the bilateral Laplace transform for the case where . 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. Some common transfer function families and their particular characteristics are: Butterworth filter – maximally flat in passband and stopband for the given order Chebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order Chebyshev filter (Type II) – maximally flat in passband, sharper cutoff than a Butterworth filter of the same order 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 always be 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. A useful representation bridging state space and transfer function methods was proposed by Howard H. Rosenbrock and is referred to as Rosenbrock system matrix. Optics In optics, modulation transfer function indicates the capability of optical contrast transmission. For example, when observing a series of black-white-light fringes drawn with a specific spatial frequency, the image quality may decay. White fringes fade while black ones turn brighter. The modulation transfer function in a specific spatial frequency is defined by where modulation (M) is computed from the following image or light brightness: Imaging In imaging, transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light. Non-linear systems Transfer functions do not properly exist for many non-linear systems. For example, they do not exist for relaxation oscillators; however, describing functions can sometimes be used to approximate such nonlinear time-invariant systems. See also References