Questions tagged [autocorrelation]
Autocorrelation is the cross-correlation of a signal with itself.
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questions
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doubt on demostration autocorrelation theorem
I’m stuck on this demonstration. It is written " If $x'$ is held constant, $dx=dy$ ". But how $x’$ can be constant? shouldn't $x'$ be a variable?
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61 views
Extremes of the integral of the Fourier transform of $\rho(\tau)$
The definition of correlation function of a velocity component, in turbulence (and, more in general, in all the signal), is on a limited domain
$\rho(\tau)=\frac {1}{T}\int_{t_{0}}^{t_ {0}+T}v(t)^{.}...
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23 views
Power spectrum is phase blind
I came accros the phrase " It is well known that the autocorrelation and power spectrum are phase blind". Is there an analytical way to prove this, so I can understand why this is correct.
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1answer
82 views
Isolate g(t) starting from the definition of power spectrum
Starting from the following definition of power spectrum of a signal $g(t)$ in time:
$$S(f)=\int_{-\infty}^{+\infty}\rho(\tau)e^{-i2\pi f \tau} d\tau$$
where
$$\rho(\tau)=\int_{-\infty}^{+\infty}g(t)^{...
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1answer
47 views
Expected Value of a sequence with two random variables
If I have a signal of the form $x\left(n\right)=Acos\left(nω+ϕ\right)$
where
$\omega \in \left[\omega -\lambda ,\omega \:+\lambda \:\right]$ is a uniform random variable
and
$\phi $ is also a uniform ...
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0answers
10 views
Calculate ACF in C++?
I would like to manually reproduce the method that authors of an article used in their research (DOI: 10.1038/s41598-017-02750-9 (Page 8. top)). It is mentioned as "ACF", so I wrote ...
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27 views
Signal “seasonality” detection
I am wanting to do a seasonality detection on a signal, but am struggling big time to find a solution.
I have tried several different things, including statsmodels STL/seasonal_decompose. It seems ...
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1answer
41 views
autocorrelation of multiple signals
Problem: I am looking at an adaptive filtering application where the eigenvaluespread of the autocorrelation matrix $R$ is important for the convergence of the algorithm. For a single channel system ...
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63 views
Ringing/Oscillation in the reconstructed Periodogram
I have an original periodogram that I need to model with autoregressive process. However the model isn't right as it is not fitted well to the original periodogram.
I am suspecting I am doing ...
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2answers
72 views
Processes/Transforms involved to get brainwave data from raw EEG? (Autocorrelation confusion)
Not clear on what the autocorrelation function of raw EEG means physically
why can't you take the FT of a the EEG itself and get frequency data?
With BCI & basic electrode setups you can ...
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1answer
87 views
Find the autocorrelation function of signal $x(t) = u(t) - u(t-1)$
I have used the energy-type signal autocorrelation function:
$$\mathcal{R}_{xx}(\tau)=\int_{-\infty}^{\infty}x(t)x^*(t+\tau)dt$$
I have rewritten the equation as:
$$\begin{align}
\int_{-\infty}^{\...
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2answers
104 views
How to find the output mean and autocorrelation of a non-linear system
I need help with this question. I am sure this is the right StackExchange forum for this type of question.
Consider a nonlinear device such that the output is $Y(t) = aX^2(t)$, where the input X(t) ...
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29 views
Cross-correlation estimation in MATLAB using a method similar to pwelch for the PSD
Is there a function in MATLAB to estimate the cross correlation function of two wide-sense stationary signals (or the autocorrelation of the signal with itself) in a way similar to the estimation of ...
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27 views
Generating data with given Auto and Cross Correlations
I have two discrete vectors $\mathbf{x}_1$ and $\mathbf{x}_2$, and I'm trying to generate more data $\mathbf{f}_1$ and $\mathbf{f}_2$ that has some basics properties of $\mathbf{x}_1$ and $\mathbf{x}...
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1answer
56 views
What are some really accurate ways to get the value of a peak (local maximum) given some points around it? (To be used for autocorrelation peaks.)
I have looked everywhere on the internet for this and, surprisingly, haven't found much useful information.
Given 3 or more points closest to a peak (local maximum) what are some of the most accurate ...
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1answer
196 views
Autocorrelation function of a triangular wave
I am interested to find the analytical expression for the autocorrelation function of a signal that comprises triangular pulses. I have followed the derivation in "Statistical Theory of ...
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0answers
78 views
Calculate periodicity of audio signal
I have to calculate the periodicity of a audio signal like this:
You can see by eye, that the volume rises the highest every second "blob".
