Questions tagged [autocorrelation]

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

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209 views

LTI filtering for wide-sense stationary process

Why is it that if $U[n]$ is wide-sense stationary and it is convolved with $h[n]$ to produce $W[n]$, the autocorrelation becomes $R_{WW}[n] = R_{UU}[n]*h[n]*h[-n]$? I know that in general $R_{WW}[n_{...
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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 ...
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1answer
440 views

finding period with autocorrelation is not correct

I have recorded a signal, which I know is periodic (apart from noise). The period length is unknown. I want to extract the last period from the signal. Before going to a noisy signal, I first tested ...
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1answer
2k views

xcorr in MATLAB for periodic function

I have a periodic signal and I want to find it's autocorrelation function. I can calculate it exactly: $$R_{uu}(h) = \frac 1M \sum_{k=0}^{M-1} u(k)\cdot u(k-h)$$ But will ...
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1answer
718 views

What's wrong with this Average Magnitude Difference algorithm?

I implemented this Average Magnitude Difference algorithm in Javascript ...
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1answer
1k views

Angle of arrival (AOA) estimation using FMCW radar using MUSIC algorithm

I am working with FMCW phased array radar with only upchirps. The range doppler matrix is obtained using the two dimensional fast Fourier transform on multiple chirps. I want to obtain the angular ...
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3answers
322 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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1answer
837 views

Time-domain cross-correlation with padded signals [closed]

I have a frequency domain cross-correlation implementation written in C (based on: https://github.com/dMaggot/libxcorr). It uses the library FFTW3 and this is the gist of it: ...
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3answers
116 views

Weighted Sum of Auto Correlation - Lower Bound

Given a vector $ v $ with elements $ {\left\{ {v}_{n} \right\}}_{n = - \infty}^{\infty} $ and denoting $ {v}_{n}^{\left( k \right)} = {v}_{n - k} $, namely, a shifted vector by $ k $ elements (Mind ...
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1answer
677 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 ...
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1answer
324 views

Fourier Transform gives unexpected results: signal reversal and conjugation

As far as I understand the math you can reverse a real valued signal by Fourier transformation, taking the complex conjugate of the result and inverse Fourier transformation, i.e. \begin{equation} \...
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3answers
207 views

How can define an indicator which measures the degree of similarity between two signals?

The similarity of two signals is calculated by cross correlation. But, how to define an indicator which quantitatively measures the degree of similarity between two signals? Thanks.
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72 views

Dependence of $\tt n$ in $\tt fft(data,n)$ in auto correlation of data

The followings are results of autocorrelation of an 'image'(1*) (subplot 221) using xcorr, n point ...
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1answer
269 views

Should I observe a signal multiple times for MUSIC algorithm?

In a paper Multiple emitter location and signal parameter estimation, the well-known MUSIC algorithm starts by calculating covariance (autocorrelation, whatever) matrix with: where overlines stand ...
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1answer
2k views

Random process $X(t)$ with autocorrelation function given find the mean and the variance

Autocorrelation function is $$R_{xx}(\tau)=\frac{20}{1+2\tau^2}$$ So at $\tau=0$$$R_{xx}(0)=20=E[X(t)X(t)]=E[X^2(t)]$$ The variance is $$\mathrm{Var}[X(t)]=E[X^2(t)]-E^2[X(t)]=20-E^2[X(t)]$$ As $X(t)$...
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246 views

Power spectral density of $\left(x(t)\right)^2$?

The relation between $x(t)$ and output $y(t)$ of a non-linear device is expressed as $$y(t) = (x(t))^2$$ Let $x(t)$ be zero-mean stationary Gaussian random process with auto-correlation $$R_x(...
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How to preprocess such signals?

