Editing Frequency response

{{Short description|Output as a function of input frequency}}
{{about|an output-to-input relationship of an [[electric circuit]]|a change in frequency in an [[electrical grid]]|Frequency response (electrical grid)}}
{{More citations needed|introduction and first section|date=August 2011}}
In [[signal processing]] and [[electronics]], the '''frequency response''' of a system is the quantitative measure of the magnitude and [[Phase (waves)|phase]] of the output as a function of input frequency.<ref>{{Cite book |last=Smith |first=Steven W. |title=The Scientist and Engineer's Guide to Digital Signal Processing |publisher=California Technical Pub |year=1997 |isbn=978-0966017632 |pages=177–180}}</ref> The frequency response is widely used in the design and analysis of systems, such as [[audio system|audio]] and [[control system]]s, where they simplify mathematical analysis by converting governing [[differential equations]] into [[algebraic equations]]. In an audio system, it may be used to minimize audible [[distortion]] by designing components (such as [[microphones]], [[Audio power amplifier|amplifiers]] and [[loudspeakers]]) so that the overall response is as flat (uniform) as possible across the system's [[Bandwidth (signal processing)|bandwidth]]. In control systems, such as a vehicle's [[cruise control]], it may be used to assess system [[Stability theory|stability]], often through the use of [[Bode plot]]s. Systems with a specific frequency response can be designed using [[analog filter|analog]] and [[digital filter]]s.

The frequency response characterizes systems in the [[frequency domain]], just as the [[impulse response]] characterizes systems in the [[time domain]]. In [[linear system]]s (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 [[Fourier transform]] of the impulse response. The frequency response allows simpler analysis of cascaded systems such as [[multistage amplifier]]s, as the response of the overall system can be found through multiplication of the individual stages' frequency responses (as opposed to [[convolution]] of the impulse response in the time domain). The frequency response is closely related to the [[transfer function]] in linear systems, which is the [[Laplace transform]] of the impulse response. They are equivalent when the real part <math>\sigma</math> of the transfer function's complex variable <math>s = \sigma + j\omega</math> is zero.<ref name="Feucht1990">{{cite book|author=Dennis L. Feucht|title=Handbook of Analog Circuit Design|year=1990|publisher=Elsevier Science|isbn=978-1-4832-5938-3|page=192}}</ref>

== Measurement and plotting ==
[[Image:Butterworth response.svg|thumb|300 px|Magnitude response of a low pass filter with 6&nbsp;dB per octave or 20&nbsp;dB per decade [[roll-off]]]]

Measuring the frequency response typically involves exciting the system with an input signal and measuring the resulting output signal, calculating the [[frequency spectrum|frequency spectra]] of the two signals (for example, using the [[fast Fourier transform]] 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:
* 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 [[steady-state]] at each point of interest
* Applying an [[Dirac delta function|impulse]] signal and taking the Fourier transform of the [[impulse response|system's response]]
* Applying a [[wide-sense stationary]] [[white noise]] signal over a long period of time and taking the Fourier transform of the system's response. With this method, the [[cross-spectral density]] (rather than the [[power spectral density]]) should be used if phase information is required

The frequency response is characterized by the ''magnitude'', typically in [[decibel]]s (dB) or as a generic [[amplitude]] of the dependent variable, and the ''[[Phase (waves)|phase]]'', in [[radian]]s or degrees, measured against frequency, in [[radians per second|radian/s]], [[Hertz]] (Hz) or as a fraction of the [[Nyquist rate|sampling frequency]].

There are three common ways of plotting response measurements:
* [[Bode plot]]s graph magnitude and phase against frequency on two rectangular plots
* [[Nyquist plot]]s graph magnitude and phase [[parametric plot|parametrically]] against frequency in polar form
* [[Nichols plot]]s graph magnitude and phase parametrically against frequency in rectangular form

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 [[digital filters]]), the frequency response often contains a main lobe with multiple periodic sidelobes, due to [[spectral leakage]] caused by digital processes such as [[sampling (signal processing)|sampling]] and [[Window function|windowing]].<ref>L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp</ref>

===Nonlinear frequency response===
If the system under investigation is [[nonlinear]], 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.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> Nonlinear frequency response methods may reveal effects such as [[resonance]], [[intermodulation]], and [[energy transfer]].

