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  "title" : "Out-of-Place Functions for 2D Real FFT"
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# Out-of-Place Functions for 2D Real FFT

Perform fast Fourier transforms out of place on 2D real data.

## Discussion

The functions in this group use the following operation for a forward real-to-complex transform:

```c
N0 = 1 << Log2N0;
N1 = 1 << Log2N1;

if (IA1 == 0) IA1 = IA0*N0/2;
if (IC1 == 0) IC1 = IC0*N0/2;

scale = 2;

// Define a real matrix, h:
for (j1 = 0; j1 < N1  ; ++j1)
for (j0 = 0; j0 < N0/2; ++j0)
{
    h[j1][2*j0+0] = A->realp[j1*IA1 + j0*IA0]
              + i * A->imagp[j1*IA1 + j0*IA0];
    h[j1][2*j0+1] = A->realp[j1*IA1 + j0*IA0]
              + i * A->imagp[j1*IA1 + j0*IA0];
}

// Perform Discrete Fourier Transform.
for (k1 = 0; k1 < N1; ++k1)
for (k0 = 0; k0 < N0; ++k0)
    H[k1][k0] = scale * sum(sum(h[j1][j0]
        * e**(-Direction*2*pi*i*j0*k0/N0), 0 <= j0 < N0)
        * e**(-Direction*2*pi*i*j1*k1/N1), 0 <= j1 < N1);

// Pack special pure-real elements into output matrix:
C->realp[0*IC1][0*IC0] = H[0   ][0   ].
C->imagp[0*IC1][0*IC0] = H[0   ][N0/2]
C->realp[1*IC1][0*IC0] = H[N1/2][0   ].
C->imagp[1*IC1][0*IC0] = H[N1/2][N0/2]

// Pack two vectors into output matrix "vertically":
// (This awkward format is due to a legacy implementation.)
for (k1 = 1; k1 < N1/2; ++k1)
{
    C->realp[(2*k1+0)*IC1][0*IC0] = Re(H[k1][0   ]);
    C->realp[(2*k1+1)*IC1][0*IC0] = Im(H[k1][0   ]);
    C->imagp[(2*k1+0)*IC1][0*IC0] = Re(H[k1][N0/2]);
    C->imagp[(2*k1+1)*IC1][0*IC0] = Im(H[k1][N0/2]);
}

// Store regular elements:
for (k1 = 0; k1 < N1  ; ++k1)
for (k0 = 1; k0 < N0/2; ++k0)
{
    C->realp[k1*IC1 + k0*IC0] = Re(H[k1][k0]);
    C->imagp[k1*IC1 + k0*IC0] = Im(H[k1][k0]);
}
```

The functions in this group use the following operation for an inverse complex-to-real transform:

```c
N0 = 1 << Log2N0;
N1 = 1 << Log2N1;

if (IA1 == 0) IA1 = IA0*N0/2;
if (IC1 == 0) IC1 = IC0*N0/2;

scale = 1. / (N1*N0);

// Define a complex matrix, h, in multiple steps:

// Unpack the special elements:
h[0   ][0   ] = A->realp[0*IA1][0*IA0];
h[0   ][N0/2] = A->imagp[0*IA1][0*IA0];
h[N1/2][0   ] = A->realp[1*IA1][0*IA0];
h[N1/2][N0/2] = A->imagp[1*IA1][0*IA0];

// Unpack the two vectors from "vertical" storage:
for (j1 = 1; j1 < N1/2; ++j1)
{
    h[j1][0   ] = A->realp[(2*j1+0)*IA1][0*IA0]
            + i * A->realp[(2*j1+1)*IA1][0*IA0]
    h[j1][N0/2] = A->imagp[(2*j1+0)*IA1][0*IA0]
            + i * A->imagp[(2*j1+1)*IA1][0*IA0]
}

// Take regular elements:
for (j1 = 0; j1 < N1  ; ++j1)
for (j0 = 1; j0 < N0/2; ++j0)
{
    h[j1][j0   ] = A->realp[j1*IA1 + j0*IA0]
             + i * A->imagp[j1*IA1 + j0*IA0];
    h[j1][N0-j0] = conj(h[j1][j0]);
}

// Perform Discrete Fourier Transform.
for (k1 = 0; k1 < N1; ++k1)
for (k0 = 0; k0 < N0; ++k0)
    H[k1][k0] = scale * sum(sum(h[j1][j0]
        * e**(-Direction*2*pi*i*j0*k0/N0), 0 <= j0 < N0)
        * e**(-Direction*2*pi*i*j1*k1/N1), 0 <= j1 < N1);

// Store result.
for (k1 = 0; k1 < N1  ; ++k1)
for (k0 = 0; k0 < N0/2; ++k0)
{
    C->realp[k1*IC1 + k0*IC0] = Re(H[k1][2*k0+0]);
    C->imagp[k1*IC1 + k0*IC0] = Im(H[k1][2*k0+1]);
}
```

The temporary buffer versions perform the same operation but use a temporary buffer for improved performance.

## Topics

### Out-of-Place FFT Functions

[`extern void vDSP_fft2d_zrop(FFTSetup __Setup, const DSPSplitComplex *__A, vDSP_Stride __IA0, vDSP_Stride __IA1, const DSPSplitComplex *__C, vDSP_Stride __IC0, vDSP_Stride __IC1, vDSP_Length __Log2N0, vDSP_Length __Log2N1, FFTDirection __Direction);`](/documentation/Accelerate/vDSP_fft2d_zrop)

Computes a 2D forward or inverse out-of-place, single-precision real FFT.

[`extern void vDSP_fft2d_zropD(FFTSetupD __Setup, const DSPDoubleSplitComplex *__A, vDSP_Stride __IA0, vDSP_Stride __IA1, const DSPDoubleSplitComplex *__C, vDSP_Stride __IC0, vDSP_Stride __IC1, vDSP_Length __Log2N0, vDSP_Length __Log2N1, FFTDirection __Direction);`](/documentation/Accelerate/vDSP_fft2d_zropD)

Computes a 2D forward or inverse out-of-place, double-precision real FFT.

### Out-of-Place FFT Functions with Temporary Buffer

[`extern void vDSP_fft2d_zropt(FFTSetup __Setup, const DSPSplitComplex *__A, vDSP_Stride __IA0, vDSP_Stride __IA1, const DSPSplitComplex *__C, vDSP_Stride __IC0, vDSP_Stride __IC1, const DSPSplitComplex *__Buffer, vDSP_Length __Log2N0, vDSP_Length __Log2N1, FFTDirection __Direction);`](/documentation/Accelerate/vDSP_fft2d_zropt)

Computes a 2D forward or inverse out-of-place, single-precision real FFT using a temporary buffer.

[`extern void vDSP_fft2d_zroptD(FFTSetupD __Setup, const DSPDoubleSplitComplex *__A, vDSP_Stride __IA0, vDSP_Stride __IA1, const DSPDoubleSplitComplex *__C, vDSP_Stride __IC0, vDSP_Stride __IC1, const DSPDoubleSplitComplex *__Buffer, vDSP_Length __Log2N0, vDSP_Length __Log2N1, FFTDirection __Direction);`](/documentation/Accelerate/vDSP_fft2d_zroptD)

Computes a 2D forward or inverse out-of-place, double-precision real FFT using a temporary buffer.


---

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