Digital filters are most often applied to discrete time series by convolving the time series with the unit impulse response, or weighting function of the filter. Each output point computed is a weighted sum of a finite number of the input points. It is also possible to recursively filter a time series. That is, we can compute each output point as a weighted sum of input points plus a weighted sum of previously computed output points. The advantage of this technique is that certain filtering operations can be performed much faster using recursion as compared to convolution. A recursive filter can be expressed as a ratio of two polynomials of the z‐transform variable. Given the rational z‐transform for a particular filter, the recursive rule or algorithm for that filter can be derived. There are various ways to determine the rational z‐transform for specific types of filtering. Using our knowledge of the z‐transform, we can design filters by specifying the roots of the two rational polynomials. Also, we can use the relationship between the Laplace transform and the z‐transform to convert desirable analog filter equations into their digital equivalents. We can use least‐square techniques to approximate a desired impulse response with a recursion formula. Often the resulting recursion formula will be considerably more efficient in use than the original convolution operation. Wiener filters also can be expressed as rational z‐transforms. In many cases the resulting recursion formulas will be quite simple, providing efficient and rapid digital processing.