==Applications==
In the audible range frequency response is usually referred to in connection with [[electronic amplifier]]s, [[microphone]]s and [[loudspeakers]]. Radio spectrum frequency response can refer to measurements of [[coaxial cable]], [[Category 6 cable|twisted-pair cable]], [[video switching]] equipment, [[wireless]] communications devices, and antenna systems. Infrasonic frequency response measurements include [[earthquakes]] and [[electroencephalography]] (brain waves).

Frequency response curves are often used to indicate the accuracy of electronic components or systems.<ref name=Stark51>Stark, 2002, p. 51.</ref> 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.<ref name=Stark51/> In other cases, 3D-form of frequency response graphs are sometimes used.

Frequency response requirements differ depending on the application.<ref name=Luther141>Luther, 1999, p. 141.</ref> In [[high fidelity]] audio, an amplifier requires a flat frequency response of at least 20–20,000&nbsp;Hz, with a tolerance as tight as ±0.1&nbsp;dB in the mid-range frequencies around 1000&nbsp;Hz; however, in [[telephony]], a frequency response of 400–4,000&nbsp;Hz, with a tolerance of ±1&nbsp;dB is sufficient for intelligibility of speech.<ref name=Luther141/>

Once a frequency response has been measured (e.g., as an impulse response), provided the system is [[LTI system theory|linear and time-invariant]], its characteristic can be approximated with arbitrary accuracy by a [[digital filter]].  Similarly, if a system is demonstrated to have a poor frequency response, a digital or [[analog filter]] 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 [[Radar jamming and deception|anti-jamming protection of radars]], 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,<ref>{{Cite journal |last1=Henry |first1=N. |last2=Pedersen |first2=M. |last3=Williams |first3=M. |last4=Donkin |first4=L. |date=2023-07-03 |title=Behavioral Posology: A Novel Paradigm for Modeling the Healthy Limits of Behaviors |journal=Advanced Theory and Simulations |volume=6 |issue=9 |language=en |doi=10.1002/adts.202300214 |issn=2513-0390|doi-access=free }}</ref> or in the optimization of drug treatment regimens.<ref>{{Cite journal |last1=Schulthess |first1=Pascal |last2=Post |first2=Teun M. |last3=Yates |first3=James |last4=van der Graaf |first4=Piet H. |date=February 2018 |title=Frequency-Domain Response Analysis for Quantitative Systems Pharmacology Models: Frequency-domain response analysis for QSP models |journal=CPT: Pharmacometrics & Systems Pharmacology |language=en |volume=7 |issue=2 |pages=111–123 |doi=10.1002/psp4.12266 |pmc=5824121 |pmid=29193852}}</ref>

==See also==
{{col div|colwidth=35em}}
*[[Audio system measurements]]
*[[Bandwidth (signal processing)]]
*[[Bode plot]]
*[[Impulse response]]
*[[Spectral sensitivity]]
*[[Steady state (electronics)]]
*[[Transient response]]
*[[Universal dielectric response]]
{{colend}}

==References==
;Notes
{{Reflist}}
;Bibliography
*Luther, Arch C.; Inglis, Andrew F. [https://books.google.com/books?id=VRailj6TKqUC ''Video engineering''], McGraw-Hill, 1999. {{ISBN|0-07-135017-9}}
*Stark, Scott Hunter. [https://books.google.com/books?id=7QOcDeGFx4UC ''Live Sound Reinforcement''], Vallejo, California, Artistpro.com, 1996–2002. {{ISBN|0-918371-07-4}}
* L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp

==External links==
*[[University of Michigan]]: [http://www.engin.umich.edu/group/ctm/freq/freq.html Frequency Response Analysis and Design Tutorial] {{Webarchive|url=https://web.archive.org/web/20121017115622/http://www.engin.umich.edu/group/ctm/freq/freq.html |date=2012-10-17 }}
*Smith, Julius O. III: [http://ccrma.stanford.edu/~jos/filters/ Introduction to Digital Filters with Audio Applications] has a nice chapter on [http://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html Frequency Response]

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[[Category:Signal processing]]
[[Category:Control theory]]
[[Category:Audio amplifier specifications]]