Collaboration diagram for Biquad Cascade IIR Filters Using Direct Form I Structure:

Functions
void	tpt_biquad_cascade_df1_f32 (f32_t aOutData, const tpt_biquad_cascade_df1_f32_t aFilter, const f32_t *aInData, uint32_t aCount)
	Processing function for the floating-point Biquad cascade filter. More...

void	tpt_biquad_cascade_df1_fast_q15 (q15_t aOutData, const tpt_biquad_cascade_df1_q15_t aFilter, const q15_t *aInData, uint32_t aCount)
	Processing function for the Q15 Biquad cascade filter. More...

void	tpt_biquad_cascade_df1_fast_q31 (q31_t aOutData, const tpt_biquad_cascade_df1_q31_t aFilter, const q31_t *aInData, uint32_t aCount)
	Processing function for the Q31 Biquad cascade filter.(fast version) More...

void	tpt_biquad_cascade_df1_init_f32 (tpt_biquad_cascade_df1_f32_t aFilter, uint16_t aStages, const f32_t aCoeffs, f32_t *aState)
	Initialization function for the floating-point Biquad cascade filter. More...

void	tpt_biquad_cascade_df1_init_q15 (tpt_biquad_cascade_df1_q15_t aFilter, uint16_t aStages, const q15_t aCoeffs, q15_t *aState, int8_t aPostShift)
	Initialization function for the Q15 Biquad cascade filter. More...

void	tpt_biquad_cascade_df1_init_q31 (tpt_biquad_cascade_df1_q31_t aFilter, uint16_t aStages, const q31_t aCoeffs, q31_t *aState, int8_t aPostShift)
	Initialization function for the Q31 Biquad cascade filter. More...

void	tpt_biquad_cascade_df1_q15 (q15_t aOutData, const tpt_biquad_cascade_df1_q15_t aFilter, const q15_t *aInData, uint32_t aCount)
	Processing function for the Q15 Biquad cascade filter. More...

void	tpt_biquad_cascade_df1_q31 (q31_t aOutData, const tpt_biquad_cascade_df1_q31_t aFilter, const q31_t *aInData, uint32_t aCount)
	Processing function for the Q31 Biquad cascade filter. More...

Detailed Description

This set of functions implements arbitrary order recursive (IIR) filters. The filters are implemented as a cascade of second order Biquad sections. The functions support Q15, Q31 and floating-point data types.

The functions 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] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]

A Direct Form I algorithm is used with 5 coefficients and 4 state variables per stage.

Single Biquad filter stage

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 use the difference equation:

    y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] - a1 * y[n-1] - a2 * y[n-2]

In this case the feedback coefficients a1 and a2 must be negated when used with the libml/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.

8th order filter using a cascade of Biquad 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).

The pState points to state variables array. Each Biquad stage has 4 state variables x[n-1], x[n-2], y[n-1], and y[n-2]. The state variables are arranged in the pState array as:

     { x[n-1], x[n-2], y[n-1], y[n-2] }

: 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 * uStages values. The state variables are updated after each block of data is processed, the coefficients are untouched.

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. There are separate instance structure declarations for each of the 3 supported data types.

Init Function

There is also an associated initialization function for each data type. 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. The code below statically initializes each of the 3 different data type filter instance structures

    tpt_biquad_cascade_df1_f32_t aFilter1 = { aStages, aState, aCoeffs };
    tpt_biquad_cascade_df1_q15_t aFilter2 = { aStages, aState, aCoeffs };
    tpt_biquad_cascade_df1_q31_t aFilter3 = { 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_df1_f32()

void tpt_biquad_cascade_df1_f32	(	f32_t *	aOutData,
		const tpt_biquad_cascade_df1_f32_t *	aFilter,
		const f32_t *	aInData,
		uint32_t	aCount
	)

Processing function for the floating-point Biquad cascade filter.

