# Arcsin of a number greater than one

I am using a DFT to sum multiple sinusoidal signals produced by an N-anteanna array.

$$X_k=\sum_{n=0}^{N-1}f(n)e^{-i2\pi \frac{n}Nk}$$ Assume $f(n)$ is the excitation of the nth antenna and is the same for all antennas aside from an angular dependent phase difference, which is given by the exponential term.

Each k value (the variable that each bin corresponds to) corresponds to a particular azimuth angle given by: $$k=\frac{Dsin\theta}{\lambda}$$ Where:

D=total array aperture

$\theta=azimuth$

$\lambda=antenna wavelength$

Each antenna is equally spaced (linear array) and hence the N values of k are equally spread from $0\rightarrow\frac{(N-1)\lambda}D$. When trying to find the azimuth angle, rearranging k gives $$sin\theta=\frac{k\lambda}D$$ Using the maximum value of k:$$sin\theta=\frac{(N-1)\lambda^2}{D^2}$$ It is clear that the right hand side of this expression can easily be >1. In such a case, how do I actually find what azimuth the bin is referring to? My calculator wont give the arcsin of a number >1. Matlab gives an imaginary solution. How do I interpret the imaginary solution as the correct azimuth angle?

• What size wavelengths do these antennas produces that this quantity exceeds 1? – Triatticus Jan 16 '18 at 23:41
• I'm not too sure about practical/realistic values, I am just looking at the theory. But I imagine there would be a number of possibilities, say 2 0.5m wavelengths in 0.5m aperture or 10 antennas with 0.15m wavelengths in a 1m aperture. Would that not work? – Sam Jan 17 '18 at 0:09