DSP/Source/FilteringFunctions/arm_biquad_cascade_df1_32x64_q31.c - third_party/github/STMicroelectronics/cmsis_core - Git at Google

 /* ----------------------------------------------------------------------
  * Project:      CMSIS DSP Library
  * Title:        arm_biquad_cascade_df1_32x64_q31.c
  * Description:  High precision Q31 Biquad cascade filter processing function
  *
  * $Date:        18. March 2019
  * $Revision:    V1.6.0
  *
  * Target Processor: Cortex-M cores
  * -------------------------------------------------------------------- */
 /*
  * Copyright (C) 2010-2019 ARM Limited or its affiliates. All rights reserved.
  *
  * SPDX-License-Identifier: Apache-2.0
  *
  * Licensed under the Apache License, Version 2.0 (the License); you may
  * not use this file except in compliance with the License.
  * You may obtain a copy of the License at
  *
  * www.apache.org/licenses/LICENSE-2.0
  *
  * Unless required by applicable law or agreed to in writing, software
  * distributed under the License is distributed on an AS IS BASIS, WITHOUT
  * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  * See the License for the specific language governing permissions and
  * limitations under the License.
  */

 #include "arm_math.h"

 /**
   @ingroup groupFilters
  */

 /**
   @defgroup BiquadCascadeDF1_32x64 High Precision Q31 Biquad Cascade Filter

   This function implements a high precision Biquad cascade filter which operates on
   Q31 data values.  The filter coefficients are in 1.31 format and the state variables
   are in 1.63 format.  The double precision state variables reduce quantization noise
   in the filter and provide a cleaner output.
   These filters are particularly useful when implementing filters in which the
   singularities are close to the unit circle.  This is common for low pass or high
   pass filters with very low cutoff frequencies.

   The function operates on blocks of input and output data
   and each call to the function processes <code>blockSize</code> samples through
   the filter. <code>pSrc</code> and <code>pDst</code> points to input and output arrays
   containing <code>blockSize</code> Q31 values.

   @par           Algorithm
                    Each Biquad stage implements a second order filter using the difference equation:
   <pre>
       y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]
   </pre>
                    A Direct Form I algorithm is used with 5 coefficients and 4 state variables per stage.
                    \image html Biquad.gif "Single Biquad filter stage"
                    Coefficients <code>b0, b1 and b2 </code> multiply the input signal <code>x[n]</code> and are referred to as the feedforward coefficients.
                    Coefficients <code>a1</code> and <code>a2</code> multiply the output signal <code>y[n]</code> 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
   <pre>
       y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] - a1 * y[n-1] - a2 * y[n-2]
   </pre>
                    In this case the feedback coefficients <code>a1</code> and <code>a2</code> must be negated when used with the CMSIS DSP Library.
   @par
                    Higher order filters are realized as a cascade of second order sections.
                    <code>numStages</code> refers to the number of second order stages used.
                    For example, an 8th order filter would be realized with <code>numStages=4</code> second order stages.
                    \image html BiquadCascade.gif "8th order filter using a cascade of Biquad stages"
                    A 9th order filter would be realized with <code>numStages=5</code> second order stages
                    with the coefficients for one of the stages configured as a first order filter
                    (<code>b2=0</code> and <code>a2=0</code>).
   @par
                    The <code>pState</code> points to state variables array.
                    Each Biquad stage has 4 state variables <code>x[n-1], x[n-2], y[n-1],</code> and <code>y[n-2]</code> and each state variable in 1.63 format to improve precision.
                    The state variables are arranged in the array as:
   <pre>
       {x[n-1], x[n-2], y[n-1], y[n-2]}
   </pre>
   @par
                    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 <code>4*numStages</code> values of data in 1.63 format.
                    The state variables are updated after each block of data is processed, the coefficients are untouched.

   @par           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.

   @par           Init Function
                    There is also an associated initialization function which performs the 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:
                    numStages, pCoeffs, postShift, pState. Also set all of the values in pState to zero.

