Frequency response

<h2 id="measurement-and-plotting">Measurement and plotting</h2>

Measuring the frequency response typically involves exciting the system with an input signal and measuring the resulting output signal, calculating the <a href="/facts/Frequency_spectrum/q3Rfs7b0">frequency spectra</a> of the two signals (for example, using the <a href="/facts/Fast_Fourier_transform/caT7UUCh">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.
Several methods using different input signals may be used to measure the frequency response of a system, including:

<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="/facts/Steady-state/qgrL4BNs">steady-state</a> at each point of interest</li>
<li>Applying an <a href="/facts/Dirac_delta_function/V8PjRP8T">impulse</a> signal and taking the Fourier transform of the <a href="/facts/Impulse_response/l4vaE5er">system's response</a></li>
<li>Applying a <a href="/facts/Wide-sense_stationary/vozdgfEX">wide-sense stationary</a> <a href="/facts/White_noise/wJ0SR21n">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="/facts/Cross-spectral_density/q3Rfs7b0">cross-spectral density</a> (rather than the <a href="/facts/Power_spectral_density/q3Rfs7b0">power spectral density</a>) should be used if phase information is required</li></ul>
The frequency response is characterized by the magnitude, typically in <a href="/facts/Decibel/0LyxAPXh">decibels</a> (dB) or as a generic <a href="/facts/Amplitude/aEckuXJM">amplitude</a> of the dependent variable, and the <a href="/facts/Phase_(waves)/cpGtuVll">phase</a>, in <a href="/facts/Radian/Srd38Sxa">radians</a> or degrees, measured against frequency, in <a href="/facts/Radians_per_second/cG0plKHD">radian/s</a>, <a href="/facts/Hertz/6Wshj5qm">Hertz</a> (Hz) or as a fraction of the <a href="/facts/Nyquist_rate/Xb1v87Bu">sampling frequency</a>.
There are three common ways of plotting response measurements:

<ul><li><a href="/facts/Bode_plot/xcF1ej98">Bode plots</a> graph magnitude and phase against frequency on two rectangular plots</li>
<li><a href="/facts/Nyquist_plot/J8rxK4yA">Nyquist plots</a> graph magnitude and phase <a href="/facts/Parametric_plot/5Uvrlgqf">parametrically</a> against frequency in polar form</li>
<li><a href="/facts/Nichols_plot/dBccIX4P">Nichols plots</a> graph magnitude and phase parametrically against frequency in rectangular form</li></ul>
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="/facts/Digital_filters/2wcxCRdh">digital filters</a>), the frequency response often contains a main lobe with multiple periodic sidelobes, due to <a href="/facts/Spectral_leakage/yFqfjdDr">spectral leakage</a> caused by digital processes such as <a href="/facts/Sampling_(signal_processing)/xxck7aLN">sampling</a> and <a href="/facts/Window_function/5Xj7h2ah">windowing</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h3>Nonlinear frequency response</h3>
If the system under investigation is <a href="/facts/Nonlinear/4aYItALC">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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> Nonlinear frequency response methods may reveal effects such as <a href="/facts/Resonance/q7VLALUz">resonance</a>, <a href="/facts/Intermodulation/R9ex1SMn">intermodulation</a>, and <a href="/facts/Energy_transfer/wd8PrQM7">energy transfer</a>.

