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<title>Transfer function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Transfer function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Transformation_(function)" title="Transformation (function)">Transformation (function)</a>.</div>

<p>In <a href="Engineering" title="Engineering">engineering</a>, a <b>transfer function</b> (also known as <b>system function</b><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> or <b>network function</b>) of a system, sub-system, or component is a <a href="Function_(mathematics)" title="Function (mathematics)">mathematical function</a> that <a href="Mathematical_model" title="Mathematical model">models</a> the system's output for each possible input.<sup id="cite_ref-LaughtonWarne2002_2-0" class="reference"><a href="#cite_note-LaughtonWarne2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Parr1993_3-0" class="reference"><a href="#cite_note-Parr1993-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SinclairDunton2007_4-0" class="reference"><a href="#cite_note-SinclairDunton2007-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It is widely used in <a href="Electronic_engineering" title="Electronic engineering">electronic engineering</a> tools like <a href="Electronic_circuit_simulation" title="Electronic circuit simulation">circuit simulators</a> and <a href="Control_system" title="Control system">control systems</a>. In simple cases, this function can be represented as a two-dimensional <a href="Graph_(function)" class="mw-redirect" title="Graph (function)">graph</a> of an independent <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> input versus the dependent scalar output (known as a <b>transfer curve</b> or <b>characteristic curve</b>). Transfer functions for components are used to design and analyze systems assembled from components, particularly using the <a href="Block_diagram" title="Block diagram">block diagram</a> technique, in electronics and <a href="Control_theory" title="Control theory">control theory</a>.

</p><p>Dimensions and units of the transfer function model the output response of the device for a range of possible inputs. The transfer function of a <a href="Two-port" class="mw-redirect" title="Two-port">two-port</a> electronic circuit, such as an <a href="Amplifier" title="Amplifier">amplifier</a>, might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical <a href="Actuator" title="Actuator">actuator</a> might be the mechanical displacement of the movable arm as a function of electric current applied to the device; the transfer function of a <a href="Photodetector" title="Photodetector">photodetector</a> might be the output voltage as a function of the <a href="Luminous_intensity" title="Luminous intensity">luminous intensity</a> of incident light of a given <a href="Wavelength" title="Wavelength">wavelength</a>.

</p><p>The term "transfer function" is also used in the <a href="Frequency_domain" title="Frequency domain">frequency domain</a> analysis of systems using transform methods, such as the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a>; it is the <a href="Amplitude" title="Amplitude">amplitude</a> of the output as a function of the <a href="Frequency" title="Frequency">frequency</a> of the input signal. The transfer function of an <a href="Electronic_filter" title="Electronic filter">electronic filter</a> is the amplitude at the output as a function of the frequency of a constant amplitude <a href="Sine_wave" title="Sine wave">sine wave</a> applied to the input. For optical imaging devices, the <a href="Optical_transfer_function" title="Optical transfer function">optical transfer function</a> is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the <a href="Point_spread_function" title="Point spread function">point spread function</a> (a function of <a href="Spatial_frequency" title="Spatial frequency">spatial frequency</a>).

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<div class="mw-heading mw-heading2"><h2 id="Linear_time-invariant_systems">Linear time-invariant systems</h2></div>

<p>Transfer functions are commonly used in the analysis of systems such as <a href="Single-input_single-output" class="mw-redirect" title="Single-input single-output">single-input single-output</a> <a href="Filter_(signal_processing)" title="Filter (signal processing)">filters</a> in <a href="Signal_processing" title="Signal processing">signal processing</a>, <a href="Communication_theory" title="Communication theory">communication theory</a>, and <a href="Control_theory" title="Control theory">control theory</a>. The term is often used exclusively to refer to <a href="Linear_time-invariant" class="mw-redirect" title="Linear time-invariant">linear time-invariant</a> (LTI) systems. Most real systems have <a href="Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> input-output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear that <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system theory</a> is an acceptable representation of their input-output behavior.