   @par
                    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 filter instance structure use
   <pre>
       arm_biquad_cas_df1_32x64_ins_q31 S1 = {numStages, pState, pCoeffs, postShift};
   </pre>
                    where <code>numStages</code> is the number of Biquad stages in the filter;
                    <code>pState</code> is the address of the state buffer;
                    <code>pCoeffs</code> is the address of the coefficient buffer;
                    <code>postShift</code> shift to be applied which is described in detail below.
   @par           Fixed-Point Behavior
                    Care must be taken while using Biquad Cascade 32x64 filter function.
                    Following issues must be considered:
                    - Scaling of coefficients
                    - Filter gain
                    - Overflow and saturation

   @par
                    Filter coefficients are represented as fractional values and
                    restricted to lie in the range <code>[-1 +1)</code>.
                    The processing function has an additional scaling parameter <code>postShift</code>
                    which allows the filter coefficients to exceed the range <code>[+1 -1)</code>.
                    At the output of the filter's accumulator is a shift register which shifts the result by <code>postShift</code> bits.
                    \image html BiquadPostshift.gif "Fixed-point Biquad with shift by postShift bits after accumulator"
                    This essentially scales the filter coefficients by <code>2^postShift</code>.
                    For example, to realize the coefficients
   <pre>
      {1.5, -0.8, 1.2, 1.6, -0.9}
   </pre>
                    set the Coefficient array to:
   <pre>
      {0.75, -0.4, 0.6, 0.8, -0.45}
   </pre>
                    and set <code>postShift=1</code>
   @par
                    The second thing to keep in mind is the gain through the filter.
                    The frequency response of a Biquad filter is a function of its coefficients.
                    It is possible for the gain through the filter to exceed 1.0 meaning that the
                    filter increases the amplitude of certain frequencies.
                    This means that an input signal with amplitude < 1.0 may result in an output > 1.0
                    and these are saturated or overflowed based on the implementation of the filter.
                    To avoid this behavior the filter needs to be scaled down such that its peak gain < 1.0
                    or the input signal must be scaled down so that the combination of input and filter are never overflowed.
   @par
                    The third item to consider is the overflow and saturation behavior of the fixed-point Q31 version.
                    This is described in the function specific documentation below.
  */

 /**
   @addtogroup BiquadCascadeDF1_32x64
   @{
  */

 /**
   @brief         Processing function for the Q31 Biquad cascade 32x64 filter.
   @param[in]     S         points to an instance of the high precision Q31 Biquad cascade filter
   @param[in]     pSrc      points to the block of input data
   @param[out]    pDst      points to the block of output data
   @param[in]     blockSize number of samples to process
   @return        none

   @par           Details
                    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 <code>postShift</code> bits and the result truncated to
                    1.31 format by discarding the low 32 bits.
   @par
                    Two related functions are provided in the CMSIS DSP library.
                    - \ref arm_biquad_cascade_df1_q31() implements a Biquad cascade with 32-bit coefficients and state variables with a Q63 accumulator.
                    - \ref arm_biquad_cascade_df1_fast_q31() implements a Biquad cascade with 32-bit coefficients and state variables with a Q31 accumulator.
  */

 void arm_biquad_cas_df1_32x64_q31(
   const arm_biquad_cas_df1_32x64_ins_q31 * S,
         q31_t * pSrc,
         q31_t * pDst,
         uint32_t blockSize)
 {
         q31_t *pIn = pSrc;                             /* input pointer initialization */
         q31_t *pOut = pDst;                            /* output pointer initialization */
         q63_t *pState = S->pState;                     /* state pointer initialization */
   const q31_t *pCoeffs = S->pCoeffs;                   /* coeff pointer initialization */
         q63_t acc;                                     /* accumulator */
         q31_t Xn1, Xn2;                                /* Input Filter state variables */
         q63_t Yn1, Yn2;                                /* Output Filter state variables */
         q31_t b0, b1, b2, a1, a2;                      /* Filter coefficients */
         q31_t Xn;                                      /* temporary input */
         int32_t shift = (int32_t) S->postShift + 1;    /* Shift to be applied to the output */
         uint32_t sample, stage = S->numStages;         /* loop counters */
         q31_t acc_l, acc_h;                            /* temporary output */
         uint32_t uShift = ((uint32_t) S->postShift + 1U);
         uint32_t lShift = 32U - uShift;                /* Shift to be applied to the output */

   do
   {
     /* Reading the coefficients */
     b0 = *pCoeffs++;
     b1 = *pCoeffs++;
     b2 = *pCoeffs++;
     a1 = *pCoeffs++;
     a2 = *pCoeffs++;

