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# NAG Toolbox: nag_sum_fft_complex_1d_sep (c06fc)

## Purpose

nag_sum_fft_complex_1d_sep (c06fc) calculates the discrete Fourier transform of a sequence of $n$ complex data values (using a work array for extra speed).

## Syntax

[x, y, ifail] = c06fc(x, y, 'n', n)
[x, y, ifail] = nag_sum_fft_complex_1d_sep(x, y, 'n', n)

## Description

Given a sequence of $n$ complex data values ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, nag_sum_fft_complex_1d_sep (c06fc) calculates their discrete Fourier transform defined by
 $z^k = ak + i bk = 1n ∑ j=0 n-1 zj × exp -i 2πjk n , k= 0, 1, …, n-1 .$
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
To compute the inverse discrete Fourier transform defined by
 $w^k = 1n ∑ j=0 n-1 zj × exp +i 2πjk n ,$
this function should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in $y$).
nag_sum_fft_complex_1d_sep (c06fc) uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)). There are some restrictions on the value of $n$ (see Arguments).

## References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
If x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the function from which nag_sum_fft_complex_1d_sep (c06fc) is called, then ${\mathbf{x}}\left(\mathit{j}\right)$ must contain ${x}_{\mathit{j}}$, the real part of ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
If y is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the function from which nag_sum_fft_complex_1d_sep (c06fc) is called, then ${\mathbf{y}}\left(\mathit{j}\right)$ must contain ${y}_{\mathit{j}}$, the imaginary part of ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
$n$, the number of data values. The largest prime factor of n must not exceed $19$, and the total number of prime factors of n, counting repetitions, must not exceed $20$.
Constraint: ${\mathbf{n}}>1$.

### Output Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The real parts ${a}_{k}$ of the components of the discrete Fourier transform. If x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the function from which nag_sum_fft_complex_1d_sep (c06fc) is called, then for $0\le k\le n-1$, ${a}_{k}$ is contained in ${\mathbf{x}}\left(k\right)$.
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The imaginary parts ${b}_{k}$ of the components of the discrete Fourier transform. If y is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the function from which nag_sum_fft_complex_1d_sep (c06fc) is called, then for $0\le k\le n-1$, ${b}_{k}$ is contained in ${\mathbf{y}}\left(k\right)$.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
At least one of the prime factors of n is greater than $19$.
${\mathbf{ifail}}=2$
n has more than $20$ prime factors.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{n}}\le 1$.
${\mathbf{ifail}}=4$
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

The time taken is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. nag_sum_fft_complex_1d_sep (c06fc) is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.

## Example

This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by nag_sum_fft_complex_1d_sep (c06fc)). It then performs an inverse transform using nag_sum_fft_complex_1d_sep (c06fc), and prints the sequence so obtained alongside the original data values.
```function c06fc_example

fprintf('c06fc example results\n\n');

z = [0.34907 - 0.37168*i;
0.54890 - 0.35669*i;
0.74776 - 0.31175*i;
0.94459 - 0.23702*i;
1.13850 - 0.13274*i;
1.32850 + 0.00074*i;
1.51370 + 0.16298*i];
x = real(z);
y = imag(z);

[ztr, zti, ifail] = c06fc(x, y);
ztrans = ztr + i*zti;
disp('Components of discrete Fourier transform')
disp(ztrans);

[xres, yres, ifail] = c06fc(ztr, -zti);
zres = xres-i*yres;
zout = [z  zres];

fprintf('Original sequence as restored by inverse transform\n');
fprintf('      Original            Restored\n')
disp(zout);

```
```c06fc example results

Components of discrete Fourier transform
2.4836 - 0.4710i
-0.5518 + 0.4968i
-0.3671 + 0.0976i
-0.2877 - 0.0586i
-0.2251 - 0.1748i
-0.1483 - 0.3084i
0.0198 - 0.5650i

Original sequence as restored by inverse transform
Original            Restored
0.3491 - 0.3717i   0.3491 - 0.3717i
0.5489 - 0.3567i   0.5489 - 0.3567i
0.7478 - 0.3118i   0.7478 - 0.3117i
0.9446 - 0.2370i   0.9446 - 0.2370i
1.1385 - 0.1327i   1.1385 - 0.1327i
1.3285 + 0.0007i   1.3285 + 0.0007i
1.5137 + 0.1630i   1.5137 + 0.1630i

```

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