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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 */
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