     /* Reading the state values */
     Xn1 = (q31_t) (pState[0]);
     Xn2 = (q31_t) (pState[1]);
     Yn1 = pState[2];
     Yn2 = pState[3];

 #if defined (ARM_MATH_LOOPUNROLL)

     /* Apply loop unrolling and compute 4 output values simultaneously. */
     /* Variable acc hold output value that is being computed and stored in destination buffer
      * acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]
      */

     /* Loop unrolling: Compute 4 outputs at a time */
     sample = blockSize >> 2U;

     while (sample > 0U)
     {
       /* Read the input */
       Xn = *pIn++;

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

       /* acc =  b0 * x[n] */
       acc = (q63_t) Xn * b0;

       /* acc +=  b1 * x[n-1] */
       acc += (q63_t) Xn1 * b1;

       /* acc +=  b[2] * x[n-2] */
       acc += (q63_t) Xn2 * b2;

       /* acc +=  a1 * y[n-1] */
       acc += mult32x64(Yn1, a1);

       /* acc +=  a2 * y[n-2] */
       acc += mult32x64(Yn2, a2);

       /* The result is converted to 1.63 , Yn2 variable is reused */
       Yn2 = acc << shift;

       /* Calc lower part of acc */
       acc_l = acc & 0xffffffff;

       /* Calc upper part of acc */
       acc_h = (acc >> 32) & 0xffffffff;

       /* Apply shift for lower part of acc and upper part of acc */
       acc_h = (uint32_t) acc_l >> lShift | acc_h << uShift;

       /* Store the output in the destination buffer in 1.31 format. */
       *pOut = acc_h;

       /* Read the second input into Xn2, to reuse the value */
       Xn2 = *pIn++;

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

       /* acc +=  b1 * x[n-1] */
       acc = (q63_t) Xn * b1;

       /* acc =  b0 * x[n] */
       acc += (q63_t) Xn2 * b0;

       /* acc +=  b[2] * x[n-2] */
       acc += (q63_t) Xn1 * b2;

       /* acc +=  a1 * y[n-1] */
       acc += mult32x64(Yn2, a1);

       /* acc +=  a2 * y[n-2] */
       acc += mult32x64(Yn1, a2);

       /* The result is converted to 1.63, Yn1 variable is reused */
       Yn1 = acc << shift;

       /* Calc lower part of acc */
       acc_l = acc & 0xffffffff;

       /* Calc upper part of acc */
       acc_h = (acc >> 32) & 0xffffffff;

       /* Apply shift for lower part of acc and upper part of acc */
       acc_h = (uint32_t) acc_l >> lShift | acc_h << uShift;

       /* Read the third input into Xn1, to reuse the value */
       Xn1 = *pIn++;

       /* The result is converted to 1.31 */
       /* Store the output in the destination buffer. */
       *(pOut + 1U) = acc_h;

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

       /* acc =  b0 * x[n] */
       acc = (q63_t) Xn1 * b0;

       /* acc +=  b1 * x[n-1] */
       acc += (q63_t) Xn2 * b1;

       /* acc +=  b[2] * x[n-2] */
       acc += (q63_t) Xn * b2;

       /* acc +=  a1 * y[n-1] */
       acc += mult32x64(Yn1, a1);

       /* acc +=  a2 * y[n-2] */
       acc += mult32x64(Yn2, a2);

       /* The result is converted to 1.63, Yn2 variable is reused  */
       Yn2 = acc << shift;

       /* Calc lower part of acc */
       acc_l = acc & 0xffffffff;

