Transfer function

For "transfer function" as used in computer graphics, see lookup table.

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system. With optical imaging devices, for example, it is the Fourier transform (hence a function of spatial frequency) of the point spread function i.e. the intensity distribution caused by a point object in the field of view.

Explanation

The transfer function is commonly used in the analysis of single-input single-output electronic filters, 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 Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)

and output Failed to parse (Missing texvc executable; please see math/README to configure.): y(t)

, the transfer function is the linear mapping of the Laplace transform of the input, Failed to parse (Missing texvc executable; please see math/README to configure.): X(s) , to the output Failed to parse (Missing texvc executable; please see math/README to configure.): Y(s)

Failed to parse (Missing texvc executable; please see math/README to configure.): Y(s) = H(s)\;X(s)

or

Failed to parse (Missing texvc executable; please see math/README to configure.): H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} }

where Failed to parse (Missing texvc executable; please see math/README to configure.): H(s)

is the transfer function of the LTI system.

In discrete-time systems, the function is similarly written as Failed to parse (Missing texvc executable; please see math/README to configure.): H(z) = \frac{Y(z)}{X(z)}

(see Z transform) and is often referred to as the pulse-transfer function.

Signal processing

Let Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) \

be the input to a general linear time-invariant system, and Failed to parse (Missing texvc executable; please see math/README to configure.):  y(t) \ 
be the output, and the Laplace transform of Failed to parse (Missing texvc executable; please see math/README to configure.):  x(t) \ 
and Failed to parse (Missing texvc executable; please see math/README to configure.):  y(t) \ 
be

Failed to parse (Missing texvc executable; please see math/README to configure.): X(s) = \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) e^{-st}\, dt

Failed to parse (Missing texvc executable; please see math/README to configure.): Y(s) = \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} y(t) e^{-st}\, dt

.

Then the output is related to the input by the transfer function Failed to parse (Missing texvc executable; please see math/README to configure.): H(s) \

as

Failed to parse (Missing texvc executable; please see math/README to configure.): Y(s) = H(s) X(s) \,

and the transfer function itself is therefore

Failed to parse (Missing texvc executable; please see math/README to configure.): H(s) = \frac{Y(s)} {X(s)}

.

In particular, if a complex harmonic signal with a sinusoidal component with amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): |X| \ , angular frequency Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \

and phase Failed to parse (Missing texvc executable; please see math/README to configure.): \arg(X) \ 

Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) = Xe^{j\omega t} = |X|e^{j(\omega t + \arg(X))}

where Failed to parse (Missing texvc executable; please see math/README to configure.): X = |X|e^{j\arg(X)}

is input to a linear time-invariant system, then the corresponding component in the output is:

Failed to parse (Missing texvc executable; please see math/README to configure.): y(t) = Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))}

and Failed to parse (Missing texvc executable; please see math/README to configure.): Y = |Y|e^{j\arg(Y)}

.

Note that, in a linear time-invariant system, the input frequency Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \

has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system.  The frequency response Failed to parse (Missing texvc executable; please see math/README to configure.):  H(j \omega) \ 
describes this change for every frequency Failed to parse (Missing texvc executable; please see math/README to configure.):  \omega \ 
in terms of gain:

Failed to parse (Missing texvc executable; please see math/README to configure.): G(\omega) = \frac{|Y|}{|X|} = |H(j \omega)| \

and phase shift:

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi(\omega) = \arg(Y) - \arg(X) = \arg( H(j \omega))

.

The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau_{\phi}(\omega) = -\begin{matrix}\frac{\phi(\omega)}{\omega}\end{matrix}

.

The group delay (i.e., 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \ ,

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau_{g}(\omega) = -\begin{matrix}\frac{d\phi(\omega)}{d\omega}\end{matrix}

.

The transfer function can also be shown using the Fourier transform which is only a special case of the bilateral Laplace transform for the case where Failed to parse (Missing texvc executable; please see math/README to configure.): s = j \omega .

Common transfer function families

While any LTI system can be described by some transfer function or another, there are certain "families" of special transfer functions that are commonly used. Typical infinite impulse response filters are designed to implement one of these special transfer functions.

Some common transfer function families and their particular characteristics are:

• matched filter -- optimum pulse response for any arbitrary pulse shape
• Butterworth filter - maximally flat pass band for the given order
• Chebyshev filter(Type I) - no gain ripple in stop band, sharper cutoff than Butterworth
• Chebyshev filter(Type II) - no gain ripple in pass band, sharper cutoff than Butterworth
• Bessel filter - best pulse response for a given order, because it has no group delay ripple
• Elliptic filter - sharpest cutoff (narrowest transition between pass band and stop band) for the given order
• Optimum "L" filter
• Gaussian filter - minimum group delay; gives no overshoot to a step function.
• Hourglass filter
• Raised-cosine filter

Control engineering

In control engineering and control theory the transfer function is derived using the Laplace transform.

The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multiple-input multiple-output (MIMO) systems, and has been largely supplanted by state space representations for such systems. In spite of this, a transfer matrix can be always obtained for any linear system, in order 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.

See also

• Bode plot
• Convolution
• Frequency response
• LTI system theory
• Nyquist plot
• Semilog graph

External links

• transfer function on PlanetMathar:دالة تحويل

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