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?