# Real FFT Functions

CMSIS-DSP: Real FFT Functions

| CMSIS-DSP Version 1.5.2 CMSIS DSP Software Library

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Real FFT Functions
Transform Functions

## Content

| Real FFT Tables

## Functions

void | arm_rfft_f32 (const arm_rfft_instance_f32 *S, float32_t *pSrc, float32_t *pDst)
| Processing function for the floating-point RFFT/RIFFT. More...
void | arm_rfft_fast_f32 (arm_rfft_fast_instance_f32 *S, float32_t *p, float32_t *pOut, uint8_t ifftFlag)
| Processing function for the floating-point real FFT. More...
arm_status | arm_rfft_fast_init_f32 (arm_rfft_fast_instance_f32 *S, uint16_t fftLen)
| Initialization function for the floating-point real FFT. More...
arm_status | arm_rfft_init_f32 (arm_rfft_instance_f32 *S, arm_cfft_radix4_instance_f32 *S_CFFT, uint32_t fftLenReal, uint32_t ifftFlagR, uint32_t bitReverseFlag)
arm_status | arm_rfft_init_q15 (arm_rfft_instance_q15 *S, uint32_t fftLenReal, uint32_t ifftFlagR, uint32_t bitReverseFlag)
| Initialization function for the Q15 RFFT/RIFFT. More...
arm_status | arm_rfft_init_q31 (arm_rfft_instance_q31 *S, uint32_t fftLenReal, uint32_t ifftFlagR, uint32_t bitReverseFlag)
| Initialization function for the Q31 RFFT/RIFFT. More...
void | arm_rfft_q15 (const arm_rfft_instance_q15 *S, q15_t *pSrc, q15_t *pDst)
| Processing function for the Q15 RFFT/RIFFT. More...
void | arm_rfft_q31 (const arm_rfft_instance_q31 *S, q31_t *pSrc, q31_t *pDst)
| Processing function for the Q31 RFFT/RIFFT. More...

## Description

  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.

  The Fast RFFT algorith relays on the mixed radix CFFT that save processor usage.

  The real length N forward FFT of a sequence is computed using the steps shown below.

  Real Fast Fourier Transform

  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.

  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.

  Real Inverse Fast Fourier Transform

  The algorithms for floating-point, Q15, and Q31 data are slightly different and we describe each algorithm in turn.

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.

  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:

  X = { real[0], imag[0], real[1], imag[1], real[2], imag[2] ... real[(N/2)-1], imag[(N/2)-1 }

  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.

  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.

Q15 and Q31
  The real algorithms are defined in a similar manner and utilize N/2 complex transforms behind the scenes.

  The complex transforms used internally include scaling to prevent fixed-point overflows. The overall scaling equals 1/(fftLen/2).

  A separate instance structure must be defined for each transform used but twiddle factor and bit reversal tables can be reused.

  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.

  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; `pTwiddleAReal`points to the A array of twiddle coefficients; `pTwiddleBReal`points 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.

end of RealFFT_Table group

## Function Documentation

void arm_rfft_f32 | ( | const arm_rfft_instance_f32 * | S,
|  | float32_t * | pSrc,
|  | float32_t * | pDst
| ) |  |

Deprecated:
  Do not use this function. It has been superceded by arm_rfft_fast_f32 and will be removed in the future.
  Parameters

    [in] | *S | points to an instance of the floating-point RFFT/RIFFT structure.
    [in] | *pSrc | points to the input buffer.
    [out] | *pDst | points to the output buffer.

  Returns
    none.

