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<title>Standard linear array</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Standard linear array</span></span>
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<p>In the context of <a href="Phased_array" title="Phased array">phased arrays</a>, a <b>standard linear array</b> (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 <a href="Grating_lobes" title="Grating lobes">grating lobes</a> in the visible region of the array.<sup id="cite_ref-Van_Trees_1-0" class="reference"><a href="#cite_note-Van_Trees-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>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.<sup id="cite_ref-Van_Trees_1-1" class="reference"><a href="#cite_note-Van_Trees-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Per Shannon's <a href="Sampling_theorem" class="mw-redirect" title="Sampling theorem">sampling theorem</a>, 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 <a href="Array_factor" title="Array factor">array factor</a>) of a linear array is the Fourier transform of the element pattern,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> the sampling theorem directly applies, but in the spatial instead of spectral domain. The <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (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 <a href="Wavenumber" title="Wavenumber">wavenumber</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {2\pi }{\lambda }}}">
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<annotation encoding="application/x-tex">{\displaystyle k={\frac {2\pi }{\lambda }}}</annotation>
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</math></span><img src="./f567225915e2b51c00573536e20eee76db99740b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.64ex; height:5.343ex;" alt="{\displaystyle k={\frac {2\pi }{\lambda }}}" loading="lazy"></span> radians per meter, in the spatial domain. Therefore, the spatial sampling rate, in samples per meter, must be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq 2{\frac {samples}{cycle}}\times {\frac {k{\frac {radians}{meter}}}{2\pi {\frac {radians}{cycle}}}}}">
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<annotation encoding="application/x-tex">{\displaystyle \geq 2{\frac {samples}{cycle}}\times {\frac {k{\frac {radians}{meter}}}{2\pi {\frac {radians}{cycle}}}}}</annotation>
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</math></span><img src="./2963bcb3f84ee8178bc1b07efedf0d6482a2c0e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:25.522ex; height:8.343ex;" alt="{\displaystyle \geq 2{\frac {samples}{cycle}}\times {\frac {k{\frac {radians}{meter}}}{2\pi {\frac {radians}{cycle}}}}}" loading="lazy"></span>. The sampling interval, which is the inverse of the sampling rate, in meters per sample, must be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq {\frac {\lambda }{2}}}">
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<annotation encoding="application/x-tex">{\displaystyle \leq {\frac {\lambda }{2}}}</annotation>
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</math></span><img src="./c5d6d4e4fc02a1cdff699c964fd7f6919d80b8cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.645ex; height:5.343ex;" alt="{\displaystyle \leq {\frac {\lambda }{2}}}" loading="lazy"></span>.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-Van_Trees-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Van_Trees_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Van_Trees_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFVan_Trees" class="citation book cs1">Van Trees, H.L. <i>Optimum Array Processing</i>. p. 51.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFMailloux2005" class="citation book cs1">Mailloux, R.J. (2005). <i>Phased Array Antenna Handbook</i>. Norwood, MA: Artech House. pp. <span class="nowrap">109–</span>111.</cite></span>
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