# Time-variant system

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A time-variant system is a system that is not time invariant (TIV). Roughly speaking, characteristics its output depend explicitly upon time.

## Overview

There are many well developed techniques for dealing with the response of linear time invariant systems, such as Laplace and Fourier transforms. However, these techniques are not strictly valid for time-varying systems. A system undergoing slow time variation in comparison to its time constants can usually be considered to be time invariant: they are close to time invariant on a small scale. An example of this is the aging and wear of electronic components, which happens on a scale of years, and thus does not result in any behaviour qualitatively different from that observed in a time invariant system: day-to-day, they are effectively time invariant, though year to year, the parameters may change. Other linear time variant systems may behave more like nonlinear systems, if the system changes quickly – significantly differently between measurements.

The following things can be said about a time-variant system:

• It has explicit dependence on time.
• It does not have an impulse response in the normal sense. The system can be characterized by an impulse response except the impulse response must be known at each and every time instant.
• It is not stationary

## Examples of time-variant systems

The following time varying systems cannot be modelled by assuming that they are time invariant:

• Aircraft – Time variant characteristics are caused by different configuration of control surfaces during take off, cruise and landing as well as constantly decreasing weight due to consumption of fuel.
• The Earth's thermodynamic response to incoming solar radiation varies with time due to changes in the Earth's albedo and the presence of greenhouse gasses in the atmosphere.
• The human vocal tract is a time variant system, with its transfer function at any given time dependent on the shape of the vocal organs. As with any fluid-filled tube, resonances (called formants) change as the vocal organs such as the tongue and velum move. Mathematical models of the vocal tract are therefore time-variant, with transfer functions often linearly interpolated between states over time.
• Linear time varying processes such as amplitude modulation occur on a time scale similar to or faster than that of the input signal. In practice amplitude modulation is often implemented using time invariant nonlinear elements such as diodes.
• The Discrete Wavelet Transform, often used in modern signal processing, is time variant because it makes use of the decimation operation.

## Time-variant system: Elaboration

By definition, the input–output characteristics vary with time. Let:

• x(t) be an excitation signal.
• T(x(t), t) describe the input–output map of a system in relaxed state.
• y(t) be the system's output response $y(t) = T(x(t), t)$ to the excitation signal.

If the excitation signal is delayed by time k (i.e., x(t-k)) and the output response $T( x(t-k), t )$ is not equivalent to a delayed version of the original output $y(t-k)=T( x(t-k), t-k )$, then the system is time variant.