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# Notes on PFFFT

We strongly recommend to read this file before using PFFFT and to always wrap the original C library within a C++ wrapper.

## Scratch buffer

The caller can optionally provide a scratch buffer. When not provided, VLA is used to provide a thread-safe option. However, it is recommended to write a C++ wrapper which allocates its own scratch buffer. Note that the scratch buffer has the same memory alignment requirements of the input and output vectors.

## Output layout

PFFFT computes the forward transform with two possible output layouts:

1. ordered
2. unordered

### Ordered layout

Calling pffft_transform_ordered produces an array of interleaved real and imaginary parts. The last Fourier coefficient is purely real and stored in the imaginary part of the DC component (which is also purely real).

### Unordered layout

Calling pffft_transform produces an array with a more complex structure, but in a more efficient way than pffft_transform_ordered. Below, the output produced by Matlab and that produced by PFFFT are compared. The comparison is made for a 32 point transform of a 16 sample buffer. A 32 point transform has been chosen as this is the minimum supported by PFFFT.

Important notes:

• In Matlab the DC (Matlab index 1 [R1, I1]]) and Nyquist (Matlab index 17 [R17, I17]) values are not repeated as complex conjugates.
• In PFFFT the Nyquist real and imaginary parts ([R17, I17]) are omitted entirely.
• In PFFFT the final 8 values (4 real and 4 imaginary) are not in the same order as all of the others.
• In PFFFT all imaginary parts are stored as negatives (like second half in Matlab).
+-------+-----------+-------+-------+
| Index |  Matlab   | Index | PFFFT |
+-------+-----------+-------+-------+
|     1 | R1 + I1   |     0 | R1    |
|     2 | R2+ I2    |     1 | R2    |
|     3 | R3 + I3   |     2 | R3    |
|     4 | R4 + I4   |     3 | R4    |
|     5 | R5 + I5   |     4 | -I1   |
|     6 | R6 + I6   |     5 | -I2   |
|     7 | R7 + I7   |     6 | -I3   |
|     8 | R8 + I8   |     7 | -I4   |
|     9 | R9 + I9   |     8 | R5    |
|    10 | R10 + I10 |     9 | R6    |
|    11 | R11 + I11 |    10 | R7    |
|    12 | R12 + I12 |    11 | R8    |
|    13 | R13 + I13 |    12 | -I5   |
|    14 | R14 + I14 |    13 | -I6   |
|    15 | R15 + I15 |    14 | -I7   |
|    16 | R16 + I16 |    15 | -I8   |
|    17 | R17 + I17 |    16 | R9    |
|    18 | R16 - I16 |    17 | R10   |
|    19 | R15 - I15 |    18 | R11   |
|    20 | R14 - I14 |    19 | R12   |
|    21 | R13 - I13 |    20 | -I9   |
|    22 | R12 - I12 |    21 | -I10  |
|    23 | R11 - I11 |    22 | -I11  |
|    24 | R10 - I10 |    23 | -I12  |
|    25 | R9 - I9   |    24 | R13   |
|    26 | R8 - I8   |    25 | R16   |
|    27 | R7 - I7   |    26 | R15   |
|    28 | R6 - I6   |    27 | R14   |
|    29 | R5 - I5   |    28 | -I13  |
|    30 | R4 - I4   |    29 | -I16  |
|    31 | R3 - I3   |    30 | -I15  |
|    32 | R2 - I2   |    31 | -I14  |
+-------+-----------+-------+-------+