Time-variant system

A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application[1]. In other words, a time delay or time advance of input not only shifts the output signal in time but also changes other parameters and behavior. Time variant systems respond differently to the same input at different times. The opposite is true for time invariant systems (TIV).

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 differing 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

Linear time-variant systems

Linear-time variant (LTV) systems are the ones whose parameters vary with time according to previously specified laws. Mathematically, there is a well defined dependence of the system over time and over the input parameters that change over time.

In order to solve time-variant systems, the algebraic methods consider initial conditions of the system i.e. whether the system is zero-input or non-zero input system.

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 irradiance varies with time due to changes in the Earth's albedo and the presence of greenhouse gases in the atmosphere[2] [3].
  • 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[4].
  • 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 system nonlinear elements such as diodes.
  • Discrete wavelet transform, often used in modern signal processing, is time variant because it makes use of the decimation operation.
  • Adaptive filters in digital signal processing (DSP) are time variant filters. They follow a time varying input signal and learn to distinguish between unwanted digital signal (usually noise) and the wanted signal buried together in input. The most typical implementation of adaptive filters is Least mean square (LMS) method[5]. The LMS algorithm is a successive-approximation technique that obtains the optimal filter coefficients required for the minimisation of error (or unwanted signal). The coefficients of filter will vary over time and update themselves as input signal varies.[5]
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See also

References
  1. Cherniakov, Mikhail (2003). An Introduction to Parametric Digital Filters and Oscillators. Wiley. pp. 47–49. ISBN 978-0470851043.
  2. Sung, Taehong; Yoon, Sang; Kim, Kyung (2015-07-13). "A Mathematical Model of Hourly Solar Radiation in Varying Weather Conditions for a Dynamic Simulation of the Solar Organic Rankine Cycle". Energies. 8 (7): 7058–7069. doi:10.3390/en8077058. ISSN 1996-1073.
  3. Alzahrani, Ahmad; Shamsi, Pourya; Dagli, Cihan; Ferdowsi, Mehdi (2017). "Solar Irradiance Forecasting Using Deep Neural Networks". Procedia Computer Science. 114: 304–313. doi:10.1016/j.procs.2017.09.045.
  4. Strube, H. (1982). "Time-varying wave digital filters and vocal-tract models". ICASSP '82. IEEE International Conference on Acoustics, Speech, and Signal Processing. Paris, France: Institute of Electrical and Electronics Engineers. 7: 923–926. doi:10.1109/ICASSP.1982.1171595.
  5. Gaydecki, Patrick (2004). Foundations of Digital Signal Processing: theory, algorithms and hardware design. MPG Book limited, Cornwall: The Institute of Electrical Engineers (IEE), UK. pp. 387–401. ISBN 978-0852964316.
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