/* ----------------------------------------------------------------------
* Copyright (C) 2010-2013 ARM Limited. All rights reserved.
*
* $Date: 17. January 2013
* $Revision: V1.4.1
*
* Project: CMSIS DSP Library
* Title: arm_rfft_f32.c
*
* Description: RFFT & RIFFT Floating point process function
*
* Target Processor: Cortex-M4/Cortex-M3/Cortex-M0
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* - Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* - Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
* - Neither the name of ARM LIMITED nor the names of its contributors
* may be used to endorse or promote products derived from this
* software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
* COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
* BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
* -------------------------------------------------------------------- */
#include "arm_math.h"
void stage_rfft_f32(
arm_rfft_fast_instance_f32 * S,
float32_t * p, float32_t * pOut)
{
uint32_t k; /* Loop Counter */
float32_t twR, twI; /* RFFT Twiddle coefficients */
float32_t * pCoeff = S->pTwiddleRFFT; /* Points to RFFT Twiddle factors */
float32_t *pA = p; /* increasing pointer */
float32_t *pB = p; /* decreasing pointer */
float32_t xAR, xAI, xBR, xBI; /* temporary variables */
float32_t t1a, t1b; /* temporary variables */
float32_t p0, p1, p2, p3; /* temporary variables */
k = (S->Sint).fftLen - 1;
/* Pack first and last sample of the frequency domain together */
xBR = pB[0];
xBI = pB[1];
xAR = pA[0];
xAI = pA[1];
twR = *pCoeff++ ;
twI = *pCoeff++ ;
// U1 = XA(1) + XB(1); % It is real
t1a = xBR + xAR ;
// U2 = XB(1) - XA(1); % It is imaginary
t1b = xBI + xAI ;
// real(tw * (xB - xA)) = twR * (xBR - xAR) - twI * (xBI - xAI);
// imag(tw * (xB - xA)) = twI * (xBR - xAR) + twR * (xBI - xAI);
*pOut++ = 0.5f * ( t1a + t1b );
*pOut++ = 0.5f * ( t1a - t1b );
// XA(1) = 1/2*( U1 - imag(U2) + i*( U1 +imag(U2) ));
pB = p + 2*k;
pA += 2;
do
{
/*
function X = my_split_rfft(X, ifftFlag)
% X is a series of real numbers
L = length(X);
XC = X(1:2:end) +i*X(2:2:end);
XA = fft(XC);
XB = conj(XA([1 end:-1:2]));
TW = i*exp(-2*pi*i*[0:L/2-1]/L).';
for l = 2:L/2
XA(l) = 1/2 * (XA(l) + XB(l) + TW(l) * (XB(l) - XA(l)));
end
XA(1) = 1/2* (XA(1) + XB(1) + TW(1) * (XB(1) - XA(1))) + i*( 1/2*( XA(1) + XB(1) + i*( XA(1) - XB(1))));
X = XA;
*/
xBI = pB[1];
xBR = pB[0];
xAR = pA[0];
xAI = pA[1];
twR = *pCoeff++;
twI = *pCoeff++;
t1a = xBR - xAR ;
t1b = xBI + xAI ;
// real(tw * (xB - xA)) = twR * (xBR - xAR) - twI * (xBI - xAI);
// imag(tw * (xB - xA)) = twI * (xBR - xAR) + twR * (xBI - xAI);
p0 = twR * t1a;
p1 = twI * t1a;
p2 = twR * t1b;
p3 = twI * t1b;
*pOut++ = 0.5f * (xAR + xBR + p0 + p3 ); //xAR
*pOut++ = 0.5f * (xAI - xBI + p1 - p2 ); //xAI
pA += 2;
pB -= 2;
k--;
} while(k > 0u);
}
/* Prepares data for inverse cfft */
void merge_rfft_f32(
arm_rfft_fast_instance_f32 * S,
float32_t * p, float32_t * pOut)
{
uint32_t k; /* Loop Counter */
float32_t twR, twI; /* RFFT Twiddle coefficients */
float32_t *pCoeff = S->pTwiddleRFFT; /* Points to RFFT Twiddle factors */
float32_t *pA = p; /* increasing pointer */
