Standard linear array

A standard linear array (SLA) is a linear array of interconnected transducer elements, e.g. microphones or antennas, where the individual elements are uniformly weighted (un-tapered) and arranged in a straight line spaced at one half of the smallest wavelength of the intended signal to be received and/or transmitted.^[1] The reason for this spacing is that it prevents grating lobes in the visible region of the array.^[2]

Intuitively one can think of a linear array of elements as spatial sampling of a signal in the same sense as time sampling of a signal. Per Shannon's sampling theorem, the sampling rate must be at least twice the highest frequency of the desired signal in order to preclude spectral aliasing. The discrete-time Fourier transform (DTFT) of a sampled signal is always periodic, producing "copies" of the spectrum at intervals of the sampling frequency. The analog of radian frequency in the time ___domain is wavenumber, ${\displaystyle k={\frac {2\pi }{\lambda }}}$ radians per meter, in the spatial ___domain. Therefore the spatial sampling rate, in samples per meter, must be ${\displaystyle \geq 2{\frac {samples}{cycle}}\times {\frac {k{\frac {radians}{meter}}}{2\pi {\frac {radians}{cycle}}}}}$. The sampling interval, which is the inverse of the sampling rate, in meters per sample, must be ${\displaystyle \leq {\frac {\lambda }{2}}}$.

1. Van Trees, H.L. Optimum Array Processing. p. 51.
2. Richards, M.A. Principles of Modern Radar: Basic Principles. Edison, NJ: Scitech Publishing. pp. 330–332.