Time Varying Filters

Time varying recursive filter structures with convenient controls are hard to find in general. For example, given an analog voltage-controlled filter (VCF) that behaves in a valued way as a function of the control voltage, how does one find a similar digital counterpart? The standard technique for digitizing an analog filter is the bilinear transform, and frequency scaling can be done in the digital domain using an allpass substitution in the digital filter transfer function. To obtain a digital VCF, one might think of implementing real-time frequency scaling by replacing each delay element of the unscaled digital filter with a first-order allpass filter; however, when this is applied to a recursive digital filter, such as the Moog Ladder mentioned earlier, a nonrealizable structure is obtained because a delay-free loop is introduced. The general way to eliminate the delay-free loop is to multiply out the filter denominator and renormalize it, but this destroys the nice control structure which led to the choice of the analog prototype filter in the first place. An ad hoc solution which preserves the control structure is to insert a unit-sample delay in the loop to make it implementable. However, this generally degrades the frequency response at high frequencies.