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<div class="mw-heading mw-heading3"><h3 id="Continuous-time">Continuous-time</h3></div>

<p>Descriptions are given in terms of a <a href="Complex_variable" class="mw-redirect" title="Complex variable">complex variable</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 s=\sigma +j\cdot \omega }">

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

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</math></span><img src="./bdc5bc85809fc5a4af728b15af62f99fb483faab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.442ex; height:2.509ex;" alt="{\displaystyle s=\sigma +j\cdot \omega }" loading="lazy"></span>. In many applications it is sufficient to set <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 =0}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =0}</annotation>

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</math></span><img src="./1eb4831f1e0ca1ba7d007dc6b973e54787e1a4b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =0}" loading="lazy"></span> (thus <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=j\cdot \omega }">

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

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</math></span><img src="./51a288711d5d19b0eecc950f8b23334b1bf05b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.272ex; height:2.509ex;" alt="{\displaystyle s=j\cdot \omega }" loading="lazy"></span>), which reduces the <a href="Laplace_transform" title="Laplace transform">Laplace transforms</a> with complex arguments to <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case in <a href="Signal_processing" title="Signal processing">signal processing</a> and <a href="Communication_theory" title="Communication theory">communication theory</a>), not the fleeting turn-on and turn-off <a href="Transient_response" title="Transient response">transient response</a> or stability issues.

</p><p>For <a href="Continuous-time" class="mw-redirect" title="Continuous-time">continuous-time</a> input signal <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 x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> and output <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 y(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

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</math></span><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"></span>, dividing the Laplace transform of the output, <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 Y(s)={\mathcal {L}}\left\{y(t)\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}}</annotation>

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</math></span><img src="./a79669ea8d67c9720f88461e092702769feaeb2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.892ex; height:2.843ex;" alt="{\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}}" loading="lazy"></span>, by the Laplace transform of the input, <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 X(s)={\mathcal {L}}\left\{x(t)\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}</annotation>

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</math></span><img src="./38c5baf3780558f202ce0b982fa02f0771a678b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.273ex; height:2.843ex;" alt="{\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}" loading="lazy"></span>, yields the system's transfer function <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 H(s)}">

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<annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation>

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</math></span><img src="./f1b91390324fc9c33ec00fe57e3924ad7118cc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {Y(s)}{X(s)}}={\frac {{\mathcal {L}}\left\{y(t)\right\}}{{\mathcal {L}}\left\{x(t)\right\}}}}</annotation>

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</math></span><img src="./c6b6ef964e38af4f4540b354028f45e085c11faf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.007ex; height:6.509ex;" alt="{\displaystyle H(s)={\frac {Y(s)}{X(s)}}={\frac {{\mathcal {L}}\left\{y(t)\right\}}{{\mathcal {L}}\left\{x(t)\right\}}}}" loading="lazy"></span></dd></dl>

<p>which can be rearranged as:

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<annotation encoding="application/x-tex">{\displaystyle Y(s)=H(s)\;X(s)\,.}</annotation>

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</math></span><img src="./ae1e5ed76344d5cb87f929b85f937635e379d28e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.294ex; height:2.843ex;" alt="{\displaystyle Y(s)=H(s)\;X(s)\,.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Discrete-time">Discrete-time</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Z-transform#Linear_constant-coefficient_difference_equation" title="Z-transform">Z-transform § Linear constant-coefficient difference equation</a></div>

<p><a href="Discrete-time" class="mw-redirect" title="Discrete-time">Discrete-time</a> signals may be notated as arrays indexed by an <a href="Integer" title="Integer">integer</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 n}">

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

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> (e.g. <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 x[n]}">

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<annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation>

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</math></span><img src="./864cbbefbdcb55af4d9390911de1bf70167c4a3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}" loading="lazy"></span> for input and <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 y[n]}">

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<annotation encoding="application/x-tex">{\displaystyle y[n]}</annotation>

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</math></span><img src="./305428e6d1fb59cd0163a7a96ace52292a262afa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.844ex; height:2.843ex;" alt="{\displaystyle y[n]}" loading="lazy"></span> for output). Instead of using the Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the <a href="Z-transform" title="Z-transform">z-transform</a> (notated with a corresponding capital letter, like <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 X(z)}">

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<annotation encoding="application/x-tex">{\displaystyle X(z)}</annotation>

