Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference

SRCFT and DRCFT--Real-to-Complex Fourier Transform

These subroutines compute a set of m complex discrete n-point Fourier transforms of real data.

Table 132. Data Types

Note:

Two invocations of this subroutine are necessary: one to prepare the working storage for the subroutine, and the other to perform the computations.

Syntax

On Entry

init

is a flag, where:

If init <> 0, trigonometric functions and other parameters, depending on arguments other than x, are computed and saved in aux1. The contents of x and y are not used or changed.

If init = 0, the discrete Fourier transforms of the given sequences are computed. The only arguments that may change after initialization are x, y, and aux2. All scalar arguments must be the same as when the subroutine was called for initialization with init <> 0.

Specified as: a fullword integer. It can have any value.

x

is the array X, consisting of m sequences of length n, which are to be transformed. The sequences are assumed to be stored with stride 1. Specified as: an array of (at least) length n+(m-1)inc2x, containing numbers of the data type indicated in Table 132. See Notes for more details. (It can be declared as X(inc2x,m).)

inc2x

is the stride between the first elements of the sequences in array X. (If m = 1, this argument is ignored.) Specified as: a fullword integer; inc2x >= n.

y

See On Return.

inc2y

is the stride between the first elements of the sequences in array Y. (If m = 1, this argument is ignored.) Specified as: a fullword integer; inc2y >= (n/2)+1.

n

is the length of each sequence to be transformed. Specified as: a fullword integer; n <= 37748736 and must be one of the values listed in Acceptable Lengths for the Transforms. For all other values specified less than 37748736, you have the option of having the next larger acceptable value returned in this argument. For details, see Providing a Correct Transform Length to ESSL.

m

is the number of sequences to be transformed. Specified as: a fullword integer; m > 0.

isign

controls the direction of the transform, determining the sign Isign of the exponent of W_n, where:

If isign = positive value, Isign = + (transforming time to frequency).

If isign = negative value, Isign = - (transforming frequency to time).

Specified as: a fullword integer; isign > 0 or isign < 0.

scale

is the scaling constant scale. See Function for its usage. Specified as: a number of the data type indicated in Table 132, where scale > 0.0 or scale < 0.0.

aux1

is the working storage for this subroutine, where:

If init <> 0, the working storage is computed.

If init = 0, the working storage is used in the computation of the Fourier transforms.

Specified as: an area of storage, containing naux1 long-precision real numbers.

naux1

is the number of doublewords in the working storage specified in aux1. Specified as: a fullword integer; naux1 > 14 and naux1 >= (minimum value required for successful processing). To determine a sufficient value, use the processor-independent formulas. For values between 14 and the minimum value, you have the option of having the minimum value returned in this argument. For details, see Using Auxiliary Storage in ESSL.

aux2

has the following meaning:

If naux2 = 0 and error 2015 is unrecoverable, aux2 is ignored.

Otherwise, it is the working storage used by this subroutine, which is available for use by the calling program between calls to this subroutine.

Specified as: an area of storage, containing naux2 long-precision real numbers. On output, the contents are overwritten.

naux2

is the number of doublewords in the working storage specified in aux2. Specified as: a fullword integer, where:

If naux2 = 0 and error 2015 is unrecoverable, SRCFT and DRCFT dynamically allocate the work area used by the subroutine. The work area is deallocated before control is returned to the calling program.

Otherwise, naux2 >= (minimum value required for successful processing). To determine a sufficient value, use the processor-independent formulas. For all other values specified less than the minimum value, you have the option of having the minimum value returned in this argument. For details, see Using Auxiliary Storage in ESSL.

aux3

this argument is provided for migration purposes only and is ignored.

Specified as: an area of storage, containing naux3 long-precision real numbers.

naux3

this argument is provided for migration purposes only and is ignored.

Specified as: a fullword integer.

On Return

y

has the following meaning, where:

If init <> 0, this argument is not used, and its contents remain unchanged.

If init = 0, this is array Y, consisting of the results of the m complex discrete Fourier transforms, each of length n. The sequences are stored with the stride 1. Due to complex conjugate symmetry, only the first (n/2) + 1 elements of each sequence are given in the output--that is, y_ki, k = 0, 1, ..., n/2, i = 1, 2, ..., m.

Returned as: an array of (at least) length n/2+1+(m-1)inc2y, containing numbers of the data type indicated in Table 132. This array must be aligned on a doubleword boundary. (It can be declared as Y(inc2y,m).)

aux1

is the working storage for this subroutine, where:

If init <> 0, it contains information ready to be passed in a subsequent invocation of this subroutine.