The spectogram (that is visualized ...
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38 views
Auto-correlation of absolute squared stochastic process
Consider the stochastic process $a(t) \in \mathbb{C}$. Its autocorrelation function is given as
$$
\phi_{aa}(\tau)=\left(a(t)\star a(t)\right)(\tau)=\int_{-\infty}^{\infty}a^*(t)\cdot a(t+\tau) \...
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1answer
71 views
Prove that a specific auto-correlation function is even
Background (from digital signal processing):
The $\delta$ function is defined as below:
$$\delta[n]=\begin{cases}
1 \quad \mbox{if} \hspace{.4em} n=0\\
0 \quad \mbox{o.w.}
\end{cases}$$
In an LTI (...
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0answers
30 views
How to compute autocorrelation with exponential IIR filter efficiently
I have a discrete dataset (called V). I want to compute the autocorrelation of V at multiple time horizons. Normally I know that I can use the IFFT of the PSD. But in my case I need it with ...
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25 views
How to apply the window function when finding autocorrelation?
I want to get frequencies from amplitudes and I'm following Chap 5.1.1 from Speech Signal Processing by Praat
As the in pictures, I will multiply the signal $s(t)$ with the Hanning window $w(t)$, ...
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28 views
PSD of linearly modulated signal using autocorrelation?
Consider a signal $v(t)$ given by
$$v(t)=\sum_{n=-\infty}^{\infty} b[n]p(t-nT).$$
Assume that $b[n]$ is uncorrelated with zero mean, i.e. $\mathbb{E}[b[n]b^*[m]]=\mathbb{E}[|b[n]|^2]\delta[n-m]$ and $\...
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24 views
Calculation of eigendecomposition of a signal in its Fourier domain?
I want to find the eigendecomposition of a 3-dimensional discretely sampled signal $X$, where each sample $X_{i,j,k}$ is treated as a vector $\langle i, j, k\rangle$ (with the origin at the middle of ...
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2answers
253 views
Autocorrelation for periodic signals
Autocorrelation for power signals is defined by $$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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1answer
60 views
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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0answers
49 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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72 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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1answer
68 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
156 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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21 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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1answer
60 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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1answer
65 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:
$$x(t) \rightarrow \boxed{\frac{d}{dt}} ...
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2answers
325 views
Power Spectral Density from Probability Density Function
The samples of a signal $x[n]$ are i.i.d. and follow a triangular pdf with $a = 0,\ b = 2,\ c = 1$:
The DC-power of the signal is
$$\mu_x^2 = \big(\mathbb{E}(X)\big)^2 = \left(\int_{-\infty}^{\infty} ...
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44 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(τ) and ...
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1answer
24 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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30 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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0answers
56 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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0answers
51 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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2answers
98 views
Discrete-time sampling of filtered white noise
I am trying to understand how I can relate a discrete-time random process to a continuous-time random process sampled at discrete times.
Suppose I have a noise source $N_\tau(t)$ which is derived ...
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0answers
19 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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3answers
594 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
62 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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54 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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0answers
43 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" ...
0
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1answer
101 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.
$$
Power Spectrum = \frac1{2\pi} \ \sum_{(k=-(N-1))}^{N-1} w[k] R[k] e^{-i\...
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0answers
52 views
What does $x^*$ mean in autocorrelation
Suppose $x[n]$ is a sequence of length $N$. Following formual defines circular autocorrelation sequence of $x[n]$:
$$r_{xx}[m] = \sum_0^{N-1}x[n]x^*[n-m \mod N]$$
What does $x^*$ mean in this equation?...
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1answer
118 views
Calculate autocorrelation of a sinus
I am a beginer in signal processing.
I am currently studying a signal $s(t) = \cos(h(t))$, and I have to calculate its autocorellation.
Since $h(t)$ is periodic of period $T_s$, I calculated the value ...
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2answers
97 views
Contradictory results from power spectral densities and autocorrelation in shot noise
I have a white noise generator circuit which involves a Zener diode meant to undergo breakdown across a resistor $R1$, and am trying to compare power spectral densities and autocorrelation of the ...
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2answers
151 views
Does a signal and its autocorrelation ever have the same Fourier spectra?
My understanding is that Fourier power spectra give the frequencies and corresponding intensities/amplitudes contained within a signal. In certain techniques, such as molecular dynamics, it is common ...
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2answers
145 views
Questions about autocorrelation, correlation, and randomness
I'm learning about autocorrelation functions and trying to calculate some from a time series. Conceptually, what is the difference between the following two autocorrelation functions?
$$g_{1}(\tau)=\...
