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I've been using F-K Filter, for a while, but I guess I never had good basic understanding about it Math. Someone asked me what is the cause of frequency wrap around in F-K Spectra plot ? I know it's because of aliasing. But if somebody please elaborate more on the cause of this wrap around? Simple Math explanation perhaps.

Thanks

:)

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    $\begingroup$ I have no idea about what an F-K Filter is, but it appears to me (correct me if I'm wrong) that it's about Electrical Engineering. In that case, you could ask the quesiton here instead: Electronics.SE $\endgroup$
    – Dimension10
    Commented Jul 23, 2013 at 17:01
  • $\begingroup$ Also, please reply to a user by using "@username", as I have done here as an example. Else, they won't be notified. For example, to reply to me, "@Dimension10", etc. Just noted that because many new users are violating this rule, not that you will,. : ) , , m. $\endgroup$
    – Dimension10
    Commented Jul 23, 2013 at 17:02

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Aliasing and frequency wrap is a consequence of violation of Nyquist-Shannon sampling theorem which states that a continuous signal must be discretely sampled at least twice the frequency of the highest frequency in the signal.Hence we need to briefly go into the mathematics of the same.

Let x(t) be a continuous signal, y(n) is discrete signal where y(n)= x(nT). Please note that x(t) will have a continuous time Fourier transform (CTFT) while y(n) will have a Discrete Time Fourier Transform (DTFT)

We can construct a mathematical model of sampling using the continuous Dirac delta function p(t) : $ p(t)= \sum_{k=-\infty}^{k= \infty}\delta(t-KT) \forall t \in Real $

Let w(t) =x(t)p(t).We can show that CTFT of continuous function w(t) is DTFT of y(n). Writing the frequency domain representation of w(t)

$W(\omega) = 1/2\pi \ X(\omega) * P(\omega) = \int_{-\infty}^{\infty} X(\Omega) P(\Omega -\omega)d\Omega \ \ \ $ , where * is convolution

now CTFT of p(t) can be written as

$P(\omega) = 2\pi /T \sum_{-\infty}^{\infty} \delta(\omega -k\ 2\pi /T) $

thus $W(\omega) = 1/2\pi \int_{-\infty}^{\infty} X(\Omega)2\pi /T \sum_{-\infty}^{\infty} \delta(\omega -k\ 2\pi /T) d\Omega \\ = 1/T \sum_{-\infty}^{\infty} \int_{-\infty}^{\infty} X(\Omega) \delta(\omega -k\ 2\pi /T) d\Omega \\ =1/T \sum_{-\infty}^{\infty}X(\omega -k\ 2\pi /T) \ \ \ \ \ \ \ \ \ ...using\ \ shifting\ property $ ,

Thus we can say that $Y(\omega) = 1/T \sum_{-\infty}^{\infty} X((\omega -2\pi k)/T) $

So we can say that DTFT of y(n): $Y(\omega) $, is a shifted and repeated version of CTFT x(t) i.e. $X(\omega)$. The DTFT is the sum of the CTFT and its copies shifted by multiples of 2π/T.This is shown in the figure bleow. The frequency axis is normalized to −π/T < ω < π/T.

enter image description here

If X(ω) = 0 outside the range −π/T < ω < π/T, i.e. x(t) has no frequency greater than nyquist, then the copies will not overlap the range −π < ω < π and there is no problem as seen in figure above.

However if X has non-zero frequency components higher than π/T (fs/2 or nyquist). There will be overlap causing a wrap around in f-k domain. Refer below:

enter image description here

Notice that in the sampled signal, the frequencies in the vicinity of π are distorted by the overlapping of frequency components above and below π/T in the original signal causing wrap in F-k domain. Now in often spatial sampling is low as in case of Seismic or MRI due to constrains on number of detectors hence there is a aliasing in wavenumber. This spatial aliasing appears as wrap around in Frequency wavenumber Spectra.

Source of Images :R. G. Lyons: Understanding Digital Signal Processing (2nd Edition)

Reference : EECS, University of California Berkeley

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@Dimension10,

My understanding of aliasing in FK is the ambiguous/wrong interpretation of apparent velocity and direction of arrival. If you see a clear line in your FK plot, the ratio of F and K (assuming K_x, a regular linear array) anywhere along the line gives you the apparent velocity for the plane wave crossing the array. If you know the actual velocity of the medium in which the wave propagates, you can find the azimuth, or the direction of wave arrival. Now, if the distance between the neighbouring sensors on your array is larger than half the wavelength, you will get aliasing as there are not enough spatial samples to uniquely identify the wave. In my field, seismic, this would give rise to wrong interpretation of the dip, that is which way the reflector (a geological layer) is tilting. Of course, in seismic analysis the medium velocity is not homogeneous which makes the interpretation more difficult.

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  • $\begingroup$ I have read the answer, I have the same query, I always asked about the mathematical explanation about the Aliasing in the FK domain, we all know that TF(f(k,k))=integarl double(s(x,t)e^-(kx+ft))dxdt, so how we can explain what happen in the fk domain, when the sampling interval failed to image the event in the kx, ky direction, and why the event is likely going to Wrap or to image in the opposite side with a negative Wavenumber value,,, Please any Ideas. $\endgroup$
    – user7597
    Commented Jan 18, 2014 at 0:42

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