# convolve: Convolution of Sequences via FFT

## Description

Use the Fast Fourier Transform to compute the several kinds of convolutions of two sequences.

## Usage

 `1` ```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 `0`s (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, …, n+m-1.

If `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.

## References

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.
 ``` 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") ```