/* ----------------------------------------------------------------------
* Copyright (C) 2010 ARM Limited. All rights reserved.
*
* $Date: 29. November 2010
* $Revision: V1.0.3
*
* Project: CMSIS DSP Library
* Title: arm_biquad_cascade_df1_32x64_q31.c
*
* Description: High precision Q31 Biquad cascade filter processing function
*
* Target Processor: Cortex-M4/Cortex-M3
*
* Version 1.0.3 2010/11/29
* Re-organized the CMSIS folders and updated documentation.
*
* Version 1.0.2 2010/11/11
* Documentation updated.
*
* Version 1.0.1 2010/10/05
* Production release and review comments incorporated.
*
* Version 1.0.0 2010/09/20
* Production release and review comments incorporated.
*
* Version 0.0.7 2010/06/10
* Misra-C changes done
* -------------------------------------------------------------------- */
#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 blockSize
samples through
* the filter. pSrc
and pDst
points to input and output arrays
* containing blockSize
Q31 values.
*
* \par 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. * \image html Biquad.gif "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 CMSIS DSP Library.
*
* \par
* 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.
* \image html BiquadCascade.gif "8th order filter using a cascade of Biquad 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
).
*
* \par
* 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]
and each state variable in 1.63 format to improve precision.
* The state variables are arranged in the array as:
* * {x[n-1], x[n-2], y[n-1], y[n-2]} ** * \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
4*numStages
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.
* \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
* * arm_biquad_cas_df1_32x64_ins_q31 S1 = {numStages, pState, pCoeffs, postShift}; ** 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; postShift
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 [-1 +1)
.
* The processing function has an additional scaling parameter postShift
* which allows the filter coefficients to exceed the range [+1 -1)
.
* At the output of the filter's accumulator is a shift register which shifts the result by postShift
bits.
* \image html BiquadPostshift.gif "Fixed-point Biquad with shift by postShift bits after accumulator"
* This essentially scales the filter coefficients by 2^postShift
.
* For example, to realize the coefficients
* * {1.5, -0.8, 1.2, 1.6, -0.9} ** set the Coefficient array to: *
* {0.75, -0.4, 0.6, 0.8, -0.45} ** and set
postShift=1
*
* \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
* @{
*/
/**
* @details
* @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
* 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 postShift
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.
* arm_biquad_cascade_df1_q31()
implements a Biquad cascade with 32-bit coefficients and state variables with a Q63 accumulator.
* 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 */
q31_t *pCoeffs = S->pCoeffs; /* coeff pointer initialization */
q63_t acc; /* accumulator */
q63_t Xn1, Xn2, Yn1, Yn2; /* Filter state variables */
q31_t b0, b1, b2, a1, a2; /* Filter coefficients */
q63_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 */
do
{
/* Reading the coefficients */
b0 = *pCoeffs++;
b1 = *pCoeffs++;
b2 = *pCoeffs++;
a1 = *pCoeffs++;
a2 = *pCoeffs++;
/* Reading the state values */
Xn1 = pState[0];
Xn2 = pState[1];
Yn1 = pState[2];
Yn2 = pState[3];
/* Apply loop unrolling and compute 4 output values simultaneously. */
/* The variable acc hold output value that is being computed and
* stored in the destination buffer
* acc = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]
*/
sample = blockSize >> 2u;
/* First part of the processing with loop unrolling. Compute 4 outputs at a time.
** a second loop below computes the remaining 1 to 3 samples. */
while(sample > 0u)
{
/* Read the input */
Xn = *pIn++;
/* The value is shifted to the MSB to perform 32x64 multiplication */
Xn = Xn << 32;
/* 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 = mult32x64(Xn, b0);
/* acc += b1 * x[n-1] */
acc += mult32x64(Xn1, b1);
/* acc += b[2] * x[n-2] */
acc += mult32x64(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;
/* Store the output in the destination buffer in 1.31 format. */
*pOut++ = (q31_t) (acc >> (32 - shift));
/* Read the second input into Xn2, to reuse the value */
Xn2 = *pIn++;
/* The value is shifted to the MSB to perform 32x64 multiplication */
Xn2 = Xn2 << 32;
/* 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 = mult32x64(Xn2, b0);
/* acc += b1 * x[n-1] */
acc += mult32x64(Xn, b1);
/* acc += b[2] * x[n-2] */
acc += mult32x64(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;
/* The result is converted to 1.31 */
/* Store the output in the destination buffer. */
*pOut++ = (q31_t) (acc >> (32 - shift));
/* Read the third input into Xn1, to reuse the value */
Xn1 = *pIn++;
/* The value is shifted to the MSB to perform 32x64 multiplication */
Xn1 = Xn1 << 32;
/* 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 = mult32x64(Xn1, b0);
/* acc += b1 * x[n-1] */
acc += mult32x64(Xn2, b1);
/* acc += b[2] * x[n-2] */
acc += mult32x64(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;
/* Store the output in the destination buffer in 1.31 format. */
*pOut++ = (q31_t) (acc >> (32 - shift));
/* Read the fourth input into Xn, to reuse the value */
Xn = *pIn++;
/* The value is shifted to the MSB to perform 32x64 multiplication */
Xn = Xn << 32;
/* 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 = mult32x64(Xn, b0);
/* acc += b1 * x[n-1] */
acc += mult32x64(Xn1, b1);
/* acc += b[2] * x[n-2] */
acc += mult32x64(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;
/* 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;
/* Store the output in the destination buffer in 1.31 format. */
*pOut++ = (q31_t) (acc >> (32 - shift));
/* decrement the loop counter */
sample--;
}
/* If the blockSize is not a multiple of 4, compute any remaining output samples here.
** No loop unrolling is used. */
sample = (blockSize & 0x3u);
while(sample > 0u)
{
/* Read the input */
Xn = *pIn++;
/* The value is shifted to the MSB to perform 32x64 multiplication */
Xn = Xn << 32;
/* 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 = mult32x64(Xn, b0);
/* acc += b1 * x[n-1] */
acc += mult32x64(Xn1, b1);
/* acc += b[2] * x[n-2] */
acc += mult32x64(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;
Yn1 = acc << shift;
/* Store the output in the destination buffer in 1.31 format. */
*pOut++ = (q31_t) (acc >> (32 - shift));
/* decrement the 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++ = Xn1;
*pState++ = Xn2;
*pState++ = Yn1;
*pState++ = Yn2;
} while(--stage);
}
/**
* @} end of BiquadCascadeDF1_32x64 group
*/