NAG Library Function Document
nag_sum_fft_complex_1d (c06pcc) calculates the discrete Fourier transform of a sequence of complex data values (using complex data type).
||nag_sum_fft_complex_1d (Nag_TransformDirection direct,
Given a sequence of
complex data values
, nag_sum_fft_complex_1d (c06pcc) calculates their (forward
) discrete Fourier transform (DFT) defined by
(Note the scale factor of
in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of nag_sum_fft_complex_1d (c06pcc) with followed by a call with will restore the original data.
nag_sum_fft_complex_1d (c06pcc) uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)
) known as the Stockham self-sorting algorithm, which is described in Temperton (1983)
is a large prime number or if
contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see Bluestein (1968)
), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.
Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23
direct – Nag_TransformDirectionInput
: if the forward transform as defined in Section 3
is to be computed, then direct
must be set equal to
If the backward transform is to be computed then direct
must be set equal to
x[n] – ComplexInput/Output
must contain , for .
On exit: the components of the discrete Fourier transform.
is contained in , for .
n – IntegerInput
On entry: , the number of data values.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
is an invalid value of direct
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
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).
8 Parallelism and Performance
nag_sum_fft_complex_1d (c06pcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_1d (c06pcc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note
for your implementation for any additional implementation-specific information.
The time taken is approximately proportional to , but also depends on the factorization of . nag_sum_fft_complex_1d (c06pcc) is faster if the only prime factors of are , or ; and fastest of all if is a power of .
This function internally allocates a workspace of Complex values.
When the Bluestein’s FFT algorithm is in use, an additional Complex workspace of size approximately is allocated.
This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by nag_sum_fft_complex_1d (c06pcc) with ). It then performs an inverse transform using nag_sum_fft_complex_1d (c06pcc) with , and prints the sequence so obtained alongside the original data values.
10.1 Program Text
Program Text (c06pcce.c)
10.2 Program Data
Program Data (c06pcce.d)
10.3 Program Results
Program Results (c06pcce.r)