# Question on spatial sampling for non-linear and/or non-uniform arrays

I am a little confused as to how to compute the nyquist criterion for a sensor array that is not linear, and/or non-uniform. For the linear, uniformly spaced case, the spatial nyquist is delimited by finding the half-wavelength that fits between the space of two adjacent sensors. Thus, if the distance between two sensors is given by $d$, then the highest frequency that is allowed is given by:

$$d < \frac{\lambda}{2} = \frac{c}{2f}$$

Where $c$ is the speed of light, and $f$ is the frequency of the signal. And so therefore, the highest frequency allowed such that no spatial aliasing occurs is given by:

$$f < \frac{c}{2d}$$

So my question is twofold:

I) What is the spatial nyquist for arrays that are uniformly spaced, but non-linear, (for example, 10 sensors evenly spaced on a circle)?

II) What is the spatial nyquist for cases where the array is non-uniform and non-linear. (for example, some odd shaped number of sensors placed arbitrarily).

I think it has to do with the largest distance between them, but I am not clear on this, because, well, what distances do you care about?

Thanks

• It seems that these rules would be analogous to time-domain sampling at an irregular rate. I'm not aware of any specific criterion that applies in that case, but that doesn't mean that it doesn't exist. – Jason R Apr 11 '13 at 3:12
• Wouldn't the spatial Nyquist depend on the shortest distance between antennas? – Jim Clay Apr 11 '13 at 12:27
• @JimClay I dont think so - I think it depends on the longest distance between two consecutive antennas. Its like a group of 10 who want to run a marathon together - they can only move as fast as the slowest person. Similarly, I believe if we have a non-uniform array, the largest distance dictates the maximum frequency that can be represented by the group of antennas. (Thats my current understanding at least. What I do not get is how is this largest measure computed, between which antennas, etc.) – Spacey Apr 11 '13 at 14:25
• @JasonR Yes, I am sure it exists. Cant seem to find anything on this for either temporal or spatial case via google though. – Spacey Apr 11 '13 at 14:27
• @Mohammad: It's possible that there could be some results in the field of compressed sensing that might apply here. I don't know much about it, but I've heard the nonuniform time-domain sampling problem formulated in that kind of framework before. You typically have to make some assumptions about the signal's sparsity in order for it to work, though. – Jason R Apr 11 '13 at 14:29