IIR Cascaded Second-Order Sections Form Structures

For an IIR filter with a Cascaded Second-Order Sections Form, you can implement one of the following section types of IIR Cascaded Second-Order Sections filter structures:

  • IIR Cascaded Second-Order Sections Form I
  • IIR Cascaded Second-Order Sections Form II
  • IIR Cascaded Second-Order Sections Form I Transposed
  • IIR Cascaded Second-Order Sections Form II Transposed

The transfer function of an Infinite Impulse Response (IIR) filter with a Cascaded Second-Order Sections Form structure is defined as follows:

H ( z ) = G a i n ⁢   ⋅ ∏ n = 0 N − 1     b n ⁢   ⁢   [ 0 ]   + b n     [ 1 ]     z − 1 + b n ⁢   [ 2 ]     z − 2 1 + a n ⁢   [ 1 ]     z − 1 + a n ⁢   [ 2 ]     z − 2

where

  • z is a complex variable
  • N is the number of sections
  • a is the set of reverse coefficients
  • b is the set of forward coefficients

IIR Cascaded Second-Order Sections Form I

Comparing with the IIR Direct Form structures, the IIR Cascaded Second-Order Sections Form structures have more computational complexity. However, the cascaded structures help alleviate finite word length effects. Refer to the Understanding Filter Structure Graphs topic for information that helps you read and understand a filter structure graph. The following figure represents the IIR Cascaded Second-Order Sections Form I structure:

IIR Cascaded Second-Order Sections Form II

Comparing with Form I, this structure uses the same number of mathematical operations but fewer delays. The following figure represents the IIR Cascaded Second-Order Sections Form II structure:

IIR Cascaded Second-Order Sections Form I Transposed

The following figure represents the IIR Cascaded Second-Order Sections Form I Transposed structure:

IIR Cascaded Second-Order Sections Form II Transposed

The following figure represents the IIR Cascaded Second-Order Sections Form II Transposed structure:

The IIR Cascaded Second-Order Sections Form I and Form II Transposed structures implement forward coefficients first. The Form I Transposed and Form II structures implement reverse coefficients first. The IIR Cascaded Second-Order Sections Form II structure has the same computational complexity as the Form I, but the Form I requires more memory for saving internal states. The Form II Transposed is the structure that you most frequently use. Using the Form I and Form II and their transposed structures has the same advantages and disadvantages as using the FIR Direct Form and FIR Direct Form Transposed structures.

Note If you want to generate LabVIEW FPGA code from an IIR filter, the manufacturer recommends you use the IIR Cascaded Second-Order Sections Form II Transposed structure for this filter.