Questions tagged [autocorrelation]

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

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47
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4answers
43k views

What is the difference between convolution and cross-correlation?

I've found on multiple sites that convolution and cross-correlation are similar (including the tag wiki for convolution), but I didn't find anywhere how they differ. What is the difference between ...
31
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1answer
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?
17
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1answer
1k views

What Does Make an Error Surface Convex? Is It Determined by the Covarinace Matrix or the Hessian?

I am currently learning about least-squares (and other) estimations for regression, and from what I am also reading in some adaptive algorithm literatures, often times the phrase "... and since the ...
14
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1answer
1k views

On using eigenvectors to estimate a signals' fundamental frequency, via MUSIC

Context: (Disclaimer: This is NOT a comm problem). I am trying to estimate the fundamental frequency of a real, periodic signal. This signal, was constructed by match filtering a raw signal, to that ...
12
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4answers
19k views

Efficiently calculating autocorrelation using FFTs

I'm trying to calculate an autocorrelation on a platform where the only accelerated primitive I have available is the (I)FFT. I'm having a problem though. I prototyped it in MATLAB. I am, however, ...
12
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4answers
2k views

Recommendation for book - Writing DSP code in C

I am looking for some good book, that simply show how you actually write a code in C, to do all the main DSP methods . FFT. Low-pass and high-pass filters. Auto-correlation. Noise processing. And ...
12
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4answers
13k views

How to 'whiten' a time domain signal?

I am trying to understand how exactly to implement what is known as a 'pre-whitening' filter or simply a 'whitening' filter. I understand that the purpose is to make it have a delta as its ...
11
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3answers
20k views

Autocorrelation in audio analysis

I'm reading up on Autocorrelation, but I'm not sure I understand exactly how it works and what output I should expect. Am I right in thinking that I should input my signal to the AC function and have ...
11
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3answers
10k 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.
11
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2answers
8k views

Covariance vs Autocorrelation

I'm trying to figure out if there is a direct relationship between these concepts. Strictly from the definitions, they appear to be different concepts in general. The more I think about it, however, ...
10
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1answer
363 views

Designing a feature vector for discriminating between different sonic waveforms

Consider the 4 following waveform signals: ...
9
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1answer
3k 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 ...
9
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1answer
6k views

Auto Correlation vs Cross Correlation vs Convolution and their applications

I know from wikipedia that auto correlation in done on the same signal while cross correlation is done on different signals.But what does this actually imply in terms of application.I can always apply ...
9
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1answer
238 views

Unit Problem in Designing a Filter for a Given Auto Correlation Function

Given a WSS process with the following Auto Correlation Function: $$ r\left ( \tau \right ) = {\sigma}^{2} {e}^{-\alpha \left | \tau \right |} $$ The Laplace Transform would be: $$ R \left ( s \...
9
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2answers
1k views

Best way to evaluate “quality” of autocorrelation?

This is a side-trip from my snoring app. I had a crack at producing an autocorrelation of the audio signal, to see if that "correlates" with snoring/breathing very well. I have a simple algorithm ...
8
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2answers
2k views

Why do we deal with the eigenvectors of the autocorrelation instead of the data itself?

How intuitively to understand why eigenvectors of the autocorrelation matrix are used, but eigenvectors of the matrix constructed from temporal samples have no sense and aren't used? For example, in ...
6
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4answers
8k views

Conceptually, how does a Fourier transform differ from an autocorrelation?

I realize the two are derived using different algorithms, and the units are different, but from a conceptual standpoint of the information they provide how do they differ? I'm thinking here about the ...
6
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2answers
443 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 ...
5
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3answers
433 views

Extraction of non-sinusiodal repetition rates

I have an auto-correlation function that was generated from a signal, and I am trying to extract its 'repetition rate' in order to calculate the dominant frequency of the pulse, but I am not exactly ...
5
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2answers
582 views

Autocorrelation Jitter

I'm doing autocorrelation (via fft) on pure, synthetic generated sinusoids and getting unexpected jitter in the results. The jitter depend on where in the sine the signal buffer is started. I get ...
5
votes
1answer
3k views

Autocorrelation of a telegraph process/constant signal

I am trying to calculate the autocorrelation function for the telegraph process, but I somehow don't get the right results. I am calculating the correlation function of a signal X which can take ...
5
votes
1answer
626 views

Echo cancelling using autocorrelation function

I was given a problem, but I couldn't solve. I did some researches but I still didn't figure it out. Here is the problem: An audio signal $ s(t) $ is generated by a speaker reflects in a wall with ...
5
votes
1answer
473 views

Assumptions for Hurst exponent calculation

Are there any general assumptions for the calculation of the Hurst exponent? Does the signal need to be stationary, for example? Does it depend on the method? What about the length of the time ...
5
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1answer
548 views

Relaxation time in a stationary stochastic process

Foreword: I'll give some background here because I'm not sure all my premises are correct. I encourage you to IGNORE THE BACKGROUND, unless you're really interested / bored :) The Question I want to ...
4
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1answer
1k views

Generating good synchronization sequences

As part of my work I am putting together a simple bursty BPSK communication system. I want to make the demodulation easy, so I am going to put a synchronization sequence at the beginning of each ...
4
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2answers
146 views

Difference between $\mathbb{E}[\mathbf{x} \mathbf{x}^{\rm{H}}]$ and $\mathbb{E}[(\mathbf{x}-\boldsymbol{\mu}) (\mathbf{x}-\boldsymbol{\mu})^{\rm{H}}]$

Let us have a random vector $\mathbf{x} \sim \mathcal{CN} (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} \neq \mathbf{0}$. What can we say about the relationship between the elements ...
4
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1answer
1k views

