Use the Fast Fourier Transform to compute the several kinds of convolutions of two sequences.
numeric sequences of the same length to be convolved.
character; partially matched to
The Fast Fourier Transform,
fft, is used for efficiency.
The input sequences
y must have the same length if
circular is true.
Note that the usual definition of convolution of two sequences
y is given by
convolve(x, rev(y), type = "o").
r <- convolve(x, y, type = "open")
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.
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.
Brillinger, D. R. (1981) Time Series: Data Analysis and Theory, Second Edition. San Francisco: Holden-Day.
nextn, and particularly
filter (from the stats package) which may be
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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")
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