Create a n-by-n discrete Fourier transform matrix.

The discrete Fourier transform (DFT) matrix for a given dimension n is calculated.

dftMatrix(n)

Arguments

n	Dimension for the DFT matrix.

Value

The n-by-n DFT matrix.

Details

The DFT matrix can be used for computing the discrete Fourier transform of a matrix or vector. dftMatrix(n) %*% testMatrix is the same as apply(testMatrix, MARGIN = 2, FUN = fft).

Examples

set.seed(987)
testMatrix <- matrix(sample(1:10, size = 25, replace = TRUE), nrow = 5)
D <- dftMatrix(5)

# Discrete Fourier transform with matrix multiplication:
D %*% testMatrix
#>                     [,1]           [,2]          [,3]                 [,4]
#> [1,] 37.000000+0.000000i 29.0+0.000000i 14+0.0000000i 28.000000+ 0.000000i
#> [2,]  0.354102-1.538842i -3.5+5.792096i -1-0.7265425i  1.190983+ 0.416272i
#> [3,] -6.354102+0.363271i -3.5-2.991927i -1-3.0776835i  2.309017+12.086220i
#> [4,] -6.354102-0.363271i -3.5+2.991927i -1+3.0776835i  2.309017-12.086220i
#> [5,]  0.354102+1.538842i -3.5-5.792096i -1+0.7265425i  1.190983- 0.416272i
#>                     [,5]
#> [1,] 34.000000+0.000000i
#> [2,]  4.854102+7.330938i
#> [3,] -1.854102-3.355198i
#> [4,] -1.854102+3.355198i
#> [5,]  4.854102-7.330938i

# Discrete Fourier transform with function fft: 
apply(testMatrix, MARGIN = 2, FUN = fft)
#>                     [,1]           [,2]          [,3]                 [,4]
#> [1,] 37.000000+0.000000i 29.0+0.000000i 14+0.0000000i 28.000000+ 0.000000i
#> [2,]  0.354102-1.538842i -3.5+5.792096i -1-0.7265425i  1.190983+ 0.416272i
#> [3,] -6.354102+0.363271i -3.5-2.991927i -1-3.0776835i  2.309017+12.086220i
#> [4,] -6.354102-0.363271i -3.5+2.991927i -1+3.0776835i  2.309017-12.086220i
#> [5,]  0.354102+1.538842i -3.5-5.792096i -1+0.7265425i  1.190983- 0.416272i
#>                     [,5]
#> [1,] 34.000000+0.000000i
#> [2,]  4.854102+7.330938i
#> [3,] -1.854102-3.355198i
#> [4,] -1.854102+3.355198i
#> [5,]  4.854102-7.330938i