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r |
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\name{convolve}
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hornik |
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\alias{convolve}
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r |
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\title{Fast Convolution}
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hornik |
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\description{
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Use the Fast Fourier Transform to compute the several kinds of
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convolutions of two sequences.
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}
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r |
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\usage{
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pd |
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convolve(x, y, conj = TRUE, type = c("circular", "open", "filter"))
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r |
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}
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\arguments{
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hornik |
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\item{x,y}{numeric sequences \emph{of the same length} to be
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convolved.}
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\item{conj}{logical; if \code{TRUE}, take the complex \emph{conjugate}
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before back-transforming (default, and used for usual convolution).}
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\item{type}{character; one of \code{"circular"}, \code{"open"},
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\code{"filter"} (beginning of word is ok). For \code{circular}, the
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two sequences are treated as \emph{circular}, i.e., periodic.
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pd |
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hornik |
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For \code{open} and \code{filter}, the sequences are padded with
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\code{0}s (from left and right) first; \code{"filter"} returns the
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middle sub-vector of \code{"open"}, namely, the result of running a
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weighted mean of \code{x} with weights \code{y}.}
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r |
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}
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pd |
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\details{
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maechler |
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The Fast Fourier Transform, \code{\link{fft}}, is used for efficiency.
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pd |
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maechler |
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The input sequences \code{x} and \code{y} must have the same length if
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\code{circular} is true.
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Note that the usual definition of convolution of two sequences
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\code{x} and \code{y} is given by \code{convolve(x, rev(y), type = "o")}.
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pd |
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}
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r |
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\value{
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maechler |
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If \code{r <- convolve(x,y, type = "open")}
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and \code{n <- length(x)}, \code{m <- length(y)}, then
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\deqn{r_k = \sum_{i} x_{k-m+i} y_{i}}{r[k] = sum(i; x[k-m+i] * y[i])}
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where the sum is over all valid indices \eqn{i}, for \eqn{k = 1,\dots, n+m-1}
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maechler |
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maechler |
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If \code{type == "circular"}, \eqn{n = m} is required, and the above is
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true for \eqn{i , k = 1,\dots,n} when
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\eqn{x_{j} := x_{n+j}}{x[j] := x[n+j]} for \eqn{j < 1}.
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r |
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}
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\references{
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ripley |
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Brillinger, D. R. (1981)
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maechler |
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\emph{Time Series: Data Analysis and Theory}, Second Edition.
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San Francisco: Holden-Day.
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r |
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}
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maechler |
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\seealso{\code{\link{fft}}, \code{\link{nextn}}, and particularly
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ripley |
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\code{\link[stats]{filter}} (from the \pkg{stats} package) which may be
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maechler |
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more appropriate.
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r |
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}
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maechler |
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\examples{
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x <- c(0,0,0,100,0,0,0)
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y <- c(0,0,1, 2 ,1,0,0)/4
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hornik |
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zapsmall(convolve(x,y)) # *NOT* what you first thought.
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pd |
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zapsmall(convolve(x, y[3:5], type="f")) # rather
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ripley |
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x <- rnorm(50)
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y <- rnorm(50)
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pd |
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# Circular convolution *has* this symmetry:
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ripley |
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all.equal(convolve(x,y, conj = FALSE), rev(convolve(rev(y),x)))
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pd |
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n <- length(x <- -20:24)
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y <- (x-10)^2/1000 + rnorm(x)/8
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Han <- function(y) # Hanning
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convolve(y, c(1,2,1)/4, type = "filter")
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plot(x,y, main="Using convolve(.) for Hanning filters")
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lines(x[-c(1 , n) ], Han(y), col="red")
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lines(x[-c(1:2, (n-1):n)], Han(Han(y)), lwd=2, col="dark blue")
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maechler |
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}
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maechler |
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\keyword{math}
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\keyword{dplot}
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