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@ -29,108 +29,103 @@
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#include "dsp/filtering_functions.h"
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#include "dsp/filtering_functions.h"
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/**
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/**
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@ingroup groupFilters
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@ingroup groupFilters
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*/
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*/
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/**
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/**
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@defgroup BiquadCascadeDF2T Biquad Cascade IIR Filters Using a Direct Form II Transposed Structure
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@defgroup BiquadCascadeDF2T Biquad Cascade IIR Filters Using a Direct Form II Transposed Structure
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This set of functions implements arbitrary order recursive (IIR) filters using a transposed direct form II structure.
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This set of functions implements arbitrary order recursive (IIR) filters using a transposed direct form II structure.
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The filters are implemented as a cascade of second order Biquad sections.
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The filters are implemented as a cascade of second order Biquad sections.
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These functions provide a slight memory savings as compared to the direct form I Biquad filter functions.
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These functions provide a slight memory savings as compared to the direct form I Biquad filter functions.
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Only floating-point data is supported.
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Only floating-point data is supported.
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This function operate on blocks of input and output data and each call to the function
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processes <code>blockSize</code> samples through the filter.
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This function operate on blocks of input and output data and each call to the function
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<code>pSrc</code> points to the array of input data and
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processes <code>blockSize</code> samples through the filter.
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<code>pDst</code> points to the array of output data.
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<code>pSrc</code> points to the array of input data and
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Both arrays contain <code>blockSize</code> values.
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<code>pDst</code> points to the array of output data.
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@par Algorithm
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Both arrays contain <code>blockSize</code> values.
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Each Biquad stage implements a second order filter using the difference equation:
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<pre>
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@par Algorithm
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y[n] = b0 * x[n] + d1
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Each Biquad stage implements a second order filter using the difference equation:
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d1 = b1 * x[n] + a1 * y[n] + d2
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<pre>
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d2 = b2 * x[n] + a2 * y[n]
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y[n] = b0 * x[n] + d1
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</pre>
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d1 = b1 * x[n] + a1 * y[n] + d2
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where d1 and d2 represent the two state values.
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d2 = b2 * x[n] + a2 * y[n]
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@par
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</pre>
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A Biquad filter using a transposed Direct Form II structure is shown below.
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where d1 and d2 represent the two state values.
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\image html BiquadDF2Transposed.gif "Single transposed Direct Form II Biquad"
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@par
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Coefficients <code>b0, b1, and b2 </code> multiply the input signal <code>x[n]</code> and are referred to as the feedforward coefficients.
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A Biquad filter using a transposed Direct Form II structure is shown below.
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Coefficients <code>a1</code> and <code>a2</code> multiply the output signal <code>y[n]</code> and are referred to as the feedback coefficients.
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\image html BiquadDF2Transposed.gif "Single transposed Direct Form II Biquad"
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Pay careful attention to the sign of the feedback coefficients.
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Coefficients <code>b0, b1, and b2 </code> multiply the input signal <code>x[n]</code> and are referred to as the feedforward coefficients.
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Some design tools flip the sign of the feedback coefficients:
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Coefficients <code>a1</code> and <code>a2</code> multiply the output signal <code>y[n]</code> and are referred to as the feedback coefficients.
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<pre>
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Pay careful attention to the sign of the feedback coefficients.
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y[n] = b0 * x[n] + d1;
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Some design tools flip the sign of the feedback coefficients:
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d1 = b1 * x[n] - a1 * y[n] + d2;
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<pre>
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d2 = b2 * x[n] - a2 * y[n];
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y[n] = b0 * x[n] + d1;
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</pre>
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d1 = b1 * x[n] - a1 * y[n] + d2;
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In this case the feedback coefficients <code>a1</code> and <code>a2</code> must be negated when used with the CMSIS DSP Library.
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d2 = b2 * x[n] - a2 * y[n];
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@par
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</pre>
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Higher order filters are realized as a cascade of second order sections.
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In this case the feedback coefficients <code>a1</code> and <code>a2</code> must be negated when used with the CMSIS DSP Library.
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<code>numStages</code> refers to the number of second order stages used.
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@par
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For example, an 8th order filter would be realized with <code>numStages=4</code> second order stages.
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Higher order filters are realized as a cascade of second order sections.
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A 9th order filter would be realized with <code>numStages=5</code> second order stages with the
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<code>numStages</code> refers to the number of second order stages used.
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coefficients for one of the stages configured as a first order filter (<code>b2=0</code> and <code>a2=0</code>).
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For example, an 8th order filter would be realized with <code>numStages=4</code> second order stages.
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@par
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A 9th order filter would be realized with <code>numStages=5</code> second order stages with the
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<code>pState</code> points to the state variable array.
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coefficients for one of the stages configured as a first order filter (<code>b2=0</code> and <code>a2=0</code>).
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Each Biquad stage has 2 state variables <code>d1</code> and <code>d2</code>.
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@par
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The state variables are arranged in the <code>pState</code> array as:
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<code>pState</code> points to the state variable array.
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<pre>
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Each Biquad stage has 2 state variables <code>d1</code> and <code>d2</code>.
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{d11, d12, d21, d22, ...}
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The state variables are arranged in the <code>pState</code> array as:
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</pre>
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<pre>
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where <code>d1x</code> refers to the state variables for the first Biquad and
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{d11, d12, d21, d22, ...}
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<code>d2x</code> refers to the state variables for the second Biquad.