       /* Calc upper part of acc */
       acc_h = (acc >> 32) & 0xffffffff;

       /* Apply shift for lower part of acc and upper part of acc */
       acc_h = (uint32_t) acc_l >> lShift | acc_h << uShift;

       /* Store the output in the destination buffer in 1.31 format. */
       *(pOut + 2U) = acc_h;

       /* Read the fourth input into Xn, to reuse the value */
       Xn = *pIn++;

       /* acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */
       /* acc =  b0 * x[n] */
       acc = (q63_t) Xn * b0;

       /* acc +=  b1 * x[n-1] */
       acc += (q63_t) Xn1 * b1;

       /* acc +=  b[2] * x[n-2] */
       acc += (q63_t) Xn2 * b2;

       /* acc +=  a1 * y[n-1] */
       acc += mult32x64(Yn2, a1);

       /* acc +=  a2 * y[n-2] */
       acc += mult32x64(Yn1, a2);

       /* The result is converted to 1.63, Yn1 variable is reused  */
       Yn1 = acc << shift;

       /* Calc lower part of acc */
       acc_l = acc & 0xffffffff;

       /* Calc upper part of acc */
       acc_h = (acc >> 32) & 0xffffffff;

       /* Apply shift for lower part of acc and upper part of acc */
       acc_h = (uint32_t) acc_l >> lShift | acc_h << uShift;

       /* Store the output in the destination buffer in 1.31 format. */
       *(pOut + 3U) = acc_h;

       /* Every time after the output is computed state should be updated. */
       /* The states should be updated as: */
       /* Xn2 = Xn1 */
       /* Xn1 = Xn  */
       /* Yn2 = Yn1 */
       /* Yn1 = acc */
       Xn2 = Xn1;
       Xn1 = Xn;

       /* update output pointer */
       pOut += 4U;

       /* decrement loop counter */
       sample--;
     }

     /* Loop unrolling: Compute remaining outputs */
     sample = blockSize & 0x3U;

 #else

     /* Initialize blkCnt with number of samples */
     sample = blockSize;

 #endif /* #if defined (ARM_MATH_LOOPUNROLL) */

     while (sample > 0U)
     {
       /* Read the input */
       Xn = *pIn++;

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

       /* acc =  b0 * x[n] */
       acc = (q63_t) Xn * b0;
       /* acc +=  b1 * x[n-1] */
       acc += (q63_t) Xn1 * b1;
       /* acc +=  b[2] * x[n-2] */
       acc += (q63_t) Xn2 * b2;
       /* acc +=  a1 * y[n-1] */
       acc += mult32x64(Yn1, a1);
       /* acc +=  a2 * y[n-2] */
       acc += mult32x64(Yn2, a2);

       /* Every time after the output is computed state should be updated. */
       /* The states should be updated as: */
       /* Xn2 = Xn1 */
       /* Xn1 = Xn  */
       /* Yn2 = Yn1 */
       /* Yn1 = acc */
       Xn2 = Xn1;
       Xn1 = Xn;
       Yn2 = Yn1;

       /* The result is converted to 1.63, Yn1 variable is reused  */
       Yn1 = acc << shift;

       /* Calc lower part of acc */
       acc_l = acc & 0xffffffff;

       /* Calc upper part of acc */
       acc_h = (acc >> 32) & 0xffffffff;

       /* Apply shift for lower part of acc and upper part of acc */
       acc_h = (uint32_t) acc_l >> lShift | acc_h << uShift;

       /* Store the output in the destination buffer in 1.31 format. */
       *pOut++ = acc_h;
       /* Yn1 = acc << shift; */

       /* Store the output in the destination buffer in 1.31 format. */
 /*    *pOut++ = (q31_t) (acc >> (32 - shift));  */

       /* decrement loop counter */
       sample--;
     }

     /* The first stage output is given as input to the second stage. */
     pIn = pDst;

     /* Reset to destination buffer working pointer */
     pOut = pDst;

