pigweed / third_party / github / STMicroelectronics / cmsis_core / refs/heads/cm4 / . / DSP / Source / FilteringFunctions / arm_biquad_cascade_df1_32x64_q31.c

/* ---------------------------------------------------------------------- | |

* 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 | |

*/ |