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

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Positive definites of correlation functions

Say I you have two time series, $x_k, k=1,2$ generated from two, possibly correlated, complex gaussian processes. The lag 0 and 1 auto-correlation estimate for the two time series is denoted $R_{0,k}, ...
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1answer
19 views

detection of periodicities in n-dimensional signals

Generally speaking, what analyses are necessary and sufficient for the detection of periodicities in an n-dimensional signal amounting to a discretely sampled density distribution over n-dimensional ...
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2answers
48 views

Why use $\chi^2$ test to determine the presence of white noise?

I want to test for the presence of broadband noise in a snapshot 1000 complex baseband samples recorded by a software defined radio. As a follow-up to this post, why was the $\chi^2$ test used? How ...
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19 views

Time Series Extrapolation from existing past values

Using the Levinson-Durbin algorithm, I am trying to predict the next value in a time series based on previous observations, but the results do not follow the trend of the time series. How can I ...
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16 views

On the confidence intervals of matrix autocorrelation

Considering the case of signal autocorrelation in 1D, I've read that the distribution of the correlation coefficients $r$ of a white noise (independent and identically distributed random variable) ...
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0answers
37 views

Find the best (seamless) looping points in an audio signal

I have a mono signal x[n] sampled at 44.1 kHz (a single sustained note of an instrument) with Attack, Decay, Sustain, Release, that lasts about 10 seconds (the ...
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1answer
42 views

Linear Predictive coding vs AR modeling

I'm looking for a suitable explanation of the circumstances in which the LPC error polynomial for a discrete time process x[n] is replaceable with an error polynomial categorized under the AR model? I ...
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0answers
30 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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53 views

Subtracting audio signal emitted - trying to use spectral subtraction to localize moving objects

I am a Software Engineer without much signal processing background and currently spending and experimenting to get use to it. My scenario: Assume a speaker and a microphone array. A speaker emits an ...
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1answer
39 views

What exactly does compression say about correlation of data?

I've been using the following formula on various empirical data $d$, to obtain a correlation factor $c_f$:- $$ c_f = { |C(d_s)| \over |C(d)|} $$ where $C$ is a compression function like bz2 or zip, ...
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1answer
57 views

ACF and impulse response

These are questions according to the system identification. I can understand some questions: Is there a relation between autocorrelation function and impulse response? Is it possible to identify the ...
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1answer
37 views

Auto-correlation of the sum of two generic signals

Be $x[n]$ and $y[n]$ two generic discrete-time signals. Given $s[n] = x[n] + y[n]$ I want to evaluate its autocorrelation $R_s[l]$. By definition (https://en.wikipedia.org/wiki/Cross-correlation): $$...
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3answers
80 views

The explanation of $|R_{XY}(\tau)| \le \sqrt{R_{XX}(0)R_{YY}(0)}$

If i said the explanation of $|R_{XX}(\tau)| \le R_{XX}(0)$ is that in the time domain,any signal wave are the same as itself when it doesn't shift.Then what is the explanation of $|R_{XY}(\tau)| \le ...
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1answer
25 views

Synchronization with a specified sequence

I'm currently creating a project in Matlab where I'm simulating a communication, based on the SSB modulation, between a transmitter and a receiver. I've added a Barker sequence in the trasmitted ...
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1answer
26 views

Null autocorrelation function and stationary

I can show that a process $X(t)$ is Wide Sense stationary (WSS) by showing that $E[X(t)]$ is constant and that its autocorrelation function is in function of $\tau=t_1-t_2$, that is, $R_X(t+\tau,t)=...
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2answers
95 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+...
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1answer
62 views

Autocorrelation of Addition of Two Independent Signals

Given a random signal $ Z \left( t \right) $ which is addition of two independent signals $ X \left( t \right) $ and $ Y \left( t \right) $ with constant parameters $ a $ and $ b $: $$ Z (t) = aX(t) +...
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2answers
57 views

Ornstein Uhlenbeck with drift

The Ornstein-Uhlenbeck (OU) process $dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t $ generates coloured noise with autocorrelation function $R(t) = \langle X_t,X_{t'}\rangle = \...
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2answers
70 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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2answers
79 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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2answers
29 views

Calculation of autocorrelation - does calculation “loop”?

