Questions tagged [autocorrelation]

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

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Difference transformation and Stationarization of Moving Average

I have a temperature sensor data, I want to denoise it. The first thing that came to my mind was to take the moving average, it was very smooth but it is still not stationary. If I take the log ...
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Blackman-Tukey PSD in Python

I am trying to calculate the Blackman-Tuckey (BT) PSD in Python to check my understanding (getting started with signal processing). I have tried making the calculation myself and compare it with Scipy'...
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Does oversampling lead to colored noise?

Suppose we receive $R(t)=X(t)+W(t)$, where $X(t)$ is band-limited to $[-B/2, B/2]$ and $W(t)$ is white Gaussian noise with autocorrelation $R_W(\tau)=\frac{N_0}2\delta(\tau)$. If we filter $R(t)$ ...
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Cross-correlation between brain signals: how to handle autocorrelation?

I've been asked to calculate the cross correlation in the 3-12 Hz band between two simultaneously recorded brain signals in two distinct brain areas. Data were acquired at 32 kHz then preprocessed by ...
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i am stuck on question 3 and isn't getting desired output. can anyone explain what should i do and provide code for this?

Given the autocorrelation function ACF(n)=cos((2/30)n) for n= -5,-4,…0,1,…, 5 use the simplified Lim & Malik iterative algorithm to extend the ACF to 256 points. Note that the simplified version ...
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Interpretation of a non-canonical Allan Variance plot

I am currently working on characterizing the noise sources of a Global Navigation Satellite System (GNSS) sensor using an Allan Variance plot, which is commonly employed to analyze frequency stability ...
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How to find the pattern of a signal or main cycle of a signal?

I have a dataset where a certain pattern is repeatating in different time interval. The repetation of data pattern can come with slight change. I want first find the pattern (or cycle) and later by ...
asteroid's user avatar
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Signal Correlation time computation for Time delay estimation

I need to compute the Signal Correlation Time of an audio signal to check wether time-delay estimations are anomalous. In paper The Signal correlation time is defined as the width of the main lobe of ...
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$i^{\text{th}}$-dimensional autocorrelation function

I am referring to the work of Stephen A. Billings on "Identification of a class of nonlinear systems using correlation analysis" from the year 1978, where it is mentioned that the $i^{\text{...
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Fundamental question about Autocorrelation

I need help! I'm trying to calculate the Power Spectral Density of a quantum operator ($\delta \hat{n}(t)$) given by: $$ \delta \hat{n}(t) = A(t)\delta\hat{a}(t)+A^{*}(t)\delta\hat{a}^{\dagger}(t) $$ ...
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Zero-mean preprocessing before calculating the autocorrelation

I am aware that if we do not subtract the mean value from the white noise at the beginning (if the mean is not equal to 0), that its autocorrelation function will be triangle shaped and not a delta ...
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Calculation of frequency correlation function based on power delay profile

We know that the frequency-domain correlation function and the power delay profile appears as a Fourier transform pair, i.e., $$A_{c}(\Delta f) = \int_{-\infty}^{\infty}PDP(\tau) \cdot e^{-j2\pi \...
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Calculation of the discrete cosine transform

I am wanting to use the discrete cosine transform to relate the autocovariance function of a process to its periodogram. Following Chris Chatfield's book (Time Series Analysis, p129), I am wanting to ...
hydrologist's user avatar
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Power Spectral Density and Total Power Calculation for Signal

I am currently working on analyzing a signal with the autocorrelation function given by: $$R(\tau) = 100 \cos( 10000\pi\tau) (Λ(2000\tau))^2$$ I need assistance in plotting the power spectral density ...
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Frequency content of a noisy signal

To find the frequency content of a noisy signal (PSD), there are two methods below: #1 Take the fourier transform of its power signal (square the noisy signal) #2 Find the autocorrelation function of ...
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Finding mean, autocorrelation, and power spectrum of $y[n]=(x∗h)[n]$ where $x[n]$ is zero mean white gaussian noise

If given $x[n]$ is zero mean white Gaussian noise with variance $\sigma_x^2$, and a filter with a known impulse response $h[n]$, how would I find the mean, autocorrelation, and power spectrum of $y[n] ...
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What's the exact definition of the power spectral density function?

