Questions tagged [autocorrelation]
Autocorrelation is the cross-correlation of a signal with itself.
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Calculation of eigendecomposition of a signal in its Fourier domain?
I want to find the eigendecomposition of a 3-dimensional discretely sampled signal $X$, where each sample $X_{i,j,k}$ is treated as a vector $\langle i, j, k\rangle$ (with the origin at the middle of ...
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93 views
Autocorrelation for periodic signals
Autocorrelation for power signals is defined by $$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tx(t)x^*(t-\tau)dt\tag{1}$$ Is it true that for periodic signals $(1)$ can be computed by $$R_x(\tau)=...
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How is time used in this autocorrelation expression?
There are two papers I'm working through that explain applications of autocorrelation.
The first - by Monti - is quite clear to me:
It runs a sum over the time axis.
The second one by Brown makes no ...
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32 views
finding power spectral density from a vector
I have been given a vector:
\begin{equation}
v= \:\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}
\end{equation}
my job is to find the power spectral density from this vector
\...
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29 views
Estimating Signal-To-Noise ratio from discrete time series
A hidden variable $x(t)$ (in my case, neuronal signal averaged over a brain area) exhibits dynamics on multiple timescales from milliseconds to seconds. An observable $y(t)$ is produced by convolving $...
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36 views
Computing Autocorrelation via FFT
I have seen several methods to calculate Autocorrelations using FFTs, and am confused about why they differ.
Zero-Pad it to double its original length.Take the FFT. Then replace all the coefficients ...
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1answer
58 views
Proving the upper bound of cross correlation
I am reading about cross-correlation from this document and equation (5) states that
The maximum value of the crosscorrelation is not always when the shift
equals zero; however, we can prove the ...
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14 views
Self-correlation vs Digital frequency: figure from paper
In the paper, is discussed a MIMO-OFDM system over the frequency self-correlation of i.i.d. Rayleigh
channels and subcarrier grouping. in this paper is shown the following figure.
it shows the self-...
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39 views
What's the relation between frequency band of $X(j\omega)$ and $\Phi_{xx}(j\omega)$?
in which:
$x_{c}(t)$ is a continuous-time signal
$X(j\Omega)$ is the Fourier Transform of $x_{c}(t)$
$\Phi_{xx}(j\Omega)$ is the Power Spectrum Density of $x_{c}(t)$ which defined as Fourier ...
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38 views
auto-correlation of differentiator
How to do this entirely in the time-domain, without using frequency domain?
Let's say I have a continuous-time system that is a differentiator, and X(t) is WSS:
$$x(t) \rightarrow \boxed{\frac{d}{dt}} ...
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152 views
Power Spectral Density from Probability Density Function
The samples of a signal $x[n]$ are i.i.d. and follow a triangular pdf with $a = 0,\ b = 2,\ c = 1$:
The DC-power of the signal is
$$\mu_x^2 = \big(\mathbb{E}(X)\big)^2 = \left(\int_{-\infty}^{\infty} ...
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31 views
correlation matrix vs. correlation function?
Can someone help me understand the difference between a 1-dim autocorrelation function and a 2-dim autocorrelation matrix of a random process aka time series?
My Leon-Garcia textbook defines CX(Ļ) and ...
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19 views
Are ACF plots sufficient to detect white noise assuming probable existence of non linear associations?
ACF plots use autocorrelations tests to determine associations in time series. Autocorrelations, however, only measure linear associations. To my understanding, certain non-linear associations can ...
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26 views
aperiodic auto correlation properties
I'm reading about PSC (primary synchronization code) in UMTS. It is a so-called Golay sequence.
It is said it has a good aperiodic auto correlation properties.
What does it mean (in terms of aperiodic)...
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53 views
Welch's method, no windowing function
I'm trying to pick a method to increase the bin-width on several discrete Fourier transforms I'm incorporating into some scientific analysis code. I have several questions about how this could work ...
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42 views
Autocorrelation not as expected
I am trying to find the autocorrelation function on the intensity of a pixel over time stored in an array "Intensity". Usually one would expect the highest point to be at t = 0 and then decay. I am ...
