Linear time-invariant systems
Transfer functions are commonly used in the analysis of systems such as
single-input single-output filters, typically within the fields of
signal processing,
communication theory, and
control theory. The term is often used exclusively to refer to
linear time-invariant (LTI) systems, 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.
The descriptions below are given in terms of a complex variable, s = σ + j ⋅ ω {\displaystyle s=\sigma +j\cdot \omega }
, which bears a brief explanation. In many applications, it is sufficient to define σ = 0 {\displaystyle \sigma =0}
(and s = j ⋅ ω {\displaystyle s=j\cdot \omega }
), which reduces the
Laplace transforms with complex arguments to
Fourier transforms with real argument ω. The applications where this is common are ones where there is interest only in the steady-state response of an LTI system, not the fleeting turn-on and turn-off behaviors or stability issues. That is usually the case for
signal processing and
communication theory.
(and s = j ⋅ ω {\displaystyle s=j\cdot \omega }
), which reduces the Laplace transforms with complex arguments to Fourier transforms with real argument ω. The applications where this is common are ones where there is interest only in the steady-state response of an LTI system, not the fleeting turn-on and turn-off behaviors or stability issues. That is usually the case for signal processing and communication theory.
