Editing Functional decomposition (section)

== Signal processing ==
Functional decomposition is used in the analysis of many [[signal processing]] systems, such as [[LTI system theory|LTI systems]]. The input signal to an LTI system can be expressed as a function, <math>f(t)</math>. Then <math>f(t)</math> can be decomposed into a linear combination of other functions, called component signals:
::<math> f(t) = a_1 \cdot g_1(t) + a_2 \cdot g_2(t) + a_3 \cdot g_3(t) + \dots + a_n \cdot g_n(t) </math>
Here, <math> \{g_1(t), g_2(t), g_3(t), \dots , g_n(t)\} </math> are the component signals. Note that <math> \{a_1, a_2, a_3, \dots , a_n\} </math> are constants. This decomposition aids in analysis, because now the output of the system can be expressed in terms of the components of the input. If we let <math>T\{\}</math> represent the effect of the system, then the output signal is <math>T\{f(t)\}</math>, which can be expressed as:
::<math> T\{f(t)\} = T\{ a_1 \cdot g_1(t) + a_2 \cdot g_2(t) + a_3 \cdot g_3(t) + \dots + a_n \cdot g_n(t)\}</math>
::<math> = a_1 \cdot T\{g_1(t)\} + a_2 \cdot T\{g_2(t)\} + a_3 \cdot T\{g_3(t)\} + \dots + a_n \cdot T\{g_n(t)\}</math>
In other words, the system can be seen as acting separately on each of the components of the input signal. Commonly used examples of this type of decomposition are the [[Fourier series]] and the [[Fourier transform]].