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11 votes

Covariance vs Autocorrelation

According to your definition of autocorrelation, the autocorrelation is simply the covariance of the two random variables $Z(n)$ and $Z(n+\tau)$. This function is also called autocovariance. As an ...
Matt L.'s user avatar
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11 votes

Why do we need to conjugate complex signals in autocorrelation and cross correlation

$\underline{\text{Prologue :}}$ Let me ask you another question. How will you compare two complex numbers $U$ ($a+jb$) and $V$ ($c+jd$)? By comparing magnitude? Subtract them and take real part? ...
abhilash's user avatar
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10 votes

What is the difference between convolution and cross-correlation?

As a student I was involved in the same problem as you are. Let me explain to you in the simplest words without any math. Convolution: It is used to convolve two functions. May sound redundant but I'...
Andres's user avatar
  • 101
9 votes

Understanding the definition of mean/autocorrelation

The definition of the autocorrelation function $R_x(\tau)$ depends on the nature of your $x$. If $x$ is a deterministic signal with finite energy then: $$R_x(\tau)=\int_{-\infty}^{+\infty}x(t)x^*(t-\...
Learn_and_Share's user avatar
9 votes

When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

The FIRST PROBLEM you ask about is how does $$R_x(\tau) = E[x(t)x(t+\tau)] $$ equal $$R_x(\tau) = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^{+T} x(t)x(t+\tau) dt$$ because (7) requires ...
Peter K.'s user avatar
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9 votes

When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

To add to Peter K.'s answer: The correct defition of the autocorrelation is $ R_x(\tau) = E[x(t)x(t+\tau)]$. If the process is stationary, that simplifes a little to $ R_x(\tau) = E[x(0) x(\tau)]$. If ...
leonbloy's user avatar
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8 votes
Accepted

Auto-correlation function, an inverse problem

Let's look at the case $x[n] \in \mathbb{R}$, where $x[n]$ is real. Autocorrelation is basically convolution of the signal with it's time inverse. This can be easily expressed in the frequency domain....
Hilmar's user avatar
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8 votes

Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches

If we can assume white noise and the signal itself is of constant frequency over the full 15 second duration, then the autocorrelation would be suboptimal in determining the frequency compared to the ...
Dan Boschen's user avatar
7 votes
Accepted

Guitar pitch detection with autocorrelation

Cons: Not as accurate This is just compared to the other methods. I was measuring frequency very accurately to look for clock drift, etc: 1000.000004 Hz for 1000 Hz, for instance. For guitar pitch ...
endolith's user avatar
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7 votes
Accepted

perfect sequences

Let $\theta_a$ and $\theta_c$ respectively denote the maximum magnitudes of the off-peak or out-of-phase periodic autocorrelation functions and the periodic crosscorrelation functions of a set of $K$ ...
Dilip Sarwate's user avatar
7 votes
Accepted

Correlation of independent random processes

No. Quoting Wikipedia's article Independence (probability theory): If $X$ and $Y$ are independent random variables, then the expectation operator $\operatorname{E}$ has the property $$\...
7 votes

Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches

Similar approach to Dan's but a different way to go about it. First lets define what exactly we mean by "peak" frequency. I suggest it is the frequency that minimizes the square error ...
Hilmar's user avatar
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7 votes

When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

The magnitude-square $\Big| X(\omega) \Big|^2$ of the Fourier Transform of energy signals is the energy spectral density. For power signals, you gotta do a little bit with normalization, so that ...
robert bristow-johnson's user avatar
6 votes
Accepted

Confusion about ensembles and averages in autocorrelation matrices

First, the output of the xcorr() function returns lag-0 of the auto-correlation sequence (ACS) estimate at its middle sample, as you recognize. The function argument scaleopt provides normalization ...
Fat32's user avatar
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6 votes
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Allan Variance vs Autocorrelation - Advantages

My current work involves the design details of atomic clocks where we use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary ...
Dan Boschen's user avatar
6 votes

Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches

I'd just add to these two fantastic answers that since the frequencies of interest are so low compared to your sampling frequency, I imagine you can also decimate your signal quite a bit before trying ...
Jdip's user avatar
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5 votes

AutoCorrelation Matrix vs Covariance Matrix for the MUSIC Algorithm

That really depends on who is asking and precisely what definitions you want to use. In signal processing, the autocovariance is usually a non-normalized, mean-corrected quantity derived from a ...
Peter K.'s user avatar
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5 votes

How is the energy of $x_1\cdot x_2$ related to the energies of $x_1$ and $x_2$?

