Questions tagged [autocorrelation]

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

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Why is there only one integration in the solution if there is two integral in the formula?

In this problem the random variable is theta and according to the formula there should be two integrations but in the solution there is only one . Nor am i able to understand the meaning of x1 and x2 ...
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Understanding Pitch Detection with Autocorrelation Methods

I've been reading through A Smarter Way to Find Pitch, describing its pitch detection algorithm using autocorrelation. I've been having trouble understanding the accuracy claims. It says: MPM runs ...
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Why autocorrelation can be more efficiently calculated using the fft

Can anyone explain why autocorrelation can be more efficiently calculated using the fft ?
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444 views

Noise and AutoCorrelation Function

I would like to find a way to identify whether some components of a signal I have are "noise components" or not. Given the fact that the (normalized) autocorrelation function of white noise is a ...
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ACF expression using FFT method

It is known that the ACF of a 2D digital image can be obtained by computing the inverse Fourier transform ...
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47 views

Relation between $X(f)$ and $S_x(f)$

We know that for a signal $x(t)$, it is related to $R_x(\tau)$ as, $$R_x(\tau) = \int_{-\infty}^{\infty}x(t)x(t-\tau)dt$$. $\\$We also know that $$R_x(\tau) \rightleftharpoons S_x(f)$$ $\\$How do we ...
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142 views

Correlation of independent random processes

Suppose $X(t)$ and $Y(t)$ be two independent random processes. Is $E(X(t_1)Y(t_2))$ necessarily zero?
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what is equal to the FT of the Autocorrelation function

I am trying to make my own demonstration in order to find what is equal to the FT of the Autocorrelation function. The autocorrelation function in some book about turbulence is defined as: r($\tau$)=...
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28 views

Is the expectation of a random process $X(t)$ with zero DC component necessarily zero?

Is the expectation of a random process $X(t)$ with zero DC component necessarily zero? Or can it be non-zero depending upon the process?
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the demonstration that states that the FT of the ACF function is the square of the DTFT of the signal

I am following the book The Intuitive Guide to Fourier Analysis & Spectral Estimation with MATLAB. I am trying to selflearn the fourier analysis in matlab. I got lost in one passage in the ...
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91 views

Mathematical description of the ACF using fft2

I computed the acf of an image with the following code: ...
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1answer
263 views

Why is the Fourier (or cosine) transform decorrelating?

The discrete Fourier transform (DFT) and the discrete cosine transform (DCT) both decompose a signal into its frequency-domain spectrum. One property that I have seen praised across various domains ...
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93 views

What kind of power spectrum implies uncorrelation in time?

I am new in signal processing field, and actually I just need some basic tools for what I am working on. I've learnt that the autocorrelation function is the inverse Fourier transform of the signal's ...
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297 views

Autocorrelation of signal with offset

If I have a stationary signal $x(t)$ with zero mean and with an auto correlation $r_{xx}(\tau)$, then what is the auto correlation of $y(t)=x(t)+b$ ? My calculations so far: $$\begin{align} r_{yy}(\...
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111 views

How to find the fundamental frequency of a discrete signal using partial autocorrelation?

I hope you can help me with this question. I am trying to calculate the fundamental frequency (to know the beats per minute) of a cardiac pulse signal using partial autocorrelation. I use a 12-bit ...
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842 views

Autoconvolution vs Autocorrelation

I have a serious doubt about Autocorrelation and AutoConvolution. My understanding is, Autocorrelation Autocorrelation is ...
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1answer
893 views

Autocorrelation of sinc function

I'm having trouble on computing the autocorrelation of the sinc function I want to compute $$R_{hh}(\tau)=\int_{-\infty}^{\infty}\operatorname{sinc}(t) \ \operatorname{sinc}(t-\tau) \ \mathrm{d}t$$ ...
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Equivalence of the Power Spectral Density definitions

I am trying to show the equivalence of the following Power Spectral Density definitions in Matlab: Definition 1: $$ P(\omega) = \sum_{k=-\infty}^{\infty} r(k)e^{-j\omega k} $$ Definition 2: $$ P(\...
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How to add noise with given autocorrelation to signal in Matlab?

