Questions tagged [autocorrelation]

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

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Autocorrelation for Stationary Signals

I'm having trouble grasping the autocorrelation function for stationary signals, both strict stationary and WSS. First for strict sense, we have $$\forall(\tau,t_1, \ldots, t_n) \in \mathbb{R} \land ...
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46 views

What is the time reversal convolution

Given a signal $r(t)$ which is a result of convolution between signal $x(t)$ and a channel $h(t)$ as below : $r(t) = h(t)*x(t); $ what I know, the time reversal convolution can be process as ...
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Power spectral density of surface roughness

I am currently working on understanding how visually and intuitively the power spectral density is the fourier transform of the autocorrelation of a signal. I was wondering if anyone had a good ...
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33 views

estimation of ARMA(1,1) with MA parameter greater than 1

I am working with the following simple ARMA(1,1) model: $$ z_{t+1} = \phi z_{t} + \theta\varepsilon_{t} + \varepsilon_{t+1} $$ In my case $\theta$ depends on some other parameters, and, therefore, I ...
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Impulsive autocorrelation sequence

I am looking for a proof that a finite length sequence can only have an impulsive autocorrelation sequence if the sequence is a single impulse itself, ie.: assume $s[n]$ is nonzero for $n=0,1,2,3$ ...
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What is the autocorrelation of a Dirac pulse?

What is the autocorrelation of $x(t) = \delta(t)$? Can you explain to me how to calculate it?
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Effect of filtering on Energy of a Digital Signal

I am trying to detect (not remove) noise from the digital signal to get meaningful information out of it. The signal I need is in the frequency range of 0.1 Hz to 10 Hz. Anything beyond this is just ...
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19 views

How to compute the energy of a NON-STATIONARY (transient) random discrete-time signal

When computing the energy of a NON-STATIONARY (transient) random discrete-time signal $x(n)$, does it make more sense to compute the energy as $ E=\sum_1^N{x^2(n)}$ over all the $N$ samples or does ...
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What is the meaning of **Sample** in **Sample ACF**

What is the meaning of Sample in Sample ACF (autocorrelation function)
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55 views

Definitions of PSD, ESD and Autocorrelation

This question is going to encompass multiple areas and may not have a single answer but here it goes. I'm trying to conceptualize 3 different concepts and how they relate to each other. These 3 ...
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1answer
131 views

Autocorrelation of a uniform random process

i am currently learning the basics of signal processing. As you may know the definition of the autocorrelation is different if you look at a random process or for example a deterministic signal My ...
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26 views

Find repeating sequences in audio file

I have a long audio file (12+ hours). I know that there are some unknown small (2 minutes each or thereabout, it varies) repeating (not bit by bit: it is recorded with a mic) chunks in it, repeating ...
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Energy definition for Autocorrelation lag 0 and lag 1 for complex signals

I am studying the role of an auto-correlation matrix for random signals and the difference of energy between a lag 0 and lag 1 matrix. Consider a complex input signal $x(k)=[x1,x2]^T$ and $x(k-1)=[x0,...
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ACF of $X_t = a + bt$

I was able to prove the following convergence results. But how can we determine the theoretical autocorrelation function of the deterministic sequence $X_t = a + bt$ and $X_t = c\ \text{cos}(\omega t)$...
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98 views

PSD from autocorrelation in MATLAB

I am trying to simulate a simple stochastic process defined by the equation: \begin{equation} \frac{1}{v}\frac{db}{dt} +\Gamma_0 b= \sqrt{\sigma}R(t), \end{equation} where $R(t)$ is a zero-mean white ...
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Autocorrelation of an AR model

When I plot the autocorrelation function of an autoregressive signal, I get this instead of a damping sinusoid. I am having trouble understanding why this is happening. My signal is an generated ...
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23 views

Linear prediction and filter stability

I am currently trying to implement an iterative block-wise algorithm in which AR coefficients are computed for each block. There is a issue with my code where values get too large, and I was ...
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46 views

SNR and Correlation

Signal to Noise ratio is essentially the ratio of the signal power to the average noise power. Correlation is the similarity of two signals. Assuming a signal travels through a channel or medium, and ...
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67 views

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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69 views

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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48 views

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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116 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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47 views

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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108 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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39 views

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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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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71 views

Mathematical description of the ACF using fft2

I computed the acf of an image with the following code: ...
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154 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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Increasing error safety in the cross correlation of two binary signals in wireless communication

I want to cross correlate two binary signals, where one is corrupted by noise and shifted by a time constant $\tau$, but has the same bit pattern (so basically an auto correlation). Similar to case 2 ...
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1answer
73 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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2answers
179 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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After fitting auto ARIMA's order, in prediction I'm getting bad result

For sunspot dataset. Below is the ARIMA code. Auto ARIMA finds the best ARMA(2,1,2)(2,0,1) model. But when I plot the prediction seems wrong: ...
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66 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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339 views

Autoconvolution vs Autocorrelation

I have a serious doubt about Autocorrelation and AutoConvolution. My understanding is, Autocorrelation Autocorrelation is ...
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24 views

Aperiodic Correlation function using FFT Method

Periodic correlation function between two binary sequences say x and y can be computed using FFT. corr_value = ifft(fft(x).*conj(fft(y)); Can some one please help on how to compute aperiodic ...
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454 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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52 views

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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39 views

Autocorrelation function and product of summations

Let $u[k]$ be a white gaussian noise with variance $\sigma^2$ and $y[k]=\sum\limits_{i=0}^q b_i u[k-i]$, that is a moving average model. I am trying to compute the autocorrelation function of $y[k]$: ...
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1answer
108 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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70 views

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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181 views

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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40 views

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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On the spectral representation of deterministic and random signals

I went back to many references in order to fix some of the confusions that I have on many concepts in signal spectral representation. I concluded that: 1) Deterministic signals may be represented ...
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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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135 views

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\...