convolve                package:stats                R Documentation

Fast Convolution

Description:

     Use the Fast Fourier Transform to compute the several kinds of
     convolutions of two sequences.

Usage:

     convolve(x, y, conj = TRUE, type = c("circular", "open", "filter"))

Arguments:

     x,y: numeric sequences _of the same length_ to be convolved.

    conj: logical; if 'TRUE', take the complex _conjugate_ before
          back-transforming (default, and used for usual convolution).

    type: character; one of '"circular"', '"open"', '"filter"'
          (beginning of word is ok).  For 'circular', the two sequences
          are treated as _circular_, i.e., periodic.

          For 'open' and 'filter', the sequences are padded with '0's
          (from left and right) first; '"filter"' returns the middle
          sub-vector of '"open"', namely, the result of running a
          weighted mean of 'x' with weights 'y'.

Details:

     The Fast Fourier Transform, 'fft', is used for efficiency.

     The input sequences 'x' and  'y' must have the same length if
     'circular' is true.

     Note that the usual definition of convolution of two sequences 'x'
     and 'y' is given by 'convolve(x, rev(y), type = "o")'.

Value:

     If 'r <- convolve(x,y, type = "open")' and 'n <- length(x)', 'm <-
     length(y)', then

                    r[k] = sum(i; x[k-m+i] * y[i])

     where the sum is over all valid indices i, for k = 1,..., n+m-1

     If 'type == "circular"', n = m is required, and the above is true
     for i , k = 1,...,n when x[j] := x[n+j] for j < 1.

References:

     Brillinger, D. R. (1981) _Time Series: Data Analysis and Theory_,
     Second Edition. San Francisco: Holden-Day.

See Also:

     'fft', 'nextn', and particularly 'filter' (from the 'stats'
     package) which may be more appropriate.

Examples:

     require(graphics)

     x <- c(0,0,0,100,0,0,0)
     y <- c(0,0,1, 2 ,1,0,0)/4
     zapsmall(convolve(x,y))         #  *NOT* what you first thought.
     zapsmall(convolve(x, y[3:5], type="f")) # rather
     x <- rnorm(50)
     y <- rnorm(50)
     # Circular convolution *has* this symmetry:
     all.equal(convolve(x,y, conj = FALSE), rev(convolve(rev(y),x)))

     n <- length(x <- -20:24)
     y <- (x-10)^2/1000 + rnorm(x)/8

     Han <- function(y) # Hanning
            convolve(y, c(1,2,1)/4, type = "filter")

     plot(x,y, main="Using  convolve(.) for Hanning filters")
     lines(x[-c(1  , n)      ], Han(y), col="red")
     lines(x[-c(1:2, (n-1):n)], Han(Han(y)), lwd=2, col="dark blue")

