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 (see Chapter 6, Transforms;). The complex vector z is computed from  via IMSL routine FFTCB (see Chapter 6, Transforms;).

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 arrary 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 -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)

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