float32_t *pB = p; /* decreasing pointer */
float32_t xAR, xAI, xBR, xBI; /* temporary variables */
float32_t t1a, t1b, r, s, t, u; /* temporary variables */
k = (S->Sint).fftLen - 1;
xAR = pA[0];
xAI = pA[1];
pCoeff += 2 ;
*pOut++ = 0.5f * ( xAR + xAI );
*pOut++ = 0.5f * ( xAR - xAI );
pB = p + 2*k ;
pA += 2 ;
while(k > 0u)
{
/* G is half of the frequency complex spectrum */
//for k = 2:N
// Xk(k) = 1/2 * (G(k) + conj(G(N-k+2)) + Tw(k)*( G(k) - conj(G(N-k+2))));
xBI = pB[1] ;
xBR = pB[0] ;
xAR = pA[0];
xAI = pA[1];
twR = *pCoeff++;
twI = *pCoeff++;
t1a = xAR - xBR ;
t1b = xAI + xBI ;
r = twR * t1a;
s = twI * t1b;
t = twI * t1a;
u = twR * t1b;
// real(tw * (xA - xB)) = twR * (xAR - xBR) - twI * (xAI - xBI);
// imag(tw * (xA - xB)) = twI * (xAR - xBR) + twR * (xAI - xBI);
*pOut++ = 0.5f * (xAR + xBR - r - s ); //xAR
*pOut++ = 0.5f * (xAI - xBI + t - u ); //xAI
pA += 2;
pB -= 2;
k--;
}
}
/**
* @ingroup groupTransforms
*/
/**
* @defgroup Fast Real FFT Functions
*
* \par
* The CMSIS DSP library includes specialized algorithms for computing the
* FFT of real data sequences. The FFT is defined over complex data but
* in many applications the input is real. Real FFT algorithms take advantage
* of the symmetry properties of the FFT and have a speed advantage over complex
* algorithms of the same length.
* \par
* The Fast RFFT algorith relays on the mixed radix CFFT that save processor usage.
* \par
* The real length N forward FFT of a sequence is computed using the steps shown below.
* \par
* \image html RFFT.gif "Real Fast Fourier Transform"
* \par
* The real sequence is initially treated as if it were complex to perform a CFFT.
* Later, a processing stage reshapes the data to obtain half of the frequency spectrum
* in complex format. Except the first complex number that contains the two real numbers
* X[0] and X[N/2] all the data is complex. In other words, the first complex sample
* contains two real values packed.
* \par
* The input for the inverse RFFT should keep the same format as the output of the
* forward RFFT. A first processing stage pre-process the data to later perform an
* inverse CFFT.
* \par
* \image html RIFFT.gif "Real Inverse Fast Fourier Transform"
* \par
* The algorithms for floating-point, Q15, and Q31 data are slightly different
* and we describe each algorithm in turn.
* \par Floating-point
* The main functions are arm_rfft_fast_f32()
* and arm_rfft_fast_init_f32(). The older functions
* arm_rfft_f32() and arm_rfft_init_f32() have been
* deprecated but are still documented.
* \par
* The FFT of a real N-point sequence has even symmetry in the frequency
* domain. The second half of the data equals the conjugate of the first half
* flipped in frequency:
*
*X[0] - real data
*X[1] - complex data
*X[2] - complex data
*...
*X[fftLen/2-1] - complex data
*X[fftLen/2] - real data
*X[fftLen/2+1] - conjugate of X[fftLen/2-1]
*X[fftLen/2+2] - conjugate of X[fftLen/2-2]
*...
*X[fftLen-1] - conjugate of X[1]
*
* Looking at the data, we see that we can uniquely represent the FFT using only
*
*N/2+1 samples:
*X[0] - real data
*X[1] - complex data
*X[2] - complex data
*...
