Transfer Function

A transfer function is a mathematical representation of the relation between the input and output of a ( Linear Time-invariant ) System . EXPLANATION The transfer function is commonly used in the analysis of single-input single-output Analog Electronic Circuits , for instance. It is mainly used in Signal Processing , Communication Theory , and Control Theory . The term is often used exclusively to refer to Linear, Time-invariant Systems (LTI), as covered in this article. Most real systems have Non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior that is close enough to linear that LTI System Theory is an acceptable representation of the input/output behavior. In its simplest form for Continuous-time input signal $x(t)\,$ and output $y(t)\,$ , the transfer function is the linear mapping of the Laplace Transform of the input, $X(s)\,$ , to the output $Y(s)\,$ : : $Y(s) = H(s) \, X(s)$ or : $H(s) = rac{Y(s)} {X(s)} = rac { \mathcal{L}\left\{y(t) ight\} } { \mathcal{L}\left\{x(t) ight\} }$ where $H(s)\,$ is the transfer function of the LTI system. In Discrete-time systems, the function is similarly written as $H(z) = rac{Y(z)}{X(z)}$ (see Z Transform ). SIGNAL PROCESSING Let $x(t) \$ be the input to a general Linear Time-invariant System , and $y(t) \$ be the output, and the Laplace Transform of $x(t) \$ and $y(t) \$ be : $X(s) = \mathcal{L}\left \{ x(t) ight \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) e^{-st}\, dt$ : $Y(s) = \mathcal{L}\left \{ y(t) ight \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} y(t) e^{-st}\, dt$ . Then the output is related to the input by the transfer function $H(s) \$ as :: $Y(s) = H(s) X(s) \,$ and the transfer function itself is therefore :: $H(s) = rac{Y(s)} {X(s)}$ .
	:<math> X(t)	Xe^{j\omega t} = Xe^{j(\omega t + \arg(X))} </math>
	:where <math> X	Xe^{j\arg(X)} </math>
	:<math> Y(t)	Ye^{j\omega t} = Ye^{j(\omega t + \arg(Y))} </math>
	:and <math> Y	Ye^{j\arg(Y)} </math>

	CATEGORIES ABOUT TRANSFER FUNCTION

	electrical circuits

	signal processing

	control theory

	cybernetics

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Transfer Function