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<p>In <a href="Signal_processing" title="Signal processing">signal processing</a>, <b>apodization</b> (from <a href="Greek_language" title="Greek language">Greek</a> "removing the foot") is the modification of the shape of a <a href="Mathematical_function" class="mw-redirect" title="Mathematical function">mathematical function</a>. The function may represent an electrical signal, an optical transmission, or a mechanical structure. In <a href="Optics" title="Optics">optics</a>, it is primarily used to remove <a href="Airy_disk" title="Airy disk">Airy disks</a> caused by <a href="Diffraction" title="Diffraction">diffraction</a> around an intensity peak, improving the focus.

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<div class="mw-heading mw-heading2"><h2 id="Apodization_in_electronics">Apodization in electronics</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Window_function" title="Window function">Window function</a></div>

<div class="mw-heading mw-heading3"><h3 id="Apodization_in_signal_processing">Apodization in signal processing</h3></div>

<p>The term apodization is used frequently in publications on <a href="Fourier-transform_infrared_spectroscopy" title="Fourier-transform infrared spectroscopy">Fourier-transform infrared (FTIR)</a> signal processing. An example of apodization is the use of the <a href="Hann_window" class="mw-redirect" title="Hann window">Hann window</a> in <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> analyzers to smooth the discontinuities at the beginning and end of the sampled time record.

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<div class="mw-heading mw-heading3"><h3 id="Apodization_in_digital_audio">Apodization in digital audio</h3></div>

<p>An apodizing filter can be used in <a href="Digital_audio_processing" class="mw-redirect" title="Digital audio processing">digital audio processing</a> instead of the more common brick-wall filters, in order to reduce the pre- and post-<a href="Ringing_(signal)" title="Ringing (signal)">ringing</a> that the latter introduces.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Apodization_in_mass_spectrometry">Apodization in mass spectrometry</h3></div>

<p>During oscillation within an <a href="Orbitrap" title="Orbitrap">Orbitrap</a>, a transient signal may not be stable until the ions settle into their oscillations. Toward the end, subtle ion collisions add up to cause noticeable dephasing. This presents a problem for the Fourier transform, as it averages the oscillatory signal across the length of the time-domain measurement. Software allows "apodization", the removal of the front and back section of the transient signal from consideration in the FT calculation. Thus, apodization improves the resolution of the resulting mass spectrum. Another way to improve the quality of the transient is to wait to collect data until ions have settled into stable oscillatory motion within the trap.<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>

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<div class="mw-heading mw-heading3"><h3 id="Apodization_in_nuclear_magnetic_resonance_spectroscopy">Apodization in nuclear magnetic resonance spectroscopy</h3></div>

<p>Apodization is applied to <a href="Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">NMR</a> signals before discrete <a href="Fourier_transform" title="Fourier transform">Fourier transformation</a>. Typically, NMR signals are truncated due to time constraints (indirect dimension) or to obtain a higher signal-to-noise ratio. In order to reduce truncation artifacts, the signals are subjected to apodization with different types of <a href="Window_function" title="Window function">window functions</a>. <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Apodization_in_optics">Apodization in optics</h2></div>

<p>In optical design jargon, an apodization function is used to purposely change the input intensity profile of an <a href="Optical_system" class="mw-redirect" title="Optical system">optical system</a>, and it may be a complicated function to tailor the system to certain properties. Usually, it refers to a non-uniform illumination or transmission profile that approaches zero at the edges.

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<div class="mw-heading mw-heading3"><h3 id="Apodization_in_imaging">Apodization in imaging</h3></div>

<p>Since side lobes of the Airy disk are responsible for degrading the image, techniques for suppressing them are used. If the imaging beam has a <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a>, then when the truncation ratio (the ratio of the diameter of the Gaussian beam to the diameter of the truncating aperture) is set to 1, the side-lobes become negligible and the beam profile becomes purely Gaussian.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>In <a href="Medical_ultrasonography" class="mw-redirect" title="Medical ultrasonography">medical ultrasonography</a>, the effect of <a href="Grating_lobe" class="mw-redirect" title="Grating lobe">grating lobes</a> can be reduced by activating <a href="Ultrasonic_transducer" title="Ultrasonic transducer">ultrasonic transducer</a> elements using variable voltages in the apodization process.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Apodization_in_photography">Apodization in photography</h3></div>

<p>Most camera lenses contain <a href="Diaphragm_(optics)" title="Diaphragm (optics)">diaphragms</a> which decrease the amount of light coming into the camera. These are not strictly an example of apodization, since the diaphragm does not produce a smooth transition to zero intensity, nor does it shape the intensity profile beyond the "top hat" transmission of its aperture.

