DFTI_PACKED

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 7/13/2023

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DFTI_PACKED_FORMAT

The result of the forward transform of real data is a conjugate-even sequence. Due to the symmetry property, only a part of the complex-valued sequence is stored in memory. The combination of the DFTI_PACKED_FORMAT and DFTI_CONJUGATE_EVEN_STORAGE configuration parameters defines how the conjugate-even sequence data is packed. If DFTI_CONJUGATE_EVEN_STORAGE is set to DFTI_COMPLEX_COMPLEX (default), the only possible value of DFTI_PACKED_FORMAT is DFTI_CCE_FORMAT; this association of configuration parameters is supported for transforms of any dimension. For a description of the corresponding packed format, see DFTI_CONJUGATE_EVEN_STORAGE. For one-dimensional transforms (only) with DFTI_CONJUGATE_EVEN_STORAGE set to DFTI_COMPLEX_REAL, the DFTI_PACKED_FORMAT configuration parameter must be DFTI_CCS_FORMAT, DFTI_PACK_FORMAT, or DFTI_PERM_FORMAT. The corresponding packed formats are explained and illustrated below.

DFTI_CCS_FORMAT for One-dimensional Transforms

The following figure illustrates the storage of a one-dimensional (1D) size-N conjugate-even sequence in a real array for the CCS, PACK, and PERM packed formats. The CCS format requires an array of size N+2, while the other formats require an array of size N. Zero-based indexing is used.

Storage of a 1D Size-N Conjugate-even Sequence in a Real Array

NOTE:

For storage of a one-dimensional conjugate-even sequence in a real array, CCS is in the same format as CCE.

The real and imaginary parts of the complex-valued conjugate-even sequence Z_k are located in a real-valued array AC as illustrated by figure "Storage of a 1D Size-N Conjugate-even Sequence in a Real Array" and can be used to reconstruct the whole conjugate-even sequence as follows:

float *AR; // malloc( sizeof(float)*N )
float *AC; // malloc( sizeof(float)*(N+2) )
...
status = DftiSetValue( desc, DFTI_PACKED_FORMAT, DFTI_CCS_FORMAT );
...
// on input:  R{k} = AR[k]  
status = DftiComputeForward( desc, AR, AC );  // real-to-complex FFT
// on output:
// for k=0…N/2:     Z{k} =   AC[2*k+0]         + I*AC[2*k+1]
// for k=N/2+1…N-1: Z{k} =   AC[2*(N-k)%N + 0] - I*AC[2*(N-k)%N + 1]

DFTI_PACK_FORMAT for One-dimensional Transforms

float *AR; // malloc( sizeof(float)*N )
float *AC; // malloc( sizeof(float)*N )
...
status = DftiSetValue( desc, DFTI_PACKED_FORMAT, DFTI_PACK_FORMAT );
...
// on input:  R{k} = AR[k]  
status = DftiComputeForward( desc, AR, AC );  // real-to-complex FFT
// on output: Z{k} = re + I*im, where
// if (k == 0) {
//     re =  AC[0];
//     im =  0;
// } else if (k == N-k) {
//     re =  AC[2*k-1];
//     im =  0;
// } else if (k <= N/2) {
//     re =  AC[2*k-1];
//     im =  AC[2*k-0];
// } else {
//     re =  AC[2*(N-k)-1];
//     im = -AC[2*(N-k)-0];
// }

DFTI_PERM_FORMAT for One-dimensional Transforms

The real and imaginary parts of the complex-valued conjugate-even sequence Z_k are located in real-valued array AC as illustrated by figure "Storage of a 1D Size-N Conjugate-even Sequence in a Real Array" and can be used to reconstruct the whole conjugate-even sequence as follows:

float *AR; // malloc( sizeof(float)*N )
float *AC; // malloc( sizeof(float)*N )
...
status = DftiSetValue( desc, DFTI_PACKED_FORMAT, DFTI_PERM_FORMAT );
...
// on input:  R{k} = AR[k]  
status = DftiComputeForward( desc, AR, AC );  // real-to-complex FFT
// on output: Z{k} = re + I*im, where
// if (k == 0) {
//     re =  AC[0];
//     im =  0;
// } else if (k == N-k) {
//     re =  AC[1];
//     im =  0;
// } else if (k <= N/2) {
//     re =  AC[2*k+0 - N%2];
//     im =  AC[2*k+1 - N%2];
// } else {
//     re =  AC[2*(N-k)+0 - N%2];
//     im = -AC[2*(N-k)+1 - N%2];
// }

Parent topic: Configuration Settings

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Developer Reference for Intel® oneAPI Math Kernel Library for C

DFTI_PACKED_FORMAT

DFTI_CCS_FORMAT for One-dimensional Transforms

DFTI_PACK_FORMAT for One-dimensional Transforms

DFTI_PERM_FORMAT for One-dimensional Transforms

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