References arm_bitreversal_f32(), arm_radix4_butterfly_f32(), arm_radix4_butterfly_inverse_f32(), arm_split_rfft_f32(), arm_split_rifft_f32(), arm_rfft_instance_f32::bitReverseFlagR, arm_cfft_radix4_instance_f32::bitRevFactor, arm_cfft_radix4_instance_f32::fftLen, arm_rfft_instance_f32::fftLenBy2, arm_rfft_instance_f32::ifftFlagR, arm_cfft_radix4_instance_f32::onebyfftLen, arm_cfft_radix4_instance_f32::pBitRevTable, arm_rfft_instance_f32::pCfft, arm_cfft_radix4_instance_f32::pTwiddle, arm_rfft_instance_f32::pTwiddleAReal, arm_rfft_instance_f32::pTwiddleBReal, arm_cfft_radix4_instance_f32::twidCoefModifier, and arm_rfft_instance_f32::twidCoefRModifier.

void arm_rfft_fast_f32 | ( | arm_rfft_fast_instance_f32 * | S,
|  | float32_t * | p,
|  | float32_t * | pOut,
|  | uint8_t | ifftFlag
| ) |  |

Parameters

  [in] | *S | points to an arm_rfft_fast_instance_f32 structure.
  [in] | *p | points to the input buffer.
  [in] | *pOut | points to the output buffer.
  [in] | ifftFlag | RFFT if flag is 0, RIFFT if flag is 1

Returns
  none.

References arm_cfft_f32(), arm_cfft_instance_f32::fftLen, arm_rfft_fast_instance_f32::fftLenRFFT, merge_rfft_f32(), arm_rfft_fast_instance_f32::Sint, and stage_rfft_f32().

arm_status arm_rfft_fast_init_f32 | ( | arm_rfft_fast_instance_f32 * | S,
|  | uint16_t | fftLen
| ) |  |

Parameters

  [in,out] | *S | points to an arm_rfft_fast_instance_f32 structure.
  [in] | fftLen | length of the Real Sequence.

Returns
  The function returns ARM_MATH_SUCCESS if initialization is successful or ARM_MATH_ARGUMENT_ERROR if `fftLen` is not a supported value.

Description:

  The parameter `fftLen` Specifies length of RFFT/CIFFT process. Supported FFT Lengths are 32, 64, 128, 256, 512, 1024, 2048, 4096.

  This Function also initializes Twiddle factor table pointer and Bit reversal table pointer.

References ARM_MATH_ARGUMENT_ERROR, ARM_MATH_SUCCESS, armBitRevIndexTable1024, armBitRevIndexTable128, armBitRevIndexTable16, armBitRevIndexTable2048, armBitRevIndexTable256, armBitRevIndexTable32, armBitRevIndexTable512, armBitRevIndexTable64, ARMBITREVINDEXTABLE_1024_TABLE_LENGTH, ARMBITREVINDEXTABLE_128_TABLE_LENGTH, ARMBITREVINDEXTABLE_16_TABLE_LENGTH, ARMBITREVINDEXTABLE_2048_TABLE_LENGTH, ARMBITREVINDEXTABLE_256_TABLE_LENGTH, ARMBITREVINDEXTABLE_32_TABLE_LENGTH, ARMBITREVINDEXTABLE_512_TABLE_LENGTH, ARMBITREVINDEXTABLE_64_TABLE_LENGTH, arm_cfft_instance_f32::bitRevLength, arm_cfft_instance_f32::fftLen, arm_rfft_fast_instance_f32::fftLenRFFT, arm_cfft_instance_f32::pBitRevTable, arm_cfft_instance_f32::pTwiddle, arm_rfft_fast_instance_f32::pTwiddleRFFT, arm_rfft_fast_instance_f32::Sint, status, twiddleCoef_1024, twiddleCoef_128, twiddleCoef_16, twiddleCoef_2048, twiddleCoef_256, twiddleCoef_32, twiddleCoef_512, twiddleCoef_64, twiddleCoef_rfft_1024, twiddleCoef_rfft_128, twiddleCoef_rfft_2048, twiddleCoef_rfft_256, twiddleCoef_rfft_32, twiddleCoef_rfft_4096, twiddleCoef_rfft_512, and twiddleCoef_rfft_64.

arm_status arm_rfft_init_f32 | ( | arm_rfft_instance_f32 * | S,
|  | arm_cfft_radix4_instance_f32 * | S_CFFT,
|  | uint32_t | fftLenReal,
|  | uint32_t | ifftFlagR,
|  | uint32_t | bitReverseFlag
| ) |  |

