Digital Signal Processing, 10th Edition

8Digital Ladder Filters

The filter structures presented in the previous chapters are deduced directly from their Z-transfer functions, with the coefficients applied to the multiplying circuits being those of the powers of Z⁻¹. More elaborate structures can be developed.

In analog filtering, structures exist which allow filters with very low ripple and excellent selectivity to be constructed using passive components of limited precision. In digital filtering, these properties can be translated into a reduction in the round-off noise and in the number of bits representing the coefficients.

Analog filter networks are based on cascading two port circuits whose properties will be considered first [1].

8.1 Properties of Two-Port Circuits

The general two-port circuit terminated by the resistors R₁ and R₂ is shown in Figure 8.1, together with the currents I and voltages V at ports 1 and 2. This circuit, assumed to be linear, is defined by its impedance matrix z, which establishes the relations between the variables, generally written in reduced form as:

Then,

(8.1)

with:

The values z₁₂ and z₂₁ are the transfer impedances of the two-port circuit. It is reciprocal if z₁₂ = z₂₁.

If ...