Parameters

[out]	aOutData	points to the block of output data.
[in]	aFilter	points to an instance of the floating-point Biquad cascade structure.
[in]	aInData	points to the block of input data.
[in]	aCount	number of samples to process

Returns: none

◆ tpt_biquad_cascade_df1_fast_q15()

void tpt_biquad_cascade_df1_fast_q15	(	q15_t *	aOutData,
		const tpt_biquad_cascade_df1_q15_t *	aFilter,
		const q15_t *	aInData,
		uint32_t	aCount
	)

Processing function for the Q15 Biquad cascade filter.

Processing function for the Q15 Biquad cascade filter.(fast version)

Parameters

[out]	aOutData	points to the location where the output result is written.
[in]	aFilter	points to an instance of the Q15 Biquad cascade structure.
[in]	aInData	points to the block of input data.
[in]	aCount	number of samples to process

Returns: none

Scaling and Overflow Behavior: This fast version uses a 32-bit accumulator with 2.30 format.The accumulator maintains full precision of the intermediate multiplication results but provides only a single guard bit.Thus, if the accumulator result overflows it wraps around and distorts the result.In order to avoid overflows completely the input signal must be scaled down by two bits and lie in the range [-0.25 +0.25).The 2.30 accumulator is then shifted by postShift bits and the result truncated to 1.15 format by discarding the low 16 bits.

Remarks: Refer to tpt_biquad_cascade_df1_q15() for a slower implementation of this function which uses 64-bit accumulation to avoid wrap around distortion. Both the slow and the fast versions use the same instance structure. Use the function tpt_biquad_cascade_df1_init_q15() to initialize the filter structure.

◆ tpt_biquad_cascade_df1_fast_q31()

void tpt_biquad_cascade_df1_fast_q31	(	q31_t *	aOutData,
		const tpt_biquad_cascade_df1_q31_t *	aFilter,
		const q31_t *	aInData,
		uint32_t	aCount
	)

Processing function for the Q31 Biquad cascade filter.(fast version)

Parameters

[out]	aOutData	points to the block of output data.
[in]	aFilter	points to an instance of the Q31 Biquad cascade structure.
[in]	aInData	points to the block of input data.
[in]	aCount	number of samples to process

Returns: none

Scaling and Overflow Behavior: This function is optimized for speed at the expense of fixed-point precision and overflow protection.The result of each 1.31 x 1.31 multiplication is truncated to 2.30 format.These intermediate results are added to a 2.30 accumulator.Finally, the accumulator is saturated and converted to a 1.31 result.The fast version has the same overflow behavior as the standard version and provides less precision since it discards the low 32 bits of each multiplication result.In order to avoid overflows completely the input signal must be scaled down by two bits and lie in the range [-0.25 +0.25). Use the intialization function tpt_biquad_cascade_df1_init_q31() to initialize filter structure.

Remarks: Refer to tpt_biquad_cascade_df1_q31() for a slower implementation of this function which uses 64-bit accumulation to provide higher precision. Both the slow and the fast versions use the same instance structure. Use the function tpt_biquad_cascade_df1_init_q31() to initialize the filter structure.

◆ tpt_biquad_cascade_df1_init_f32()

void tpt_biquad_cascade_df1_init_f32	(	tpt_biquad_cascade_df1_f32_t *	aFilter,
		uint16_t	aStages,
		const f32_t *	aCoeffs,
		f32_t *	aState
	)

Initialization function for the floating-point Biquad cascade filter.

Parameters

[in,out]	aFilter	points to an instance of the floating-point Biquad cascade 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 pState is a pointer to state array. Each Biquad stage has 4 state variables x[n-1], x[n-2], y[n-1], and y[n-2]. The state variables are arranged in the pState array as:

    { x[n-1], x[n-2], y[n-1], y[n-2] }

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.

◆ tpt_biquad_cascade_df1_init_q15()

void tpt_biquad_cascade_df1_init_q15	(	tpt_biquad_cascade_df1_q15_t *	aFilter,
		uint16_t	aStages,
		const q15_t *	aCoeffs,
		q15_t *	aState,
		int8_t	aPostShift
	)

Initialization function for the Q15 Biquad cascade filter.

