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}
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}
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\usage{
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\usage{
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convolve(x, y, conj = TRUE, type = c("circular", "open", "filter"))
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convolve(x, y, conj = TRUE, type = c("circular", "open", "filter"))
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}
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}
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\arguments{
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\arguments{
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  \item{x,y}{numeric sequences \emph{of the same length} to be
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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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    convolved.}
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  \item{conj}{logical; if \code{TRUE}, take the complex \emph{conjugate}
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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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    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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  \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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    \code{"filter"} (beginning of word is ok).  For \code{circular}, the
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  Note that the usual definition of convolution of two sequences
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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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  \code{x} and \code{y} is given by \code{convolve(x, rev(y), type = "o")}.
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}
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}
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\value{
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\value{
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  If \code{r <- convolve(x,y, type = "open")}
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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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  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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  \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
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  where the sum is over all valid indices \eqn{i}, for
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  \eqn{k = 1, \dots, n+m-1}.
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  \eqn{k = 1, \dots, n+m-1}.
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\examples{
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\examples{
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require(graphics)
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require(graphics)
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x <- c(0,0,0,100,0,0,0)
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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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y <- c(0,0,1, 2 ,1,0,0)/4
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zapsmall(convolve(x,y))         #  *NOT* what you first thought.
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zapsmall(convolve(x, y))         #  *NOT* what you first thought.
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zapsmall(convolve(x, y[3:5], type = "f")) # rather
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zapsmall(convolve(x, y[3:5], type = "f")) # rather
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x <- rnorm(50)
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x <- rnorm(50)
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y <- rnorm(50)
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y <- rnorm(50)
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# Circular convolution *has* this symmetry:
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# Circular convolution *has* this symmetry:
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all.equal(convolve(x,y, conj = FALSE), rev(convolve(rev(y),x)))
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all.equal(convolve(x, y, conj = FALSE), rev(convolve(rev(y),x)))
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n <- length(x <- -20:24)
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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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y <- (x-10)^2/1000 + rnorm(x)/8
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Han <- function(y) # Hanning
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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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       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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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  , 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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lines(x[-c(1:2, (n-1):n)], Han(Han(y)), lwd = 2, col = "dark blue")
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}
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}
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\keyword{math}
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\keyword{math}
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\keyword{dplot}
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\keyword{dplot}