</p><p>Some lenses use other methods to reduce the amount of light let in. For example, the Minolta/Sony STF 135mm f/2.8 T4.5 lens has a special design which accomplishes this by using a concave neutral-gray-tinted lens element as an apodization filter, thereby producing a pleasant <a href="Bokeh" title="Bokeh">bokeh</a>. The same optical effect can be achieved by combining <a href="Depth-of-field_bracketing" class="mw-redirect" title="Depth-of-field bracketing">depth-of-field bracketing</a> with <a href="Multiple_exposure" title="Multiple exposure">multiple exposure</a>, as implemented in the <a href="Minolta_Maxxum_7" class="mw-redirect" title="Minolta Maxxum 7">Minolta Maxxum 7</a>'s <a href="STF_function" class="mw-redirect" title="STF function">STF function</a>. In 2014, <a href="Fujifilm" title="Fujifilm">Fujifilm</a> announced a lens utilizing a similar apodization filter in the <a href="Fujinon_XF_56mm_F1.2_R_APD_lens" class="mw-redirect" title="Fujinon XF 56mm F1.2 R APD lens">Fujinon XF 56mm F1.2 R APD lens</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In 2017, <a href="Sony" title="Sony">Sony</a> introduced the <a href="Sony_E-mount" title="Sony E-mount">E-mount</a> full-frame lens <a href="Sony_FE_100mm_F2.8_STF_GM_OSS" title="Sony FE 100mm F2.8 STF GM OSS">Sony FE 100mm F2.8 STF GM OSS</a> (<a href="SEL-100F28GM" class="mw-redirect" title="SEL-100F28GM">SEL-100F28GM</a>) based on the same optical <a href="Smooth_Trans_Focus" title="Smooth Trans Focus">Smooth Trans Focus</a> principle.<sup id="cite_ref-Sony_SEL-100F28GM_7-0" class="reference"><a href="#cite_note-Sony_SEL-100F28GM-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Simulation of a <a href="Gaussian_beam" title="Gaussian beam">Gaussian</a> laser beam input profile is also an example of apodization.

</p><p><a href="Photon_sieve" title="Photon sieve">Photon sieves</a> provide a relatively easy way to achieve tailored optical apodization.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Apodization_in_astronomy">Apodization in astronomy</h3></div>

<p>Apodization is used in telescope optics to improve the dynamic range of the image. For example, stars with low intensity in the close vicinity of very bright stars can be made visible using this technique, and even images of planets can be obtained when otherwise obscured by the bright atmosphere of the star they orbit.<sup id="cite_ref-hecht_9-0" class="reference"><a href="#cite_note-hecht-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Generally, apodization reduces the resolution of an optical image; however, because it reduces diffraction edge effects, it can actually enhance certain small details. In fact, the notion of resolution, as it is commonly defined with the <a href="Angular_resolution#The_Rayleigh_criterion" title="Angular resolution">Rayleigh criterion</a>, is in this case partially irrelevant. One has to understand that the image formed in the focal plane of a lens (or a mirror) is modeled through the <a href="Fresnel_diffraction" title="Fresnel diffraction">Fresnel diffraction</a> formalism. The classical diffraction pattern, the <a href="Airy_disk" title="Airy disk">Airy disk</a>, is connected to a circular pupil, without any obstruction, and with a uniform transmission. Any change in the shape of the pupil (for example a square instead of a circle), or its transmission, results in an alteration to the associated diffraction pattern.

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<ul><li><a href="Apodization_function" class="mw-redirect" title="Apodization function">Apodization function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFRobjohns2016" class="citation web cs1">Robjohns, Hugh (August 2016). <a rel="nofollow" class="external text" href="https://www.soundonsound.com/techniques/mqa-time-domain-accuracy-digital-audio-quality">"MQA Time-domain Accuracy & Digital Audio Quality"</a>. <i>soundonsound.com</i>. Sound On Sound. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230310175409/https://www.soundonsound.com/techniques/mqa-time-domain-accuracy-digital-audio-quality">Archived</a> from the original on 10 March 2023.</cite></span>

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<li id="cite_note-hecht-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-hecht_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFE._Hecht1987" class="citation book cs1">E. Hecht (1987). <i>Optics</i> (2nd ed.). Addison Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-11609-0</bdi>.</cite> Section 11.3.3.</span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://iopscience.iop.org/2041-8205/722/1/L49/"><i>FIRST RESULTS FROM VERY LARGE TELESCOPE NACO APODIZING PHASE PLATE: 4 μm IMAGES OF THE EXOPLANET β PICTORIS b*</i></a> The Astrophysical Journal (Letter)</span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://spacefellowship.com/news/art23185/planet-hunters-no-longer-blinded-by-the-light.html">Planet hunters no longer blinded by the light.</a> spacefellowship.com Note: this article includes several images of such a phase plate</span>

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