Standard linear array: Difference between revisions

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{{Short description|Standard linear arrays}}

In the context of [[phased array]]s, a standard linear array (SLA) is a [[uniform linear array]] (ULA) of interconnected transducer elements, e.g. microphones or antennas, where the individual elements are arranged in a straight line spaced at one half of the smallest wavelength of the intended signal to be received and/or transmitted. Therefore, an SLA is a subset of the ULA category. The reason for this spacing is that it prevents [[grating lobes]] in the [[visible region]] of the array.<ref name="Van Trees">{{cite book |last1=Van Trees |first1=H.L. |title=Optimum Array Processing |page=51}}</ref>

Intuitively one can think of a ULA as spatial sampling of a signal in the same sense as time sampling of a signal. Grating lobes are identical to aliasing that occurs in time series analysis for an under-sampled signal.<ref name="Van Trees" /> 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. Because the beam pattern (or [[array factor]]) of a linear array is the Fourier transform of the element pattern,<ref>{{cite book |last1=Mailloux |first1=R.J. |title=Phased Array Antenna Handbook |date=2005 |publisher=Artech House |___location=Norwood, MA |pages=109–111}}</ref> the sampling theorem directly applies, but in the spatial instead of spectral ___domain. The [[discrete-time Fourier transform]] (DTFT) of a sampled signal is always periodic, producing "copies" of the spectrum at intervals of the sampling frequency. In the spatial ___domain, these copies are the grating lobes. The analog of radian frequency in the time ___domain is [[wavenumber]], <math>k = \frac{2\pi}{\lambda}</math> radians per meter, in the spatial ___domain. Therefore the spatial sampling rate, in samples per meter, must be <math>\geq 2 \frac{samples}{cycle} \times \frac{k \frac{radians}{meter}}{2\pi \frac{radians}{cycle}}</math>. The sampling interval, which is the inverse of the sampling rate, in meters per sample, must be <math>\leq \frac{\lambda}{2}</math>.