I am interested in denoising accerelation measurements, recorded in ambient vibration tests. Such tests consist in recording the vibrations of a mechanical structure, say a table for example. So say I ...
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1answer
38 views

Testing for changes in auto-covariance

I am working with uniformly-spaced time series data where I am interested in knowing whether there are changes in temporal auto-covariance. The mean can be assumed constant. Visually, there are no ...
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1answer
820 views

How to estimate the autocorrelation from nonuniformly spaced data

Assume a continues-time random process $X(t)$ sampled nonuniformely in time to acquire discrete signal $x[n]$. The sampling times are known but the autocorrelation is not. Is there an accurate ...
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1answer
330 views

Help in interpreting the auto correlation graph

I want to check if a time series is (a) random (b) independent. For these I am using the autocorrelation (AC). Autocorrelation refers to the correlation of a time series with its own past and future ...
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2answers
1k views

Power spectral density of sinusoidal signal in noise

In a literature I face with this input and power spectral density (PSD) $$x(t)=s(t)+n(t)=A\cos\left(\omega_c t +\phi\right) + n(t)$$ first I want to know How can we find PSD of $\cos\left(\omega_c t ...
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3answers
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Calculating the fundamental frequency using Autocorrelation often gives half the expected value

I'm currently writing a mobile app which needs to analyse musical notes and find the fundamental frequency to determine the pitch. To do this I'm reading in audio data, taking an FFT, taking the auto-...
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1answer
767 views

Does the Wiener–Khinchin-Einstein theorem hold for non-Gaussian processes? If so are there any assumptions?

The Wiener–Khinchin-Einstein theorem states that the auto-correlation $(r_{xx}(\tau))$ and spectral density $(S(f))$ are Fourier duals, i.e. $$r_{xx}(\tau) = \int^{+\infty}_{-\infty} S(f) \exp\left( ...
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1answer
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Autocorrelation matrix derivation

Hi I am trying to derive the autocorrelation matrix and I am unsure about how exactly it works. I can see that the $4\times 1$ matrices result in the Hermitiain and Toeplitz matrix? Surely the only ...
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1answer
444 views

Normalized autocorrelation of a sum of two signals

I obtained the autocorrelation function of a sum of two signals: as but I want to know what is the normalized autocorrelation of above equation?
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308 views

Event detection in a running person's acceleration data

I am trying to detect some events on this signal : This is the acceleration of someone running. By eyes we can detect many blocs, sometimes separated by a little interval. I would like to be able to ...
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3answers
344 views

Could there be any reason to prefer convolution-based calculation of autocorrelation?

Theoretically both of ways of calculating autocorrelation function are identical: strightforward convolution and Fourier-based method where we use FFT/iFFT in practice. And as it is well known, the ...
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2answers
419 views

What does the frequency axis of a Power Spectral Density mean?

I have never really understood what the frequency axis meant when we plot the Power Spectral Density(PSD). Does it correspond to frequency as we get after we take the Fourier Transform of a time ...
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1answer
673 views

Finding the fundamental frequency from Autocorrelated data

I'm writing an app in which I need to find the fundamental frequency of a note produced by a trombone. To do this I'm taking the FFT of audio data from a microphone and then using autocorrelation code ...
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5answers
10k views

Where does the delta function come from if we derive autocorrelation directly?

I am reading a book "Creating Noise" written by Hollos & Hollos and have a question about the autocorrelation function of the Gaussain white noise when reading the following passage: From the ...
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Autocorrelation of a $L$-point moving average system

The $L$-point moving system is: $$y[n]=\frac 1L\sum_{k=0}^{L-1}x[n-k]$$ $x[n]$ is a Bernoulli random signal with $\beta=0.5$ (equal probability) The autocorrelation of $x[n]$: $\psi_{xx}[m]=\delta[...
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1answer
532 views

Finding the deterministic autocorrelation function (ACF) from its power spectrum

The power spectrum of a stationary discrete-time random signal is $$\Phi_{xx}(e^{j\omega})=\begin{cases} 1 & |\omega|<\pi/2 \\ 0 & \pi/2 <|\omega| \le\pi \end{cases} $$ (a) ...
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2answers
321 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: ...
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1k views

AutoCorrelation Matrix vs Covariance Matrix for the MUSIC Algorithm [closed]

What is the difference between an autocovariance matrix and autocorrelation matrix?
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48 views

How is the energy of $x_1\cdot x_2$ related to the energies of $x_1$ and $x_2$?