My question is a mix of signal processing and Matlab coding. I have an FIR filter with added noise $w_n$ $$x_n=\sum_{m=0}^{N_h-1}h_mg_{n-m}+w_n.$$ Now the noise $w_n$ has the autocorrelation $$E[...
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Zero-padding vs. nonzero-padding in computation of auto-correlation with FFT

Isn't the usual zero-padding in the computation of the auto-correlation function with FFT just one of many possible extrapolations of the original signal? If I have a measured signal which has good ...
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185 views

Finding period of a square wave with varying sampling frequency

I have a square wave (0-1.8V) with a varying sampling frequency (from a circuit simulator). It is also not a perfect square wave (the high and low signal could be very close to but not precisely zero ...
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Is there an algorithm for computing auto-correlation in a frame-based manner?

I am trying to find out if an algorithm similar to the Overlap-And-Add method that enables frame-based computation of convolution and cross-correlation exists. When I say frame-based computation I am ...
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Pitch detection, YIN, pYIN

So here is the "seminal" YIN paper: Cheveigne A, Kawahara H. - YIN, a fundamental frequency estimator for speech and music and the new, improved probabilistic YIN: Mauch M, Dixon S. - PYIN: A ...
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Interpreting Auto correlation results

I used xcorr function in Matlab to compute the autocorrelation of my signal. I obtained a graph, which is shown below. How to interpret this? I read that autocorrelation helps in finding the patterns ...
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Relationship between the autocorrelations of X(t) and X(nt)

Defining: $X(t)$ WSS random process with autocorrelation function $R_{X}(\tau) = \mathbb{E}[X(t)X(t+\tau)]$. $Y[n] = X(nT)$ (sampling of $X$ at a rate $\frac1T$) with autocorrelation function $R_Y(\...
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48 views

Are there any signals with brickwall autocorrelation?

Are there any signals whose autocorrelation $R(\tau)$ has the following form? Assuming $\tau_c > 0$ and $R_0 > 0$ a constant, $$R(\tau) = \begin{cases}R_0, \text{ for $|\tau| < \tau_c$} \\ 0,...
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How to find a variance of sample sequence

I have a sequence such as $$r[n] = y[n]v[n]$$ $y[n]$ and $v[n]$ are zero-mean and statistically independent. I need to find a variance of $r[n]$ and show that it is white and equal to $\sigma ^2_y\...
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About using Walsh matrix as spreading sequence

I'm asking about some details about using the Walsh matrix as spreading sequence code. For example, suppose I'm using a $4\times 4$ Walsh matrix given by $$H = \begin{pmatrix} 1 & 1 & 1 & ...
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46 views

Particular Correlation formula

I'm reading a book where the autocorrelation of white noise is expressed as: What is the term $Q(k)$ and why is is expressed as an average value of a dot product ?
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59 views

Difference between autocorrelations

M. H. Hayes calls in his book "Statistical digital signal processing" autocorrelation sequences $r[k]$. For optimal filters the desired autocorrelation has the index d -> $r_d[k]$. However, often ...
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95 views

Correlation between 2 signals of uneven dimensions

As a part of my work, I am trying to correlate the audio signal in a video with the pixels of each frame. The steps I follow are: Audio sampling rate and frame rate of the video are known. So, ...
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78 views

Simulate time series given temporal auto-correlation functions

Given a random process $x[n] \in \mathbb{R}$ (say of length $N$) and all correlation functions such as: \begin{align} \langle x[i]\rangle\\ \langle x[i]x[j]\rangle\\ \langle x[i]x[j]x[k]\rangle\\ \...
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252 views

Confusion about ensembles and averages in autocorrelation matrices

I just noticed that until now I often don't cared about the scaling of the autocorrelation Matrix in Matlab. I then checked with the book Statistical Digital Signal Processing from M. H. Hayes. There ...
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Autocorrelation

What is autocorrelation in real life? Any example? How does autocorrelation dependent on time difference? Here is Xt1 and Xt2 how comes \phi{tau}?
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2k views

