I'm working with a transition matrix that is 1-p, p q, 1-q And I need to find E(X_n X_{n+\tau}) with E = {-1,1}. The first row is probabilities given currently in state -1. And this needs to be a function of tau. I started by initiializing my M.C. with the steady-state...
  36. L

    Determining the autocorrelation function

    Homework Statement Stress described as: S(t) = a0 + a1X(t) + a2X2(t) where X(t) is a the random displacement, a Gaussian random process and is stationary. Determine the autocorrelation function of S(t) (hint: remember a nice formula for the evaluation of high order moments of Gaussian random...
  37. O

    MHB Mean and Autocorrelation of a Deterministic Function

    Hello everyone! I have a couple of questions related to random processes: (1) Isn't the mean of a process $X(t)$ defined as $E[X(t)]$ which, for example, if $X(t)$ belongs of a countable and finite set, would defined as the some of the elements of this set weighted by their corresponding...
  38. R

    System Resonse's Autocorrelation Function (using integral)

    Homework Statement x(t) is wss. R_x(\tau) = e^{-a|\tau|} The process goes through a system with the impulse response h(t) = u(t)e^{-bt} What is the output's autocorrelation function?The Attempt at a Solution For starters, I already found a solution by finding the output power spectral...
  39. L

    Autocorrelation vs Hurst exponent. differences and similarities

    What is the relation between autocorrelation and Hurst exponent in time series analysis? which are the differences and which are the similarities? thanx
  40. A

    The differences between autocorrelation and convolution

    Hi, This is something that has appeared in a module, we've had a lab session in it but I am still not sure what it is. I don't understand the formulas given in lecture notes so I was hoping someone could explain it? Autocorrelation R1(τ) = ∫f(t)f(t+τ)dt = f V f Convolution C12(τ)=...
  41. mnb96

    Why E[x^2] = autocorrelation of x evaluated in 0 ?

    Hi, I was studying the derivation of the solution of Wiener filter from the http://en.wikipedia.org/wiki/Wiener_filter#Wiener_filter_problem_setup". There is a step I don't quite understand. First, we define the square error between the estimated signal \hat{s}(t) and the original true signal...
  42. K

    Time Series: Partial Autocorrelation Function (PACF)

    Consider a stationary AR(2) process: Xt - Xt-1 + 0.3Xt-2 = 6 + at where {at} is white noise with mean 0 and variance 1. Find the partial autocorrelation function (PACF). I searched a number of time series textbooks, but all of them only described how to find the PACF for an ARMA process...
  43. S

    Autocorrelation and pitch detection

    I have heard that autocorrelation is used in pitch detection, but no proper or convincing explanation of how it is used. I had previously worked on echo removal using autocorrelation. But what puzzles me is that autocorrelation is done in time domain but how can this be used to detect...
  44. M

    Calculating Autocorrelations for Large Data Sets in Excel: Tips and Alternatives

    I have a set of 9,999 equi-spaced data points, and I would like to calculate the autocorrelations for ALL lags up to 5,000. In Excel, I want to have this code: =CORREL(B1:B9998,B2:B9999) =CORREL(B1:B9997,B3:B9999) =CORREL(B1:B9996,B4:B9999) ... =CORREL(B1:B5000,B5000:B9999)...
  45. V

    Pulse duration from interferometric autocorrelation

    Hi, I have some interferometric autocorrelation traces of ~20fs pulses. Does anyone know how to convert to or calculate the actual pulse duration from the fringes of the trace? Thanks, Richard
  46. S

    Autocorrelation of intensity fluctuations

    Hi Guys, I am working on some fluorescence correlation spectroscopy and I have some data generated by the machine I use. The data has the correlation plot of G(tau) vs the delay time (tau) which is an exponential decay. The data also has the count rate given for the duration for the...
  47. A

    Autocorrelation - How to implement step by step

    Hola, my first time in this forum. Two microphones at diferent distances from an audio source are receiving the same steady signal (frequency/amplitude unchanged). A microprocessor takes succesive samples of both channels in batches of N samples. Time between samples is ts microseconds...
  48. M

    Why is the autocorrelation equal to the expected value?

    Homework Statement This isn't really a homework question as much as something that I just couldn't figure out. I just noticed in an exam I was working on that they at one point converted the expected value of a signal multipled by itself to the crosscorelation of the signal at l = 0, and a...
  49. N

    Autocorrelation of white noise

    I am really new to this and so I am trying to understand some basic stuff here. Autocorrelation of a univariate white noise is 0. Am I correct?
  50. F

    Autocorrelation Function Question

    Hi, we're learning about the autocorrelation function at uni, and I know it's meant to show similarities between a function and a delayed version of that function. But how does the autocorrelation show these similarities? For example, ifx(t)=Asinc(2Wt) then R_x(\tau...
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