     /*  Store the updated state variables back into the pState array */
     *pState++ = (q63_t) Xn1;
     *pState++ = (q63_t) Xn2;
     *pState++ = Yn1;
     *pState++ = Yn2;

   } while (--stage);

 }

 /**
   @} end of BiquadCascadeDF1_32x64 group
  */
	/* ----------------------------------------------------------------------
	* Project: CMSIS DSP Library
	* Title: arm_biquad_cascade_df1_32x64_q31.c
	* Description: High precision Q31 Biquad cascade filter processing function
	*
	* $Date: 18. March 2019
	* $Revision: V1.6.0
	*
	* Target Processor: Cortex-M cores
	* -------------------------------------------------------------------- */
	/*
	* Copyright (C) 2010-2019 ARM Limited or its affiliates. All rights reserved.
	*
	* SPDX-License-Identifier: Apache-2.0
	*
	* Licensed under the Apache License, Version 2.0 (the License); you may
	* not use this file except in compliance with the License.
	* You may obtain a copy of the License at
	*
	* www.apache.org/licenses/LICENSE-2.0
	*
	* Unless required by applicable law or agreed to in writing, software
	* distributed under the License is distributed on an AS IS BASIS, WITHOUT
	* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
	* See the License for the specific language governing permissions and
	* limitations under the License.
	*/

	#include "arm_math.h"

	/**
	@ingroup groupFilters
	*/

	/**
	@defgroup BiquadCascadeDF1_32x64 High Precision Q31 Biquad Cascade Filter

	This function implements a high precision Biquad cascade filter which operates on
	Q31 data values. The filter coefficients are in 1.31 format and the state variables
	are in 1.63 format. The double precision state variables reduce quantization noise
	in the filter and provide a cleaner output.
	These filters are particularly useful when implementing filters in which the
	singularities are close to the unit circle. This is common for low pass or high
	pass filters with very low cutoff frequencies.

	The function operates on blocks of input and output data
	and each call to the function processes <code>blockSize</code> samples through
	the filter. <code>pSrc</code> and <code>pDst</code> points to input and output arrays
	containing <code>blockSize</code> Q31 values.

	@par Algorithm
	Each Biquad stage implements a second order filter using the difference equation:
	<pre>
	y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]
	</pre>
	A Direct Form I algorithm is used with 5 coefficients and 4 state variables per stage.
	\image html Biquad.gif "Single Biquad filter stage"
	Coefficients <code>b0, b1 and b2 </code> multiply the input signal <code>x[n]</code> and are referred to as the feedforward coefficients.
	Coefficients <code>a1</code> and <code>a2</code> multiply the output signal <code>y[n]</code> 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
	<pre>
	y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] - a1 * y[n-1] - a2 * y[n-2]
	</pre>
	In this case the feedback coefficients <code>a1</code> and <code>a2</code> must be negated when used with the CMSIS DSP Library.
	@par
	Higher order filters are realized as a cascade of second order sections.
	<code>numStages</code> refers to the number of second order stages used.
	For example, an 8th order filter would be realized with <code>numStages=4</code> second order stages.
	\image html BiquadCascade.gif "8th order filter using a cascade of Biquad stages"
	A 9th order filter would be realized with <code>numStages=5</code> second order stages
	with the coefficients for one of the stages configured as a first order filter
	(<code>b2=0</code> and <code>a2=0</code>).
	@par
	The <code>pState</code> points to state variables array.
	Each Biquad stage has 4 state variables <code>x[n-1], x[n-2], y[n-1],</code> and <code>y[n-2]</code> and each state variable in 1.63 format to improve precision.
	The state variables are arranged in the array as:
	<pre>
	{x[n-1], x[n-2], y[n-1], y[n-2]}
	</pre>
	@par
	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 <code>4*numStages</code> values of data in 1.63 format.
	The state variables are updated after each block of data is processed, the coefficients are untouched.

	@par 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.

	@par Init Function
	There is also an associated initialization function which performs the 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:
	numStages, pCoeffs, postShift, pState. Also set all of the values in pState to zero.

