arm_biquad_cascade_df1_32x64_q31.c

00001 /* ----------------------------------------------------------------------  
00002 * Copyright (C) 2010 ARM Limited. All rights reserved.  
00003 *  
00004 * $Date:        29. November 2010  
00005 * $Revision:    V1.0.3  
00006 *  
00007 * Project:      CMSIS DSP Library  
00008 * Title:        arm_biquad_cascade_df1_32x64_q31.c  
00009 *  
00010 * Description:  High precision Q31 Biquad cascade filter processing function  
00011 *  
00012 * Target Processor: Cortex-M4/Cortex-M3
00013 *  
00014 * Version 1.0.3 2010/11/29 
00015 *    Re-organized the CMSIS folders and updated documentation.  
00016 *   
00017 * Version 1.0.2 2010/11/11  
00018 *    Documentation updated.   
00019 *  
00020 * Version 1.0.1 2010/10/05   
00021 *    Production release and review comments incorporated.  
00022 *  
00023 * Version 1.0.0 2010/09/20   
00024 *    Production release and review comments incorporated.  
00025 *  
00026 * Version 0.0.7  2010/06/10   
00027 *    Misra-C changes done  
00028 * -------------------------------------------------------------------- */ 
00029  
00030 #include "arm_math.h" 
00031  
00032 /**  
00033  * @ingroup groupFilters  
00034  */ 
00035  
00036 /**  
00037  * @defgroup BiquadCascadeDF1_32x64 High Precision Q31 Biquad Cascade Filter  
00038  *  
00039  * This function implements a high precision Biquad cascade filter which operates on  
00040  * Q31 data values.  The filter coefficients are in 1.31 format and the state variables  
00041  * are in 1.63 format.  The double precision state variables reduce quantization noise  
00042  * in the filter and provide a cleaner output.  
00043  * These filters are particularly useful when implementing filters in which the  
00044  * singularities are close to the unit circle.  This is common for low pass or high  
00045  * pass filters with very low cutoff frequencies.  
00046  *  
00047  * The function operates on blocks of input and output data  
00048  * and each call to the function processes <code>blockSize</code> samples through  
00049  * the filter. <code>pSrc</code> and <code>pDst</code> points to input and output arrays  
00050  * containing <code>blockSize</code> Q31 values.  
00051  *  
00052  * \par Algorithm  
00053  * Each Biquad stage implements a second order filter using the difference equation:  
00054  * <pre>  
00055  *     y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]  
00056  * </pre>  
00057  * A Direct Form I algorithm is used with 5 coefficients and 4 state variables per stage.  
00058  * \image html Biquad.gif "Single Biquad filter stage"  
00059  * Coefficients <code>b0, b1, and b2 </code> multiply the input signal <code>x[n]</code> and are referred to as the feedforward coefficients.  
00060  * Coefficients <code>a1</code> and <code>a2</code> multiply the output signal <code>y[n]</code> and are referred to as the feedback coefficients.  
00061  * Pay careful attention to the sign of the feedback coefficients.  
00062  * Some design tools use the difference equation  
00063  * <pre>  
00064  *     y[n] = b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] - a1 * y[n-1] - a2 * y[n-2]  
00065  * </pre>  
00066  * In this case the feedback coefficients <code>a1</code> and <code>a2</code> must be negated when used with the CMSIS DSP Library.  
00067  *  
00068  * \par  
00069  * Higher order filters are realized as a cascade of second order sections.  
00070  * <code>numStages</code> refers to the number of second order stages used.  
00071  * For example, an 8th order filter would be realized with <code>numStages=4</code> second order stages.  
00072  * \image html BiquadCascade.gif "8th order filter using a cascade of Biquad stages"  
00073  * 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>).  
00074  *  
00075  * \par  
00076  * The <code>pState</code> points to state variables array .  
00077  * 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.  
00078  * The state variables are arranged in the array as:  
00079  * <pre>  
00080  *     {x[n-1], x[n-2], y[n-1], y[n-2]}  
00081  * </pre>  
00082  *  
00083  * \par  
00084  * The 4 state variables for stage 1 are first, then the 4 state variables for stage 2, and so on.  
00085  * The state array has a total length of <code>4*numStages</code> values of data in 1.63 format.  
00086  * The state variables are updated after each block of data is processed; the coefficients are untouched.  
00087  *  
00088  * \par Instance Structure  
00089  * The coefficients and state variables for a filter are stored together in an instance data structure.  
00090  * A separate instance structure must be defined for each filter.  
00091  * Coefficient arrays may be shared among several instances while state variable arrays cannot be shared.  
00092  *  
00093  * \par Init Function  
00094  * There is also an associated initialization function which performs the following operations:  
00095  * - Sets the values of the internal structure fields.  