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</math></span><img src="./727fb275ca22820bf91e526120c4939a1d38a2b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.877ex; height:2.843ex;" alt="{\displaystyle X(z)}" loading="lazy"></span> and <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 Y(z)}">

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<annotation encoding="application/x-tex">{\displaystyle Y(z)}</annotation>

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</math></span><img src="./1529e80a525ae5701df402042a02b40f0bd7d1a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.671ex; height:2.843ex;" alt="{\displaystyle Y(z)}" loading="lazy"></span>), so a discrete-time system's transfer function can be written as:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Direct_derivation_from_differential_equations">Direct derivation from differential equations</h3></div>

<p>A <a href="Linear_differential_equation" title="Linear differential equation">linear differential equation</a> with constant coefficients

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<annotation encoding="application/x-tex">{\displaystyle L[u]={\frac {d^{n}u}{dt^{n}}}+a_{1}{\frac {d^{n-1}u}{dt^{n-1}}}+\dotsb +a_{n-1}{\frac {du}{dt}}+a_{n}u=r(t)}</annotation>

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</math></span><img src="./4f1504bd7d731876d14d76c5d4746a68e98d9f12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:53.483ex; height:6.009ex;" alt="{\displaystyle L[u]={\frac {d^{n}u}{dt^{n}}}+a_{1}{\frac {d^{n-1}u}{dt^{n-1}}}+\dotsb +a_{n-1}{\frac {du}{dt}}+a_{n}u=r(t)}" loading="lazy"></span></dd></dl>

<p>where <i>u</i> and <i>r</i> are suitably smooth functions of <i>t</i>, has <i>L</i> as the operator defined on the relevant function space that transforms <i>u</i> into <i>r</i>. That kind of equation can be used to constrain the output function <i>u</i> in terms of the <i>forcing</i> function <i>r</i>. The transfer function can be used to define an operator <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 F[r]=u}">

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<annotation encoding="application/x-tex">{\displaystyle F[r]=u}</annotation>

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</math></span><img src="./876c77d0dadd2671e078abf7c6d7532580222970.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle F[r]=u}" loading="lazy"></span> that serves as a right inverse of <i>L</i>, meaning that <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 L[F[r]]=r}">

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<annotation encoding="application/x-tex">{\displaystyle L[F[r]]=r}</annotation>

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</math></span><img src="./d4513a4d4543e0afac921bb6153fd3a8b93dcf60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.107ex; height:2.843ex;" alt="{\displaystyle L[F[r]]=r}" loading="lazy"></span>.

</p><p>Solutions of the homogeneous <a href="Linear_differential_equation#Homogeneous_equations_with_constant_coefficients" title="Linear differential equation">constant-coefficient differential equation</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 L[u]=0}">

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<annotation encoding="application/x-tex">{\displaystyle L[u]=0}</annotation>

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</math></span><img src="./990bc74c22611d6fea8f4e1ef3d2bc1a16260f78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.467ex; height:2.843ex;" alt="{\displaystyle L[u]=0}" loading="lazy"></span> can be found by trying <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 u=e^{\lambda t}}">

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<annotation encoding="application/x-tex">{\displaystyle u=e^{\lambda t}}</annotation>

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</math></span><img src="./584a2d0e5b79b2dc3c8cf0f19edd6d9c667c12e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.296ex; height:2.676ex;" alt="{\displaystyle u=e^{\lambda t}}" loading="lazy"></span>. That substitution yields the <a href="Characteristic_equation_(calculus)" title="Characteristic equation (calculus)">characteristic polynomial</a>

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<annotation encoding="application/x-tex">{\displaystyle p_{L}(\lambda )=\lambda ^{n}+a_{1}\lambda ^{n-1}+\dotsb +a_{n-1}\lambda +a_{n}\,}</annotation>

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</math></span><img src="./e6762393f3439806c1aee7e100a3f451dfb727a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:41.23ex; height:3.176ex;" alt="{\displaystyle p_{L}(\lambda )=\lambda ^{n}+a_{1}\lambda ^{n-1}+\dotsb +a_{n-1}\lambda +a_{n}\,}" loading="lazy"></span></dd></dl>