	@par
	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 filter instance structure use
	<pre>
	arm_biquad_cas_df1_32x64_ins_q31 S1 = {numStages, pState, pCoeffs, postShift};
	</pre>
	where <code>numStages</code> is the number of Biquad stages in the filter;
	<code>pState</code> is the address of the state buffer;
	<code>pCoeffs</code> is the address of the coefficient buffer;
	<code>postShift</code> shift to be applied which is described in detail below.
	@par Fixed-Point Behavior
	Care must be taken while using Biquad Cascade 32x64 filter function.
	Following issues must be considered:
	- Scaling of coefficients
	- Filter gain
	- Overflow and saturation

	@par
	Filter coefficients are represented as fractional values and
	restricted to lie in the range <code>[-1 +1)</code>.
	The processing function has an additional scaling parameter <code>postShift</code>
	which allows the filter coefficients to exceed the range <code>[+1 -1)</code>.
	At the output of the filter's accumulator is a shift register which shifts the result by <code>postShift</code> bits.
	\image html BiquadPostshift.gif "Fixed-point Biquad with shift by postShift bits after accumulator"
	This essentially scales the filter coefficients by <code>2^postShift</code>.
	For example, to realize the coefficients
	<pre>
	{1.5, -0.8, 1.2, 1.6, -0.9}
	</pre>
	set the Coefficient array to:
	<pre>
	{0.75, -0.4, 0.6, 0.8, -0.45}
	</pre>
	and set <code>postShift=1</code>
	@par
	The second thing to keep in mind is the gain through the filter.
	The frequency response of a Biquad filter is a function of its coefficients.
	It is possible for the gain through the filter to exceed 1.0 meaning that the
	filter increases the amplitude of certain frequencies.
	This means that an input signal with amplitude < 1.0 may result in an output > 1.0
	and these are saturated or overflowed based on the implementation of the filter.
	To avoid this behavior the filter needs to be scaled down such that its peak gain < 1.0
	or the input signal must be scaled down so that the combination of input and filter are never overflowed.
	@par
	The third item to consider is the overflow and saturation behavior of the fixed-point Q31 version.
	This is described in the function specific documentation below.
	*/

	/**
	@addtogroup BiquadCascadeDF1_32x64
	@{
	*/

	/**
	@brief Processing function for the Q31 Biquad cascade 32x64 filter.
	@param[in] S points to an instance of the high precision Q31 Biquad cascade filter
	@param[in] pSrc points to the block of input data
	@param[out] pDst points to the block of output data
	@param[in] blockSize number of samples to process
	@return none

	@par Details
	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 <code>postShift</code> bits and the result truncated to
	1.31 format by discarding the low 32 bits.
	@par
	Two related functions are provided in the CMSIS DSP library.
	- \ref arm_biquad_cascade_df1_q31() implements a Biquad cascade with 32-bit coefficients and state variables with a Q63 accumulator.
	- \ref arm_biquad_cascade_df1_fast_q31() implements a Biquad cascade with 32-bit coefficients and state variables with a Q31 accumulator.
	*/

	void arm_biquad_cas_df1_32x64_q31(
	const arm_biquad_cas_df1_32x64_ins_q31 * S,
	q31_t * pSrc,
	q31_t * pDst,
	uint32_t blockSize)
	{
	q31_t pIn = pSrc; / input pointer initialization */
	q31_t pOut = pDst; / output pointer initialization */
	q63_t pState = S->pState; / state pointer initialization */
	const q31_t pCoeffs = S->pCoeffs; / coeff pointer initialization */
	q63_t acc; /* accumulator */
	q31_t Xn1, Xn2; /* Input Filter state variables */
	q63_t Yn1, Yn2; /* Output Filter state variables */
	q31_t b0, b1, b2, a1, a2; /* Filter coefficients */
	q31_t Xn; /* temporary input */
	int32_t shift = (int32_t) S->postShift + 1; /* Shift to be applied to the output */
	uint32_t sample, stage = S->numStages; /* loop counters */
	q31_t acc_l, acc_h; /* temporary output */
	uint32_t uShift = ((uint32_t) S->postShift + 1U);
	uint32_t lShift = 32U - uShift; /* Shift to be applied to the output */

	do
	{
	/* Reading the coefficients */
	b0 = *pCoeffs++;
	b1 = *pCoeffs++;
	b2 = *pCoeffs++;
	a1 = *pCoeffs++;
	a2 = *pCoeffs++;

