Sun Performance Library Reference Manual: 6 - Fast Fourier Transform Subroutines (FFTPACK and VFFTPACK)

Cosine Quarter-Wave Transform

The subroutines described in this section compute the Fourier coefficients in a cosine series representation with only odd wave numbers. The transform is defined as:

For i=1,...,n

Note - The xCOSQ operations are unnormalized inverses of themselves, so a call to xCOSQF followed by a call to xCOSQB will multiply the input sequence by 4 × N. The VxCOSQ operations are normalized, so a call of VxCOSQF followed by a call of VxCOSQB will return the original sequence.

Calling Sequence


CALL COSQF	(N, RX, RWSAVE)
CALL DCOSQF	(N, DX, RWSAVE)
CALL VCOSQF	(M, N, RX, RXT, MDIMX, RWSAVE)
CALL VDCOSQF	(M, N, DX, DXT, MDIMX, DWSAVE)

void cosqf	(int n, float rx, float rwsave)
void dcosqf	(int n, double dx, double dwsave)
void vcosqf	(int m, int n, float rx, int mdimx, float rwsave)
void vdcosqf	(int m, int n, double dx, int mdimx, double dwsave)

Arguments

M

(For vector operations only.)
The number of sequences to be transformed. M 0.

N

Length of the sequence to be transformed. These subroutines are most efficient when N is a product of small primes. N 0.

xX

On entry, an array of length N containing the sequence to be transformed. For VxCOSQF, a real two-dimensional array with dimensions of (MDIMX × N) whose rows contain the sequences to be transformed. On exit, the quarter-wave cosine transform of the input.

xXT

(For vector operations only.)
A real two-dimensional work array with dimensions of (MDIMX × N).

MDIMX

(For vector operations only.)
Leading dimension of the arrays xX and xXT as specified in a dimension or type statement. MDIMX M.

xWSAVE

On entry, an array with dimension of at least (3 × N + 15) for scalar subroutines or (2 × N + 15) for vector subroutines, initialized by xCOSTI or VxCOSTI.

M	(For vector operations only.) The number of sequences to be transformed. M 0.
N	Length of the sequence to be transformed. These subroutines are most efficient when N is a product of small primes. N 0.
xX	On entry, an array of length N containing the sequence to be transformed. For VxCOSQF, a real two-dimensional array with dimensions of (MDIMX × N) whose rows contain the sequences to be transformed. On exit, the quarter-wave cosine transform of the input.
xXT	(For vector operations only.) A real two-dimensional work array with dimensions of (MDIMX × N).
MDIMX	(For vector operations only.) Leading dimension of the arrays xX and xXT as specified in a dimension or type statement. MDIMX M.
xWSAVE	On entry, an array with dimension of at least (3 × N + 15) for scalar subroutines or (2 × N + 15) for vector subroutines, initialized by xCOSTI or VxCOSTI.

Sample Program



PROGRAM TEST
IMPLICIT NONE
C
INTEGER M, N
PARAMETER (M = 4)
PARAMETER (N = 6)
C
INTEGER I, J
REAL PI, WSAVE(N + 15), X(M, N), XT(M, N)
C
EXTERNAL VCOSQB, VCOSQF, VCOSQI
INTRINSIC ACOS, COS
C
C Initialize the array to m real even quarter-wave sequences,
C that is, they can be expanded in terms of a cosine series
C with only odd wave numbers.
C
PI = ACOS (-1.0)
DO 110, J=1, M
DO 100, I=1, N
X(J,I) = 40.0 * J * COS ((I-1) * PI / 2.0 / N )
100 CONTINUE
110 CONTINUE
C
CALL VCOSQI (N, WSAVE)
PRINT 1000
DO 120, J=1, M
PRINT 1010, J, (X(J, I), I = 1, N)
120 CONTINUE
CALL VCOSQF (M, N, X, XT, M, WSAVE)
PRINT 1020
DO 130, J=1, M
PRINT 1010, J, (X(J, I), I = 1, N)
130 CONTINUE
CALL VCOSQB (M, N, X, XT, M, WSAVE)
PRINT 1030
DO 140, J=1, M
PRINT 1010, J, (X(J, I), I = 1, N)
140 CONTINUE
C
1000 FORMAT (1X, 'Original Sequence: ')
1010 FORMAT(1X, ' Sequence', I2, ': ', 100(F5.1, 1X))
1020 FORMAT (1X, 'Transformed Sequence: ')
1030 FORMAT (1X, 'Recovered Sequence: ')
C
END

Sample Output



Original Sequence:
Sequence 1: 40.0 38.6 34.6 28.3 20.0 10.4
Sequence 2: 80.0 77.3 69.3 56.6 40.0 20.7
Sequence 3: 120.0 115.9 103.9 84.9 60.0 31.1
Sequence 4: 160.0 154.5 138.6 113.1 80.0 41.4
Transformed Sequence:
Sequence 1: 49.0 0.0 0.0 0.0 0.0 0.0
Sequence 2: 98.0 0.0 0.0 0.0 0.0 0.0
Sequence 3: 147.0 0.0 0.0 0.0 0.0 0.0
Sequence 4: 196.0 0.0 0.0 0.0 0.0 0.0
Recovered Sequence:
Sequence 1: 40.0 38.6 34.6 28.3 20.0 10.4
Sequence 2: 80.0 77.3 69.3 56.6 40.0 20.7
Sequence 3: 120.0 115.9 103.9 84.9 60.0 31.1
Sequence 4: 160.0 154.5 138.6 113.1 80.0 41.4