With a delay $l$, autocorrelation is defined as: $$r_{xx}(l) = \sum_{n=-\infty}^{\infty}x(n)x(n-l) = \sum_{n=-\infty}^{\infty}x(n)x(n+l).$$ I want to calculate the autocorrelation of a signal $$x(n)...
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37 views

Autocorrelation Matrix of a AR process

i'm beggining to study signal processing, and i'm having trouble on writting it ( specially defining the signal x(n) ) in matlab and finding the solution for this problem, if anyone could help I would ...
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0answers
28 views

Computing Power spectral Density

Acronyms: Power spectral density, PSD Autocorrelation, AC Hey so I'm in my first ever DSP class and thoroughly enjoy the material, but absolutely suffer in the HW. I have this question ...
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1answer
32 views

Is there a closed form solution for a temporal calibration of two different signals?

Given two signals $f_1(t),f_2(t+dt)$ where $t$ is time and $dt$ is the time delay between the two signals, is there any closed-form solution with respect to $dt$? what are the efficient global ...
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0answers
19 views

Correlation of the amplitude and angle, and various modulations

I am trying to determine if a signal exists at a frequency by sampling, shifting and downsampling (to get the baseband), and autocorrelating the result. The signal is IQ samples from an SDR. I have ...
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2answers
135 views

Autocorrelation of discrete signal $\sin{2\pi fn}$

Determine the autocorrelation $r_{xx}[m]$ of the discrete signal $$x[n] = (\sin2\pi fn).$$ where $n$ and $m$ are integers. Using the definition I get $$\begin{align} r_{xx}[m] &= \...
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0answers
11 views

computing auto correlation for some images

would you please help me how to compute auto correlation for some images (in the form of some 2D matrices)? for example, how can I compute auto correlation for these 3 matrices manually?
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2answers
66 views

Derivation of $ R_{N(t)}(\tau) $ from its $f_{N(t)}(\eta)$

How can we prove the auto-correlation function of white gaussian noise $\{ R_{N(t)}(\tau) \}$ is $\frac{N_0}{2} \delta(\tau)$ from its p.d.f in equation below? $$ f_{N(t)}(\eta)=\frac{1}{\sqrt{2 \pi \...
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2answers
65 views

How to calculate time of arrival using room impulse response

I need to find the timestamp of arrival of a particular signal at a receiver location within a room. The known parameters are the room impulse response h for a ...
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1answer
32 views

Finding number of independent samples using autocorrelation

I have a pressure signal (y) with 512000 samples and with a sampling frequency ...
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2answers
63 views

Autocorrelation of a Shifted Sequence

Suppose I have a sequence $x[k]$ with $\mathcal{Z}$-transform $$ X(z) = x_{0} + x_{1}z^{-1} + x_{2}z^{-2} + \ldots + x_{N-1}z^{N-1}$$ I know that for real-valued $x[k]$ the $\mathcal{Z}$-transform ...
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57 views

Average of autocorrelation functions

Let $X$ and $Y$ are two non-random time series of length $N$ with $\rho_i^{xx}$ and $\rho_i^{yy}$ are the autocorrelation functions of lag $i$, respectively. What is the autocorrelation function of $Z$...
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1answer
66 views

Doubt about wide sense stationary random process

I have white Gaussian noise $F[n]$ with zero mean and autocorrelation $R_F[n_1,n_2]=\delta[n_1-n_2]$. If now I consider the random process defined as $$X[n]=u[n]e^{-kn}F[n]$$ Is $X[n]$ a wide-ense ...
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1answer
357 views
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42 views

Autocorrelation BPSK Transmit with Binary Sequence (PRN Code)

I am working to design the Transceiver using a SDR hardware. In the transmitter side, the BPSK modulation was used to modulate a binary sequence number which has good autocorrelation properties. The ...
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32 views

what is the relation between autocorrelation lags and fundamental frequency?

