A LTI system is both Linear and Time Invariant.

Linear

A linear system is additive and homogenous.

Additive

  • Requires that for any two inputs \(x_{1}[n]\) and \(x_{2}[n]\), where \(Lx_1[n]\rightarrow y_1[n]\) and given system L \(Lx_2[n]\rightarrow y_2[n]\):
    \(\(Lx=y\rightarrow L(x_1+x_2)\ne y_1+y_2\)\)
  • Aka, replace x with the two inputs, then set them equal to the sum of the two inputs.
  • Basically, this test fails if the [[Difference Equation]] is not a linear one (i.e. sinusoidal or exponential) or if there is a intercept that does not scale with the input.

Homogenous

  • Requires that for any scalar k and input x, \(kx\rightarrow ky\). This will fail when there is an intercept or input invariant part.
    \(\(kx[n]+1\ne k(x[n]+1)\)\)
Time Invariant
  • A delay in the input results in a corresponding delay in the output.
  • E.g. given system L, input x, and output y,
    \(\(Lx=y \rightarrow Lx[n-m]=y[n-m]\)\)

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