NAG Library Function Document
nag_fft_hermitian (c06ebc) calculates the discrete Fourier transform of a Hermitian sequence of complex data values.
||nag_fft_hermitian (Integer n,
Given a Hermitian sequence of
complex data values
(i.e., a sequence such that
is real and
is the complex conjugate of
), nag_fft_hermitian (c06ebc) calculates their discrete Fourier transform defined by
(Note the scale factor of
in this definition.) The transformed values
are purely real.
To compute the inverse discrete Fourier transform defined by
this function should be preceded by a call of nag_conjugate_hermitian (c06gbc)
to form the complex conjugates of the
nag_fft_hermitian (c06ebc) uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)
). There are some restrictions on the value of
(see Section 5
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
n – IntegerInput
On entry: , 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.
x[n] – doubleInput/Output
: the sequence to be transformed stored in Hermitian form. If the data values
are written as
, then for
is contained in
, and for
is contained in
. It is not necessary for other elements of the sequence to be explicitly stored. (See also Section 9
On exit: the components of the discrete Fourier transform . is stored in , for .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
At least one of the prime factors of n
is greater than 19.
has more than 20 prime factors.
On entry, .
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 , but also depends on the factorization of . nag_fft_hermitian (c06ebc) is somewhat faster than average if the only prime factors of are 2, 3 or 5; and fastest of all if is a power of 2.
On the other hand, nag_fft_hermitian (c06ebc) is particularly slow if has several unpaired prime factors, i.e., if the ‘square-free’ part of has several factors.
This example reads in a sequence of real data values which is assumed to be a Hermitian sequence of complex data values stored in Hermitian form. The input sequence is expanded into a full complex sequence and printed alongside the original sequence. The discrete Fourier transform (as computed by nag_fft_hermitian (c06ebc)) is printed out. It then performs an inverse transform using nag_fft_real (c06eac)
and nag_conjugate_hermitian (c06gbc)
, and prints the sequence so obtained alongside the original data values.
9.1 Program Text
Program Text (c06ebce.c)
9.2 Program Data
Program Data (c06ebce.d)
9.3 Program Results
Program Results (c06ebce.r)