# How to find discrete Z-transform of a list of complex numbers

Can we find the discrete Z-transform of a list of complex numbers in Mathematica?

Similar to the function Fourier for the discrete Fourier transform, do we have a function for Z-transform?

Yes, to find the Z-transform, use the function ZTransform[]. To compute the inverse Z-transform, of a function, InverseZTransform[]. Works with complex numbers: And explicit lists of complex numbers! • The right function to use is ListZTransform[], as the OP mentioned "a list of complex numbers"; ZTransform[] is intended for functions. – J. M.'s technical difficulties Mar 15 '19 at 12:49
• Thanks, but for ZTransform[expr,k,z], we need expr to be expressed in terms of k, I cannot use if for a series of complex numbers. I want something similar to Fourier[list] – Pojj Mar 15 '19 at 12:51
• @J.M.isslightlypensive, Great! This is it. Thank you – Pojj Mar 15 '19 at 12:53
• The Z-transform is linear so $\cal{Z} (a+ i b) = \cal{Z} (a) + i \cal{Z} (b)$. Also, ZTransform[] takes complex input, for example ZTransform[Exp[I k],k,z] works as intended. But you are right! I did not pay enough attention to your request for working with complex numbers! – mjw Mar 15 '19 at 12:57
• @J.M., Yes, I see, ... I guess it depends what we want to use the list for. If we simply want the Z-transform of a list, then the result is a list! If we want the list to imply a sequence, then ListZTransform[] is the way to go. Thank you! – mjw Mar 15 '19 at 13:06