# Deriving finite impulse response for polylogarithm

As a part of my research i have to use the following z-transform in matlab 'filter' function so as to derive the convoluted signal from the original one.

$$\frac{1}{{\rm Li}_{k}(z^{-1}e^{-b})}$$

But since i read that the polylogarithm can be expressed as a function only for specific values of k (k can take many values, not necessarily integers). Can anyone guide me as to how can i use it for my matlab codes. Will i have to derive the FIR. Can it be derived in this case ?

• You have to use some approximation because the polylogarithm is an infinite series of powers of its argument. Furthermore, the filter is IIR, not FIR. Jul 8, 2014 at 13:09
• It would be good if you could give us some more background information, because you will not able to realize this transfer function exactly. And since you need an approximation, it is important to know what type of approximation is best suited to your problem. Jul 9, 2014 at 9:40
• @MattL. Thanks for the response. Basically this is z-transform of the inverse filter of a more general model of RIR. For the schroeder's model the z-transform of the inverse filter is straight forward but here it isn't so. For schroeder's model k=0 in the above equation. This model is a more general one. If i could get a compact form for the general case where k is arbitrary i will be able to implement the inverse filter in matlab. Thanks in advance. Jul 10, 2014 at 14:58
• I think that there's no way of an exact implementation of this transfer function. That's why I was talking about an approximation. The best type of approximation, however, depends on the application. Jul 10, 2014 at 15:06