A system is linear if it follows the superposition property:

  • Additivity If and , then
  • Homogeneity (scaling) If , then for all real or imaginary k,

If a linear system is also time invariant, it is called a linear time-invariant system.

Response of a Linear System

A system’s output for is the result of two independent causes:

  • the initial conditions of the system (or the system state) at
  • the input for . If the system is linear, the we can decompose the output into the sum of those two components:
  • zero-input response: result only from initial system state
  • zero-state response: result only from the system input This property of linear systems, which allows the separation of an output into components from the initial conditions and from the input, is called the decomposition property.

Significance of a Linear System

Even though almost all systems observed in practice is nonlinear, it is possible to approximate most of them by linear systems for small-signal analysis. The analysis of non-linear system is much more difficult. By contrast, the analysis of linear systems is greatly simplified by the superposition property.

In particular, if input can be broken down into simple functionsThen the response is given by

For example, for an arbitrary input , we can approximate it with a sum of rectangular pulses of width and of varying heights. Alternatively, we can also approximate it with unit step functions: signal representation in terms of impulse and step components.png

Time-domain analysis of linear systems uses the above approach. And when performing frequency domain analysis using Laplace transform or Fourier transform, we instead use sinusoids or exponentials as basis.

References

  • Linear Systems and Signals, 3rd Edition, Chapter 1.7