Questions tagged [autocorrelation]

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

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130 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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1answer
65 views

Correlation of independent random processes

Suppose $X(t)$ and $Y(t)$ be two independent random processes. Is $E(X(t_1)Y(t_2))$ necessarily zero?
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what is equal to the FT of the Autocorrelation function

I am trying to make my own demonstration in order to find what is equal to the FT of the Autocorrelation function. The autocorrelation function in some book about turbulence is defined as: r($\tau$)=...
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274 views

Derivation of PSD of sampled bandlimited random process

When a bandlimited random process whose PSD \begin{equation} S(\omega) = \begin{cases} \frac{N_0}{2} & -10B<\omega<10B\\[2ex] 0 & \text{otherwise.} \end{cases} \end{equation} is ...
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Is the expectation of a random process $X(t)$ with zero DC component necessarily zero?

Is the expectation of a random process $X(t)$ with zero DC component necessarily zero? Or can it be non-zero depending upon the process?
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1answer
48 views

the demonstration that states that the FT of the ACF function is the square of the DTFT of the signal

I am following the book The Intuitive Guide to Fourier Analysis & Spectral Estimation with MATLAB. I am trying to selflearn the fourier analysis in matlab. I got lost in one passage in the ...
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1answer
29 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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Autocorrelation of vector of images

I have a set of images (think a ball getting bigger and smaller). I want to assess the periodicity in this string of images (each data point in my signal is an image). How can I go about doing this in ...
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3answers
610 views

Least Squares and Auto and Cross Correlation

I am trying to understand why auto and cross correlation helps find the best fit line in least squares. I have an equation as stated here: $Ax=b$ -- I have not exact solution, so I use the least ...
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1answer
61 views

Mathematical description of the ACF using fft2

I computed the acf of an image with the following code: ...
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0answers
17 views

What is definition of cross correlation function?

I mean there are three different formulas for cross correlation that I have studied f and g are two functions whose correlation has to be found 1.For Energy signals 2.For Power signals 3.For ...
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1answer
567 views

Autocorrelation: numpy versus FFT

I have a series a of values (0 and 1) coming from a Brownian process with drift for which I am studying the autocorrelation. I used two methods: 1) numpy autocorrelation: ...
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2answers
102 views

Why is the Fourier (or cosine) transform decorrelating?

The discrete Fourier transform (DFT) and the discrete cosine transform (DCT) both decompose a signal into its frequency-domain spectrum. One property that I have seen praised across various domains ...
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Increasing error safety in the cross correlation of two binary signals in wireless communication

I want to cross correlate two binary signals, where one is corrupted by noise and shifted by a time constant $\tau$, but has the same bit pattern (so basically an auto correlation). Similar to case 2 ...
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2answers
416 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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1answer
66 views

What kind of power spectrum implies uncorrelation in time?

I am new in signal processing field, and actually I just need some basic tools for what I am working on. I've learnt that the autocorrelation function is the inverse Fourier transform of the signal's ...
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7 views

After fitting auto ARIMA's order, in prediction I'm getting bad result

For sunspot dataset. Below is the ARIMA code. Auto ARIMA finds the best ARMA(2,1,2)(2,0,1) model. But when I plot the prediction seems wrong: ...
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1answer
50 views

How to find the fundamental frequency of a discrete signal using partial autocorrelation?

I hope you can help me with this question. I am trying to calculate the fundamental frequency (to know the beats per minute) of a cardiac pulse signal using partial autocorrelation. I use a 12-bit ...
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2answers
2k 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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1answer
110 views

Autoconvolution vs Autocorrelation

I have a serious doubt about Autocorrelation and AutoConvolution. My understanding is, Autocorrelation Autocorrelation is ...
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1answer
56 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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1answer
190 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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1answer
634 views

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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2answers
73 views

Image Interpolation Using the Yule Walker Equations

I have been studying about the Yule-Walker equations for prediction of a time series data from knowledge of past values of the series. Is there any way I can use the same in an image to exploit the ...
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1answer
265 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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1answer
41 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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Aperiodic Correlation function using FFT Method

Periodic correlation function between two binary sequences say x and y can be computed using FFT. corr_value = ifft(fft(x).*conj(fft(y)); Can some one please help on how to compute aperiodic ...
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2answers
237 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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1k views

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
211 views

Autocorrelation of sinc function

I'm having trouble on computing the autocorrelation of the sinc function I want to compute $$R_{hh}(\tau)=\int_{-\infty}^{\infty}\operatorname{sinc}(t) \ \operatorname{sinc}(t-\tau) \ \mathrm{d}t$$ ...
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37 views

Equivalence of the Power Spectral Density definitions

I am trying to show the equivalence of the following Power Spectral Density definitions in Matlab: Definition 1: $$ P(\omega) = \sum_{k=-\infty}^{\infty} r(k)e^{-j\omega k} $$ Definition 2: $$ P(\...
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How to add noise with given autocorrelation to signal in Matlab?

