Wave digital filters result from the mapping of a lumped analog electrical network (usually made up of the elements mentioned in the previous section connected using Kirchoff's Laws, and which is intended for use as a filter) into the discrete-time domain. In the linear time-invariant case, this translation is carried out using a particular type of spectral mapping between the analog frequency variable and a new discrete frequency variable which will be a rational function of
which is interpreted as a unit delay, of duration ; the mapping affects only reactive -ports, i.e., those whose behavior is frequency-dependent, such as the inductor and capacitor. *Memoryless* elements, such as the transformer, gyrator and resistor (as well as the parallel or series connection, interpreted as an -port) are frequency-independent, and will be unaffected by such a transformation.

The frequency mapping proposed by Fettweis^{} in [41] is a particular type of bilinear transform, given by

We can then write

so clearly

This implies that stable, causal transfer functions in will be mapped to stable causal transfer functions in the discrete variable , and moreover that positive real functions will be mapped to functions which are

In particular, for a harmonic state--that is, for real frequencies such that and , we have that

so that the entire analog frequency spectrum is mapped to the discrete frequency spectrum exactly once. In particular, we have that the analog DC frequency is mapped to discrete DC , and that analog infinite frequency is mapped to the Nyquist frequency. It should be clear that there will be significant warping of the spectrum away from either extreme.

It is also worthwhile examining the mapping (2.11) on the unit circle in the low-frequency limit, in which case we can expand the right side of the mapping about , to get

The mapping (2.12) can be rewritten as

The frequency mapping thus becomes more accurate near in the limit as . The order of this approximation (namely to ) will play an important role in numerical integration methods, because it defines the accuracy of a numerical scheme [65,131,176].

It is important to mention that the time-domain interpretation of the bilinear mapping (2.11) is called the *trapezoid rule* for numerical integration. That is, treating as the unit delay, the right-hand side of (2.11) serves as an approximation to the derivative in a discrete-time setting. For example, in the case of the inductor, application of the mapping yields the following difference equation relating the voltage and current:

It should be understood here that and in (2.13) now represent discrete approximations to the voltage and current of (2.7) at time , for integer

For the rest of this section, so as to avoid unnecessary extra notation, we will assume that we have discrete time voltages and currents. Thus and now refer to sequences and , for integer, and the steady state quantities and are complex amplitudes of a sequence at the discrete frequency .