Autocorrelation: numpy versus FFT

I have a series a of values (0 and 1) coming from a Brownian process with drift for which I am studying the autocorrelation. I used two methods: 1) numpy autocorrelation: ...
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230 views

Autocorrelation sequence in terms of Fourier transform of the underlying signal

Let $x(n)$ be a sequence of length $N$, which is zero outside the interval $(0,N-1)$. Let $X(k), k=0,1,\cdots,N-1$ be the FFT coefficients of $x(n)$, that is, $X(k)=\sum_{n=0}^{N-1}x(n) \exp\left( -\...
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Co-prediction of signals

Is there any plausible method to estimate or predict one signal, on the basis of known value of another signal, provided the two signals bear a strong correlation? For instance, the audio amplitude ...
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1answer
101 views

Can spectral density be a complex quantity?

I have a signal ($S(t)$) which is product of a Gaussian ($G(t)$) and a random phase function ($e^{i\theta(t)}$, here $\theta(t)$ is a random function), as shown below $S(t)=G(t).e^{i\theta(t)}$ If I ...
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How to deduce the shape of autocorrelation graphically?

I'm asking for your help in understanding if I have a rectangular function, so the autocorrelation for it will be triangular... and I can prove that graphically but this is time-consuming especially ...
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$E(X(t)X(t))=\sigma^2\delta(\tau)$ or $E(X(t)X(t))=\sigma^2$

(Question already asked on Math StackExchange) Let's say we have a white noise process $x(t)$ such that: $E(X(t)X(t+\tau))=N\delta(\tau)$ $E(X(t))=0$ In particular, with $\tau=0$, $E(X(t)X(t))=E(X^...
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Practical insightful time domain functions [closed]

This question might be a little bit different, but i am having hard time to find some sort of a list/book/website/advice, with very practical time domain functions that provides useful insgihts for a ...
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1answer
65 views

What does a star shaped autocorrelogram mean?

Using the data in the upper graph, I get the weird autocorrelogram of the lower graph:- I've never seen a autocorrelogram of this shape. They're usually flat along the x axis. Testing the code with ...
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How can faux correlated data be generated for testing or training?

There are one or two questions here that ask how to assess data correlation. The data tends to be empirical. Is there some tool or standardised technique for generating correlated data from scratch? ...
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Interpolation of missing audio signal in a video sequence

Suppose there is a video sequence and there are some frames for which the audio data is missing. I want to interpolate the missing audio data on the basis of the correlation of the audio signal with ...
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1answer
283 views

How to determine fundamental frequencies (beats/minute) of heartbeat? (matlab)

read = matfile('ECG.mat'); [cor, lags] = xcorr(read.ecg, read.ecg); read.Fs; So i have read.ecg (signal) and ...
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Why is the sound field intensity due to $K$ point sources given by $ I(p,\omega) = \sum_{k=1}^K \sigma_k^2(\omega) \delta(p - p_k)$?

I am trying to understand the following piece of text. I am not used to dealing with sound intensity and power so I'm not familiar with the derivation of the formula $(*)$ below. Statement: 1. We ...
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275 views

Problem understanding the Expectation Operator

I know that the Expectation Operator $E\{x\}$ four discrete values is $$ \sum_k \alpha Pr(x = \alpha_k)$$ and its very intuitive when speaking out a formula which contains the Expectation Operator. ...
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494 views

How to compute autocorrelation of signal defined by difference equations?

I have no experience with difference equations and I want to learn how to compute the following, but I found no resource online. Any help would be greatly appreciated. Find: $$\mathbb{E}\left[d[n]d[...
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1answer
186 views

Correlation of a signal

I have one sample for a signal. This sample is a vector of length 384. I need to calculate the correlation matrix for this signal,So I need many samples for the same signal. How can i generate these ...
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How to visualize the autocorrelation matrix and it's properties

Having a hard time wrapping my head around autocorrelation matrix as it applies to a spectral estimation problem like MUSIC or ESPRIT. If the signal vector contains a summation of sinusoids in noise, ...

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