from numpy.fft import fft$,$ ifft

from numpy import real, imag

#fft $-->$ computes fft of a list or numpy array

#ifft $->$ computes inverse fft of list or numpy array

# Create a list

lst$0=[1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1]$

# Compute its fft

fft$\_$lst$0=$ fft$($lst$0)$

print$($f$\text{'}$FFT of $\{$lst$0\}$ is

$\backslash$t $\{$fft$\_$lst$0\}\text{'})$

# Compute iverse fft

ifft$\_$lst$0=$ ifft$($fft$\_$lst$0)$

print$($f$\text{'}$After ifft: $\{$ifft$\_$lst$0\}\text{'})$

# Check that all the imaginary parts are tiny in the ifft result

# Note that they will not be zero due to floating point error

assert$($all$([$abs$($imag$($x$))<=1$E$-10$ for x in ifft$\_$lst$0])),$ 'Something went wrong $--$ we should not have complex parts to the ifft result'

# Extract the real parts

print$(\text{'}$Note that we can suppress the vanishingly small complex cofficients'$)$

fix$\_$ifft$\_$lst$0=[$real$($x$)$ for x in ifft$\_$lst$0]$

print$($f$\text{'}$After converting back to float: $\{$fix$\_$ifft$\_$lst$0\}\text{'})$