Let's say first signal x1 = [1 2 3 4], second signal x2 = [0.08 0.77 0.77 0.08] (Hamming window), third signal ...
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1answer
1k views

How to prove that the peak of the autocorrelation function is at zero lag?

Show that for a signal $f(\tau)$ with finite energy and energy autocorrelation function $\phi^e_{ff} (\tau),$$$|\phi_{ff}^e (\tau)| \leq \phi_{ff}^e (0), \ \ \forall \tau.$$ According to my textbook ...
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1answer
727 views

Golay complementary sequences in 802.11ad

The Golay complementary sequences consist of two sequences of the same length $N$ such that there acyclic autocorrelation sequences have sidelobes equal in magnitude but opposite in sign. So, when the ...
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1answer
316 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 ...
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1answer
2k views

Determining the fundamental frequency/pitch of a note

What I am trying to achieve is getting the fundamental frequency of a note played by an instrument. What I have already done is performing an FFT on a samples of audio file, and here's what I get: ...
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1answer
148 views

How is the matrix $\mathbf R_x$ not Toeplitz in case of a signal missing one term?

I am solving a question that says if we have sequence $x(n)$ of a signal missing one term then we have to find autocorrelation matrix $R_x$ as follows: $$R_x = E\{\mathbf {xx^H}\}$$ Now if I take $...
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353 views

Finding phase of fundamental from autocorrelation

I have some code, shown below that I have been using to find the fundamental of a guitar string. After this function is called, I go thru the real buffer and find the peak, whose location gives me ...
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1answer
739 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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1answer
884 views

Calculation of an autocorrelation function

A sample of a random process is given as: $$ x(t) = A\cos(2\pi f_0t) + Bw(t) $$ where $w(t)$ is a white noise process with $0$ mean and a power spectral density of $\frac{N_0}{2}$, and $f_0$, $A$ ...
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1answer
236 views

Evaluating the spectral density of generated noise through the autocovariance

I'm working on generating noise signals $X(t)$ (with $t \in \left[0,T\right]$ with step size $\delta t$) with a prescribed power spectral density $S_{XX}(f)$ and I'm figuring out how well I am ...
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4answers
2k views

Finding the Amplitude of a Sinusoid in Noise

I'm trying to solve for the amplitude and frequency of a sinusoid embedded in zero-mean gaussian white noise. I am supplied a sample file of a 40000 element array. I first took the autocorrelation ...
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0answers
23 views

Power Spectral Density Via Rank 1 Update

I know that I can find the autocorrelation matrix of a series of finite length sequences via rank-1 updates using the relation: $\mathbf{R}[k] = \frac{1}{k}\mathbf{R}[k-1] + \frac{1}{k+1}\mathbf{r}[k]...
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1answer
580 views

Evaluation of Autocorrelation and Power Spectral Density of white noise through a filter

So say there's a filter with an impulse response of $h(t) = (0.8)^t u(t)$. I'd like to pass white noise through this and figure out the autocorrelation and power spectral density of the output. I'm ...
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1answer
175 views

Autocorrelation question

I am trying to get an understanding of autocorrelation and I am having some issues with trying to understand the process. I have a Bernoulli process called $X[t]$. In this process, $P(X[t] = 1) = p$ ...
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1answer
220 views

Autocorrelation of a noisy linear map

I am interested in calculating the autocorrelation function of a linear map with some noise (model given below) but am slightly confused in doing so. At first, I did not realize there were two ...
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1answer
188 views

Auto correlation definition

My question has to do with the definition of auto correlation/cross-correlation for random processes. Oppenheim/Schafer (Discrete time Signal Processing, Pg. 815 (Appendix A.2),2nd ed.) define auto ...

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