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Post Filter Theory Laplacian Space  Z Space  Z Transform Stability Analog values of the postfilter coefficients are produced as parts of a Laplace Transform:
The amplitude and phase of the filter can be derived from the above by:
is similar to except that the returned angle can be in the range from to . From here we can calculate the gain (in dB) of the filter: The filter types are designed as follows:
Additional Notes: For the resonator filter, the maximum and minimum phase changes will occur at: These frequencies also happen to be the halfgain points measured in dB (or the rootamplitude gain points). For the lead/lag filters, the maximal phase change will occur at: This frequency also happens to be the dB gain mean point measured (or the amplitude gain geometric mean point).
Though Laplacian Space is useful for designing or quickly analyzing a biquad filter's design, it does not accurately model digital biquad filters. Digital filters are described naturally by Z transforms. It is possible to convert a filter from a Laplace transform to a Z transform, as will be described below, while maintaining the same general characteristics. The amplitude and phase information will be slightly warped by moving into Z space. One should note, however, that for the filters listed above the characteristics of gains, bandwidths, and center or breakpoint frequencies are unchanged. Biquad filters are described by the following Z transform: One should note that only filters where the roots of the denominator lie within the unit circle are stable. Though digital filters can be constructed where the equations for amplitude and phase for both the Z transform version and the Laplace transform version may converge, the filter itself will be unstable, continually adding energy to the system. Please see the Z Transform Stability Section below. The equations for amplitude, phase and dB gain can be derived from the above Z transform: The equations for converting between the analog (Laplace transform) coefficients and the digital (Z transform) coefficients are handled internally by the MPI, but are listed below so that one can accurately analyze the performance of the biquad filters.
As briefly described in the last section, it is possible for the digital filters constructed from analog filters to be unstable. One needs to ensure that:
To guarantee a filter does not continually add energy to a system, the following relationship must be satisfied by the Z transform coefficients: To guarantee a filter has no phase lag at 0 frequency, the following relationship must be satisfied by the Z transform coefficients: If it is found that this last condition is not true, then one should change the sign on all B_{n} coefficients. Equivalently, one can change the sign of all b_{n} coefficients for the Laplace (analog) transform.

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