References arm_cfft_radix4_init_f32(), ARM_MATH_ARGUMENT_ERROR, ARM_MATH_SUCCESS, arm_rfft_instance_f32::bitReverseFlagR, arm_rfft_instance_f32::fftLenBy2, arm_rfft_instance_f32::fftLenReal, arm_rfft_instance_f32::ifftFlagR, arm_rfft_instance_f32::pCfft, arm_rfft_instance_f32::pTwiddleAReal, arm_rfft_instance_f32::pTwiddleBReal, realCoefA, realCoefB, status, and arm_rfft_instance_f32::twidCoefRModifier.

Referenced by arm_dct4_init_f32().

arm_status arm_rfft_init_q15 | ( | arm_rfft_instance_q15 * | S,
|  | uint32_t | fftLenReal,
|  | uint32_t | ifftFlagR,
|  | uint32_t | bitReverseFlag
| ) |  |

Parameters

  [in,out] | *S | points to an instance of the Q15 RFFT/RIFFT structure.
  [in] | fftLenReal | length of the FFT.
  [in] | ifftFlagR | flag that selects forward (ifftFlagR=0) or inverse (ifftFlagR=1) transform.
  [in] | bitReverseFlag | flag that enables (bitReverseFlag=1) or disables (bitReverseFlag=0) bit reversal of output.

Returns
  The function returns ARM_MATH_SUCCESS if initialization is successful or ARM_MATH_ARGUMENT_ERROR if `fftLenReal` is not a supported value.

Description:

  The parameter `fftLenReal` Specifies length of RFFT/RIFFT Process. Supported FFT Lengths are 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.

  The parameter `ifftFlagR` controls whether a forward or inverse transform is computed. Set(=1) ifftFlagR to calculate RIFFT, otherwise RFFT is calculated.

  The parameter `bitReverseFlag` controls whether output is in normal order or bit reversed order. Set(=1) bitReverseFlag for output to be in normal order otherwise output is in bit reversed order.

  This function also initializes Twiddle factor table.

References arm_cfft_sR_q15_len1024, arm_cfft_sR_q15_len128, arm_cfft_sR_q15_len16, arm_cfft_sR_q15_len2048, arm_cfft_sR_q15_len256, arm_cfft_sR_q15_len32, arm_cfft_sR_q15_len4096, arm_cfft_sR_q15_len512, arm_cfft_sR_q15_len64, ARM_MATH_ARGUMENT_ERROR, ARM_MATH_SUCCESS, arm_rfft_instance_q15::bitReverseFlagR, arm_rfft_instance_q15::fftLenReal, arm_rfft_instance_q15::ifftFlagR, arm_rfft_instance_q15::pCfft, arm_rfft_instance_q15::pTwiddleAReal, arm_rfft_instance_q15::pTwiddleBReal, realCoefAQ15, realCoefBQ15, status, and arm_rfft_instance_q15::twidCoefRModifier.

Referenced by arm_dct4_init_q15().

arm_status arm_rfft_init_q31 | ( | arm_rfft_instance_q31 * | S,
|  | uint32_t | fftLenReal,
|  | uint32_t | ifftFlagR,
|  | uint32_t | bitReverseFlag
| ) |  |

Parameters

  [in,out] | *S | points to an instance of the Q31 RFFT/RIFFT structure.
  [in] | fftLenReal | length of the FFT.
  [in] | ifftFlagR | flag that selects forward (ifftFlagR=0) or inverse (ifftFlagR=1) transform.
  [in] | bitReverseFlag | flag that enables (bitReverseFlag=1) or disables (bitReverseFlag=0) bit reversal of output.

Returns
  The function returns ARM_MATH_SUCCESS if initialization is successful or ARM_MATH_ARGUMENT_ERROR if `fftLenReal` is not a supported value.

