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<title>Frequency response</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Frequency response</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about an output-to-input relationship of an <a href="Electric_circuit" class="mw-redirect" title="Electric circuit">electric circuit</a>. For a change in frequency in an <a href="Electrical_grid" title="Electrical grid">electrical grid</a>, see <a href="Frequency_response_(electrical_grid)" class="mw-redirect" title="Frequency response (electrical grid)">Frequency response (electrical grid)</a>.</div>

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<p>In <a href="Signal_processing" title="Signal processing">signal processing</a> and <a href="Electronics" title="Electronics">electronics</a>, the <b>frequency response</b> of a system is the quantitative measure of the magnitude and <a href="Phase_(waves)" title="Phase (waves)">phase</a> of the output as a function of input frequency.<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> The frequency response is widely used in the design and analysis of systems, such as <a href="Audio_system" class="mw-redirect" title="Audio system">audio</a> and <a href="Control_system" title="Control system">control systems</a>, where they simplify mathematical analysis by converting governing <a href="Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> into <a href="Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a>. In an audio system, it may be used to minimize audible <a href="Distortion" title="Distortion">distortion</a> by designing components (such as <a href="Microphones" class="mw-redirect" title="Microphones">microphones</a>, <a href="Audio_power_amplifier" title="Audio power amplifier">amplifiers</a> and <a href="Loudspeakers" class="mw-redirect" title="Loudspeakers">loudspeakers</a>) so that the overall response is as flat (uniform) as possible across the system's <a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a>. In control systems, such as a vehicle's <a href="Cruise_control" title="Cruise control">cruise control</a>, it may be used to assess system <a href="Stability_theory" title="Stability theory">stability</a>, often through the use of <a href="Bode_plot" title="Bode plot">Bode plots</a>. Systems with a specific frequency response can be designed using <a href="Analog_filter" class="mw-redirect" title="Analog filter">analog</a> and <a href="Digital_filter" title="Digital filter">digital filters</a>.

</p><p>The frequency response characterizes systems in the <a href="Frequency_domain" title="Frequency domain">frequency domain</a>, just as the <a href="Impulse_response" title="Impulse response">impulse response</a> characterizes systems in the <a href="Time_domain" title="Time domain">time domain</a>. In <a href="Linear_system" title="Linear system">linear systems</a> (or as an approximation to a real system neglecting second order non-linear properties), either response completely describes the system and thus there is a one-to-one correspondence: the frequency response is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the impulse response. The frequency response allows simpler analysis of cascaded systems such as <a href="Multistage_amplifier" title="Multistage amplifier">multistage amplifiers</a>, as the response of the overall system can be found through multiplication of the individual stages' frequency responses (as opposed to <a href="Convolution" title="Convolution">convolution</a> of the impulse response in the time domain). The frequency response is closely related to the <a href="Transfer_function" title="Transfer function">transfer function</a> in linear systems, which is the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> of the impulse response. They are equivalent when the real part <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 \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> of the transfer function's complex variable <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 s=\sigma +j\omega }">

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<annotation encoding="application/x-tex">{\displaystyle s=\sigma +j\omega }</annotation>

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</math></span><img src="./52b9a99cfe16a2003294604462f41acc7863fe66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.763ex; height:2.509ex;" alt="{\displaystyle s=\sigma +j\omega }" loading="lazy"></span> is zero.<sup id="cite_ref-Feucht1990_2-0" class="reference"><a href="#cite_note-Feucht1990-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Measurement_and_plotting">Measurement and plotting</h2></div>

<p>Measuring the frequency response typically involves exciting the system with an input signal and measuring the resulting output signal, calculating the <a href="Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">frequency spectra</a> of the two signals (for example, using the <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> for discrete signals), and comparing the spectra to isolate the effect of the system. In linear systems, the frequency range of the input signal should cover the frequency range of interest.

</p><p>Several methods using different input signals may be used to measure the frequency response of a system, including:

</p>

<ul><li>Applying constant amplitude sinusoids stepped through a range of frequencies and comparing the amplitude and phase shift of the output relative to the input. The frequency sweep must be slow enough for the system to reach its <a href="Steady-state" class="mw-redirect" title="Steady-state">steady-state</a> at each point of interest</li>

<li>Applying an <a href="Dirac_delta_function" title="Dirac delta function">impulse</a> signal and taking the Fourier transform of the <a href="Impulse_response" title="Impulse response">system's response</a></li>

<li>Applying a <a href="Wide-sense_stationary" class="mw-redirect" title="Wide-sense stationary">wide-sense stationary</a> <a href="White_noise" title="White noise">white noise</a> signal over a long period of time and taking the Fourier transform of the system's response. With this method, the <a href="Cross-spectral_density" class="mw-redirect" title="Cross-spectral density">cross-spectral density</a> (rather than the <a href="Power_spectral_density" class="mw-redirect" title="Power spectral density">power spectral density</a>) should be used if phase information is required</li></ul>

