Questions tagged [autocorrelation]

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

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58 views

How to calculate the coefficients of Auto regression model of multiple data?

I have ACF of a time series data, I want to interpolate the ACF using AR model for which I need to calculate the coefficient of the AR model The problem is I have (6000x6000) matrix of ACF and I cant ...
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40 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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47 views

Kronecker delta instead of dirac delta as correlation function of white noise

From my understanding, if you have a sample $x_{t_1},\dots,x_{t_n}$ of $X_{t_1},\dots,X_{t_n}$ which are iid $N(0,1)$, then $x_{t_1},\dots,x_{t_n}$ is a sample path of Gaussian white noise. However, ...
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Help with obtaining the power spectral density of a simple continuous cosine (using both forms of the definition for PSD)

I am trying to go through a simple example to teach myself about Parseval's theorem and calculating power spectral density (PSD) in practice and would be very grateful if someone could check my ...
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1answer
20 views

Autocorrelation - Understanding reduced correlation at periodic time shifts using np.correlate (versus statistical autocorrelation)?

I'm going through the Think DSP by Allen B. Downey, and I'm struggling to understand a specific aspect of np.correlate and how it differs from statistic autocorrelation. The question is at the bottom, ...
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1answer
53 views

Power spectrum of uniform white noise

Given a white noise image $W_{i,j} \sim U[a,b]$ where each pixel is distributed uniformly in $[a,b]$, how would I go about computing its power spectral density? That is, I want to find $E[|\hat{W}_{i,...
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27 views

Mean of input based on acf of filtered output

By looking at the ACF of the output of a filter whose impulse response I know, can I approximate the mean of the input? For example ...
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1answer
47 views

Autocorrelation of LTI system output

I'm having trouble showing the following relation: The autocorrelation of a LTI system with impulse response $h[n]$, input $x[n]$ and output $y[n]=h[n]*x[n]$ is given as: $r_{yy} = r_{hh}*r_{xx}$ ...
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Why are the Fourier transforms of autocorrelation and cross-correlation different?

I'm almost definitely misunderstanding this, but as far as I can tell we can equate the Fourier transform of the auto-correlation function $R_{xx}(\tau)$ with power spectral density (PSD) like so: $\...
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36 views

Get autocorrelation function from the power-spectrum (python)

I am trying to compute the autocorrelation function of a signal for which I only know the power-spectrum. In order to test my approach I wanted to try it out on the spectrum of $1/f^2$ noise for ...
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198 views

Power Spectral Density (PSD) from autocorrelation

I have a 1D fractional Brownian motion (fBm) signal, $u(x)$, of size $N$, generated through a random number generator with a gaussian mean, $\mu=0$, and standard deviation, $\sigma=\sqrt{N}$. I want ...
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48 views

Autocorrelation to diagnose faults

I'm attending a very practical course on signals and i have some doubts, i hope to receive answers in layman terms. 1) My prof said i can use the autocorrelation of the output of a process to ...
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59 views

Negative bias in sample auto-correlation of uncorrelated time-series

I am using the following definition for the sample auto-correlation: $AC_x(l)=∑_{i=1}^{N-l}(x_i-μ)(x_{i+l}-μ)$ ; $μ=\frac{1}{N}∑_{i=1}^Nx_i$ I have noticed that this definition introduce a ...
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1answer
61 views

The expectation in power spectral density

I'm a bit confused with the definition of the power spectral density (PSD). From Wiki https://en.wikipedia.org/wiki/Spectral_density , I found the definition is: $$ S_{xx}(\omega) = \lim_{T\...
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36 views

Auto-correlation of C/A code with 30dB noise

I've a C# program that generates a GPS C/A code consisting of -1s and +1s. Auto correlating the code with rotated versions of itself shows that the peak is 1023 and the next maximum value is 63 which ...
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34 views

Autocorrelation Computation with MATLAB

I have done the computation of the autocorrelation of a sequence (vector), here called "frame", and I have plotted it. Here the code: The resulting graph is: I do not understand the position of the ...
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23 views

Stereo echo cancellation for stationary source locations: why does adaption stagnate?

In a stereo echo cancellation system, the echo path for the far-end-signal in the near-end room is estimated by an adaptive filter. Since the same signal is transmitted by two similar far-end paths, ...
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1answer
61 views

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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1answer
168 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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18 views

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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1answer
43 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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78 views

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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161 views

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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20 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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58 views

What is the meaning of **Sample** in **Sample ACF**

What is the meaning of Sample in Sample ACF (autocorrelation function)
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146 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
357 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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38 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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1answer
123 views

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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28 views

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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1answer
330 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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30 views

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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1answer
35 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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121 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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2answers
92 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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148 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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1answer
54 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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325 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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105 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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2answers
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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1answer
129 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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52 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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27 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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1answer
86 views

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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1answer
79 views

Mathematical description of the ACF using fft2

I computed the acf of an image with the following code: ...
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223 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
87 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
258 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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