Description:

  The parameter `fftLenReal` Specifies length of RFFT/RIFFT Process. Supported FFT Lengths are 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.

  The parameter `ifftFlagR` controls whether a forward or inverse transform is computed. Set(=1) ifftFlagR to calculate RIFFT, otherwise RFFT is calculated.

  The parameter `bitReverseFlag` controls whether output is in normal order or bit reversed order. Set(=1) bitReverseFlag for output to be in normal order otherwise output is in bit reversed order.

7
  This function also initializes Twiddle factor table.

References arm_cfft_sR_q31_len1024, arm_cfft_sR_q31_len128, arm_cfft_sR_q31_len16, arm_cfft_sR_q31_len2048, arm_cfft_sR_q31_len256, arm_cfft_sR_q31_len32, arm_cfft_sR_q31_len4096, arm_cfft_sR_q31_len512, arm_cfft_sR_q31_len64, ARM_MATH_ARGUMENT_ERROR, ARM_MATH_SUCCESS, arm_rfft_instance_q31::bitReverseFlagR, arm_rfft_instance_q31::fftLenReal, arm_rfft_instance_q31::ifftFlagR, arm_rfft_instance_q31::pCfft, arm_rfft_instance_q31::pTwiddleAReal, arm_rfft_instance_q31::pTwiddleBReal, realCoefAQ31, realCoefBQ31, status, and arm_rfft_instance_q31::twidCoefRModifier.

Referenced by arm_dct4_init_q31().

void arm_rfft_q15 | ( | const arm_rfft_instance_q15 * | S,
|  | q15_t * | pSrc,
|  | q15_t * | pDst
| ) |  |

Parameters

  [in] | *S | points to an instance of the Q15 RFFT/RIFFT structure.
  [in] | *pSrc | points to the input buffer.
  [out] | *pDst | points to the output buffer.

Returns
  none.

Input an output formats:

  Internally input is downscaled by 2 for every stage to avoid saturations inside CFFT/CIFFT process. Hence the output format is different for different RFFT sizes. The input and output formats for different RFFT sizes and number of bits to upscale are mentioned in the tables below for RFFT and RIFFT:

  Input and Output Formats for Q15 RFFT

  Input and Output Formats for Q15 RIFFT

References arm_cfft_q15(), arm_split_rfft_q15(), arm_split_rifft_q15(), arm_rfft_instance_q15::bitReverseFlagR, arm_rfft_instance_q15::fftLenReal, arm_rfft_instance_q15::ifftFlagR, arm_rfft_instance_q15::pCfft, arm_rfft_instance_q15::pTwiddleAReal, arm_rfft_instance_q15::pTwiddleBReal, and arm_rfft_instance_q15::twidCoefRModifier.

void arm_rfft_q31 | ( | const arm_rfft_instance_q31 * | S,
|  | q31_t * | pSrc,
|  | q31_t * | pDst
| ) |  |

Parameters

  [in] | *S | points to an instance of the Q31 RFFT/RIFFT structure.
  [in] | *pSrc | points to the input buffer.
  [out] | *pDst | points to the output buffer.

Returns
  none.

Input an output formats:

  Internally input is downscaled by 2 for every stage to avoid saturations inside CFFT/CIFFT process. Hence the output format is different for different RFFT sizes. The input and output formats for different RFFT sizes and number of bits to upscale are mentioned in the tables below for RFFT and RIFFT:

  Input and Output Formats for Q31 RFFT

  Input and Output Formats for Q31 RIFFT

References arm_cfft_q31(), arm_split_rfft_q31(), arm_split_rifft_q31(), arm_rfft_instance_q31::bitReverseFlagR, arm_rfft_instance_q31::fftLenReal, arm_rfft_instance_q31::ifftFlagR, arm_rfft_instance_q31::pCfft, arm_rfft_instance_q31::pTwiddleAReal, arm_rfft_instance_q31::pTwiddleBReal, and arm_rfft_instance_q31::twidCoefRModifier.

Referenced by arm_dct4_q31().