Questions tagged [autocorrelation]

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

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Determining the autocorrelation sequence from an AR model

I have the following equation: $$x(n)=\frac{14}{24}x(n-1)+\frac{9}{24}x(n-2)-\frac{1}{24}x(n-3)+w(n)$$ where, $w(n)$ is a stationary white noise process with variance $\sigma^2_w$ Now, I want to ...
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553 views

Confusion about PSD and RMS

Let's say I have a noise power-spectral-density (PSD) which is not flat and ranges from 0 to $f_1$ Hz in frequency. As we know, the total area under the PSD is equal to the total average power of the ...
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78 views

Interpolation of missing audio signal in a video sequence

Suppose there is a video sequence and there are some frames for which the audio data is missing. I want to interpolate the missing audio data on the basis of the correlation of the audio signal with ...
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1answer
53 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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Advice on autocorrelation via FFT

I have been creating an application in Java that transforms an audio signal and writes it to a midi file. At first I tried using autocorrelation to find the fundamental frequency. However, I have ...
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1answer
33 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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2answers
127 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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1answer
33 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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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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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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2answers
305 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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46 views

Particular Correlation formula

I'm reading a book where the autocorrelation of white noise is expressed as: What is the term $Q(k)$ and why is is expressed as an average value of a dot product ?
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25 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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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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50 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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39 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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1answer
91 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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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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1answer
8k views

Does the autocorrelation function completely describe a stochastic process?

Is a stochastic process completely described by its autocorrelation function? If not, which additional properties would be needed?
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1answer
50 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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2answers
479 views

Efficient calculation of correlation function every $N^{\rm th}$ point

I would like to calculate a long correlation function of length say, 1e6 points. I have a prior knowledge that the correlation peak will be in point k*1000. Is there an efficient way to apply this ...
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22 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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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
52 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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58 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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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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70 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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116 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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3k views

Correlation Using FFT / IFFT (Convolution in Frequency Domain) in Java

I try to find about the delay between two audio files using Cross Correlation in Java. I've already done this algorithm so far that i get a idea about how many samples is the delay. FFT x1 -> Zero ...
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2answers
92 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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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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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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80 views

Fluctuation of autocorrelation of a signal due to signal's noise

I have a question about the fluctuation of autocorrelation of a signal due to signal's noise. I have a signal defined in $-1\leq t \leq 1$ as the following: $V(t)=kt+R(t)$, where $R(t)$ is the random ...
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3answers
296 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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2answers
55 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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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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1answer
78 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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337 views

Harmonics to Noise Ratio Estimation

I'm willing to estimate the Harmonics to Noise Ratio (HNR) of a speech signal x[k] and using autocorrelation method. Theoretically, HNR is given as, $$ \ HNR = \frac{R_{xx}[T_0] }{R_{xx}[0]-R_{xx}[...
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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
98 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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1answer
266 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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524 views

How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where $\chi\...
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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
50 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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3answers
243 views

Interpreting this plot cross-correlation

I have an input signal $x(t)$ that is a white Gaussian random signal with mean 0 and variance 1. The signal $y(t)$ is the output of a linear filter with impulse response $$h(t) = |\operatorname{sinc}(...
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1answer
91 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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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
38 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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283 views

How to determine fundamental frequencies (beats/minute) of heartbeat? (matlab)

read = matfile('ECG.mat'); [cor, lags] = xcorr(read.ecg, read.ecg); read.Fs; So i have read.ecg (signal) and ...

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