Real FFT 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. Source buffer is modified by this function. More...
void	arm_rfft_fast_f16 (const arm_rfft_fast_instance_f16 *S, float16_t *p, float16_t *pOut, uint8_t ifftFlag)
	Processing function for the floating-point real FFT. More...
void	arm_rfft_fast_f32 (const arm_rfft_fast_instance_f32 *S, float32_t *p, float32_t *pOut, uint8_t ifftFlag)
	Processing function for the floating-point real FFT. More...
void	arm_rfft_fast_f64 (arm_rfft_fast_instance_f64 *S, float64_t *p, float64_t *pOut, uint8_t ifftFlag)
	Processing function for the Double Precision floating-point real FFT. More...
arm_status	arm_rfft_fast_init_f16 (arm_rfft_fast_instance_f16 *S, uint16_t fftLen)
	Initialization 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_fast_init_f64 (arm_rfft_fast_instance_f64 *S, uint16_t fftLen)
	Initialization function for the Double Precision 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)
	Initialization function for the floating-point RFFT/RIFFT. More...
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 algorithm 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_f16() and arm_rfft_fast_init_f16().

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). Due to the use of complex transform internally, the source buffer is modified by the rfft.

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 except for MVE versions where it is mandatory. If you don't use the initialization functions, then the structures should be initialized with code similar to the one below:

      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 (fftLenReal/2). 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.

Note that with MVE versions you can't initialize instance structures directly and must use the initialization function.

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 algorithm 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). Due to the use of complex transform internally, the source buffer is modified by the rfft.

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 except for MVE versions where it is mandatory. If you don't use the initialization functions, then the structures should be initialized with code similar to the one below:

      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 (fftLenReal/2). 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.

Note that with MVE versions you can't initialize instance structures directly and must use the initialization function.

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

For the RIFFT, the source buffer must at least have length fftLenReal + 2. The last two elements must be equal to what would be generated by the RFFT: (pSrc[0] - pSrc[1]) and 0.0f

void arm_rfft_fast_f16	(	const arm_rfft_fast_instance_f16 *	S,
		float16_t *	p,
		float16_t *	pOut,
		uint8_t	ifftFlag
	)

Parameters

[in]	S	points to an arm_rfft_fast_instance_f16 structure
[in]	p	points to input buffer (Source buffer is modified by this function.)
[in]	pOut	points to output buffer
[in]	ifftFlag	value = 0: RFFT value = 1: RIFFT

Returns

none

void arm_rfft_fast_f32	(	const 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 input buffer (Source buffer is modified by this function.)
[in]	pOut	points to output buffer
[in]	ifftFlag	value = 0: RFFT value = 1: RIFFT

Returns

none

void arm_rfft_fast_f64	(	arm_rfft_fast_instance_f64 *	S,
		float64_t *	p,
		float64_t *	pOut,
		uint8_t	ifftFlag
	)

Parameters

[in]	S	points to an arm_rfft_fast_instance_f64 structure
[in]	p	points to input buffer (Source buffer is modified by this function.)
[in]	pOut	points to output buffer
[in]	ifftFlag	value = 0: RFFT value = 1: RIFFT

Returns

none

arm_status arm_rfft_fast_init_f16	(	arm_rfft_fast_instance_f16 *	S,
		uint16_t	fftLen
	)

Parameters

[in,out]	S	points to an arm_rfft_fast_instance_f16 structure
[in]	fftLen	length of the Real Sequence

Returns

execution status

• ARM_MATH_SUCCESS : Operation successful
• ARM_MATH_ARGUMENT_ERROR : fftLen is not a supported length

Description

The parameter fftLen specifies the 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.

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

execution status

• ARM_MATH_SUCCESS : Operation successful
• ARM_MATH_ARGUMENT_ERROR : fftLen is not a supported length

Description

The parameter fftLen specifies the 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.

arm_status arm_rfft_fast_init_f64	(	arm_rfft_fast_instance_f64 *	S,
		uint16_t	fftLen
	)

Parameters

[in,out]	S	points to an arm_rfft_fast_instance_f64 structure
[in]	fftLen	length of the Real Sequence

Returns

execution status

• ARM_MATH_SUCCESS : Operation successful
• ARM_MATH_ARGUMENT_ERROR : fftLen is not a supported length

Description

The parameter fftLen specifies the 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.

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
	)

Deprecated:

Do not use this function. It has been superceded by arm_rfft_fast_init_f32 and will be removed in the future.

Parameters

[in,out]	S	points to an instance of the floating-point RFFT/RIFFT structure
[in,out]	S_CFFT	points to an instance of the floating-point CFFT/CIFFT structure
[in]	fftLenReal	length of the FFT.
[in]	ifftFlagR	flag that selects transform direction value = 0: forward transform value = 1: inverse transform
[in]	bitReverseFlag	flag that enables / disables bit reversal of output value = 0: disables bit reversal of output value = 1: enables bit reversal of output

Returns

execution status

• ARM_MATH_SUCCESS : Operation successful
• ARM_MATH_ARGUMENT_ERROR : fftLenReal is not a supported length

Description

The parameter fftLenRealspecifies length of RFFT/RIFFT Process. Supported FFT Lengths are 128, 512, 2048.

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.

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 transform direction value = 0: forward transform value = 1: inverse transform
[in]	bitReverseFlag	flag that enables / disables bit reversal of output value = 0: disables bit reversal of output value = 1: enables bit reversal of output

Returns

execution status

• ARM_MATH_SUCCESS : Operation successful
• ARM_MATH_ARGUMENT_ERROR : fftLenReal is not a supported length

Details

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.

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 transform direction value = 0: forward transform value = 1: inverse transform
[in]	bitReverseFlag	flag that enables / disables bit reversal of output value = 0: disables bit reversal of output value = 1: enables bit reversal of output

Returns

execution status

• ARM_MATH_SUCCESS : Operation successful
• ARM_MATH_ARGUMENT_ERROR : fftLenReal is not a supported length

Details

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.

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 input buffer (Source buffer is modified by this function.)
[out]	pDst	points to 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

If the input buffer is of length N, the output buffer must have length 2*N. The input buffer is modified by this function.

For the RIFFT, the source buffer must at least have length fftLenReal + 2. The last two elements must be equal to what would be generated by the RFFT: (pSrc[0] - pSrc[1]) >> 1 and 0

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 input buffer (Source buffer is modified by this function)
[out]	pDst	points to 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

If the input buffer is of length N, the output buffer must have length 2*N. The input buffer is modified by this function.

For the RIFFT, the source buffer must at least have length fftLenReal + 2. The last two elements must be equal to what would be generated by the RFFT: (pSrc[0] - pSrc[1]) >> 1 and 0