VCONC

 

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Computes the convolution of two complex vectors.

Required Arguments

X — Complex vector of length NX. (Input)

Y — Complex vector of length NY. (Input)

Z — Complex vector of length NZ containing the convolution Z = X * Y. (Output)

Optional Arguments

NX — Length of the vector X. (Input)
Default: NX = SIZE (X,1).

NY — Length of the vector Y. (Input)
Default: NY = SIZE (Y,1).

NZ — Length of the vector Z. (Input)
NZ must be at least NX + NY 1.
Default: NZ = SIZE (Z,1).

FORTRAN 90 Interface

Generic: CALL VCONC (X, Y, Z [, …])

Specific: The specific interface names are S_VCONC and D_VCONC.

FORTRAN 77 Interface

Single: CALL VCONC (NX, X, NY, Y, NZ, Z)

Double: The double precision name is DVCONC.

Description

The routine VCONC computes the convolution z of two complex vectors x and y. Let nx = NX, then ny = NY and nz = NZ. The vector z is defined to be

 

where nz = nx + ny ‑  1. If the index j ‑  k + 1 is outside the range 1, 2, …, nx, then xj‑ k+1 is taken to be zero.

The fast Fourier transform is used to compute the convolution. Define the complex vector u of length nz = nx + ny ‑  1 to be

 

The complex vector v, also of length nz, is defined similarly using y. Then, by the Fourier convolution theorem,

 

where the indicates the Fourier transform of u computed using IMSL routine FFTCF. The complex vector z is computed from via IMSL routine FFTCB. See Chapter 6, “Transforms” for more information on these functions.

Comments

Workspace may be explicitly provided, if desired, by use of V2ONC/DV2ONC. The reference is

CALL V2ONC (NX, X, NY, Y, NZ, Z, XWK, YWK, WK)

The additional arguments are as follows:

XWK — Complex work array of length NX + NY 1.

YWK — Complex work array of length NX + NY 1.

WK — Real work array of length 6 * (NX + NY ‑ 1) + 15.

Example

In this example, the convolution of a vector x of length 4 and a vector y of length 3 is computed. The resulting vector z is of length 4 + 3 ‑ 1. (y is sometimes called a filter.)

 

USE VCONC_INT

USE WRCRN_INT

 

IMPLICIT NONE

INTEGER NX, NY, NZ

PARAMETER (NX=4, NY=3, NZ=NX+NY-1)

!

COMPLEX X(NX), Y(NY), Z(NZ)

! Set values for X

! X = ( 1.0+2.0i 3.0+4.0i 5.0+6.0i 7.0+8.0i )

! Set values for Y

! Y = (0.0+0i 0.0+0i 1.0+0i )

!

DATA X/(1.0,2.0), (3.0,4.0), (5.0,6.0), (7.0,8.0)/

DATA Y/(0.0,0.0), (0.0,0.0), (1.0,1.0)/

! Compute vector convolution

! Z = X * Y

CALL VCONC (X,Y,Z)

! Print results

CALL WRCRN ('Z = X (*) Y', Z, 1, NZ, 1)

END

Output

 

Z = X (*) Y

1 2 3 4

( 0.00, 0.00) ( 0.00, 0.00) ( -1.00, 3.00) ( -1.00, 7.00)

 

5 6

( -1.00, 11.00) ( -1.00, 15.00)