<p>The inhomogeneous case can be easily solved if the input function <i>r</i> is also of the form <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 r(t)=e^{st}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r(t)=e^{st}}</annotation>

</semantics>

</math></span><img src="./633583032db6bc1ad227f438cd84249abde2f0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.477ex; height:3.009ex;" alt="{\displaystyle r(t)=e^{st}}" loading="lazy"></span>. By substituting <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 u=H(s)e^{st}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=H(s)e^{st}}</annotation>

</semantics>

</math></span><img src="./aa9614c995441ccc2dbe7b013e09750f11a40a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.072ex; height:3.009ex;" alt="{\displaystyle u=H(s)e^{st}}" loading="lazy"></span>, <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 L[H(s)e^{st}]=e^{st}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L[H(s)e^{st}]=e^{st}}</annotation>

</semantics>

</math></span><img src="./3bcdcfff13fb52c9dd094cbac10390cce1e50977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.299ex; height:3.009ex;" alt="{\displaystyle L[H(s)e^{st}]=e^{st}}" loading="lazy"></span> if we define

</p>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mtext>wherever </mtext>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {1}{p_{L}(s)}}\qquad {\text{wherever }}\quad p_{L}(s)\neq 0.}</annotation>

</semantics>

</math></span><img src="./2e5bc212bebc14311da654c65489b906635a858a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.314ex; height:6.009ex;" alt="{\displaystyle H(s)={\frac {1}{p_{L}(s)}}\qquad {\text{wherever }}\quad p_{L}(s)\neq 0.}" loading="lazy"></span></dd></dl>

<p>Other definitions of the transfer function are used, for example <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 1/p_{L}(ik).}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/p_{L}(ik).}</annotation>

</semantics>

</math></span><img src="./a98f2fa53746c9b7baa11ff5285c3e9898d1abaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.316ex; height:2.843ex;" alt="{\displaystyle 1/p_{L}(ik).}" loading="lazy"></span><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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Gain,_transient_behavior_and_stability">Gain, transient behavior and stability</h3></div>

<p>A general sinusoidal input to a system of frequency <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 \omega _{0}/(2\pi )}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{0}/(2\pi )}</annotation>

</semantics>

</math></span><img src="./3e5542a3053fca1aae13bbf978a9042df27adadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.966ex; height:2.843ex;" alt="{\displaystyle \omega _{0}/(2\pi )}" loading="lazy"></span> may be written <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 \exp(j\omega _{0}t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(j\omega _{0}t)}</annotation>

</semantics>

</math></span><img src="./88c3e91066730c22c893afa793f994cfba1534c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.659ex; height:2.843ex;" alt="{\displaystyle \exp(j\omega _{0}t)}" loading="lazy"></span>. The response of a system to a sinusoidal input beginning at time <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 t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math></span><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"></span> will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady-state response; it corresponds to the homogeneous solution of the <a href="Differential_equation" title="Differential equation">differential equation</a>. The transfer function for an LTI system may be written as the product:

</p>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)=\prod _{i=1}^{N}{\frac {1}{s-s_{P_{i}}}}}</annotation>

</semantics>

</math></span><img src="./26e85dd8993226df8589c3711f4689747d6cbb52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.188ex; height:7.343ex;" alt="{\displaystyle H(s)=\prod _{i=1}^{N}{\frac {1}{s-s_{P_{i}}}}}" loading="lazy"></span></dd></dl>

<p>where <i>s<sub>P<sub>i</sub></sub></i> are the <i>N</i> roots of the characteristic polynomial and will be the <a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of the transfer function. In a transfer function with a single pole <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 H(s)={\frac {1}{s-s_{P}}}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {1}{s-s_{P}}}}</annotation>

</semantics>

</math></span><img src="./642b755d27eb69066f920837a919bd2d37d18796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.386ex; height:5.509ex;" alt="{\displaystyle H(s)={\frac {1}{s-s_{P}}}}" loading="lazy"></span> where <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_{P}=\sigma _{P}+j\omega _{P}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9040b9d6959247607be5fdf9f637454623e15244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.16ex; height:2.509ex;" alt="{\displaystyle s_{P}=\sigma _{P}+j\omega _{P}}" loading="lazy"></span>, the Laplace transform of a general sinusoid of unit amplitude will 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 {\frac {1}{s-j\omega _{0}}}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s-j\omega _{0}}}}</annotation>

