11
votes
Covariance vs Autocorrelation
According to your definition of autocorrelation, the autocorrelation is simply the covariance of the two random variables $Z(n)$ and $Z(n+\tau)$. This function is also called autocovariance.
As an ...
- 92.4k
11
votes
Why do we need to conjugate complex signals in autocorrelation and cross correlation
$\underline{\text{Prologue :}}$
Let me ask you another question. How will you compare two complex numbers $U$ ($a+jb$) and $V$ ($c+jd$)? By comparing magnitude? Subtract them and take real part? ...
- 460
10
votes
What is the difference between convolution and cross-correlation?
As a student I was involved in the same problem as you are.
Let me explain to you in the simplest words without any math.
Convolution:
It is used to convolve two functions. May sound redundant but I'...
- 101
10
votes
I am an imaginary variance
So, we seem to have a power $R_X(0)$ equal to zero, and a mean $\mu_X=\sqrt{\lim_{\tau\to\infty}R_X(\tau)}=1/3$, resulting in a variance
$$\sigma^2_X=R_X(0)-\mu_X^2=-\frac19\tag{1}$$
That would mean ...
- 92.4k
9
votes
Understanding the definition of mean/autocorrelation
The definition of the autocorrelation function $R_x(\tau)$ depends on the nature of your $x$.
If $x$ is a deterministic signal with finite energy then: $$R_x(\tau)=\int_{-\infty}^{+\infty}x(t)x^*(t-\...
- 458
9
votes
When is it true that "the Fourier transform of the autocorrelation is the spectral density"?
The FIRST PROBLEM you ask about is how does
$$R_x(\tau) = E[x(t)x(t+\tau)] $$
equal
$$R_x(\tau) = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^{+T} x(t)x(t+\tau) dt$$
because (7) requires ...
- 26k
9
votes
When is it true that "the Fourier transform of the autocorrelation is the spectral density"?
To add to Peter K.'s answer:
The correct defition of the autocorrelation is $ R_x(\tau) = E[x(t)x(t+\tau)]$.
If the process is stationary, that simplifes a little to $ R_x(\tau) = E[x(0) x(\tau)]$.
If ...
- 254
8
votes
Accepted
Auto-correlation function, an inverse problem
Let's look at the case $x[n] \in \mathbb{R}$, where $x[n]$ is real.
Autocorrelation is basically convolution of the signal with it's time inverse. This can be easily expressed in the frequency domain....
- 48.2k
8
votes
Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches
If we can assume white noise and the signal itself is of constant frequency over the full 15 second duration, then the autocorrelation would be suboptimal in determining the frequency compared to the ...
- 55k
7
votes
Accepted
Guitar pitch detection with autocorrelation
Cons: Not as accurate
This is just compared to the other methods. I was measuring frequency very accurately to look for clock drift, etc: 1000.000004 Hz for 1000 Hz, for instance. For guitar pitch ...
- 16k
7
votes
Accepted
perfect sequences
Let $\theta_a$ and $\theta_c$ respectively denote the maximum magnitudes of the off-peak or out-of-phase periodic autocorrelation functions and the periodic crosscorrelation functions of a set of $K$ ...
- 20.9k
7
votes
Accepted
Correlation of independent random processes
No. Quoting Wikipedia's article Independence (probability theory):
If $X$ and $Y$ are independent random variables,
then the expectation operator $\operatorname{E}$ has the property
$$\...
7
votes
Accepted
Hard time figuring out whether the following random process is wide sense stationary
Let $\{\mathcal Y(t): -\infty < t < \infty\}$ denote a random process defined by
$$\mathcal Y(t) = \sum_{k=-\infty}^\infty \mathcal B_k p\left({t-kT}\right), \ -\infty < t < \infty\tag{1}$$...
- 20.9k
7
votes
Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches
Similar approach to Dan's but a different way to go about it. First lets define what exactly we mean by "peak" frequency. I suggest it is the frequency that minimizes the square error ...
- 48.2k
7
votes
When is it true that "the Fourier transform of the autocorrelation is the spectral density"?