00096  * - Zeros out the values in the state buffer.  
00097  * \par  
00098  * Use of the initialization function is optional.  
00099  * However, if the initialization function is used, then the instance structure cannot be placed into a const data section.  
00100  * To place an instance structure into a const data section, the instance structure must be manually initialized.  
00101  * Set the values in the state buffer to zeros before static initialization.  
00102  * For example, to statically initialize the filter instance structure use  
00103  * <pre>  
00104  *     arm_biquad_cas_df1_32x64_ins_q31 S1 = {numStages, pState, pCoeffs, postShift};  
00105  * </pre>  
00106  * where <code>numStages</code> is the number of Biquad stages in the filter; <code>pState</code> is the address of the state buffer;  
00107  * <code>pCoeffs</code> is the address of the coefficient buffer; <code>postShift</code> shift to be applied which is described in detail below.  
00108  * \par Fixed-Point Behavior  
00109  * Care must be taken while using Biquad Cascade 32x64 filter function.  
00110  * Following issues must be considered:  
00111  * - Scaling of coefficients  
00112  * - Filter gain  
00113  * - Overflow and saturation  
00114  *  
00115  * \par  
00116  * Filter coefficients are represented as fractional values and  
00117  * restricted to lie in the range <code>[-1 +1)</code>.  
00118  * The processing function has an additional scaling parameter <code>postShift</code>  
00119  * which allows the filter coefficients to exceed the range <code>[+1 -1)</code>.  
00120  * At the output of the filter's accumulator is a shift register which shifts the result by <code>postShift</code> bits.  
00121  * \image html BiquadPostshift.gif "Fixed-point Biquad with shift by postShift bits after accumulator"  
00122  * This essentially scales the filter coefficients by <code>2^postShift</code>.  
00123  * For example, to realize the coefficients  
00124  * <pre>  
00125  *    {1.5, -0.8, 1.2, 1.6, -0.9}  
00126  * </pre>  
00127  * set the Coefficient array to:  
00128  * <pre>  
00129  *    {0.75, -0.4, 0.6, 0.8, -0.45}  
00130  * </pre>  
00131  * and set <code>postShift=1</code>  
00132  *  
00133  * \par  
00134  * The second thing to keep in mind is the gain through the filter.  
00135  * The frequency response of a Biquad filter is a function of its coefficients.  
00136  * It is possible for the gain through the filter to exceed 1.0 meaning that the filter increases the amplitude of certain frequencies.  
00137  * 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.  
00138  * 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.  
00139  *  
00140  * \par  
00141  * The third item to consider is the overflow and saturation behavior of the fixed-point Q31 version.  
00142  * This is described in the function specific documentation below.  
00143  */ 
00144  
00145 /**  
00146  * @addtogroup BiquadCascadeDF1_32x64  
00147  * @{  
00148  */ 
00149  
00150 /**  
00151  * @details  
00152   
00153  * @param[in]  *S points to an instance of the high precision Q31 Biquad cascade filter.  
00154  * @param[in]  *pSrc points to the block of input data.  
00155  * @param[out] *pDst points to the block of output data.  
00156  * @param[in]  blockSize number of samples to process.  
00157  * @return none.  
00158  *  
00159  * \par  
00160  * The function is implemented using an internal 64-bit accumulator.  
00161  * The accumulator has a 2.62 format and maintains full precision of the intermediate multiplication results but provides only a single guard bit.  
00162  * Thus, if the accumulator result overflows it wraps around rather than clip.  
00163  * 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).  
00164  * After all 5 multiply-accumulates are performed, the 2.62 accumulator is shifted by <code>postShift</code> bits and the result truncated to  
00165  * 1.31 format by discarding the low 32 bits.  
00166  *  
00167  * \par  
00168  * Two related functions are provided in the CMSIS DSP library.  
00169  * <code>arm_biquad_cascade_df1_q31()</code> implements a Biquad cascade with 32-bit coefficients and state variables with a Q63 accumulator.  
00170  * <code>arm_biquad_cascade_df1_fast_q31()</code> implements a Biquad cascade with 32-bit coefficients and state variables with a Q31 accumulator.  
