DFTI_PACKED_FORMAT

The result of the forward transform (that is, in the frequency domain) of real data is represented in several possible packed formats: Pack, Perm, CCS, or CCE. The data can be packed due to the symmetry property of the FFT of a real data.

The following setting is recommended for real transforms, though not default:

• The configuration parameter DFTI_CONJUGATE_EVEN_STORAGE has the value of DFTI_COMPLEX_COMPLEX.
• Elements of the result in the conjugate-even domain have a complex type.
• The configuration parameter DFTI_PACKED_FORMAT has the value of DFTI_CCE_FORMAT.

This setting is applicable to all ranks of real transform.

The following setting for one-dimensional and two-dimensional real transforms is default:

• The configuration parameter DFTI_CONJUGATE_EVEN_STORAGE has the value of DFTI_COMPLEX_REAL.
• Data elements in the frequency domain have a real type.
• The value of DFTI_PACKED_FORMAT defines how real and imaginary parts of the data are laid out in the result.

This setting is not applicable to three-dimensional and higher-rank transforms. Though not recommended, it is default for backward compatibility.

The CCE format stores the values of the first half of the output complex conjugate-even signal resulted from the forward FFT. Note that the one-dimensional signal stored in CCE format is one complex element longer. For multi-dimensional real transform, n1 * n2 * n3 * ... * nk the size of complex matrix in CCE format is (n1/2+1)* n2 * n3 * ... * nk for Fortran and n1 * n2 * ... * (nk/2+1) for C.

The CCS format looks like the CCE format. It is the same format as CCE for one-dimensional transform. The CCS format is slightly different for multi-dimensional real transform. In CCS format, the output samples of the FFT are arranged as shown in Table "Packed Format Output Samples" for one-dimensional FFT and in Table "CCS Format Output Samples (Two-Dimensional Matrix (m+2)-by-(n+2))" for two-dimensional FFT.

The Pack format is a compact representation of a complex conjugate-symmetric sequence. The disadvantage of this format is that it is not the natural format used by the real FFT algorithms ("natural" in the sense that array is natural for complex FFTs). In Pack format, the output samples of the FFT are arranged as shown in Table "Packed Format Output Samples" for one-dimensional FFT and in Table "Pack Format Output Samples (Two-Dimensional Matrix m-by-n)" for two-dimensional FFT.

The Perm format is a permutation of the Pack format for even lengths and one is the same as the Pack format for odd lengths. In Perm format, the output samples of the FFT are arranged as shown in Table "Packed Format Output Samples" for one-dimensional FFT and in Table "Perm Format Output Samples (Two-Dimensional Matrix m-by-n)" for two-dimensional FFT.

Note that Table "Packed Format Output Samples" uses the following notation for complex data entries:

R_j = Re z_j

I_j = Im z_j

See also Table "Comparison of the Storage Effects of Complex-to-Complex and Real-to-Complex FFTs for Forward Transform" and Table "Comparison of the Storage Effects of Complex-to-Complex and Complex-to-Real FFTs for Backward Transform".

* n/u - not used.

Note that in the Table "CCS Format Output Samples (Two-Dimensional Matrix (m+2)-by-(n+2))", (n+2) columns are used for even n = k*2, while n columns are used for odd n = k*2+1. In the latter case, the first row is

z(1,1) 0 REz(1,2) IMz(1,2) ... REz(1,k) IMz(1,k)

If m is even, the (m+1)-th row is

z(m/2+1,1) 0 REz(m/2+1,2) IMz(m/2+1,2) ... REz(m/2+1,k) IMz(m/2+1,k)

Note that in the Table "Pack Format Output Samples (Two-Dimensional Matrix m-by-n)" and Table "Perm Format Output Samples (Two-Dimensional Matrix m-by-n)", for even number of columns n = k*2, while for odd number of columns n = k*2+1 and the first row is

z(1,1) REz(1,2) IMz(1,2) ... REz(1,k) IMz(1,k)

If m is even, the last row in Pack format and the second row in Perm format is

z(m/2+1,1) REz(m/2+1,2) IMz(m/2+1,2) ... REz(m/2+1,k) IMz(m/2+1,k)

The tables for two-dimensional FFT use Fortran-interface conventions. For C-interface specifics in storing packed data, see DFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, DFTI_CONJUGATE_EVEN_STORAGE. See also Table "Fortran-interface Data Layout for a 6-by-4 Matrix" and Table "C-interface Data Layout for a 6-by-4 Matrix" for examples of Fortran-interface and C-interface formats.

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