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</pre>
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The state array has a total length of <code>2*numStages</code> values.
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where <code>d1x</code> refers to the state variables for the first Biquad and
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The state variables are updated after each block of data is processed; the coefficients are untouched.
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<code>d2x</code> refers to the state variables for the second Biquad.
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@par
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The state array has a total length of <code>2*numStages</code> values.
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The CMSIS library contains Biquad filters in both Direct Form I and transposed Direct Form II.
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The state variables are updated after each block of data is processed; the coefficients are untouched.
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The advantage of the Direct Form I structure is that it is numerically more robust for fixed-point data types.
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@par
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That is why the Direct Form I structure supports Q15 and Q31 data types.
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The CMSIS library contains Biquad filters in both Direct Form I and transposed Direct Form II.
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The transposed Direct Form II structure, on the other hand, requires a wide dynamic range for the state variables <code>d1</code> and <code>d2</code>.
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The advantage of the Direct Form I structure is that it is numerically more robust for fixed-point data types.
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Because of this, the CMSIS library only has a floating-point version of the Direct Form II Biquad.
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That is why the Direct Form I structure supports Q15 and Q31 data types.
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The advantage of the Direct Form II Biquad is that it requires half the number of state variables, 2 rather than 4, per Biquad stage.
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The transposed Direct Form II structure, on the other hand, requires a wide dynamic range for the state variables <code>d1</code> and <code>d2</code>.
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@par Instance Structure
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Because of this, the CMSIS library only has a floating-point version of the Direct Form II Biquad.
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The coefficients and state variables for a filter are stored together in an instance data structure.
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The advantage of the Direct Form II Biquad is that it requires half the number of state variables, 2 rather than 4, per Biquad stage.
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A separate instance structure must be defined for each filter.
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Coefficient arrays may be shared among several instances while state variable arrays cannot be shared.
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@par Instance Structure
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@par Init Functions
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The coefficients and state variables for a filter are stored together in an instance data structure.
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There is also an associated initialization function.
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A separate instance structure must be defined for each filter.
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The initialization function performs following operations:
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Coefficient arrays may be shared among several instances while state variable arrays cannot be shared.
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- Sets the values of the internal structure fields.
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- Zeros out the values in the state buffer.
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@par Init Functions
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To do this manually without calling the init function, assign the follow subfields of the instance structure:
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There is also an associated initialization function.
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numStages, pCoeffs, pState. Also set all of the values in pState to zero.
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The initialization function performs following operations:
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@par
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- Sets the values of the internal structure fields.
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Use of the initialization function is optional.
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- Zeros out the values in the state buffer.
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However, if the initialization function is used, then the instance structure cannot be placed into a const data section.
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To do this manually without calling the init function, assign the follow subfields of the instance structure:
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To place an instance structure into a const data section, the instance structure must be manually initialized.
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numStages, pCoeffs, pState. Also set all of the values in pState to zero.
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Set the values in the state buffer to zeros before static initialization.
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@par
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For example, to statically initialize the instance structure use
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Use of the initialization function is optional.
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<pre>
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However, if the initialization function is used, then the instance structure cannot be placed into a const data section.
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arm_biquad_cascade_df2T_instance_f64 S1 = {numStages, pState, pCoeffs};
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To place an instance structure into a const data section, the instance structure must be manually initialized.
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arm_biquad_cascade_df2T_instance_f32 S1 = {numStages, pState, pCoeffs};
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Set the values in the state buffer to zeros before static initialization.
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</pre>
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For example, to statically initialize the instance structure use
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where <code>numStages</code> is the number of Biquad stages in the filter;
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<pre>
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<code>pState</code> is the address of the state buffer.
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arm_biquad_cascade_df2T_instance_f64 S1 = {numStages, pState, pCoeffs};
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<code>pCoeffs</code> is the address of the coefficient buffer;
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arm_biquad_cascade_df2T_instance_f32 S1 = {numStages, pState, pCoeffs};
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*/
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</pre>
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where <code>numStages</code> is the number of Biquad stages in the filter;
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<code>pState</code> is the address of the state buffer.
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<code>pCoeffs</code> is the address of the coefficient buffer;
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*/
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/**
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/**
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@addtogroup BiquadCascadeDF2T
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@addtogroup BiquadCascadeDF2T
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@{
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@{
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*/
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*/
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/**
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/**
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@brief Processing function for the floating-point transposed direct form II Biquad cascade filter.
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@brief Processing function for the floating-point transposed direct form II Biquad cascade filter.
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@param[in] S points to an instance of the filter data structure
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@param[in] S points to an instance of the filter data structure
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@param[in] pSrc points to the block of input data
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@param[in] pSrc points to the block of input data
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@param[out] pDst points to the block of output data
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@param[out] pDst points to the block of output data
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@param[in] blockSize number of samples to process
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@param[in] blockSize number of samples to process
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@return none
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@return none
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*/
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*/
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#if defined(ARM_MATH_NEON)
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#if defined(ARM_MATH_NEON)
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void arm_biquad_cascade_df2T_f64(
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void arm_biquad_cascade_df2T_f64(
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Jean-Baptiste Rolland
3 years ago