Negative time

Notice that X(Z) and Y(Z) need not strictly be polynomials; they may contain both positive and negative powers of Z, such as

$$\begin{eqnarray} X(Z) &=& \cdots + {x_{-2} \over Z^2} + {x_{-1} \over Z} + x_0 +... ...dots + {y_{-2} \over Z^2} + {y_{-1} \over Z} + y_0 + y_1 Z +\cdots\end{eqnarray}$$ (11) (12)

The negative powers of Z in X(Z) and Y(Z) show that the data is defined before t = 0. The effect of using negative powers of Z in the filter is different. Inspection of (9) shows that the output y_k that occurs at time k is a linear combination of current and previous inputs; that is, $(x_i,\, i \leq k)$.If the filter B(Z) had included a term like b_-1/Z, then the output y_k at time k would be a linear combination of current and previous inputs and x_k+1, an input that really has not arrived at time k. Such a filter is called a ``nonrealizable'' filter,

because it could not operate in the real world where nothing can respond now to an excitation that has not yet occurred. However, nonrealizable filters are occasionally useful in computer simulations where all the data is prerecorded.