IFFT Scaling

Does anyone know if there is a 1/N or 1/sqrt(N) scaling in the IFFT function of FFTW.jl? I am taking the IFFT of an array of complex numbers sampled from a continuous frequency domain function. The resulting time domain array seems too low in magnitude, but I don’t have actual numbers to compare it to. Have not been able to make heads or tails of source code. Below is my code. Thanks!

using Plots
using SpecialFunctions
using FFTW

#physics constants and input data
μ0 = 4*π*1e-7
σw = 3.5e7
rw = 5.6/1000
ω = collect(1:1000.0:1.0e8) #define frequency range (1 to 10^8 in increment of 1000 rad/s)

#initialize arrays
γi = zeros(ComplexF64,size(ω)[1])
Zi = copy(γi)

#Evaluate frequency domain function
γi .= sqrt.(1im.*ω*μ0*σw)
Zi .= (1/(2*π*σw*rw)) .* γi .* besseli.(0,γi*rw)./(besseli.(1,γi*rw))

fig1 = plot(ω,abs.(Zi),xaxis=:log, yaxis=:log,legend=false,lw=4)
plot!(xlabel=("Frequency [rad/s]"),ylabel=("|Z|"),fg=:black,bg=:white)
plot!(gridcolor=:black,gridwidth=2,gridstyle=:dot,gopacity = 0.8)
plot!(minorgrid=true,minorgridcolor=:black,minorgridlinewidth=1,minorgridstyle=:dot,minorgridalpha = 0.75)
plot!(xlim = (1e0,1e8),ylim = (1e-4,1e-1),size=(800,400))

#Inverse FFT
Zitime1 = ifft(Zi)

fig2 = plot(abs.(Zitime1),xlims=(1,1e8),xaxis=:log, yaxis=:log,legend=false,lw=4)
plot!(gridcolor=:black,gridwidth=2,gridstyle=:dot,gopacity = 0.8)


1/N. The forward transform (fft) is unscaled, and the inverse transform (ifft) therefore has a 1/N scaling.


Is it necessary to create and prepend the mirror complex conjugate of the frequency domain array if all I have is the data set for positive frequency?

Not if you use the irfft function.

(Note also that you can use the bfft or brfft functions if you want an unscaled transform.)

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Thanks. It is my understanding that FFT or IFFT functions generally require uniform sampling. I have a situation with logarithmic sampling of the frequency domain data set (for example 100 points per decade). Is there an ifft function that can handle that kind of input? I saw some NUFFT packages, but nothing that would handle non-integer, non-uniform spacing.