Autocorrelation of the product of deterministic and random signal

I was wondering how to calculate the autocorrelation of a deterministic signal $x(t)$ multiplied by a stochastic process $M(t)$, whose autocorellation $R_M(\tau)$ is known a priori. In my case, $x(t)$ ...
4
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1answer
993 views

Fourier transform of Autocorrelation function, what am I missing here? (Power spectrum)

Given this problem I know that the autcorrelation and power spectrum are fourier pairs, so when taking taking fourier transform of Rxx, one should end up with Sxx However, when I take the transform ...
4
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2answers
309 views

Auto Correlation for Time Frequency Analysis

Given a signal $x(t)$, how do I implement a form of autocorrelation function defined as $a(t,T) = x(t-T)x(t+T)$, where $T$ is an arbitrary constant? (a fast implementation would be ideal) Edit: ...
3
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4answers
2k views

Auto correlation correspondence to original time domain window

I'm trying to figure out how exactly an auto correlation corresponds to a time domain signal. Now I'm trying to find pitch periods in an audio file and 99% of the time I'm getting it spot on. ...
3
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3answers
1k views

Guitar pitch detection with autocorrelation

This link provides code for an autocorrelation-based pitch detection algorithm but says: Cons: Not as accurate, doesn't work for inharmonic things like musical instruments, this implementation has ...
3
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2answers
221 views

Auto-correlation function, an inverse problem

$x[n]$ is a complex function $n=0,1,2,\cdots,L-1 $ we assume $x[n]$ is periodic in its index: $x[n+L]=x[n]$ Its auto-correlation function $C[n]$ is uniquely defined as: $$ C[n]=\sum_{i=0}^{L-1} x[i+...
3
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2answers
327 views

How do we know that CTFT of the autocorrelation function is the PSD?

I know that the Fourier Transform of the autocorrelation function is the Power Spectral Density. But how can we arrive at such a result intuitively? Is it just a theorem?
3
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4answers
594 views

Can this be considered wide sense stationary?

I was discussing this problem with one of my classmates. The picture shows a recording of the heart rate during before and after sleep. Can the whole process be considered wide sense stationary? (I ...
3
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2answers
3k views

Choosing Codes or sequences with excellent Auto-Correlation properties

The Auto-Correlation function of Walsh-Hadamard codewords does not have a good characteristics. It can have more than one peak and thus, the Walsh-Hadamard codes do not have the best spreading ...
3
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2answers
135 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\...
3
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2answers
12k views

Applications of Correlation in Signal Processing

There have been discussions on the differences between convolution and correlation, autocorrelation, cross-correlation, et al. In spite of looking at the visualizations, and understanding the formula ...
3
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1answer
5k views

AutoCorrelation using FFT of power spectrum

I have a chunk of data (samples) that need to do autocorrelation using FFT method. All data are real and lies between -1 and +1. What I have done follows: Zero pad the window with the length equals ...
3
votes
1answer
954 views

Python different autocorrelation with FFT and non-FFT

Im trying to calculate the autocorrelation of soundwaves when I noticed that I get different results with scipys FFT based and with numpys methods. The 4 functions Im using: ...
3
votes
1answer
638 views

Finding System transfer function, $H(z)$ or, equivalently, its impulse response, $h[n]$

I just started on DSP and I have a question that I would like to ask. I have a zero-mean uncorrelated wide sense stationary discrete-time random process {$x[n]$, $n$ is a set of integers}. $x[n]$ is ...
3
votes
1answer
653 views

How to get the autocorrelation of this finite discrete function?

This is the function: $$ x(n) = \begin{cases} 1, & \text{-N $\leq$ n $\leq$ N ;}\\ 0, & \text{otherwise.} \end{cases} $$ And I tried to solve it, using sigma notation or summation: \begin{...
3
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1answer
3k views

Autocorrelation - Stochastic vs deterministic processes

On one hand, there is the statistical definition: $R_x(\tau) = \frac{\operatorname{E}[(X_t - \mu)(X_{t+\tau} - \mu)]}{\sigma^2}$ and on the other hand, there is this other definition (e.g. see Dilip'...
3
votes
1answer
133 views

PSD from autocorrelation in MATLAB

I am trying to simulate a simple stochastic process defined by the equation: \begin{equation} \frac{1}{v}\frac{db}{dt} +\Gamma_0 b= \sqrt{\sigma}R(t), \end{equation} where $R(t)$ is a zero-mean white ...
3
votes
1answer
160 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 ...
3
votes
1answer
297 views

Decorrelating Stationary Colored Gaussian Noise — Effect On The Desired Signal

So given stationary colored gaussian noise $\mathbf{n}$, I know that I can decorrelate it by first finding it's autocorrelation $R_{nn}$ and performing $R^{-\frac{1}{2}}_{nn} \mathbf{n}$. In ...
3
votes
2answers
2k views

Algorithms for finding fundamental frequency based on ACF result

I am developing a software for fundamental frequency tracking. For this purpose, I have designed a function which calculates autocorrelation over the signal and a second function which, based on ...
3
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0answers
336 views

What is the autocorrelation equivalent of the spectrogram called?

I'm very knowledgeable about the differences between the Fourier transform, and the autocorrelation; mainly that one converts the time domain to the frequency domain, and the other finds periodicities ...
3
votes
4answers
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 ...
3
votes
2answers
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 ...
3
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0answers
337 views

Remove the extra peaks by lag

Suppose that we have some peaks of a signal and we know the time lag between dominant peaks. How is it possible to remove the extra peaks by applying time lag ? Lets say it in MATLAB ...