<p>The frequency response is characterized by the <i>magnitude</i>, typically in <a href="Decibel" title="Decibel">decibels</a> (dB) or as a generic <a href="Amplitude" title="Amplitude">amplitude</a> of the dependent variable, and the <i><a href="Phase_(waves)" title="Phase (waves)">phase</a></i>, in <a href="Radian" title="Radian">radians</a> or degrees, measured against frequency, in <a href="Radians_per_second" class="mw-redirect" title="Radians per second">radian/s</a>, <a href="Hertz" title="Hertz">Hertz</a> (Hz) or as a fraction of the <a href="Nyquist_rate" title="Nyquist rate">sampling frequency</a>.

</p><p>There are three common ways of plotting response measurements:

</p>

<ul><li><a href="Bode_plot" title="Bode plot">Bode plots</a> graph magnitude and phase against frequency on two rectangular plots</li>

<li><a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plots</a> graph magnitude and phase <a href="Parametric_plot" class="mw-redirect" title="Parametric plot">parametrically</a> against frequency in polar form</li>

<li><a href="Nichols_plot" title="Nichols plot">Nichols plots</a> graph magnitude and phase parametrically against frequency in rectangular form</li></ul>

<p>For the design of control systems, any of the three types of plots may be used to infer closed-loop stability and stability margins from the open-loop frequency response. In many frequency domain applications, the phase response is relatively unimportant and the magnitude response of the Bode plot may be all that is required. In digital systems (such as <a href="Digital_filters" class="mw-redirect" title="Digital filters">digital filters</a>), the frequency response often contains a main lobe with multiple periodic sidelobes, due to <a href="Spectral_leakage" title="Spectral leakage">spectral leakage</a> caused by digital processes such as <a href="Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a> and <a href="Window_function" title="Window function">windowing</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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Nonlinear_frequency_response">Nonlinear frequency response</h3></div>

<p>If the system under investigation is <a href="Nonlinear" class="mw-redirect" title="Nonlinear">nonlinear</a>, linear frequency domain analysis will not reveal all the nonlinear characteristics. To overcome these limitations, generalized frequency response functions and nonlinear output frequency response functions have been defined to analyze nonlinear dynamic effects.<sup id="cite_ref-SAB1_4-0" class="reference"><a href="#cite_note-SAB1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Nonlinear frequency response methods may reveal effects such as <a href="Resonance" title="Resonance">resonance</a>, <a href="Intermodulation" title="Intermodulation">intermodulation</a>, and <a href="Energy_transfer" class="mw-redirect" title="Energy transfer">energy transfer</a>.

</p>

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

<p>In the audible range frequency response is usually referred to in connection with <a href="Electronic_amplifier" class="mw-redirect" title="Electronic amplifier">electronic amplifiers</a>, <a href="Microphone" title="Microphone">microphones</a> and <a href="Loudspeakers" class="mw-redirect" title="Loudspeakers">loudspeakers</a>. Radio spectrum frequency response can refer to measurements of <a href="Coaxial_cable" title="Coaxial cable">coaxial cable</a>, <a href="Category_6_cable" title="Category 6 cable">twisted-pair cable</a>, <a href="Video_switching" class="mw-redirect" title="Video switching">video switching</a> equipment, <a href="Wireless" title="Wireless">wireless</a> communications devices, and antenna systems. Infrasonic frequency response measurements include <a href="Earthquakes" class="mw-redirect" title="Earthquakes">earthquakes</a> and <a href="Electroencephalography" title="Electroencephalography">electroencephalography</a> (brain waves).

</p><p>Frequency response curves are often used to indicate the accuracy of electronic components or systems.<sup id="cite_ref-Stark51_5-0" class="reference"><a href="#cite_note-Stark51-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve.<sup id="cite_ref-Stark51_5-1" class="reference"><a href="#cite_note-Stark51-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In other cases, 3D-form of frequency response graphs are sometimes used.

