1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247
|
SUBROUTINE DG01ND( INDI, N, XR, XI, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To compute the discrete Fourier transform, or inverse Fourier
C transform, of a real signal.
C
C ARGUMENTS
C
C Mode Parameters
C
C INDI CHARACTER*1
C Indicates whether a Fourier transform or inverse Fourier
C transform is to be performed as follows:
C = 'D': (Direct) Fourier transform;
C = 'I': Inverse Fourier transform.
C
C Input/Output Parameters
C
C N (input) INTEGER
C Half the number of real samples. N must be a power of 2.
C N >= 2.
C
C XR (input/output) DOUBLE PRECISION array, dimension (N+1)
C On entry with INDI = 'D', the first N elements of this
C array must contain the odd part of the input signal; for
C example, XR(I) = A(2*I-1) for I = 1,2,...,N.
C On entry with INDI = 'I', the first N+1 elements of this
C array must contain the the real part of the input discrete
C Fourier transform (computed, for instance, by a previous
C call of the routine).
C On exit with INDI = 'D', the first N+1 elements of this
C array contain the real part of the output signal, that is
C of the computed discrete Fourier transform.
C On exit with INDI = 'I', the first N elements of this
C array contain the odd part of the output signal, that is
C of the computed inverse discrete Fourier transform.
C
C XI (input/output) DOUBLE PRECISION array, dimension (N+1)
C On entry with INDI = 'D', the first N elements of this
C array must contain the even part of the input signal; for
C example, XI(I) = A(2*I) for I = 1,2,...,N.
C On entry with INDI = 'I', the first N+1 elements of this
C array must contain the the imaginary part of the input
C discrete Fourier transform (computed, for instance, by a
C previous call of the routine).
C On exit with INDI = 'D', the first N+1 elements of this
C array contain the imaginary part of the output signal,
C that is of the computed discrete Fourier transform.
C On exit with INDI = 'I', the first N elements of this
C array contain the even part of the output signal, that is
C of the computed inverse discrete Fourier transform.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C Let A(1),....,A(2*N) be a real signal of 2*N samples. Then the
C first N+1 samples of the discrete Fourier transform of this signal
C are given by the formula:
C
C 2*N ((m-1)*(i-1))
C FA(m) = SUM ( A(i) * W ),
C i=1
C 2
C where m = 1,2,...,N+1, W = exp(-pi*j/N) and j = -1.
C
C This transform can be computed as follows. First, transform A(i),
C i = 1,2,...,2*N, into the complex signal Z(i) = (X(i),Y(i)),
C i = 1,2,...,N. That is, X(i) = A(2*i-1) and Y(i) = A(2*i). Next,
C perform a discrete Fourier transform on Z(i) by calling SLICOT
C Library routine DG01MD. This gives a new complex signal FZ(k),
C such that
C
C N ((k-1)*(i-1))
C FZ(k) = SUM ( Z(i) * V ),
C i=1
C
C where k = 1,2,...,N, V = exp(-2*pi*j/N). Using the values of
C FZ(k), the components of the discrete Fourier transform FA can be
C computed by simple linear relations, implemented in the DG01NY
C subroutine.
C
C Finally, let
C
C XR(k) = Re(FZ(k)), XI(k) = Im(FZ(k)), k = 1,2,...,N,
C
C be the contents of the arrays XR and XI on entry to DG01NY with
C INDI = 'D', then on exit XR and XI contain the real and imaginary
C parts of the Fourier transform of the original real signal A.
C That is,
C
C XR(m) = Re(FA(m)), XI(m) = Im(FA(m)),
C
C where m = 1,2,...,N+1.
C
C If INDI = 'I', then the routine evaluates the inverse Fourier
C transform of a complex signal which may itself be the discrete
C Fourier transform of a real signal.
C
C Let FA(m), m = 1,2,...,2*N, denote the full discrete Fourier
C transform of a real signal A(i), i=1,2,...,2*N. The relationship
C between FA and A is given by the formula:
C
C 2*N ((m-1)*(i-1))
C A(i) = SUM ( FA(m) * W ),
C m=1
C
C where W = exp(pi*j/N).
C
C Let
C
C XR(m) = Re(FA(m)) and XI(m) = Im(FA(m)) for m = 1,2,...,N+1,
C
C be the contents of the arrays XR and XI on entry to the routine
C DG01NY with INDI = 'I', then on exit the first N samples of the
C complex signal FZ are returned in XR and XI such that
C
C XR(k) = Re(FZ(k)), XI(k) = Im(FZ(k)) and k = 1,2,...,N.
C
C Next, an inverse Fourier transform is performed on FZ (e.g. by
C calling SLICOT Library routine DG01MD), to give the complex signal
C Z, whose i-th component is given by the formula:
C
C N ((k-1)*(i-1))
C Z(i) = SUM ( FZ(k) * V ),
C k=1
C
C where i = 1,2,...,N and V = exp(2*pi*j/N).
C
C Finally, the 2*N samples of the real signal A can then be obtained
C directly from Z. That is,
C
C A(2*i-1) = Re(Z(i)) and A(2*i) = Im(Z(i)), for i = 1,2,...N.
C
C Note that a discrete Fourier transform, followed by an inverse
C transform will result in a signal which is a factor 2*N larger
C than the original input signal.
C
C REFERENCES
C
C [1] Rabiner, L.R. and Rader, C.M.
C Digital Signal Processing.
C IEEE Press, 1972.
C
C NUMERICAL ASPECTS
C
C The algorithm requires 0( N*log(N) ) operations.
C
C CONTRIBUTORS
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1997.
C Supersedes Release 2.0 routine DG01BD by R. Dekeyser, and
C F. Dumortier, State University of Gent, Belgium.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Complex signals, digital signal processing, fast Fourier
C transform, real signals.
C
C ******************************************************************
C
C .. Scalar Arguments ..
CHARACTER INDI
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION XI(*), XR(*)
C .. Local Scalars ..
INTEGER J
LOGICAL LINDI
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DG01MD, DG01NY, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MOD
C .. Executable Statements ..
C
INFO = 0
LINDI = LSAME( INDI, 'D' )
C
C Test the input scalar arguments.
C
IF( .NOT.LINDI .AND. .NOT.LSAME( INDI, 'I' ) ) THEN
INFO = -1
ELSE
J = 0
IF( N.GE.2 ) THEN
J = N
C WHILE ( MOD( J, 2 ).EQ.0 ) DO
10 CONTINUE
IF ( MOD( J, 2 ).EQ.0 ) THEN
J = J/2
GO TO 10
END IF
C END WHILE 10
END IF
IF ( J.NE.1 ) INFO = -2
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'DG01ND', -INFO )
RETURN
END IF
C
C Compute the Fourier transform of Z = (XR,XI).
C
IF ( .NOT.LINDI ) CALL DG01NY( INDI, N, XR, XI )
C
CALL DG01MD( INDI, N, XR, XI, INFO )
C
IF ( LINDI ) CALL DG01NY( INDI, N, XR, XI )
C
RETURN
C *** Last line of DG01ND ***
END
|