00171  */ 
00172  
00173 void arm_biquad_cas_df1_32x64_q31 ( 
00174   const arm_biquad_cas_df1_32x64_ins_q31 * S, 
00175   q31_t * pSrc, 
00176   q31_t * pDst, 
00177   uint32_t blockSize) 
00178 { 
00179   q31_t *pIn = pSrc;                             /*  input pointer initialization  */ 
00180   q31_t *pOut = pDst;                            /*  output pointer initialization */ 
00181   q63_t *pState = S->pState;                     /*  state pointer initialization  */ 
00182   q31_t *pCoeffs = S->pCoeffs;                   /*  coeff pointer initialization  */ 
00183   q63_t acc;                                     /*  accumulator                   */ 
00184   q63_t Xn1, Xn2, Yn1, Yn2;                      /*  Filter state variables        */ 
00185   q31_t b0, b1, b2, a1, a2;                      /*  Filter coefficients           */ 
00186   q63_t Xn;                                      /*  temporary input               */ 
00187   int32_t shift = (int32_t) S->postShift + 1;    /*  Shift to be applied to the output */ 
00188   uint32_t sample, stage = S->numStages;         /*  loop counters                     */ 
00189  
00190  
00191   do 
00192   { 
00193     /* Reading the coefficients */ 
00194     b0 = *pCoeffs++; 
00195     b1 = *pCoeffs++; 
00196     b2 = *pCoeffs++; 
00197     a1 = *pCoeffs++; 
00198     a2 = *pCoeffs++; 
00199  
00200     /* Reading the state values */ 
00201     Xn1 = pState[0]; 
00202     Xn2 = pState[1]; 
00203     Yn1 = pState[2]; 
00204     Yn2 = pState[3]; 
00205  
00206     /* Apply loop unrolling and compute 4 output values simultaneously. */ 
00207     /* The variable acc hold output value that is being computed and  
00208      * stored in the destination buffer  
00209      * acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2]  
00210      */ 
00211  
00212     sample = blockSize >> 2u; 
00213  
00214     /* First part of the processing with loop unrolling. Compute 4 outputs at a time.  
00215      ** a second loop below computes the remaining 1 to 3 samples. */ 
00216     while(sample > 0u) 
00217     { 
00218       /* Read the input */ 
00219       Xn = *pIn++; 
00220  
00221       /* The value is shifted to the MSB to perform 32x64 multiplication */ 
00222       Xn = Xn << 32; 
00223  
00224       /* acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */ 
00225  
00226       /* acc =  b0 * x[n] */ 
00227       acc = mult32x64(Xn, b0); 
00228       /* acc +=  b1 * x[n-1] */ 
00229       acc += mult32x64(Xn1, b1); 
00230       /* acc +=  b[2] * x[n-2] */ 
00231       acc += mult32x64(Xn2, b2); 
00232       /* acc +=  a1 * y[n-1] */ 
00233       acc += mult32x64(Yn1, a1); 
00234       /* acc +=  a2 * y[n-2] */ 
00235       acc += mult32x64(Yn2, a2); 
00236  
00237       /* The result is converted to 1.63 , Yn2 variable is reused */ 
00238       Yn2 = acc << shift; 
00239  
00240       /* Store the output in the destination buffer in 1.31 format. */ 
00241       *pOut++ = (q31_t) (acc >> (32 - shift)); 
00242  
00243       /* Read the second input into Xn2, to reuse the value */ 
00244       Xn2 = *pIn++; 
00245  
00246       /* The value is shifted to the MSB to perform 32x64 multiplication */ 
00247       Xn2 = Xn2 << 32; 
00248  
00249       /* acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */ 
00250  
00251       /* acc =  b0 * x[n] */ 
00252       acc = mult32x64(Xn2, b0); 
00253       /* acc +=  b1 * x[n-1] */ 
00254       acc += mult32x64(Xn, b1); 
00255       /* acc +=  b[2] * x[n-2] */ 
00256       acc += mult32x64(Xn1, b2); 
00257       /* acc +=  a1 * y[n-1] */ 
00258       acc += mult32x64(Yn2, a1); 
00259       /* acc +=  a2 * y[n-2] */ 
00260       acc += mult32x64(Yn1, a2); 
00261  
00262       /* The result is converted to 1.63, Yn1 variable is reused */ 
00263       Yn1 = acc << shift; 
00264  
00265       /* The result is converted to 1.31 */ 
00266       /* Store the output in the destination buffer. */ 
00267       *pOut++ = (q31_t) (acc >> (32 - shift)); 
00268  
00269       /* Read the third input into Xn1, to reuse the value */ 
00270       Xn1 = *pIn++; 
00271  
00272       /* The value is shifted to the MSB to perform 32x64 multiplication */ 