My question is a mix of signal processing and Matlab coding. I have an FIR filter with added noise $w_n$ $$x_n=\sum_{m=0}^{N_h-1}h_mg_{n-m}+w_n.$$ Now the noise $w_n$ has the autocorrelation $$E[...
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40 views

Zero-padding vs. nonzero-padding in computation of auto-correlation with FFT

Isn't the usual zero-padding in the computation of the auto-correlation function with FFT just one of many possible extrapolations of the original signal? If I have a measured signal which has good ...
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7k 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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30 views

Autocorrelation function and product of summations

Let $u[k]$ be a white gaussian noise with variance $\sigma^2$ and $y[k]=\sum\limits_{i=0}^q b_i u[k-i]$, that is a moving average model. I am trying to compute the autocorrelation function of $y[k]$: ...
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1answer
56 views

Finding period of a square wave with varying sampling frequency

I have a square wave (0-1.8V) with a varying sampling frequency (from a circuit simulator). It is also not a perfect square wave (the high and low signal could be very close to but not precisely zero ...
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1answer
2k views

What is an AMDF?

The wikipedia page for Average Magnitude Difference Function/Formula (AMDF) appears to be empty. What is an AMDF? What are AMDF's properties? What are AMDF's strengths and weaknesses, as compared ...
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56 views

Is there an algorithm for computing auto-correlation in a frame-based manner?

I am trying to find out if an algorithm similar to the Overlap-And-Add method that enables frame-based computation of convolution and cross-correlation exists. When I say frame-based computation I am ...
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1answer
238 views

Matrix cross correlation in python

I'm currently performing matrix cross correlation in python using : C = scipy.signal.correlate2d(A,A) where A is a 2D matrix, typically a picture. As you can ...
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Pitch detection, YIN, pYIN

So here is the "seminal" YIN paper: Cheveigne A, Kawahara H. - YIN, a fundamental frequency estimator for speech and music and the new, improved probabilistic YIN: Mauch M, Dixon S. - PYIN: A ...
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38 views

Interpreting Auto correlation results

I used xcorr function in Matlab to compute the autocorrelation of my signal. I obtained a graph, which is shown below. How to interpret this? I read that autocorrelation helps in finding the patterns ...
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59 views

On the spectral representation of deterministic and random signals

I went back to many references in order to fix some of the confusions that I have on many concepts in signal spectral representation. I concluded that: 1) Deterministic signals may be represented ...
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1answer
64 views

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

Defining: $X(t)$ WSS random process with autocorrelation function $R_{X}(\tau) = \mathbb{E}[X(t)X(t+\tau)]$. $Y[n] = X(nT)$ (sampling of $X$ at a rate $\frac1T$) with autocorrelation function $R_Y(\...
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1answer
48 views

Are there any signals with brickwall autocorrelation?

Are there any signals whose autocorrelation $R(\tau)$ has the following form? Assuming $\tau_c > 0$ and $R_0 > 0$ a constant, $$R(\tau) = \begin{cases}R_0, \text{ for $|\tau| < \tau_c$} \\ 0,...
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2answers
131 views

How to find a variance of sample sequence

I have a sequence such as $$r[n] = y[n]v[n]$$ $y[n]$ and $v[n]$ are zero-mean and statistically independent. I need to find a variance of $r[n]$ and show that it is white and equal to $\sigma ^2_y\...
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2answers
64 views

About using Walsh matrix as spreading sequence

I'm asking about some details about using the Walsh matrix as spreading sequence code. For example, suppose I'm using a $4\times 4$ Walsh matrix given by $$H = \begin{pmatrix} 1 & 1 & 1 & ...
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1answer
39 views

Difference between autocorrelations

M. H. Hayes calls in his book "Statistical digital signal processing" autocorrelation sequences $r[k]$. For optimal filters the desired autocorrelation has the index d -> $r_d[k]$. However, often ...
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1answer
71 views

Simulate time series given temporal auto-correlation functions

Given a random process $x[n] \in \mathbb{R}$ (say of length $N$) and all correlation functions such as: \begin{align} \langle x[i]\rangle\\ \langle x[i]x[j]\rangle\\ \langle x[i]x[j]x[k]\rangle\\ \...
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9k views

Why does the autocorrelation get its peak at zero?

I know that zero shifting in the autocorrelation function is equal to its energy, yet, I would like to understand why the peak is at zero.
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1answer
81 views

Confusion about ensembles and averages in autocorrelation matrices

I just noticed that until now I often don't cared about the scaling of the autocorrelation Matrix in Matlab. I then checked with the book Statistical Digital Signal Processing from M. H. Hayes. There ...