Signal Processing

Questions tagged [autocorrelation]

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

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115 views

Difference between reference and measurement signal?

Which differences can we find on various signals? We assume that we have a reference signal (determined theoretically) and measured signal. Which methods exist for determining the similarities ...
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28 views

Identify and remove repeated audio chunks

I have some audio produced by a buggy recorder, at some point it may start two recording streams, each stream captures chunks of audio and write to a file. The buffer queue is processed asynchronously,...
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25 views

2D space and 1D time evolution of a random field

I also asked this on math stack-exchange, but it is also relevant for the signal processing community. I want to develop a 2D random field and its change with time with constant velocity. My process: ...
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1answer
48 views

How to properly band-pass filter an autocorrelated signal?

Let's consider an example ECG time series, sampled at 360Hz: I verified that autocorrelating a signal preserves its frequency content. For instance, both original and autocorrelated data show the ...
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36 views

Auto/cross correlation for data set with multiple realizations

Context For an assignment, I need to find the correlation matrix of a dataset with 100 random variables and 1000 realizations for each. (It's a stock market example). Problem I don't understand how ...
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1answer
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Kay Statistical Signal Processing Estimation Theory Example 7.14

Example 7.14 from Kay Estimation Theory A common WSS random process has the ACF $$r_{xx}[k]= \begin{cases} 1+b[1]^{2}+b[2]^{2} & \text{k = 0}\\ b[1] +b[1]b[2] & \text{k=1}\\ ...
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1answer
40 views

Noise model as Sinc function

In "Fundamentals of signal processing: estimation theory", example 3.13 Kay has used a model of band-limited Gaussian noise with a uniform PSD and a sinc autocorrelation function. I expected ...
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1answer
20 views

Beep Signal for Latency Calculation

I want to play audios on two mobiles with sync. But mobile phones are not in sync. I am trying to find two Mobile phones playing latency so I can pad latency on each phone so they can play together. ...
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1answer
75 views

Symmetries of analyticity / zero self-correlation

I seek to understand symmetry properties of analytic sequences, without referring to frequency domain: what criteria must a complex sequence $x[n]$ satisfy to be analytic? Framed alternatively, such a ...
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1answer
52 views

How this convolution integral is solved to obtain autocorrelation function of current for shot noise?

I am studying shot noise characteristics from this source: Here the author writes that autocorrelation function is given by: $$R_I(\tau)=\bar{h}*h*R_Z(\tau)$$ where $R_Z(\tau)=q^2(\lambda^2+\lambda \...
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33 views

Relation between bandwidth and amplitude of auto correlation of chirp signal

I am trying to understand the properties of chirp signals and their characteristics. I plotted the autocorrelation of a chirp signal in MATLAB. How can I compute the relation between the amplitude of ...
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2answers
33 views

Should upsampling impact the amplitude of an autocorrelated signal?

I want to autocorrelate a signal using numpy's correlate method. Let us consider a 10 minutes long signal sampled at 2,000SPS: ...
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51 views

Restrictiveness of the condition that different signals give same autocorrelation sequence

I'm having a set of N real data points that correspond to the autocorrelation sequence at N different lags. Suppose I know that the original set of data from which the autocorrelation was computed ...
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75 views

Create a Range Doppler Map using "Arbitrary" phase coded Waveforms

This is my first questions I am asking to this community! I want to understand how to create a Range-Doppler Map (RDM) for a phase/frequency coded radar return using an arbitrary waveform generator (...
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1answer
59 views

Method to generate binary sequences with desired cross-correlation and autocorrelation properties

I am looking for a method to generate two binary sequences of length $N$ with the following properties: Very good cross-correlation for all shifts. Good enough autocorrelation for all shifts except ...
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1answer
201 views

Allan Variance vs Autocorrelation - Advantages

I am currently studying oscillator stability and have come across the Allan variance. I gather that it was developed as an alternative to the standard variance as it doesn't necessarily converge for ...
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How to select the ARMA model parameters?

I have a series of data containing 120,000 points. The mean of each N(=60) point is zero. I want to forecast the next 60 points using the ARMA model. My question is, specificaly, how to choose the ...
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1answer
163 views

Extremes of integral of the autocorrelation theorem

I am trying to apply the correlation theorem (here attached) to this definition of correlation of a finite continuous signal $g(t)$: $\rho(\tau)=\frac{1}{T} \int_{t_{0}}^{ t_{0}+T}g(t)^{.}g(t+\tau)dt ...
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Question about a demonstration of the autocorrelation theorem

I’m stuck on this demonstration. It says: If $x'$ is held constant, $dx=dy$ How can $x’$ be constant? Shouldn't $x’$ be a variable?
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1answer
86 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
58 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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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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29 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
60 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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0answers
66 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
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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
264 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
116 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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43 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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28 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
74 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
489 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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97 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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39 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
181 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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43 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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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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39 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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633 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
130 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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75 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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147 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
232 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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1answer
68 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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95 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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556 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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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
32 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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33 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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73 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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