The magnitude-square $\Big| X(\omega) \Big|^2$ of the Fourier Transform of energy signals is the energy spectral density.
For power signals, you gotta do a little bit with normalization, so that ...
- 21.4k
6
votes
Accepted
Confusion about ensembles and averages in autocorrelation matrices
First, the output of the xcorr() function returns lag-0 of the auto-correlation sequence (ACS) estimate at its middle sample, as you recognize. The function argument scaleopt provides normalization ...
- 28.4k
6
votes
Accepted
Allan Variance vs Autocorrelation - Advantages
My current work involves the design details of atomic clocks where we use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary ...
- 55k
6
votes
Estimating period of low frequency oscillations: autocorrelation vs. Frequency approaches
I'd just add to these two fantastic answers that since the frequencies of interest are so low compared to your sampling frequency, I imagine you can also decimate your signal quite a bit before trying ...
- 6,814
5
votes
How can define an indicator which measures the degree of similarity between two signals?
The general topic of finding similarities between signals is wide ranging:
are the signals of same sampling, length, offset, shift or scale?
where do they take their values (discrete, real, complex)?...
- 32.3k
5
votes
Accepted
variance in the time domain versus variance in frequency domain
Variance is never defined as power. For a wide-sense stationary random process $X(t)$ with zero mean
$$\mu_X=E\{X(t)\}=0\tag{1}$$
the variance of $X(t)$ equals its power.
The autocorrelation of $X(...
- 92.4k
5
votes
Accepted
Derivation of $ R_{N(t)}(\tau) $ from its $f_{N(t)}(\eta)$
As @MattL points out in a comment, a Gaussian pdf does not imply whiteness. Indeed, it can be argued that the assumption that the process is a continuous-time white noise process is contrary to the ...
- 20.9k
5
votes
Auto-correlation function, an inverse problem
There is in general, as @Hilmar's answer points out, no unique solution to the question of a sequence that has the given perodic autocorrelation function. In the simplest case, that a shifted ...
- 20.9k
5
votes
Does the auto-correlation function of stationary random process always converge?
No it does not necessarily.
For example the following discrete-time, WSS random process
$$x[n] = A \sin(\omega_0 n + \phi) $$
which is called the random phase sinusoid, where $A$ and $\omega_0 \...
- 28.4k
5
votes
Accepted
Relationship between the autocorrelations of X(t) and X(nt)
The answer to the OP's question is more straightforward than rb-j's comments make it out to be.
$\{X(t)\colon -\infty < t < \infty\}$ is a continuous-time WSS random process with ...
- 20.9k
5
votes
Accepted
Autoconvolution vs Autocorrelation
Are two signals are the same if their auto-convolution functions are the same?
Not quite. Look at the autoconvolution in the frequency domain where the ...
- 20.9k
5
votes
Why autocorrelation can be more efficiently calculated using the fft
The cross-correlation between two functions $f(t)$ and $g(t)$ can be seen as a convolution of $f(t)$ and $g(-t)$. The auto-correlation is of course a special case where $f=g$. This operation in the ...
- 393
5
votes
What is the meaning of **Sample** in **Sample ACF**
It's the empirical ACF computed using the sample.
- 1,117
5
votes
auto-correlation of differentiator
The derivative of the Dirac delta impulse is written as $\delta'(\tau)$. This helps with notation because the mistake you made is to write $h(-\tau)=\frac{d}{d\tau}\delta(-\tau)$, which is not the ...
- 92.4k
5
votes
Method to generate binary sequences with desired cross-correlation and autocorrelation properties
Flattered though I am that a short note (D.V. Sarwate, "Bounds on crosscorrelation and autocorrelation of sequences", IEEE Trans. Information Theory, vol. IT-25, pp. 720-724, 1979) that I ...
- 20.9k
5
votes
Accepted
Confusion In DC power,Average Power,Ac Power and total Power
What you have studied doesn't seem to include a definition of Average Power. For deterministic signal $x(t)$, the instantaneous power delivered at time $t$ is $x^2(t)$ ${\big (}$yeh, yeh, nitpickers ...
- 20.9k
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