Discrete-time systems modeled by difference equations relate the output \(y[n]\) to the input \(x[n]\). The general form of an \(N\)th-order difference equation may be written as
\(\boxed{y[n] + A_{1}\,y[n - 1] + \cdots + A_{N}\,y[n - N] = B_{0}\,x[n] + B_{1}\,x[n - 1] + \cdots + B_{M}\,x[n - M]}\)
The order \(N\) describes the output term with the largest delay. It is customary to normalize the leading coefficient to unity.
Difference equations involve signals and their shifted versions, which can be conveniently expressed using operator notation.
For integer values of \(k\),
\(\boxed{z^{\,k}\{x[n]\} = x[n + k]}\)
where \(z^{\,k}\) is shift operator.
Linear Time-Invariant Systems (LTI)
An LTI system is described by difference equations with constant coefficients.
\(\boxed{y[n] + A_{1}\,y[n - 1] + \cdots + A_{N}\,y[n - N] = B_{0}\,x[n] + B_{1}\,x[n - 1] + \cdots + B_{M}\,x[n - M]}\)
where coefficients \(A\) and \(B\) are constants, and all terms contain \(x[n]\) or \(y[n]\).
To test for linearity or time-invariance, the following can be applied:
- Terms containing products of the input and/or output make a system equation non-linear. A constant term also makes a system equation non-linear.
- Coefficients of the input or output that are explicit functions of \(n\) make a system equation time-variant (time varying). Time-scaled (w/o reflection) inputs or outputs also make a system equation time varying.
Causality and Memory
In a causal system, the present response \(y[n]\) cannot depend on future values of the input \(x[n]\). Systems whose present response \(y[n]\) requires future inputs \(x[n + K]\) \((K > 0)\) are termed non-causal. Causality can be checked by examining the operational transfer function \(H(z)\) derived from the difference equation in operator form.
\(\boxed{H(z) = \frac{B_{0}z^M + B_{1}z^{M - 1} + \cdots + B_{M - 1}z + B_M}{A_{0}z^N + A_{1}z^{N - 1} + \cdots + A_{N - 1}z + A_N} = \frac{P_{M}(z)}{Q_{N}(z)}}\)
If the order \(M\) of the numerator polynomial \(P_{M}(z)\) exceeds the order \(N\) of the denominator polynomial \(Q_{N}(z)\), the system is non-causal.
Instantaneous and Dynamic Systems
If the response \(y[n]\) of a system at \(n = n_0\) depends only on the input \(x[n]\) at \(n = n_0\) and not at any other times (past or future), the system is called instantaneous or static. If the response \(y[n]\) of a system depends on past (and/or future) inputs \(x[n]\), the system is called dynamic.
Dynamic systems are usually described by (but not limited to) difference equations.