</semantics>

</math></span><img src="./28ce17fb3a3564d12f887a71f3a187fea9c7769b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.225ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{s-j\omega _{0}}}}" loading="lazy"></span>. The Laplace transform of the output will 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 {\frac {H(s)}{s-j\omega _{0}}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {H(s)}{s-j\omega _{0}}}}</annotation>

</semantics>

</math></span><img src="./244eb1869f6b62a2b3fb93822e6a9427f49d6f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.225ex; height:6.176ex;" alt="{\displaystyle {\frac {H(s)}{s-j\omega _{0}}}}" loading="lazy"></span>, and the temporal output will be the inverse Laplace transform of that function:

</p>

<mfrac>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t)={\frac {e^{j\,\omega _{0}\,t}-e^{(\sigma _{P}+j\,\omega _{P})t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}</annotation>

</semantics>

</math></span><img src="./34790f910e35995f9f2ad644b5d85fdfcd39c9ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.162ex; height:6.509ex;" alt="{\displaystyle g(t)={\frac {e^{j\,\omega _{0}\,t}-e^{(\sigma _{P}+j\,\omega _{P})t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}" loading="lazy"></span></dd></dl>

<p>The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if <i>σ<sub>P</sub></i> is positive. For a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be:

</p>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(\infty )={\frac {e^{j\,\omega _{0}\,t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}</annotation>

</semantics>

</math></span><img src="./0c81a081921ff4e13fb2193e8a3bc3938fc33114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.646ex; height:6.509ex;" alt="{\displaystyle g(\infty )={\frac {e^{j\,\omega _{0}\,t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}" loading="lazy"></span></dd></dl>

<p>The <a href="Frequency_response" title="Frequency response">frequency response</a> (or "gain") <i>G</i> of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude:

</p>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msqrt>

</mrow>

</mrow>

</msubsup>

<msub>

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</msub>

<msub>

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</msub>

<msup>

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</msqrt>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\omega _{i})=\left|{\frac {1}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}\right|={\frac {1}{\sqrt {\sigma _{P}^{2}+(\omega _{P}-\omega _{0})^{2}}}},}</annotation>

</semantics>

</math></span><img src="./62dad3adf90546fe83b51d6898d89ef4fc028d81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:53.228ex; height:8.509ex;" alt="{\displaystyle G(\omega _{i})=\left|{\frac {1}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}\right|={\frac {1}{\sqrt {\sigma _{P}^{2}+(\omega _{P}-\omega _{0})^{2}}}},}" loading="lazy"></span></dd></dl>

<p>which is the absolute value of the transfer function <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 H(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation>

</semantics>

</math></span><img src="./f1b91390324fc9c33ec00fe57e3924ad7118cc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}" loading="lazy"></span> evaluated at <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 j\omega _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\omega _{i}}</annotation>

</semantics>

</math></span><img src="./81ca5d6a7d81c631caf5aadc8e264c97c0d73134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.23ex; height:2.509ex;" alt="{\displaystyle j\omega _{i}}" loading="lazy"></span>. This result is valid for any number of transfer-function poles.

</p>

<div class="mw-heading mw-heading2"><h2 id="Signal_processing">Signal processing</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> is the input to a general <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">linear time-invariant system</a>, and <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 y(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

</semantics>

</math></span><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"></span> is the output, and the <a href="Bilateral_Laplace_transform" class="mw-redirect" title="Bilateral Laplace transform">bilateral Laplace transform</a> of <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 x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> and <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 y(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

</semantics>

</math></span><img src="./397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy"></span> is

