convolve {stats} | R Documentation |
Use the Fast Fourier Transform to compute the several kinds of convolutions of two sequences.
convolve(x, y, conj = TRUE, type = c("circular", "open", "filter"))
x, y |
numeric sequences of the same length to be convolved. |
conj |
logical; if |
type |
character; partially matched to For |
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")
.
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, \dots, n+m-1
.
If type == "circular"
, n = m
is required, and the above is
true for i , k = 1,\dots,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.
fft
, nextn
, and particularly
filter
(from the stats package) which may be
more appropriate.
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")