00273       Xn1 = Xn1 << 32; 
00274  
00275       /* acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */ 
00276       /* acc =  b0 * x[n] */ 
00277       acc = mult32x64(Xn1, b0); 
00278       /* acc +=  b1 * x[n-1] */ 
00279       acc += mult32x64(Xn2, b1); 
00280       /* acc +=  b[2] * x[n-2] */ 
00281       acc += mult32x64(Xn, b2); 
00282       /* acc +=  a1 * y[n-1] */ 
00283       acc += mult32x64(Yn1, a1); 
00284       /* acc +=  a2 * y[n-2] */ 
00285       acc += mult32x64(Yn2, a2); 
00286  
00287       /* The result is converted to 1.63, Yn2 variable is reused  */ 
00288       Yn2 = acc << shift; 
00289  
00290       /* Store the output in the destination buffer in 1.31 format. */ 
00291       *pOut++ = (q31_t) (acc >> (32 - shift)); 
00292  
00293       /* Read the fourth input into Xn, to reuse the value */ 
00294       Xn = *pIn++; 
00295  
00296       /* The value is shifted to the MSB to perform 32x64 multiplication */ 
00297       Xn = Xn << 32; 
00298  
00299       /* acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */ 
00300       /* acc =  b0 * x[n] */ 
00301       acc = mult32x64(Xn, b0); 
00302       /* acc +=  b1 * x[n-1] */ 
00303       acc += mult32x64(Xn1, b1); 
00304       /* acc +=  b[2] * x[n-2] */ 
00305       acc += mult32x64(Xn2, b2); 
00306       /* acc +=  a1 * y[n-1] */ 
00307       acc += mult32x64(Yn2, a1); 
00308       /* acc +=  a2 * y[n-2] */ 
00309       acc += mult32x64(Yn1, a2); 
00310  
00311       /* The result is converted to 1.63, Yn1 variable is reused  */ 
00312       Yn1 = acc << shift; 
00313  
00314       /* Every time after the output is computed state should be updated. */ 
00315       /* The states should be updated as:  */ 
00316       /* Xn2 = Xn1    */ 
00317       /* Xn1 = Xn     */ 
00318       /* Yn2 = Yn1    */ 
00319       /* Yn1 = acc    */ 
00320       Xn2 = Xn1; 
00321       Xn1 = Xn; 
00322  
00323       /* Store the output in the destination buffer in 1.31 format. */ 
00324       *pOut++ = (q31_t) (acc >> (32 - shift)); 
00325  
00326       /* decrement the loop counter */ 
00327       sample--; 
00328     } 
00329  
00330     /* If the blockSize is not a multiple of 4, compute any remaining output samples here.  
00331      ** No loop unrolling is used. */ 
00332     sample = (blockSize & 0x3u); 
00333  
00334     while(sample > 0u) 
00335     { 
00336       /* Read the input */ 
00337       Xn = *pIn++; 
00338  
00339       /* The value is shifted to the MSB to perform 32x64 multiplication */ 
00340       Xn = Xn << 32; 
00341  
00342       /* acc =  b0 * x[n] + b1 * x[n-1] + b2 * x[n-2] + a1 * y[n-1] + a2 * y[n-2] */ 
00343       /* acc =  b0 * x[n] */ 
00344       acc = mult32x64(Xn, b0); 
00345       /* acc +=  b1 * x[n-1] */ 
00346       acc += mult32x64(Xn1, b1); 
00347       /* acc +=  b[2] * x[n-2] */ 
00348       acc += mult32x64(Xn2, b2); 
00349       /* acc +=  a1 * y[n-1] */ 
00350       acc += mult32x64(Yn1, a1); 
00351       /* acc +=  a2 * y[n-2] */ 
00352       acc += mult32x64(Yn2, a2); 
00353  
00354       /* Every time after the output is computed state should be updated. */ 
00355       /* The states should be updated as:  */ 
00356       /* Xn2 = Xn1    */ 
00357       /* Xn1 = Xn     */ 
00358       /* Yn2 = Yn1    */ 
00359       /* Yn1 = acc    */ 
00360       Xn2 = Xn1; 
00361       Xn1 = Xn; 
00362       Yn2 = Yn1; 
00363       Yn1 = acc << shift; 
00364  
00365       /* Store the output in the destination buffer in 1.31 format. */ 
00366       *pOut++ = (q31_t) (acc >> (32 - shift)); 
00367  
00368       /* decrement the loop counter */ 
00369       sample--; 
00370     } 
00371  
00372     /*  The first stage output is given as input to the second stage. */ 
00373     pIn = pDst; 
00374  
00375     /* Reset to destination buffer working pointer */ 
00376     pOut = pDst; 
00377  
00378     /*  Store the updated state variables back into the pState array */ 
00379     *pState++ = Xn1; 
00380     *pState++ = Xn2; 
00381     *pState++ = Yn1; 
00382     *pState++ = Yn2; 
00383  
00384   } while(--stage); 
00385 } 
00386  
00387   /**  
00388    * @} end of BiquadCascadeDF1_32x64 group  
00389    */