If init = 0, its contents are unchanged.

Returned as: the contents are not relevant.

Notes

1. aux1 should not be used by the calling program between calls to this subroutine with init <> 0 and init = 0. However, it can be reused after intervening calls to this subroutine with different arguments.
2. When using the ESSL SMP library, for optimal performance, the number of threads specified should be the same for init <> 0 and init = 0.
3. In these subroutines, the elements in each sequence in x and y are assumed to be stored in contiguous storage locations, using a stride of 1; therefore, inc1x and inc1y values are not a part of the argument list. For optimal performance, the inc2x and inc2y values should be close to their respective minimum values, which are given below:

   min(inc2x) = n

   min(inc2y) = n/2+1

   If you specify the same array for X and Y, then inc2x must equal 2(inc2y). In this case, output overwrites input. If m = 1, the inc2x and inc2y values are not used by the subroutine. If you specify different arrays for X and Y, they must have no common elements; otherwise, results are unpredictable. See Concepts.

4. Be sure to align array X on a doubleword boundary, and specify an even number for inc2x, if possible.

Processor-Independent Formulas for SRCFT for NAUX1 and NAUX2

NAUX1 Formulas

If n <= 16384, use naux1 = 25000.

If n > 16384, use naux1 = 20000+0.82n.

NAUX2 Formulas

If n <= 16384, use naux2 = 20000.

If n > 16384, use naux2 = 20000+0.57n.

Processor-Independent Formulas for DRCFT for NAUX1 and NAUX2

NAUX1 Formulas

If n <= 4096, use naux1 = 22000.

If n > 4096, use naux1 = 20000+1.64n.

NAUX2 Formulas

If n <= 4096, use naux2 = 20000.

If n > 4096, use naux2 = 20000+1.14n.

Function

The set of m complex conjugate even discrete n-point Fourier transforms of real data in array X, with results going into array Y, is expressed as follows:

for:

k = 0, 1, ..., n-1

i = 1, 2, ..., m

where:

and where:

x_ji are elements of the sequences in array X.

y_ki are elements of the sequences in array Y.

Isign is + or - (determined by argument isign).

scale is a scalar value.

The output in array Y is complex. For scale = 1.0 and isign being positive, you obtain the discrete Fourier transform, a function of frequency. The inverse Fourier transform is obtained with scale = 1.0/n and isign being negative. See references [1], [4], [19], and [20].

Two invocations of this subroutine are necessary:

1. With init <> 0, the subroutine tests and initializes arguments of the program, setting up the aux1 working storage.
2. With init = 0, the subroutine checks that the initialization arguments in the aux1 working storage correspond to the present arguments, and if so, performs the calculation of the Fourier transforms.

Error Conditions

Resource Errors

Error 2015 is unrecoverable, naux2 = 0, and unable to allocate work area.

Computational Errors

None

Input-Argument Errors

1. n > 37748736
2. m <= 0
3. inc2x < n
4. inc2y < n/2+1
5. isign = 0
6. scale = 0.0
7. The subroutine has not been initialized with the present arguments.
8. The length of the transform in n is not an allowable value. Return code 1 is returned if error 2030 is recoverable.
9. naux1 <= 14
10. naux1 is too small--that is, less than the minimum required value. Return code 1 is returned if error 2015 is recoverable.
11. Error 2015 is recoverable or naux2<>0, and naux2 is too small--that is, less than the minimum required value. Return code 1 is returned if error 2015 is recoverable.

Example 1

This example shows an input array X with a set of m cosine sequences cos(2pijk/n), j = 0, 1, ..., 15 with the single frequencies k = 0, 1, 2, 3. The Fourier transform of the cosine sequence with frequency k = 0 or n/2 has 1.0 in the 0 or n/2 position, respectively, and zeros elsewhere. For all other k, the Fourier transform has 0.5 in the k position and zeros elsewhere. The arrays are declared as follows:

     REAL*4     X(0:65535)
     COMPLEX*8  Y(0:32768)
     REAL*8     AUX1(41928), AUX2(35344), AUX3(1)

First, initialize AUX1 using the calling sequence shown below with INIT <> 0. Then use the same calling sequence with INIT = 0 to do the calculation.

