The Z-transform of an arbitrary, time-discretized function xt is defined by
Going on to consider numerical values for the delay operator Z, we discover that it is useful to ask whether X(Z) is finite or infinite. Numerical values of Z that are of particular interest are Z = +1, Z = -1, and all those complex values of Z which are unit magnitude, say, |Z| = 1 or
The most straightforward way to say that a filter is causal is to say that its time domain coefficients vanish before zero lag, that is ut = 0 for t<0. Another way to say it is to say that U(Z) is finite for Z=0. At Z=0 the Z-transform would be infinite if the coefficients etc. were not zero. For a causal function, each term in |U(Z)| will be smaller if Z is taken inside the disk |Z|<1 rather than on it. Thus convergence at Z=0 and on the circle |Z|=1 implies convergence everywhere inside the unit disk. So boundedness combined with causality means convergence in the unit disk. Convergence at Z = 0 but not on the circle |Z| = 1 would refer to a causal function with infinite energy, a case of no practical interest. What kind of function converges on the circle, at ,but not at Z = 0? What function converges at all three places, Z = 0, , and |Z| = 1?
The filter can be expanded into powers of Z in (at least) two different ways. These are
Let bt denote a filter. Then at is its inverse filter if the convolution of at with bt is a delta function. In the Fourier domain, we would say that filters are inverse to one another if their Fourier transforms are inverse to one another. Z-transforms can be used to define the inverse filter, say, A(Z) = 1/B(Z). Whether the filter A(Z) is causal depends on whether it is finite everywhere inside the unit disk, or really on whether B(Z) vanishes anywhere inside the disk. For example, B(Z) = 1 - 2Z vanishes at Z = 1/2. There A(Z) = 1/B(Z) must be infinite, that is to say, the series A(Z) must be nonconvergent at Z = 1/2. Thus--as we have just seen--at is noncausal. A most interesting case, called minimum phase, occurs when both a filter B(Z) and its inverse are causal. In summary:
|minimum phase||both above conditions|