*X[fftLen/2-1] - complex data
*X[fftLen/2] - real data
*
* Looking more closely we see that the first and last samples are real valued.
* They can be packed together and we can thus represent the FFT of an N-point
* real sequence by N/2 complex values:
*
*X[0],X[N/2] - packed real data: X[0] + jX[N/2]
*X[1] - complex data
*X[2] - complex data
*...
*X[fftLen/2-1] - complex data
*
* The real FFT functions pack the frequency domain data in this fashion. The
* forward transform outputs the data in this form and the inverse transform
* expects input data in this form. The function always performs the needed
* bitreversal so that the input and output data is always in normal order. The
* functions support lengths of [32, 64, 128, ..., 4096] samples.
* \par
* The forward and inverse real FFT functions apply the standard FFT scaling; no
* scaling on the forward transform and 1/fftLen scaling on the inverse
* transform.
* \par Q15 and Q31
* The real algorithms are defined in a similar manner and utilize N/2 complex
* transforms behind the scenes. In the case of fixed-point data, a radix-4
* complex transform is performed and this limits the allows sequence lengths to
* 128, 512, and 2048 samples.
* \par
* TBD. We need to document input and output order of data.
* \par
* The complex transforms used internally include scaling to prevent fixed-point
* overflows. The overall scaling equals 1/(fftLen/2).
* \par
* A separate instance structure must be defined for each transform used but
* twiddle factor and bit reversal tables can be reused.
* \par
* There is also an associated initialization function for each data type.
* The initialization function performs the following operations:
* - Sets the values of the internal structure fields.
* - Initializes twiddle factor table and bit reversal table pointers.
* - Initializes the internal complex FFT data structure.
* \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 should be manually
* initialized as follows:
*
*arm_rfft_instance_q31 S = {fftLenReal, fftLenBy2, ifftFlagR, bitReverseFlagR, twidCoefRModifier, pTwiddleAReal, pTwiddleBReal, pCfft};
*arm_rfft_instance_q15 S = {fftLenReal, fftLenBy2, ifftFlagR, bitReverseFlagR, twidCoefRModifier, pTwiddleAReal, pTwiddleBReal, pCfft};
*
* where fftLenReal is the length of the real transform;
* fftLenBy2 length of the internal complex transform.
* ifftFlagR Selects forward (=0) or inverse (=1) transform.
* bitReverseFlagR Selects bit reversed output (=0) or normal order
* output (=1).
* twidCoefRModifier stride modifier for the twiddle factor table.
* The value is based on the FFT length;
* pTwiddleARealpoints to the A array of twiddle coefficients;
* pTwiddleBRealpoints to the B array of twiddle coefficients;
* pCfft points to the CFFT Instance structure. The CFFT structure
* must also be initialized. Refer to arm_cfft_radix4_f32() for details regarding
* static initialization of the complex FFT instance structure.
*/
/**
* @addtogroup RealFFT
* @{
*/
/**
* @brief Processing function for the floating-point real FFT.
* @param[in] *S points to an arm_rfft_fast_instance_f32 structure.
* @param[in] *p points to the input buffer.
* @param[in] *pOut points to an arm_rfft_fast_instance_f32 structure.
* @param[in] ifftFlag RFFT if flag is 0, RIFFT if flag is 1
* @return none.
*/
void arm_rfft_fast_f32(
arm_rfft_fast_instance_f32 * S,
float32_t * p, float32_t * pOut,
uint8_t ifftFlag)
{
arm_cfft_instance_f32 * Sint = &(S->Sint);
Sint->fftLen = S->fftLenRFFT / 2;
/* Calculation of Real FFT */
if(ifftFlag)
{
/* Real FFT comression */
merge_rfft_f32(S, p, pOut);
/* Complex radix-4 IFFT process */
arm_cfft_f32( Sint, pOut, ifftFlag, 1);
}
else
{
/* Calculation of RFFT of input */
arm_cfft_f32( Sint, p, ifftFlag, 1);
/* Real FFT extraction */
stage_rfft_f32(S, p, pOut);
}
}