<h2 id="applications">Applications</h2>
In the audible range frequency response is usually referred to in connection with <a href="/facts/Electronic_amplifier/pnRgxncC">electronic amplifiers</a>, <a href="/facts/Microphone/7YWGlXCl">microphones</a> and <a href="/facts/Loudspeakers/tqEjnJ2z">loudspeakers</a>. Radio spectrum frequency response can refer to measurements of <a href="/facts/Coaxial_cable/aG9FfCpL">coaxial cable</a>, <a href="/facts/Category_6_cable/qv2G0J8p">twisted-pair cable</a>, <a href="/facts/Video_switching/e5tpgBMU">video switching</a> equipment, <a href="/facts/Wireless/GfDoFwfo">wireless</a> communications devices, and antenna systems. Infrasonic frequency response measurements include <a href="/facts/Earthquakes/NGjIzagw">earthquakes</a> and <a href="/facts/Electroencephalography/XGvpcBl5">electroencephalography</a> (brain waves).
Frequency response curves are often used to indicate the accuracy of electronic components or systems.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 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.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> In other case, we can be use 3D-form of frequency response surface.
Frequency response requirements differ depending on the application.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In <a href="/facts/High_fidelity/K6kaTHSw">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="/facts/Telephony/2GwTTepw">telephony</a>, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
Once a frequency response has been measured (e.g., as an impulse response), provided the system is <a href="/facts/LTI_system_theory/kEmj0QDC">linear and time-invariant</a>, its characteristic can be approximated with arbitrary accuracy by a <a href="/facts/Digital_filter/2wcxCRdh">digital filter</a>. Similarly, if a system is demonstrated to have a poor frequency response, a digital or <a href="/facts/Analog_filter/Vb9JRK8I">analog filter</a> can be applied to the signals prior to their reproduction to compensate for these deficiencies.
The form of a frequency response curve is very important for <a href="/facts/Radar_jamming_and_deception/sGoEJ6Bv">anti-jamming protection of radars</a>, communications and other systems.
Frequency response analysis can also be applied to biological domains, such as the detection of hormesis in repeated behaviors with opponent process dynamics,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> or in the optimization of drug treatment regimens.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Audio_system_measurements/9hnslzGd">Audio system measurements</a></li>
<li><a href="/facts/Bandwidth_(signal_processing)/7eMDzfJI">Bandwidth (signal processing)</a></li>
<li><a href="/facts/Bode_plot/xcF1ej98">Bode plot</a></li>
<li><a href="/facts/Impulse_response/l4vaE5er">Impulse response</a></li>
<li><a href="/facts/Spectral_sensitivity/F4V3Kkww">Spectral sensitivity</a></li>
<li><a href="/facts/Steady_state_(electronics)/U3ylWEtp">Steady state (electronics)</a></li>
<li><a href="/facts/Transient_response/m4Nr61Px">Transient response</a></li>
<li><a href="/facts/Universal_dielectric_response/FC7CuDjQ">Universal dielectric response</a></li></ul>

Notes

Bibliography
<ul><li>Luther, Arch C.; Inglis, Andrew F. <a href="https://books.google.com/books?id=VRailj6TKqUC">Video engineering</a>, McGraw-Hill, 1999. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-135017-9</li>
<li>Stark, Scott Hunter. <a href="https://books.google.com/books?id=7QOcDeGFx4UC">Live Sound Reinforcement</a>, Vallejo, California, Artistpro.com, 1996–2002. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-918371-07-4</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>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/University_of_Michigan/6e1Ea5xh">University of Michigan</a>: <a href="http://www.engin.umich.edu/group/ctm/freq/freq.html">Frequency Response Analysis and Design Tutorial</a> <a 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="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li>
<li>Smith, Julius O. III: <a href="http://ccrma.stanford.edu/~jos/filters/">Introduction to Digital Filters with Audio Applications</a> has a nice chapter on <a href="http://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html">Frequency Response</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Smith, Steven W. (1997). The Scientist and Engineer's Guide to Digital Signal Processing. California Technical Pub. pp. 177–180. ISBN 978-0966017632. <a href="978-0966017632" target="_blank">978-0966017632</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dennis L. Feucht (1990). Handbook of Analog Circuit Design. Elsevier Science. p. 192. ISBN 978-1-4832-5938-3. <a href="978-1-4832-5938-3" target="_blank">978-1-4832-5938-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Stark, 2002, p. 51. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Stark, 2002, p. 51. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Luther, 1999, p. 141. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Luther, 1999, p. 141. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Henry, N.; Pedersen, M.; Williams, M.; Donkin, L. (2023-07-03). "Behavioral Posology: A Novel Paradigm for Modeling the Healthy Limits of Behaviors". Advanced Theory and Simulations. 6 (9). doi:10.1002/adts.202300214. ISSN 2513-0390. <a href="https://doi.org/10.1002%2Fadts.202300214" target="_blank">https://doi.org/10.1002%2Fadts.202300214</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Schulthess, Pascal; Post, Teun M.; Yates, James; van der Graaf, Piet H. (February 2018). "Frequency-Domain Response Analysis for Quantitative Systems Pharmacology Models: Frequency-domain response analysis for QSP models". CPT: Pharmacometrics & Systems Pharmacology. 7 (2): 111–123. doi:10.1002/psp4.12266. PMC 5824121. PMID 29193852. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5824121" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5824121</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Frequency response open-in-new

Frequency response