Biquad Cascade IIR Filters Using a Direct Form II Transposed Structure

Functions
void	arm_biquad_cascade_df2T_f32 (const arm_biquad_cascade_df2T_instance_f32 S, float32_t pSrc, float32_t *pDst, uint32_t blockSize)
	Processing function for the floating-point transposed direct form II Biquad cascade filter.
void	arm_biquad_cascade_df2T_init_f32 (arm_biquad_cascade_df2T_instance_f32 S, uint8_t numStages, float32_t pCoeffs, float32_t *pState)
	Initialization function for the floating-point transposed direct form II Biquad cascade filter.

Description

This set of functions implements arbitrary order recursive (IIR) filters using a transposed direct form II structure. The filters are implemented as a cascade of second order Biquad sections. These functions provide a slight memory savings as compared to the direct form I Biquad filter functions. Only floating-point data is supported.

This function operate on blocks of input and output data and each call to the function processes blockSize samples through the filter. pSrc points to the array of input data and pDst points to the array of output data. Both arrays contain blockSize values.

Algorithm

Each Biquad stage implements a second order filter using the difference equation:

       
    y[n] = b0 * x[n] + d1       
    d1 = b1 * x[n] + a1 * y[n] + d2       
    d2 = b2 * x[n] + a2 * y[n]

where d1 and d2 represent the two state values.

A Biquad filter using a transposed Direct Form II structure is shown below.

Single transposed Direct Form II Biquad

Coefficients b0, b1, and b2 multiply the input signal x[n] and are referred to as the feedforward coefficients. Coefficients a1 and a2 multiply the output signal y[n] and are referred to as the feedback coefficients. Pay careful attention to the sign of the feedback coefficients. Some design tools flip the sign of the feedback coefficients:

       
    y[n] = b0 * x[n] + d1;       
    d1 = b1 * x[n] - a1 * y[n] + d2;       
    d2 = b2 * x[n] - a2 * y[n];

In this case the feedback coefficients a1 and a2 must be negated when used with the CMSIS DSP Library.

: Higher order filters are realized as a cascade of second order sections. numStages refers to the number of second order stages used. For example, an 8th order filter would be realized with numStages=4 second order stages. A 9th order filter would be realized with numStages=5 second order stages with the coefficients for one of the stages configured as a first order filter (b2=0 and a2=0).

pState points to the state variable array. Each Biquad stage has 2 state variables d1 and d2. The state variables are arranged in the pState array as:

       
     {d11, d12, d21, d22, ...}

where d1x refers to the state variables for the first Biquad and d2x refers to the state variables for the second Biquad. The state array has a total length of 2*numStages values. The state variables are updated after each block of data is processed; the coefficients are untouched.

: The CMSIS library contains Biquad filters in both Direct Form I and transposed Direct Form II. The advantage of the Direct Form I structure is that it is numerically more robust for fixed-point data types. That is why the Direct Form I structure supports Q15 and Q31 data types. The transposed Direct Form II structure, on the other hand, requires a wide dynamic range for the state variables d1 and d2. Because of this, the CMSIS library only has a floating-point version of the Direct Form II Biquad. The advantage of the Direct Form II Biquad is that it requires half the number of state variables, 2 rather than 4, per Biquad stage.

Instance Structure: The coefficients and state variables for a filter are stored together in an instance data structure. A separate instance structure must be defined for each filter. Coefficient arrays may be shared among several instances while state variable arrays cannot be shared.

Init Functions

There is also an associated initialization function. The initialization function performs following operations:

Sets the values of the internal structure fields.
Zeros out the values in the state buffer.

Use of the initialization function is optional. However, if the initialization function is used, then the instance structure cannot be placed into a const data section. To place an instance structure into a const data section, the instance structure must be manually initialized. Set the values in the state buffer to zeros before static initialization. For example, to statically initialize the instance structure use

       
     arm_biquad_cascade_df2T_instance_f32 S1 = {numStages, pState, pCoeffs};

where numStages is the number of Biquad stages in the filter; pState is the address of the state buffer. pCoeffs is the address of the coefficient buffer;

Function Documentation

void arm_biquad_cascade_df2T_f32	(	const arm_biquad_cascade_df2T_instance_f32 *	S,
		float32_t *	pSrc,
		float32_t *	pDst,
		uint32_t	blockSize
	)

Parameters:

[in]	*S	points to an instance of the filter data structure.
[in]	*pSrc	points to the block of input data.
[out]	*pDst	points to the block of output data
[in]	blockSize	number of samples to process.

Returns:: none.

References blockSize, arm_biquad_cascade_df2T_instance_f32::numStages, arm_biquad_cascade_df2T_instance_f32::pCoeffs, and arm_biquad_cascade_df2T_instance_f32::pState.

void arm_biquad_cascade_df2T_init_f32	(	arm_biquad_cascade_df2T_instance_f32 *	S,
		uint8_t	numStages,
		float32_t *	pCoeffs,
		float32_t *	pState
	)

Parameters:

[in,out]	*S	points to an instance of the filter data structure.
[in]	numStages	number of 2nd order stages in the filter.
[in]	*pCoeffs	points to the filter coefficients.
[in]	*pState	points to the state buffer.

Returns:: none

Coefficient and State Ordering:

The coefficients are stored in the array pCoeffs in the following order:

    
     {b10, b11, b12, a11, a12, b20, b21, b22, a21, a22, ...}

: where b1x and a1x are the coefficients for the first stage, b2x and a2x are the coefficients for the second stage, and so on. The pCoeffs array contains a total of 5*numStages values.

: The pState is a pointer to state array. Each Biquad stage has 2 state variables d1, and d2. The 2 state variables for stage 1 are first, then the 2 state variables for stage 2, and so on. The state array has a total length of 2*numStages values. The state variables are updated after each block of data is processed; the coefficients are untouched.

References arm_biquad_cascade_df2T_instance_f32::numStages, arm_biquad_cascade_df2T_instance_f32::pCoeffs, and arm_biquad_cascade_df2T_instance_f32::pState.