I learned from my signal processing course that PSD is the Fourier transform of the autocorrelation function. $$\mathscr{F}\Big\{\mathrm{E}\big[x(t)x(t+\tau)\big]\Big\}$$ Today I took my statistic ...
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Autocorrelation-properties

How do we go about proving the property that entails; If X(t) is ergodic with no periodic components the autocorrelation converges to square of the mean as the time difference(τ) approaches infinity 𝝁...
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Proving that the autocorrelation of a signal is the convolution with its time-reversed complex conjugate

I need to prove the property $$f(x)⊕f(x)=f(x)⊗f^*(-x)$$ That is the autocorrelation of a function is the convolution with its time-reversed complex conjugate. I have constructed most of the proof but ...
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Why is power level area under the autocorrelation function of the white-noise signal?

A paper I am reading (Linear and Nonlinear Encoding Properties of an Identified Mechanoreceptor on the Fly Wing Measured with Mechanical Noise Stimuli) defines power level for a white-noise signal as ...
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Confusion with Complex Gaussian process with Auto-covariance

I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
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Input (Auto)Correlation Matrix for LMS Adaptive Algorithm

I read quite a few posts regarding the autocorrelation matrix and although some of them were of the LMS adaptive algorithm application, I didn't find an answer to my question. How do I calculate R; ...
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Inferring properties of a signal from its autocorrelation

In a previous thread, it was asked whether we can infer a signal from its autocorrelation, and the answer was a clear no, since the autocorrelation process is lossy. Still, is there anything we can ...
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Autocorrelation: why is the lagging necessary?

I am currently studying for my last exam. I wanted to ask why it is necessary or useful to perform the autocorrelation of a signal with a lagged version of itself Why does it have to be lagged?
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How is the PSD the Fourier Transform of the Autocorrelation function?

I am getting very confused with the textbook I am reading through: Modern Digital and Analogue Communication Systems: 3.8.1 Parseval's Theorem says that: The total power in a power signal is its time ...
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Proof of the Wiener-Khintchine theorem in time domain

In a proof for the Wiener-Khintchine theorem (See p. 572-573) I have seen the following operation being done: \begin{align*} S_{xx}(f) &= \lim_{T \rightarrow \infty} \int_{-2T}^{2T} Rxx(\tau) e^{-...
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Autocorrelation of a random process

I have some doubt about the following exercise. Let's consider the signal $X(t)= \operatorname{rect}\left(\frac{t}{2A}\right) $ , where $A$ is a discrete random variable which can assume one value ...
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Symmetric Autocorrelation Function vs Asymmetric Autocorrelation Function

I am trying to work through the Cyclostationary Blog to create a cyclic autocorrelation function: I have been given the above equation to determine the cyclic autocorrelation function where tau is ...
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Autocorrelation function of a filtered periodic signal

I'm having trouble with this exercise. The signal $x(t)= \cos(2\pi f_0 t)$ is sent in input to a non-linear system with input-output relationship equal to: $g(x)= 0$ if $x<0$ and $g(x)=3x$ if $ x &...
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Correlated phase noise

I am trying to find a mathematical model to understand the correlated phase noise. Here, I see the reference phase noise is almost the same after up-conversion and then down-conversion. How does ...
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When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

I am currently struggling on the book "System Identification: Theory for the User" by Lenart Ljung (freely available here) about the definition of the spectrum (around page 33). As I have ...
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Emergence - what constitutes a minimally coherent source that "just begins" to produce stationary interference patterns?

A superposition of two signals with different frequencies will never produce visible (i.e. stationary) interference patterns. Such waveforms will produce spatiotemporal beat patterns, but they rapidly ...
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how to find type of a binary sequence

I want to know how I can find the type (algorithm) of a binary sequence at hand. For instance, I'd like to know whether the following (binary) sequence is m-sequence, Barker, Gold, etc. AC DD A4 E2 F2 ...
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Continuous autocorrelation

Is there a known algorithm or an existing implementation that performs a "continuous autocorrelation"? Meaning, in a situation where I sample a signal and every time I get a new sample (or ...
Arik Yavilevich's user avatar
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scipy convolve and fftconvolve return different output [closed]

I'm writing an autocorrelation function using and I noticed a difference between fftconvolve and convolve that should use fft if it's faster. This is my function with convolve: ...
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Reducing or removing autocorrelation in spatially correlated data