</p>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}X(s)&={\mathcal {L}}\left\{x(t)\right\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }x(t)e^{-st}\,dt,\\Y(s)&={\mathcal {L}}\left\{y(t)\right\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }y(t)e^{-st}\,dt.\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./6a7e80dde6188c84112122d593a04294b7b42ba1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:37.076ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}X(s)&={\mathcal {L}}\left\{x(t)\right\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }x(t)e^{-st}\,dt,\\Y(s)&={\mathcal {L}}\left\{y(t)\right\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }y(t)e^{-st}\,dt.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>The output is related to the input by the transfer function <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 H(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation>

</semantics>

</math></span><img src="./f1b91390324fc9c33ec00fe57e3924ad7118cc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}" loading="lazy"></span> as

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y(s)=H(s)X(s)}</annotation>

</semantics>

</math></span><img src="./52d03acf1580643b32259a9fc2aad66a9de6f1fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.615ex; height:2.843ex;" alt="{\displaystyle Y(s)=H(s)X(s)}" loading="lazy"></span></dd></dl>

<p>and the transfer function itself is

</p>

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {Y(s)}{X(s)}}.}</annotation>

</semantics>

</math></span><img src="./235f5bbcc16fe659b18b7101de324f66fca1d0ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.425ex; height:6.509ex;" alt="{\displaystyle H(s)={\frac {Y(s)}{X(s)}}.}" loading="lazy"></span></dd></dl>

<p>If a <a href="Complex_number" title="Complex number">complex</a> <a href="Harmonic" title="Harmonic">harmonic</a> <a href="Signal_(information_theory)" class="mw-redirect" title="Signal (information theory)">signal</a> with a <a href="Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> component with <a href="Amplitude" title="Amplitude">amplitude</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 |X|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ffa803ac0386dbd3c04054a71776b4edc435151c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.274ex; height:2.843ex;" alt="{\displaystyle |X|}" loading="lazy"></span>, <a href="Angular_frequency" title="Angular frequency">angular frequency</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 \omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> and <a href="Phase_(waves)" title="Phase (waves)">phase</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 \arg(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arg(X)}</annotation>

</semantics>

</math></span><img src="./bb80e08ffc71f5c54f7a4137ca462cb009852b72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.026ex; height:2.843ex;" alt="{\displaystyle \arg(X)}" loading="lazy"></span>, where arg is the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a>

</p>

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<annotation encoding="application/x-tex">{\displaystyle x(t)=Xe^{j\omega t}=|X|e^{j(\omega t+\arg(X))}}</annotation>

</semantics>

</math></span><img src="./923029bae4c1f212f77b0782d73529e3e4251cd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.174ex; height:3.343ex;" alt="{\displaystyle x(t)=Xe^{j\omega t}=|X|e^{j(\omega t+\arg(X))}}" loading="lazy"></span></dd></dl>

<dl><dd>where <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 X=|X|e^{j\arg(X)}}">

</mrow>

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<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=|X|e^{j\arg(X)}}</annotation>

</semantics>

</math></span><img src="./7b1f88439dd7a00453d614d4aeafbda17e8b3ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.7ex; height:3.343ex;" alt="{\displaystyle X=|X|e^{j\arg(X)}}" loading="lazy"></span></dd></dl>

<p>is input to a <a href="Linear" class="mw-redirect" title="Linear">linear</a> time-invariant system, the corresponding component in the output is:

</p>

<mtr>

<mtd>

</mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y(t)&=Ye^{j\omega t}=|Y|e^{j(\omega t+\arg(Y))},\\Y&=|Y|e^{j\arg(Y)}.\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./b3b271c000c617e78987e0d715fd4a19111b7640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.838ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}y(t)&=Ye^{j\omega t}=|Y|e^{j(\omega t+\arg(Y))},\\Y&=|Y|e^{j\arg(Y)}.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>In a linear time-invariant system, the input frequency <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 \omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. The <a href="Frequency_response" title="Frequency response">frequency response</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 H(j\omega )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(j\omega )}</annotation>

</semantics>

</math></span><img src="./54b9cf918573394f5d6d888c7c8519bfc73eb7a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.277ex; height:2.843ex;" alt="{\displaystyle H(j\omega )}" loading="lazy"></span> describes this change for every frequency <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 \omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> in terms of gain

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {|Y|}{|X|}}=|H(j\omega )|}</annotation>

