– maximally flat in passband and stopband for the given order

Butterworth filter

– maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order

Chebyshev filter (Type I)

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

– maximally constant group delay for a given order

Bessel filter

– sharpest cutoff (narrowest transition between passband and stopband) for the given order

Elliptic filter

Optimum "L" filter

– minimum group delay; gives no overshoot to a step function

Gaussian filter

Raised-cosine filter

Control engineering[edit]

In control engineering and control theory, the transfer function is derived with the Laplace transform. The transfer function was the primary tool used in classical control engineering. A transfer matrix 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 state space and transfer function methods was proposed by Howard H. Rosenbrock, and is known as the Rosenbrock system matrix.

Non-linear systems[edit]

Transfer functions do not exist for many non-linear systems, such as relaxation oscillators;^[6] however, describing functions can sometimes be used to approximate such nonlinear time-invariant systems.

— Short primer on the mathematical analysis of (electrical) LTI systems.