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 |

`conj` |
logical; if |

`type` |
character; partially matched to For |

### 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")
```

*stats*version 4.4.1 Index]