</semantics>

</math></span><img src="./1f34b36876f68c240606d3ea8d1073722db45235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.959ex; height:6.509ex;" alt="{\displaystyle G(\omega )={\frac {|Y|}{|X|}}=|H(j\omega )|}" loading="lazy"></span></dd></dl>

<p>and phase shift

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (\omega )=\arg(Y)-\arg(X)=\arg(H(j\omega )).}</annotation>

</semantics>

</math></span><img src="./576cb1d01253ebfd1e01ce9a6adcf36efbcfe0fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.492ex; height:2.843ex;" alt="{\displaystyle \phi (\omega )=\arg(Y)-\arg(X)=\arg(H(j\omega )).}" loading="lazy"></span></dd></dl>

<p>The <a href="Phase_delay" class="mw-redirect" title="Phase delay">phase delay</a> (the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is

</p>

<msub>

</mrow>

</msub>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}.}</annotation>

</semantics>

</math></span><img src="./b4a4d08a6c5cfa63b538b9e8b82d51aaf1f836db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.513ex; height:5.676ex;" alt="{\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}.}" loading="lazy"></span></dd></dl>

<p>The <a href="Group_delay" class="mw-redirect" title="Group delay">group delay</a> (the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency <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 \omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>,

</p>

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<annotation encoding="application/x-tex">{\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}.}</annotation>

</semantics>

</math></span><img src="./989cae39e39c08a3b9525161c37fcc5c6477550b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.539ex; height:5.843ex;" alt="{\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}.}" loading="lazy"></span></dd></dl>

<p>The transfer function can also be shown using the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>, a special case of <a href="Bilateral_Laplace_transform" class="mw-redirect" title="Bilateral Laplace transform">bilateral Laplace transform</a> where <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=j\omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./114e5dcfb07325a06ed4329e3690d8225eadf422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.593ex; height:2.509ex;" alt="{\displaystyle s=j\omega }" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Common_transfer-function_families">Common transfer-function families</h3></div>

<p>Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used:

</p>

<ul><li><a href="Butterworth_filter" title="Butterworth filter">Butterworth filter</a> – maximally flat in passband and stopband for the given order</li>

<li><a href="Chebyshev_filter" title="Chebyshev filter">Chebyshev filter (Type I)</a> – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order</li>

<li>Chebyshev filter (Type II) – maximally flat in passband, sharper cutoff than a Butterworth filter of the same order</li>

<li><a href="Bessel_filter" title="Bessel filter">Bessel filter</a> – maximally constant <a href="Group_delay" class="mw-redirect" title="Group delay">group delay</a> for a given order</li>

<li><a href="Elliptic_filter" title="Elliptic filter">Elliptic filter</a> – sharpest cutoff (narrowest transition between passband and stopband) for the given order</li>

<li><a href="Optimum_%22L%22_filter" title="Optimum "L" filter">Optimum "L" filter</a></li>

<li><a href="Gaussian_filter" title="Gaussian filter">Gaussian filter</a> – minimum group delay; gives no overshoot to a step function</li>

<li><a href="Raised-cosine_filter" title="Raised-cosine filter">Raised-cosine filter</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Control_engineering">Control engineering</h2></div>

<p>In <a href="Control_engineering" title="Control engineering">control engineering</a> and <a href="Control_theory" title="Control theory">control theory</a>, the transfer function is derived with the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a>. The transfer function was the primary tool used in classical control engineering. A <a href="Transfer_function_matrix" title="Transfer function matrix">transfer matrix</a> can be obtained for any linear system to analyze its dynamics and other properties; each element of a transfer matrix is a transfer function relating a particular input variable to an output variable. A representation bridging <a href="State_space" class="mw-redirect" title="State space">state space</a> and transfer function methods was proposed by <a href="Howard_Harry_Rosenbrock" title="Howard Harry Rosenbrock">Howard H. Rosenbrock</a>, and is known as the <a href="Rosenbrock_system_matrix" title="Rosenbrock system matrix">Rosenbrock system matrix</a>.

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Transfer_functions_in_imaging" title="Transfer functions in imaging">Transfer functions in imaging</a></div>

<p>In <a href="Imaging" title="Imaging">imaging</a>, transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light.

