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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]. The state variables are arranged in the pState 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. 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. There are separate instance structure declarations for each of the 3 supported data types. @par Init Function There is also an associated initialization function for each data type. The initialization function performs 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, 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. The code below statically initializes each of the 3 different data type filter instance structures
arm_biquad_casd_df1_inst_f32 S1 = {numStages, pState, pCoeffs};
arm_biquad_casd_df1_inst_q15 S2 = {numStages, pState, pCoeffs, postShift};
arm_biquad_casd_df1_inst_q31 S3 = {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. @par Fixed-Point Behavior Care must be taken when using the Q15 and Q31 versions of the Biquad Cascade filter functions. Following issues must be considered: - Scaling of coefficients - Filter gain - Overflow and saturation @par Scaling of coefficients Filter coefficients are represented as fractional values and coefficients are restricted to lie in the range [-1 +1). The fixed-point functions have an additional scaling parameter postShift which allow 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 pCoeffs array to:
{0.75, -0.4, 0.6, 0.8, -0.45}