/* ----------------------------------------------------------------------
* Project: CMSIS DSP Library
* Title: arm_rfft_f32.c
* Description: RFFT & RIFFT Floating point process function
*
* $Date: 27. January 2017
* $Revision: V.1.5.1
*
* Target Processor: Cortex-M cores
* -------------------------------------------------------------------- */
/*
* Copyright (C) 2010-2017 ARM Limited or its affiliates. All rights reserved.
*
* SPDX-License-Identifier: Apache-2.0
*
* Licensed under the Apache License, Version 2.0 (the License); you may
* not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an AS IS BASIS, WITHOUT
* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
#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 RealFFT 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. Looking at the data, we see that we can
* uniquely represent the FFT using only N/2 complex numbers. These are
* packed into the output array in alternating real and imaginary
* components:
* \par
* X = { real[0], imag[0], real[1], imag[1], real[2], imag[2] ...
* real[(N/2)-1], imag[(N/2)-1 }
* \par
* It happens that the first complex number (real[0], imag[0]) is actually
* all real. real[0] represents the DC offset, and imag[0] should be 0.
* (real[1], imag[1]) is the fundamental frequency, (real[2], imag[2]) is
* the first harmonic and so on.
* \par
* 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 Q15 and Q31
* The real algorithms are defined in a similar manner and utilize N/2 complex
* transforms behind the scenes.
* \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 the output buffer.
* @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 compression */
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);
}
}
/**
* @} end of RealFFT group
*/