I am studying autocorrelation based feature paper. In that they extracted autocorrelation feature and after that they plotted graph lags vs fundamental fundamental frequency...I am not getting ...
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1answer
258 views

proof of Autocorrelation property of DFT

I am facing problem in proving the Auto-correlation property of Discrete Fourier Transform (DFT), that is $$\mathcal{DFT}\left\{\sum_{r=0}^{N-1}x[r]x^*[r+n]\right\} = X[k]X^*[k]= |X[k]|^2$$ where $X[...
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1answer
718 views

variance in the time domain versus variance in frequency domain

Hi All: I'm trying to better understand the connection between variance of a time series and the integral of the spectral density over all frequencies. Rather than going through all of the relations, ...
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2answers
305 views

Definition of average power?

There are two kind of average power I encountered in random signal class and textbook: definition 1: average power =$$E[|x(t)|^2]=R_{xx}(0)=\int^\infty_{-\infty} S_{xx}(f)\,df$$ definition 2: average ...
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1answer
70 views

Autocorrelation and PSD

Let $X(t)$ and $Y(t)$ be two orthogonal processes with power spectral densities $$S_{xx}(f) = S_{yy}(f)=\begin{cases} 1-\lvert f\rvert, & \lvert f\rvert<1 \\[1ex] 0,& \text{otherwise} \end{...
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4answers
177 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 ...
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1answer
164 views

Solving Wiener Hopf integral equation for causal filter of predictor

Given a stochastic signal $x(t)$ with autocorrelation function $R_{xx}(\tau)=\mathrm{exp}(- \alpha|\tau|)$, $\alpha>0$. I want to predict $x(t+\lambda)$,$\lambda>0$ by $x(t-\tau)$, $\tau\ge0$ ...
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0answers
28 views

Calculate Autocorrelation-Sequence In A System Identification Scenario With Sensor Noise

The system below is given. ${\bf c}$ are the coefficients of the adaptive filter (filter order 3), which shall approximate the system $$ H(z) = 3 + z^{-1} $$ $S_1(z)$ and $S_2(z)$ are transfer ...
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30 views

Is it possible to find out the mean of a WSS process purely from the autocorrelation? [duplicate]

Suppose that I have a wide-sense stationary random process $X(t)$ with auto-correlation $$R_X(\tau) = \frac{1}{\sqrt{4\pi}}e^{-\tau^2/4}$$ Is it possible to find out the mean $\mathbb{E}[X(t)]$ of ...
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1answer
119 views

Autocorrelation function

For what signal applies, that its autocorrelation function at point $0$ is zero, i.e. $R_f(0) = 0\ ?$ I know that autocorrelation is (RootMeanSquare)^2 and for $R_f(0)$ this equals: $$ R_f(0) = \frac{...
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0answers
47 views

Web Audio convolver to detect power and phase of a known but noisy signal?

I need to determine the power and phase shift of a known audio signal that is reflecting in a chamber and I would like to do it using the Web Audio API. From a dated signal processing background, I ...
5
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2answers
135 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 ...
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0answers
35 views

Doing Autocorrelation and Spectral Analysis on Sparse Time Tagged Data

The system I am working with should extract periodic signals using very sparse samples. Multiple independent signals might be present at the same time, all with their own time-domain behavior. Because ...
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0answers
47 views

Visualizing audio FFT - LEDs

I am trying to visualize an audio file with LEDs (RGB). So, first thing I will do is an FFT with 1024 samples on 44100 Hz signal, which will give me 512 buckets of ~86 Hz per bucket. Depending on the ...
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28 views

Bandwith of the Gaussian Random Process

$X(t)$ is a zero-mean WSS Gaussian random process. Its power spectral density is $$S_x(f)= (4 \times 10^{-5}) \text{tri}(\frac{f}{10^{5}})$$ Where $\text{tri}$ is the triangle function. What is the ...