	/* Reading the state values */
	Xn1 = (q31_t) (pState[0]);
	Xn2 = (q31_t) (pState[1]);
	Yn1 = pState[2];
	Yn2 = pState[3];

	#if defined (ARM_MATH_LOOPUNROLL)

	/* Apply loop unrolling and compute 4 output values simultaneously. */
	/* Variable acc hold output value that is being computed and stored in destination buffer
	* acc = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]
	*/

	/* Loop unrolling: Compute 4 outputs at a time */
	sample = blockSize >> 2U;

	while (sample > 0U)
	{
	/* Read the input */
	Xn = *pIn++;

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

	/* acc = b0 * x[n] */
	acc = (q63_t) Xn * b0;

	/* acc += b1 * x[n-1] */
	acc += (q63_t) Xn1 * b1;

	/* acc += b[2] * x[n-2] */
	acc += (q63_t) Xn2 * b2;

	/* acc += a1 * y[n-1] */
	acc += mult32x64(Yn1, a1);

	/* acc += a2 * y[n-2] */
	acc += mult32x64(Yn2, a2);

	/* The result is converted to 1.63 , Yn2 variable is reused */
	Yn2 = acc << shift;

	/* Calc lower part of acc */
	acc_l = acc & 0xffffffff;

	/* Calc upper part of acc */
	acc_h = (acc >> 32) & 0xffffffff;

	/* Apply shift for lower part of acc and upper part of acc */
	acc_h = (uint32_t) acc_l >> lShift \| acc_h << uShift;

	/* Store the output in the destination buffer in 1.31 format. */
	*pOut = acc_h;

	/* Read the second input into Xn2, to reuse the value */
	Xn2 = *pIn++;

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

	/* acc += b1 * x[n-1] */
	acc = (q63_t) Xn * b1;

	/* acc = b0 * x[n] */
	acc += (q63_t) Xn2 * b0;

	/* acc += b[2] * x[n-2] */
	acc += (q63_t) Xn1 * b2;

	/* acc += a1 * y[n-1] */
	acc += mult32x64(Yn2, a1);

	/* acc += a2 * y[n-2] */
	acc += mult32x64(Yn1, a2);

	/* The result is converted to 1.63, Yn1 variable is reused */
	Yn1 = acc << shift;

	/* Calc lower part of acc */
	acc_l = acc & 0xffffffff;

	/* Calc upper part of acc */
	acc_h = (acc >> 32) & 0xffffffff;

	/* Apply shift for lower part of acc and upper part of acc */
	acc_h = (uint32_t) acc_l >> lShift \| acc_h << uShift;

	/* Read the third input into Xn1, to reuse the value */
	Xn1 = *pIn++;

	/* The result is converted to 1.31 */
	/* Store the output in the destination buffer. */
	*(pOut + 1U) = acc_h;

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

	/* acc = b0 * x[n] */
	acc = (q63_t) Xn1 * b0;

	/* acc += b1 * x[n-1] */
	acc += (q63_t) Xn2 * b1;

	/* acc += b[2] * x[n-2] */
	acc += (q63_t) Xn * b2;

	/* acc += a1 * y[n-1] */
	acc += mult32x64(Yn1, a1);

	/* acc += a2 * y[n-2] */
	acc += mult32x64(Yn2, a2);

	/* The result is converted to 1.63, Yn2 variable is reused */
	Yn2 = acc << shift;

	/* Calc lower part of acc */
	acc_l = acc & 0xffffffff;

	/* Calc upper part of acc */
	acc_h = (acc >> 32) & 0xffffffff;

	/* Apply shift for lower part of acc and upper part of acc */
	acc_h = (uint32_t) acc_l >> lShift \| acc_h << uShift;