</p><p>Frequency response requirements differ depending on the application.<sup id="cite_ref-Luther141_6-0" class="reference"><a href="#cite_note-Luther141-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In <a href="High_fidelity" title="High fidelity">high fidelity</a> audio, an amplifier requires a flat frequency response of at least 20–20,000 Hz, with a tolerance as tight as ±0.1 dB in the mid-range frequencies around 1000 Hz; however, in <a href="Telephony" title="Telephony">telephony</a>, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.<sup id="cite_ref-Luther141_6-1" class="reference"><a href="#cite_note-Luther141-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>Once a frequency response has been measured (e.g., as an impulse response), provided the system is <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">linear and time-invariant</a>, its characteristic can be approximated with arbitrary accuracy by a <a href="Digital_filter" title="Digital filter">digital filter</a>. Similarly, if a system is demonstrated to have a poor frequency response, a digital or <a href="Analog_filter" class="mw-redirect" title="Analog filter">analog filter</a> can be applied to the signals prior to their reproduction to compensate for these deficiencies.

</p><p>The form of a frequency response curve is very important for <a href="Radar_jamming_and_deception" title="Radar jamming and deception">anti-jamming protection of radars</a>, communications and other systems.

</p><p>Frequency response analysis can also be applied to biological domains, such as the detection of hormesis in repeated behaviors with opponent process dynamics,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> or in the optimization of drug treatment regimens.<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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<ul><li><a href="Audio_system_measurements" title="Audio system measurements">Audio system measurements</a></li>

<li><a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">Bandwidth (signal processing)</a></li>

<li><a href="Impulse_response" title="Impulse response">Impulse response</a></li>

<li><a href="Spectral_sensitivity" title="Spectral sensitivity">Spectral sensitivity</a></li>

<li><a href="Steady_state_(electronics)" title="Steady state (electronics)">Steady state (electronics)</a></li>

<li><a href="Transient_response" title="Transient response">Transient response</a></li>

<li><a href="Universal_dielectric_response" title="Universal dielectric response">Universal dielectric response</a></li></ul>

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

<dl><dt>Notes</dt></dl>

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</style><cite id="CITEREFSmith1997" class="citation book cs1">Smith, Steven W. (1997). <i>The Scientist and Engineer's Guide to Digital Signal Processing</i>. California Technical Pub. pp. <span class="nowrap">177–</span>180. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0966017632</bdi>.</cite></span>

</li>

<li id="cite_note-Feucht1990-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Feucht1990_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDennis_L._Feucht1990" class="citation book cs1">Dennis L. Feucht (1990). <i>Handbook of Analog Circuit Design</i>. Elsevier Science. p. 192. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4832-5938-3</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp</span>

</li>

<li id="cite_note-SAB1-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-SAB1_4-0">^</a></b></span> <span class="reference-text">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</span>

</li>

<li id="cite_note-Stark51-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stark51_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stark51_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Stark, 2002, p. 51.</span>

</li>

<li id="cite_note-Luther141-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Luther141_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Luther141_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Luther, 1999, p. 141.</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFHenryPedersenWilliamsDonkin2023" class="citation journal cs1">Henry, N.; Pedersen, M.; Williams, M.; Donkin, L. (2023-07-03). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fadts.202300214">"Behavioral Posology: A Novel Paradigm for Modeling the Healthy Limits of Behaviors"</a>. <i>Advanced Theory and Simulations</i>. <b>6</b> (9). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fadts.202300214">10.1002/adts.202300214</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2513-0390">2513-0390</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFSchulthessPostYatesvan_der_Graaf2018" class="citation journal cs1">Schulthess, Pascal; Post, Teun M.; Yates, James; van der Graaf, Piet H. (February 2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5824121">"Frequency-Domain Response Analysis for Quantitative Systems Pharmacology Models: Frequency-domain response analysis for QSP models"</a>. <i>CPT: Pharmacometrics & Systems Pharmacology</i>. <b>7</b> (2): <span class="nowrap">111–</span>123. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fpsp4.12266">10.1002/psp4.12266</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5824121">5824121</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/29193852">29193852</a>.</cite></span>

</li>

</ol></div></div>

<dl><dt>Bibliography</dt></dl>

<ul><li>Luther, Arch C.; Inglis, Andrew F. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VRailj6TKqUC"><i>Video engineering</i></a>, McGraw-Hill, 1999. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-135017-9</bdi></li>

<li>Stark, Scott Hunter. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7QOcDeGFx4UC"><i>Live Sound Reinforcement</i></a>, Vallejo, California, Artistpro.com, 1996–2002. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-918371-07-4</bdi></li>

<li>L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a href="University_of_Michigan" title="University of Michigan">University of Michigan</a>: <a rel="nofollow" class="external text" href="http://www.engin.umich.edu/group/ctm/freq/freq.html">Frequency Response Analysis and Design Tutorial</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121017115622/http://www.engin.umich.edu/group/ctm/freq/freq.html">Archived</a> 2012-10-17 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li>Smith, Julius O. III: <a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/~jos/filters/">Introduction to Digital Filters with Audio Applications</a> has a nice chapter on <a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html">Frequency Response</a></li></ul>

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