Parameters

[in,out]	aFilter	points to an instance of the Q15 Biquad cascade 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.
[in]	aPostShift	Shift to be applied to the accumulator result. Varies according to the coefficients format.

Returns: none

Coefficient and State Ordering

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

    { b10, 0, b11, b12, a11, a12, b20, 0, 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 6 * aStages values.

The state variables are stored in the array aState. Each Biquad stage has 4 state variables x[n-1], x[n-2], y[n-1], and y[n-2]. The state variables are arranged in the aState array as:

    { x[n-1], x[n-2], y[n-1], y[n-2] }

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.

◆ tpt_biquad_cascade_df1_init_q31()

void tpt_biquad_cascade_df1_init_q31	(	tpt_biquad_cascade_df1_q31_t *	aFilter,
		uint16_t	aStages,
		const q31_t *	aCoeffs,
		q31_t *	aState,
		int8_t	aPostShift
	)

Initialization function for the Q31 Biquad cascade filter.

Parameters

[in,out]	aFilter	points to an instance of the Q31 Biquad cascade 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.
[in]	aPostShift	Shift to be applied after the accumulator. Varies according to the coefficients format

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 points to state variables array. Each Biquad stage has 4 state variables x[n - 1], x[n - 2], y[n - 1], and y[n - 2]. The state variables are arranged in the aState array as:

    { x[n - 1], x[n - 2], y[n - 1], y[n - 2] }

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.

◆ tpt_biquad_cascade_df1_q15()

void tpt_biquad_cascade_df1_q15	(	q15_t *	aOutData,
		const tpt_biquad_cascade_df1_q15_t *	aFilter,
		const q15_t *	aInData,
		uint32_t	aCount
	)

Processing function for the Q15 Biquad cascade filter.

Parameters

[out]	aOutData	points to the location where the output result is written.
[in]	aFilter	points to an instance of the Q15 Biquad cascade structure.
[in]	aInData	points to the block of input data.
[in]	aCount	number of samples to process

Returns: none

Scaling and Overflow Behavior: The function is implemented using a 64-bit internal accumulator. Both coefficients and state variables are represented in 1.15 format and multiplications yield a 2.30 result. The 2.30 intermediate results are accumulated in a 64-bit accumulator in 34.30 format. There is no risk of internal overflow with this approach and the full precision of intermediate multiplications is preserved. The accumulator is then shifted by iPostShift bits to truncate the result to 1.15 format by discarding the low 16 bits. Finally, the result is saturated to 1.15 format.

◆ tpt_biquad_cascade_df1_q31()

void tpt_biquad_cascade_df1_q31	(	q31_t *	aOutData,
		const tpt_biquad_cascade_df1_q31_t *	aFilter,
		const q31_t *	aInData,
		uint32_t	aCount
	)

Processing function for the Q31 Biquad cascade filter.

Parameters

[out]	aOutData	points to the block of output data.
[in]	aFilter	points to an instance of the Q31 Biquad cascade structure.
[in]	aInData	points to the block of input data.
[in]	aCount	number of samples to process

Returns: none

Scaling and Overflow Behavior: The function is implemented using an internal 64-bit accumulator. The accumulator has a 2.62 format and maintains full precision of the intermediate multiplication results but provides only a single guard bit. Thus, if the accumulator result overflows it wraps around rather than clip. In order to avoid overflows completely the input signal must be scaled down by 2 bits and lie in the range [-0.25 +0.25). After all 5 multiply-accumulates are performed, the 2.62 accumulator is shifted by iPostShift bits and the result truncated to 1.31 format by discarding the low 32 bits.

Functions

Detailed Description

Function Documentation

◆ tpt_biquad_cascade_df1_f32()

◆ tpt_biquad_cascade_df1_fast_q15()

◆ tpt_biquad_cascade_df1_fast_q31()

◆ tpt_biquad_cascade_df1_init_f32()

◆ tpt_biquad_cascade_df1_init_q15()

◆ tpt_biquad_cascade_df1_init_q31()

◆ tpt_biquad_cascade_df1_q15()

◆ tpt_biquad_cascade_df1_q31()