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16 views
Error in measuring wavelength/frequency using autocorrelation
I am currently using these two fonctions to compute to autocorrelation of a signal in order to measure the wavelength:
...
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145 views
Autocorrelation function and correlation integral
I am confused by the definition of autocorrelation function. It is originally defined as the expected value
$$R_{XX}(\tau) = E[(X(t)X(t+\tau)] = \langle X(t)X(t+\tau)\rangle\tag{1}$$
where $\langle\...
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1answer
53 views
Evaluating discrete spectral density at only a few frequencies
I'm trying to obtain the spectral density at three particular frequencies for a computational chemistry problem that I'm working on (if you are curious, it has to do with the estimation of Nuclear ...
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31 views
argmax of the cross correlation
I want to find the shift between 2 signals $\sin a$ and $\sin b$ where (with Matlab code)
$$
a = \{0, 1,2,3,4\}
$$
$$
b = a +1
$$
$$
\implies b = \{1,2,3,4,5\}
$$
then
$$
\arg\max \sin a = 2
$$
and ...
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31 views
Calculation of channel autocorrelation for LMMSE
I am currently trying to implement an LMMSE estimator/equalizer for a python based ofdm receiver. The actual reception is done with a SDR and iq-samples are stored in a file for the "ofdm-receiver" ...
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1answer
60 views
Blackman-Tukey Autopower equation
I am looking over Blackman-Tukey method for Autopower.
It uses the DFT after a window is applied on a autocorrelation.
$$
Power Spectrum = \frac1{2\pi} \ \sum_{(k=-(N-1))}^{N-1} w[k] R[k] e^{-i\...
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43 views
What does $x^*$ mean in autocorrelation
Suppose $x[n]$ is a sequence of length $N$. Following formual defines circular autocorrelation sequence of $x[n]$:
$$r_{xx}[m] = \sum_0^{N-1}x[n]x^*[n-m \mod N]$$
What does $x^*$ mean in this equation?...
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66 views
Calculate autocorrelation of a sinus
I am a beginer in signal processing.
I am currently studying a signal $s(t) = \cos(h(t))$, and I have to calculate its autocorellation.
Since $h(t)$ is periodic of period $T_s$, I calculated the value ...
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2answers
93 views
Contradictory results from power spectral densities and autocorrelation in shot noise
I have a white noise generator circuit which involves a Zener diode meant to undergo breakdown across a resistor $R1$, and am trying to compare power spectral densities and autocorrelation of the ...
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2answers
105 views
Does a signal and its autocorrelation ever have the same Fourier spectra?
My understanding is that Fourier power spectra give the frequencies and corresponding intensities/amplitudes contained within a signal. In certain techniques, such as molecular dynamics, it is common ...
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62 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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66 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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1answer
88 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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454 views
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
50 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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81 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
151 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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73 views
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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147 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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1answer
299 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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1answer
52 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)=ā_{i=1}^{N-l}(x_i-Ī¼)(x_{i+l}-Ī¼)$ ; $Ī¼=\frac{1}{N}ā_{i=1}^Nx_i$
I have noticed that this definition introduce a ...
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1answer
115 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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1answer
40 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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1answer
79 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
77 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 $$\forall(\tau,t_1, \ldots, t_n) \in \mathbb{R} \land ...
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1answer
339 views
What is the time reversal convolution
Given a signal $r(t)$ which is a result of convolution between signal $x(t)$ and a channel $h(t)$ as below :
$r(t) = h(t)*x(t); $
what I know, the time reversal convolution can be process as ...
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21 views
Power spectral density of surface roughness
I am currently working on understanding how visually and intuitively the power spectral density is the fourier transform of the autocorrelation of a signal. I was wondering if anyone had a good ...
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1answer
56 views
estimation of ARMA(1,1) with MA parameter greater than 1
I am working with the following simple ARMA(1,1) model:
$$
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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3answers
88 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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2answers
256 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 ...