Knowing the energies of $x_1$ and $x_2$ is not sufficient for determining the energy of $x_3=x_1x_2$. What you can do is determine an upper bound for the energy of $x_3$ given the energies of $x_1$ ...
Matt L.'s user avatar
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5 votes
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How to prove that the peak of the autocorrelation function is at zero lag?

The Cauchy Schwarz inequality states that: $$ \left|\int_{-\infty}^{\infty}g_1(t)g_2(t) dt\right|^2 \leq \int_{-\infty}^{\infty}|g_1(t)|^2 dt \int_{-\infty}^{\infty}|g_2(t)|^2 dt $$ I'm going to ...
David's user avatar
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5 votes

How to estimate the autocorrelation from nonuniformly spaced data

As suggested by Marcus Müller, interpolation in the time domain could be a solution. I never had to perform such a task, and the outcomes may depend in the nonuniformity of your sampling. I propose a ...
Laurent Duval's user avatar
5 votes
Accepted

Power spectral density of $\left(x(t)\right)^2$?

Since the question has been raised as to whether the hint that I had given to the OP in a comment on the original question was appropriate for a newcomer to signal processing, here goes. Stripped of ...
Dilip Sarwate's user avatar
5 votes

How can define an indicator which measures the degree of similarity between two signals?

The general topic of finding similarities between signals is wide ranging: are the signals of same sampling, length, offset, shift or scale? where do they take their values (discrete, real, complex)?...
Laurent Duval's user avatar
5 votes
Accepted

variance in the time domain versus variance in frequency domain

Variance is never defined as power. For a wide-sense stationary random process $X(t)$ with zero mean $$\mu_X=E\{X(t)\}=0\tag{1}$$ the variance of $X(t)$ equals its power. The autocorrelation of $X(...
Matt L.'s user avatar
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5 votes

Auto-correlation function, an inverse problem

There is in general, as @Hilmar's answer points out, no unique solution to the question of a sequence that has the given perodic autocorrelation function. In the simplest case, that a shifted ...
Dilip Sarwate's user avatar
5 votes

Does the auto-correlation function of stationary random process always converge?

No it does not necessarily. For example the following discrete-time, WSS random process $$x[n] = A \sin(\omega_0 n + \phi) $$ which is called the random phase sinusoid, where $A$ and $\omega_0 \...
Fat32's user avatar
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5 votes
Accepted

Relationship between the autocorrelations of X(t) and X(nt)

The answer to the OP's question is more straightforward than rb-j's comments make it out to be. $\{X(t)\colon -\infty < t < \infty\}$ is a continuous-time WSS random process with ...
Dilip Sarwate's user avatar
5 votes
Accepted

Autoconvolution vs Autocorrelation

Are two signals are the same if their auto-convolution functions are the same? Not quite. Look at the autoconvolution in the frequency domain where the ...
Dilip Sarwate's user avatar
5 votes

Why autocorrelation can be more efficiently calculated using the fft

The cross-correlation between two functions $f(t)$ and $g(t)$ can be seen as a convolution of $f(t)$ and $g(-t)$. The auto-correlation is of course a special case where $f=g$. This operation in the ...
EmThorns's user avatar
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5 votes

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

It's the empirical ACF computed using the sample.
mark leeds's user avatar
  • 1,117
5 votes

auto-correlation of differentiator

The derivative of the Dirac delta impulse is written as $\delta'(\tau)$. This helps with notation because the mistake you made is to write $h(-\tau)=\frac{d}{d\tau}\delta(-\tau)$, which is not the ...
Matt L.'s user avatar
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