</p>

<div class="mw-heading mw-heading2"><h2 id="Non-linear_systems">Non-linear systems</h2></div>

<p>Transfer functions do not exist for many <a href="Nonlinear_control" title="Nonlinear control">non-linear systems</a>, such as <a href="Relaxation_oscillator" title="Relaxation oscillator">relaxation oscillators</a>;<sup id="cite_ref-Dehaene_6-0" class="reference"><a href="#cite_note-Dehaene-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> however, <a href="Describing_function" title="Describing function">describing functions</a> can sometimes be used to approximate such nonlinear time-invariant systems.

</p>

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<ul><li><a href="Analog_computer" title="Analog computer">Analog computer</a></li>

<li><a href="Black_box" title="Black box">Black box</a></li>

<li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Duhamel's_principle" title="Duhamel's principle">Duhamel's principle</a></li>

<li><a href="Frequency_response" title="Frequency response">Frequency response</a></li>

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

<li><a href="Laplace_transform" title="Laplace transform">Laplace transform</a></li>

<li><a href="Linear_time-invariant_system" title="Linear time-invariant system">LTI system theory</a></li>

<li><a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a></li>

<li><a href="Operational_amplifier" title="Operational amplifier">Operational amplifier</a></li>

<li><a href="Optical_transfer_function" title="Optical transfer function">Optical transfer function</a></li>

<li><a href="Proper_transfer_function" title="Proper transfer function">Proper transfer function</a></li>

<li><a href="Rosenbrock_system_matrix" title="Rosenbrock system matrix">Rosenbrock system matrix</a></li>

<li><a href="Signal-flow_graph" title="Signal-flow graph">Signal-flow graph</a></li>

<li><a href="Signal_transfer_function" title="Signal transfer function">Signal transfer function</a></li></ul></div>

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

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Bernd_Girod" title="Bernd Girod">Bernd Girod</a>, Rudolf Rabenstein, Alexander Stenger, <i>Signals and systems</i>, 2nd ed., Wiley, 2001, <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-98800-6</bdi> p. 50</span>

</li>

<li id="cite_note-LaughtonWarne2002-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-LaughtonWarne2002_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFM._A._LaughtonD.F._Warne2002" class="citation book cs1">M. A. Laughton; D.F. Warne (27 September 2002). <i>Electrical Engineer's Reference Book</i> (16 ed.). Newnes. pp. 14/9–14/10. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-08-052354-5</bdi>.</cite></span>

</li>

<li id="cite_note-Parr1993-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Parr1993_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFE._A._Parr1993" class="citation book cs1">E. A. Parr (1993). <i>Logic Designer's Handbook: Circuits and Systems</i> (2nd ed.). Newness. pp. <span class="nowrap">65–</span>66. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4832-9280-9</bdi>.</cite></span>

</li>

<li id="cite_note-SinclairDunton2007-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-SinclairDunton2007_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFIan_SinclairJohn_Dunton2007" class="citation book cs1">Ian Sinclair; John Dunton (2007). <i>Electronic and Electrical Servicing: Consumer and Commercial Electronics</i>. Routledge. p. 172. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-6988-7</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFBirkhoffRota,_Gian-Carlo1978" class="citation book cs1">Birkhoff, Garrett; Rota, Gian-Carlo (1978). <i>Ordinary differential equations</i>. New York: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-05224-1</bdi>.</cite></span>

</li>

<li id="cite_note-Dehaene-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dehaene_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFValentijn_De_Smedt,_Georges_Gielen_and_Wim_Dehaene2015" class="citation book cs1">Valentijn De Smedt, Georges Gielen and Wim Dehaene (2015). <i>Temperature- and Supply Voltage-Independent Time References for Wireless Sensor Networks</i>. Springer. p. 47. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-09003-0</bdi>.</cite></span>

</li>

</ol></div>

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

<ul><li><a rel="nofollow" class="external text" href="http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf">ECE 209: Review of Circuits as LTI Systems</a> — Short primer on the mathematical analysis of (electrical) LTI systems.</li></ul>

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