text: /* ----------------------------------------------------------------------   
  * Copyright (C) 2010 ARM Limited. All rights reserved.   
  *   
  * $Date:        15. July 2011  
  * $Revision: 	V1.0.10  
  *   
  * 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/Cortex-M0
  *  
  * Version 1.0.10 2011/7/15 
  *    Big Endian support added and Merged M0 and M3/M4 Source code.  
  *   
  * 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 <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.   
   * \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   
   * @{   
   */
  
  /**   
   * @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 <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.   
   * <code>arm_biquad_cascade_df1_q31()</code> implements a Biquad cascade with 32-bit coefficients and state variables with a Q63 accumulator.   
   * <code>arm_biquad_cascade_df1_fast_q31()</code> 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                     */
  
  
  #ifndef ARM_MATH_CM0
  
    /* Run the below code for Cortex-M4 and Cortex-M3 */
  
    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);
  
  #else
  
    /* Run the below code for Cortex-M0 */
  
    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];
  
      /* 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;
  
      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);
  
  #endif /*    #ifndef ARM_MATH_CM0     */
  }
  
    /**   
     * @} end of BiquadCascadeDF1_32x64 group   
     */
  