I am trying to figure out how one can reduce or preferably remove autocorrelation in spatially correlated data. Using the R code below, one generates spatially correlated data that is normally ...
Show's user avatar
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Expected value and autocorrelation

I have a wide sense stationary stochastic process ${{X_t;t\in \Re}}$ with mean 0 and autocorrelation function $R_x(\tau)=1-\frac{1}{4}|\tau|$ for $|\tau|=0,1,2,3;$ and zero anywhere else. I'm supposed ...
A couple of three things's user avatar
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Find $E[Z^2(t)]$ when $Z(t) = X(t) - Y(t)$ where $Y$ is the output of a LTI system with WSS process $X$ as its input

I received this as a practice problem (part b only). I was able to figure out that $E[Z^2(t)]$ = $R_X(0)+R_Y(0)-2R_\text{XY}(0)$ but did not see how to continue. Checking the answers, I saw this line ...
Robert Moore's user avatar
2 votes
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153 views

Effect of down-sampling to PSD from auto-correlation?

I have a problem to evaluate the PSD from auto-correlation. As you know, PSD is the Fourier Transform of the auto-correlation. But I observe the spur at PSD when I calculate the auto-correlation with ...
David's user avatar
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What happens if you use the Fourier transform of the autocorrelation of a non-WSS process to compute power spectral density?

The Wiener–Khinchin theorem states that the power spectral density of a wide-sense stationary stochastic process can be obtained through the Fourier transform of the autocorrelation of the signal i.e. ...
Tommy Wolfheart's user avatar
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4 answers
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Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches

I have a noisy signal around $15\text{s}$ long sampled a $32\,\text{Hz}$. I am trying to estimate the peak frequency or period for a low frequency component with expected frequency in between $0.08\,\...
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The normalization of the autocorrelation function and how it changes the definitions you've learned about signal analysis in communication systems

Since this question is book-oriented, I will kindly ask you to accompany it with a book that is considered by many researchers the bible of Digital Communication: Proakis - Digital Communications, one ...
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How to analyse signal where one sample (timestamp) captures multiple data points

I have a process that captures interactions between a source and a sink. Unfortunately, the timestamp on the sample is one day. This means that regardless of the number of interactions (flows of ...
MikeB2019x's user avatar
1 vote
2 answers
209 views

How does Power Spectrum remain symmetric in Z domain?

Can you tell me how the $P_x(z)=P_x^*(1/z^*)$ is mathematically correct. I can understand the $P_x(e^{jw})=P_x^*(e^{jw})$ as $P_x$ is real value. But why take the Z domain representation in this way ($...
Kuchi Yashwanth's user avatar
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2 answers
225 views

Why is Autocorrelation between a Zero-mean Random process and a finite deterministic sequence zero?

The Solution is given above: The Question is, how did the $\mathbb{E}{[x(k)f(l)]}$ and $\mathbb{E}{[x(l)f(k)]}$ become zero? is there some rule that correlation between Random Process and ...
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How to show that the autocorrelation function of the given discrete function is this for autoregressive model(AR(2))?

This question is related to white noise representation of WSS sequences using AR(2) (autoregressive) model The function is given as: $$x(k)=\frac{1}{p_1-p_2}(p_1^{k+1}-p_2^{k+1})w[k]u[k]$$ where $w[k]$...
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Autocorrelation of random walk

I want to analyze the auto-correlation of a received power signal that I captured. Unfortunately, I cannot publish the data but I found the same problem arises for a random walk, that's why I used the ...
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The autocorrelation vector of frequency data

I am trying to find the correlation of audio frequency data resulting from an FFT with itself. It would be interesting to hear what you all think on the following question : Should one use the complex ...
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What is the Laplace Transform of the output power spectrum if the input signal is a white noise?

Let us consider a random wide-sense stationary process $n(t)$, which passes through a filter $h(t, \tau)$. Its autocorrelation function is $$R_{n^\prime}(t)=\int_{-\infty}^{\infty} \int_{-\infty}^{\...
Mark's user avatar
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2 answers
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Extract a signal from noise, so I can view noise spectrum only?

I have a perfect modulated signal. I have the same signal but with noise, this noise is from non linear amplification. Is it possible, to extract the clean signal so it's just leaving the noise? The ...
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