R: Convolution of Sequences via FFT

convolve {stats} | R Documentation

## 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 `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.

### 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")

[Package stats version 3.5.1 Index]