Convolution of Sequences via FFT
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; partially matched to "circular", "open",
"filter". For "circular", the
two sequences are treated as circular, i.e., periodic.
For "open" and "filter", the sequences are padded with
0s (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, \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.
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")