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? ...
abhilash's user avatar
  • 460
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 $K$ ...
Dilip Sarwate's user avatar
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

Can white noise be correlated to a random signal

Can white noise be correlated to a random singal Something can be correlated to something else but still be white. For example, let $X(t)$ be a white noise process, and let $Y(t) = 2X(t)$, then ...
Marcus Müller's user avatar
6 votes
Accepted

Cross correlate a 2D array

What you have (conceptually) is not a 2D array but a collection of 1D arrays. correlate2D is designed to perform a 2D correlation calculation, so that's not what ...
lxop's user avatar
  • 1,369
6 votes
Accepted

How does DCT decorrelate images?

[EDIT] In 1991, Nasir Ahmed wrote: "How I Came Up with the Discrete Cosine Transform". Interesting to read, on how he was inspired by Chebyshev polynomials, and on how he didn't get funding,...
Laurent Duval's user avatar
5 votes
Accepted

Cross-correlation or cross-covariance of non-zero mean signals

What are reasons to choose for cross-correlation or cross-covariance when comparing signals with non-zero mean? Well, part of the issue is that cross-correlation as defined in your equation: $$(f \...
Peter K.'s user avatar
  • 25.7k
5 votes

what does it mean to have a decorrelated colour space?

edit: to be clear this answer describes why Lab can be described as a decorrelated color space. This does not imply that decorrelation is the main benefit of using Lab (see many answers on why Lab is ...
Chandler's user avatar
  • 204
5 votes
Accepted

Maximum cross-correlation coefficient value for time delay estimation

As your plot shows, the second form allows for the correlation peak to be negative. Now, what does a strong negative cross correlation mean? It means the signals are very similar, except one has a ...
Florian's user avatar
  • 2,443
5 votes
Accepted

convolution vs correlation?

Convolution: $$ y(t) = h(t) \circledast x(t) = \int\limits_{-\infty}^{\infty} h(u) \, x(t-u) \ \mathrm{d}u $$ Cross Correlation: $$ R_{xy}(t) = \int\limits_{-\infty}^{\infty} y(u) \, x(t+u) \ \...
robert bristow-johnson's user avatar
4 votes

Why correlation property of $\mathcal Z$-transform contains a time reversal operation

Note that (discrete-time) convolution is defined as $$x_1[n]\star x_2[n]=\sum_kx_1[k]x_2[n-k]\tag{1}$$ and correlation is defined as $$r_{x_1,x_2}[n]=\sum_kx_1[k]x_2[k-n]\tag{2}$$ Comparing $(1)$ ...
Matt L.'s user avatar
  • 89.5k
4 votes
Accepted

Cross Correlation, how can any signals except the trivial cases be uncorrelated?

Tl;DR version: You are not missing anything; finite-duration signals cannot be uncorrelated signals. If the crosscorrelation function $x\star y$ of $x$ and $y$ is zero everywhere, then the Fourier ...
Dilip Sarwate's user avatar
4 votes
Accepted

What is definition of independent random variable

No, NO and $\mathbf{NO}$. Knowing that $E[X]=E[Y]=0$ and $E[XY]=0$ (or more generally that $E[XY]=E[X]E[Y]$ or equivalently, $E[XY]-E[X]E[Y]=0$) does not help in the least in proving or deducing that ...
Dilip Sarwate's user avatar
4 votes

Identify which of the three signals is closest to a sinusoidal curve

Is it sufficient to identify the «most sinuoidal» of those 3, or would you also want linear projections of those (consistent with a IMU sensor tilted vs the plane of motion)? A simple solution might ...
Knut Inge's user avatar
  • 3,350
4 votes
Accepted

Estimate the time delay of two signals

The issue is we are looking for likeness but the values are scaled beyond that metric. In the OP’s construct all symbols used should have equal weight toward the correlation determination; for example,...
Dan Boschen's user avatar
  • 50.2k
4 votes

How to interpret values of the autocorrelation sequence?

The autocorrelation signals you have shown are biased autocorrelation. The issue is that the higher lags have fewer data points to be used to estimate the correlation at those lags. Another issue is ...
Peter K.'s user avatar
  • 25.7k
4 votes

How to interpret values of the autocorrelation sequence?

