Computes the convolution of two real vectors.

Required Arguments

X — Vector of length NX.   (Input)

Y — Vector of length NY.   (Input)

Z — 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 VCONR (X, Y, Z [,…])

Specific:         The specific interface names are S_VCONR and D_VCONR.

FORTRAN 77 Interface

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

Double:           The double precision name is DVCONR.

Description

The routine VCONR computes the convolution z of two real vectors x and y. Let nx = NX, 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 via IMSL routines FFTCF and FFTCB (see Chapter 6, Transforms) is used to compute the complex vector w from . The vector z is then found by taking the real part of the vector w.

Comments

Workspace may be explicitly provided, if desired, by use of V2ONR/DV2ONR. The reference is

CALL V2ONR (NX, X, NY, Y, NZ, Z, XWK, YWK, ZWK, 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.

ZWK — 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 8 and a vector y of length 3 is computed. The resulting vector z is of length 8 + 3 - 1 = 10. (The vector y is sometimes called a filter.)

 

      USE VCONR_INT

      USE WRRRN_INT

 

      IMPLICIT   NONE

      INTEGER    NX, NY, NZ

      PARAMETER  (NX=8, NY=3, NZ=NX+NY-1)

!

      REAL       X(NX), Y(NY), Z(NZ)

!                                 Set values for X

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

!                                 Set values for Y

!                       Y = (0.0  0.0  1.0)

!

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

      DATA Y/0.0, 0.0, 1.0/

!                                 Compute vector convolution

!                                 Z = X * Y

      CALL VCONR (X,Y,Z)

!                                 Print results

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

      END

Output

 

                                  Z = X (*) Y
    1      2      3       4       5       6       7       8       9      10
0.000  0.000  1.000   2.000   3.000   4.000   5.000   6.000   7.000   8.000

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