	/* Store the output in the destination buffer in 1.31 format. */
	*(pOut + 2U) = acc_h;

	/* Read the fourth input into Xn, to reuse the value */
	Xn = *pIn++;

	/* acc = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */
	/* acc = b0 * x[n] */
	acc = (q63_t) Xn * b0;

	/* acc += b1 * x[n-1] */
	acc += (q63_t) Xn1 * b1;

	/* acc += b[2] * x[n-2] */
	acc += (q63_t) Xn2 * b2;

	/* acc += a1 * y[n-1] */
	acc += mult32x64(Yn2, a1);

	/* acc += a2 * y[n-2] */
	acc += mult32x64(Yn1, a2);

	/* The result is converted to 1.63, Yn1 variable is reused */
	Yn1 = acc << shift;

	/* Calc lower part of acc */
	acc_l = acc & 0xffffffff;

	/* Calc upper part of acc */
	acc_h = (acc >> 32) & 0xffffffff;

	/* Apply shift for lower part of acc and upper part of acc */
	acc_h = (uint32_t) acc_l >> lShift \| acc_h << uShift;

	/* Store the output in the destination buffer in 1.31 format. */
	*(pOut + 3U) = acc_h;

	/* Every time after the output is computed state should be updated. */
	/* The states should be updated as: */
	/* Xn2 = Xn1 */
	/* Xn1 = Xn */
	/* Yn2 = Yn1 */
	/* Yn1 = acc */
	Xn2 = Xn1;
	Xn1 = Xn;

	/* update output pointer */
	pOut += 4U;

	/* decrement loop counter */
	sample--;
	}

	/* Loop unrolling: Compute remaining outputs */
	sample = blockSize & 0x3U;

	#else

	/* Initialize blkCnt with number of samples */
	sample = blockSize;

	#endif /* #if defined (ARM_MATH_LOOPUNROLL) */

	while (sample > 0U)
	{
	/* Read the input */
	Xn = *pIn++;

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

	/* acc = b0 * x[n] */
	acc = (q63_t) Xn * b0;
	/* acc += b1 * x[n-1] */
	acc += (q63_t) Xn1 * b1;
	/* acc += b[2] * x[n-2] */
	acc += (q63_t) Xn2 * b2;
	/* acc += a1 * y[n-1] */
	acc += mult32x64(Yn1, a1);
	/* acc += a2 * y[n-2] */
	acc += mult32x64(Yn2, a2);

	/* Every time after the output is computed state should be updated. */
	/* The states should be updated as: */
	/* Xn2 = Xn1 */
	/* Xn1 = Xn */
	/* Yn2 = Yn1 */
	/* Yn1 = acc */
	Xn2 = Xn1;
	Xn1 = Xn;
	Yn2 = Yn1;

	/* The result is converted to 1.63, Yn1 variable is reused */
	Yn1 = acc << shift;

	/* Calc lower part of acc */
	acc_l = acc & 0xffffffff;

	/* Calc upper part of acc */
	acc_h = (acc >> 32) & 0xffffffff;

	/* Apply shift for lower part of acc and upper part of acc */
	acc_h = (uint32_t) acc_l >> lShift \| acc_h << uShift;

	/* Store the output in the destination buffer in 1.31 format. */
	*pOut++ = acc_h;
	/* Yn1 = acc << shift; */

	/* Store the output in the destination buffer in 1.31 format. */
	/* pOut++ = (q31_t) (acc >> (32 - shift)); /

	/* decrement loop counter */
	sample--;
	}

	/* The first stage output is given as input to the second stage. */
	pIn = pDst;

	/* Reset to destination buffer working pointer */
	pOut = pDst;

	/* Store the updated state variables back into the pState array */
	*pState++ = (q63_t) Xn1;
	*pState++ = (q63_t) Xn2;
	*pState++ = Yn1;
	*pState++ = Yn2;

	} while (--stage);

	}

	/**
	@} end of BiquadCascadeDF1_32x64 group
	*/