I think of auto correlation as «self similarity». In practice: line the signal up with a copy of itself. Step the copy one sample at a time. Multiply corresponding pair of samples from the two and ...
Knut Inge's user avatar
  • 3,350
3 votes
Accepted

Correlation : How is correlation analogous to dot product?

So using your formula for the cross-correlation: $$R_{xy}(n)=\sum_m x(m)y(n+m)$$ we see that $$\begin{align} R_{xy}[n]&=\sum_m x[m]y[n+m]\\ &= \sum_m x[m] \tilde y_n[m]\\ &= x \cdot \...
Marcus Müller's user avatar
3 votes

How to measure the time dependent correlation of two signals

Your signals look like they have about the same ordinal scale, and little lag. Hence, you can perform local correlations: take a segment of both signals centered at a time location, you can normalize ...
Laurent Duval's user avatar
3 votes

Phase shifting a noisy signal

With a real cosine a phase shift is equivalent to a time delay. If you delay the signal by $\phi_0$ then you will get the result you want. The number of samples that you will need to delay by will ...
Jim Clay's user avatar
  • 12.1k
3 votes

Why correlation property of $\mathcal Z$-transform contains a time reversal operation

Let's look at convolving: $$ x_1 = [1, 2, 3];\\ x_2 = [3, 2, 2]; $$ So: $$ Y(z) = X_1(z) X_2(z) = (1 + 2 z^{-1} + 3z^{-2})(3 + 2z^{-1} + 2 z^{-2})\\ = 3 + 2z^{-1} + 2 z^{-2} + 6z^{-1} + 4z^{-2} + 4 z^...
Peter K.'s user avatar
  • 25.7k
3 votes

Is a wavelet-based correlation measure worth any additional computational overhead?

This is very late, but maybe it's worth it anyway... The time-scale plane is not the same as the time-frequency plane, although it might be useful as well. Signals at different places in the time ...
Rodney Price's user avatar
3 votes
Accepted

Why correlation is not associative in the context of image processing?

Subtraction is also not associative: (a-b)-c does not equal a-(b-c) in general. The order of the flipping of the sign matters, just like it does for the flipping of the kernel.
Jazzmaniac's user avatar
  • 4,574
3 votes

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

One must be careful when asking questions about the relationships between the elements of a complex random vector. The short answer to your question is that you cannot say much for either cases ...
Learn_and_Share's user avatar
3 votes

Why do we need to conjugate complex signals in autocorrelation and cross correlation

HINT: What is the physical meaning of the autocorrelation evaluated at lag $\tau=0$?
Matt L.'s user avatar
  • 89.5k
3 votes
Accepted

Designing an efficient curve-matching algorithm

The two most obvious things you can try are: Fitting a Gaussian to your data and then clustering their parameters Estimate the similarity of waveforms directly and then try to cluster that Since you ...
A_A's user avatar
  • 10.7k
3 votes
Accepted

What is the time reversal convolution

I believe that h(-t) means a "time-reversed" version of h(t). Your command: 'y = conv(r,-h);' computes the convolution of 'r' and negative 'h', and you don't want that. I think you want: y = conv(r,...
Richard Lyons's user avatar
3 votes
Accepted

Understanding correlation in image processing using examples

As it was already posted multiple times: The problem comes from an inaccurate definition of correlation in your application. The Pearson correlation coefficient does require the data to be ...
M529's user avatar
  • 1,736
3 votes

Kronecker delta instead of dirac delta as correlation function of white noise

Dirac delta function has a continuous argument, but Kronecker delta function has a discrete argument. Your example is a discrete signal so Kronecker delta is used.
Mohammad M's user avatar
  • 1,327
3 votes

Identify which of the three signals is closest to a sinusoidal curve

Why not just take the FFT and see which one has the highest peak? The code below generates example data: and then takes the FFT of it: which yields: X sum: 0.9999999999999987 Y sum:0....
Peter K.'s user avatar
  • 25.7k

Only top scored, non community-wiki answers of a minimum length are eligible