Inverse Fast Fourier Transform¶

SnFFT support three types of inverse fast Fourier transforms. The dense inverse fast Fourier transform can take the inverse of the output of either the sparse or dense fast Fourier transform. The bandlimited inverse fast Fourier transform can only take the inverse of the output of the bandlimited fast Fourier transform, but benefits greatly from the restriction. The partial inverse fast Fourier transform can take the inverse of the output of either the sparse or dense fast Fourier transform.

Dense Inverse Fast Fourier Transform¶

sn_ifft(N, FFT, YOR, PT)¶

# Parameters:
#       N::Int
#       - the problem size
#       FFT::Array{Array{Float64, 2}, 1}
#       - a Fast Fourier Transform of size N
#       - should be the output of sn_fft() or sn_fft_sp()
#       YOR::Array{Array{Array{SparseMatrixCSC, 1}, 1}, 1}
#       - output1 from yor()
#       PT::Array{Array{Array{Int, 1}, 1}, 1}
#       - output2 from yor()
# Return Values:
#       SNF::Array{Float64, 1}
#       - the function over Sn that corresponds to FFT

FFT is the output of sn_ftt() or sn_fft_sp()
YOR and PT are the outputs of yor(N)
See the code for example4() for an example of the complete process to compute a dense inverse fast Fourier transform

Bandlimited Inverse Fast Fourier Transform¶

sn_ifft_bl(N, K, FFT, YOR, PT, ZFI)¶

# Parameters:
#       N::Int
#       - the problem size
#       K::Int
#       - the problem is homogenous at N-K
#       FFT::Array{Array{Float64, 2}, 1}
#       - a Fast Fourier Transform of size N
#       - should be the output of sn_fft_bl()
#       YOR::Array{Array{Array{SparseMatrixCSC, 1}, 1}, 1}
#       - output1 from yor_bl()
#       PT::Array{Array{Array{Int, 1}, 1}, 1}
#       - output2 from yor_bl()
#       ZFI::Array{Int, 1}
#       - output3 from yor_bl()
# Return Values:
#       SNF::Array{Float64, 1}
#       - the bandlimited function over Sn that corresponds to FFT

FFT is the output of sn_fft_bl()
YOR, PT, and ZFI are the outputs of yor_bl(N, K)
See the code for example7() for an example of the complete process to compute a bandlimited inverse fast Fourier transform

Partial Inverse Fast Fourier Transform¶

sn_ifft_p(N, M, FFT, YOR, PT)¶

# Parameters:
#       N::Int
#       - the problem size
#       M::Int
#       - the number of the top components of FFT to use
#       FFT::Array{Array{Float64, 2}, 1}
#       - a Fast Fourier Transform of size N
#       - should be the output of sn_fft() or sn_fft_sp()
#       YOR::Array{Array{Array{SparseMatrixCSC, 1}, 1}, 1}
#       - output1 from yor()
#       PT::Array{Array{Array{Int, 1}, 1}, 1}
#       - output2 from yor()
# Return Values:
#       SNF::Array{Float64, 1}
#       - the function over Sn that corresponds to FFT that has been smoothed to only use the top M componenets

FFT is the output of sn_ftt() or sn_fft_sp()
YOR and PT are the outputs of yor(N)
See the code for example8() for an example of the complete process to compute a partial inverse fast Fourier transform