Call Statement and Input

           INIT  X  INC2X Y INC2Y  N   M ISIGN SCALE  AUX1   NAUX1   AUX2   NAUX2   AUX3  NAUX3
            |    |    |   |   |    |   |   |     |     |       |      |       |      |      |
CALL SRCFT(INIT, X , 16 , Y , 9 , 16 , 4 , 1 , SCALE, AUX1 , 41928 , AUX2 , 35344 , AUX3 ,  0 )

X contains the following four sequences:

1.0000   1.0000   1.0000   1.0000
1.0000   0.9239   0.7071   0.3827
1.0000   0.7071   0.0000  -0.7071
1.0000   0.3827  -0.7071  -0.9239
1.0000   0.0000  -1.0000   0.0000
1.0000  -0.3827  -0.7071   0.9239
1.0000  -0.7071   0.0000   0.7071
1.0000  -0.9239   0.7071  -0.3827
1.0000  -1.0000   1.0000  -1.0000
1.0000  -0.9239   0.7071  -0.3827
1.0000  -0.7071   0.0000   0.7071
1.0000  -0.3827  -0.7071   0.9239
1.0000   0.0000  -1.0000   0.0000
1.0000   0.3827  -0.7071  -0.9239
1.0000   0.7071   0.0000  -0.7071
1.0000   0.9239   0.7071   0.3827

Output

Y contains the following four sequences:

(1.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.5000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.5000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.5000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)

Example 2

This example shows another transform computation with different data using the same initialized array AUX1 as in Example 1. The input is also a set of four cosine sequences cos(2pijk/n), j = 0, 1, ..., 15 with the single frequencies k = 8, 9, 10, 11, thus including the middle frequency k = 8. The middle frequency has the value 1.0. For other frequencies, the transform has zeros, except for frequencies k and n-k. Only the values for j = n-k are given in the output.

Call Statement and Input

           INIT X  INC2X Y INC2Y  N   M ISIGN SCALE  AUX1   NAUX1   AUX2   NAUX2   AUX3  NAUX3
            |   |    |   |   |    |   |   |     |     |       |      |       |      |
CALL SRCFT( 0 , X , 16 , Y , 9 , 16 , 4 , 1 , SCALE, AUX1 , 41928 , AUX2 , 35344 , AUX3 ,  0  )

SCALE    =  1.0/16

X contains the following four sequences:

 1.0000   1.0000   1.0000   1.0000
-1.0000  -0.9239  -0.7071  -0.3827
 1.0000   0.7071   0.0000  -0.7071
-1.0000  -0.3827   0.7071   0.9239
 1.0000   0.0000  -1.0000   0.0000
-1.0000   0.3827   0.7071  -0.9239
 1.0000  -0.7071   0.0000   0.7071
-1.0000   0.9239  -0.7071   0.3827
 1.0000  -1.0000   1.0000  -1.0000
-1.0000   0.9239  -0.7071   0.3827
 1.0000  -0.7071   0.0000   0.7071
-1.0000   0.3827   0.7071  -0.9239
 1.0000   0.0000  -1.0000   0.0000
-1.0000  -0.3827   0.7071   0.9239
 1.0000   0.7071   0.0000  -0.7071
-1.0000  -0.9239  -0.7071  -0.3827

Output

Y contains the following four sequences:

(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.5000, 0.0000)
(0.0000, 0.0000)  (0.0000, 0.0000)  (0.5000, 0.0000)  (0.0000, 0.0000)
(0.0000, 0.0000)  (0.5000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)
(1.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)  (0.0000, 0.0000)

Example 3

This example uses the mixed-radix capability. The arrays are declared as follows:

     REAL*8     X(0:11)
     COMPLEX*16 Y(0:6)
     REAL*8     AUX1(50),AUX2(50)

Arrays X and Y are made equivalent by the following statement, making them occupy the same storage:

     EQUIVALENCE (X,Y)

First, initialize AUX1 using the calling sequence shown below with INIT <> 0. Then use the same calling sequence with INIT = 0 to do the calculation.

Call Statement and Input

           INIT  X INC2X Y INC2Y  N   M ISIGN SCALE   AUX1  NAUX1 AUX2  NAUX2
            |    |   |   |   |    |   |   |     |      |      |    |      |
CALL DRCFT(INIT, X , 0 , Y , 0 , 12 , 1 , 1 , SCALE , AUX1 , 50 , AUX2 , 50)

Output

Y contains the following sequence:

(12.0000 , 0.0000)
 (0.0000 , 0.0000)
 (0.0000 , 0.0000)
 (0.0000 , 0.0000)
 (0.0000 , 0.0000)
 (0.0000 , 0.0000)
 (0.0000 , 0.0000)