Collaboration diagram for Biquad Cascade IIR Filters Using a Direct Form II Transposed Structure:

Functions
void	tpt_biquad_cascade_df2T_f32 (f32_t aOutData, const tpt_biquad_cascade_df2T_f32_t aFilter, const f32_t *aInData, uint32_t aCount)
	Processing function for the floating-point transposed direct form II Biquad cascade filter. More...

void	tpt_biquad_cascade_df2T_init_f32 (tpt_biquad_cascade_df2T_f32_t aFilter, uint16_t aStages, const f32_t aCoeffs, f32_t *aState)
	Initialization function for the floating-point transposed direct form II Biquad cascade filter. More...

void	tpt_biquad_cascade_stereo_df2T_f32 (f32_t aOutData, const tpt_biquad_cascade_stereo_df2T_f32_t aFilter, const f32_t *aInData, uint32_t aCount)
	Processing function for the floating-point transposed direct form II Biquad cascade filter. More...

void	tpt_biquad_cascade_stereo_df2T_init_f32 (tpt_biquad_cascade_stereo_df2T_f32_t aFilter, uint16_t aStages, const f32_t aCoeffs, f32_t *aState)
	Initialization function for the floating-point transposed direct form II Biquad cascade filter. More...

Detailed 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 aCount samples through the filter. aInData points to the array of input data and aOutData points to the array of output data. Both arrays contain aCount 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 DSP Library.

: Higher order filters are realized as a cascade of second order sections. uStages refers to the number of second order stages used. For example, an 8th order filter would be realized with uStages = 4 second order stages. A 9th order filter would be realized with uStages = 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 * uStages values. The state variables are updated after each block of data is processed; the coefficients are untouched.

: The DSP 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 DSP 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. To do this manually without calling the init function, assign the follow subfields of the instance structure: uStages, pCoeffs, pState. Also set all of the values in pState to zero.

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

    tpt_biquad_cascade_df2T_f32_t aFilter1 = { aStages, aState, aCoeffs };

where aStages is the number of Biquad stages in the filter; aState is the address of the state buffer. aCoeffs is the address of the coefficient buffer;

Function Documentation

◆ tpt_biquad_cascade_df2T_f32()

void tpt_biquad_cascade_df2T_f32	(	f32_t *	aOutData,
		const tpt_biquad_cascade_df2T_f32_t *	aFilter,
		const f32_t *	aInData,
		uint32_t	aCount
	)

Processing function for the floating-point transposed direct form II Biquad cascade filter.

Parameters

[out]	aOutData	points to the block of output data.
[in]	aFilter	points to an instance of the filter data structure.
[in]	aInData	points to the block of input data.
[in]	aCount	number of samples to process

Returns: none

◆ tpt_biquad_cascade_df2T_init_f32()

void tpt_biquad_cascade_df2T_init_f32	(	tpt_biquad_cascade_df2T_f32_t *	aFilter,
		uint16_t	aStages,
		const f32_t *	aCoeffs,
		f32_t *	aState
	)

Initialization function for the floating-point transposed direct form II Biquad cascade filter.

Parameters

[in,out]	aFilter	points to an instance of the filter data structure.
[in]	aStages	number of 2nd order stages in the filter.
[in]	aCoeffs	points to the filter coefficients.
[in]	aState	points to the state buffer.

Returns: none

Coefficient and State Ordering

The coefficients are stored in the array aCoeffs in the following order version.

    { 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 aCoeffs array contains a total of 5 * aStages values.

: The aState 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 * aStages values. The state variables are updated after each block of data is processed; the coefficients are untouched.

◆ tpt_biquad_cascade_stereo_df2T_f32()

void tpt_biquad_cascade_stereo_df2T_f32	(	f32_t *	aOutData,
		const tpt_biquad_cascade_stereo_df2T_f32_t *	aFilter,
		const f32_t *	aInData,
		uint32_t	aCount
	)

Processing function for the floating-point transposed direct form II Biquad cascade filter.

Parameters

[in]	aInData	points to the block of input data.
[in]	aFilter	points to an instance of the filter data structure.
[out]	aOutData	points to the block of output data.
[in]	aCount	number of samples to process

Returns: none

◆ tpt_biquad_cascade_stereo_df2T_init_f32()

void tpt_biquad_cascade_stereo_df2T_init_f32	(	tpt_biquad_cascade_stereo_df2T_f32_t *	aFilter,
		uint16_t	aStages,
		const f32_t *	aCoeffs,
		f32_t *	aState
	)

Initialization function for the floating-point transposed direct form II Biquad cascade filter.

Parameters

[in,out]	aFilter	points to an instance of the filter data structure.
[in]	aStages	number of 2nd order stages in the filter.
[in]	aCoeffs	points to the filter coefficients.
[in]	aState	points to the state buffer.

Returns: none

Coefficient and State Ordering

The coefficients are stored in the array aCoeffs 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 aCoeffs array contains a total of 5 * aStages values.

: The aState is a pointer to state array. Each Biquad stage has 4 state variables d1a, d2a, d1b and d2b for each channel. The 4 state variables for stage 1 are first, then the 4 state variables for stage 2, and so on. The state array has a total length of 4 * aStages values. The state variables are updated after each block of data is processed; the coefficients are untouched.