# Recursive form representation of FIR filter

Do we have methods to represent FIR filter in recursive form?

One of the methods that I know is frequency sampling method.

Can someone provide me any method?

• What do you mean with "in recursive form"? Are you referring to the way you evaluate the response of the FIR filter or how to find the coefficients of the filter? Do you also want to find a FIR filter given a desired frequency response, similar to the frequency sampling method? – fibonatic Nov 4 '18 at 13:50
• Its like how can we implement the FIR filter in recursive form,like we do in direct form representation, cascaded representation etc. – Kartikay Mani Tripathi Nov 4 '18 at 14:04
• Have you compared the difference equations for recursive filters (we typically call these IIR) and FIRs? What have you noticed? – Marcus Müller Nov 4 '18 at 14:23
• A FIR filter can also be represented in recursive form. We can not generalize a recursive filter as IIR or FIR.Both filters FIR and IIR can be implemeted as recursive or non-recursive form. Please see lecture of prof. Dutta Roy about this. cosmolearning.com/video-lectures/… or the same: youtube.com/watch?v=GpqMAzGEXXk – Kartikay Mani Tripathi Nov 4 '18 at 14:25
• @KartikayManiTripathi exactly; you mean what Prof. Roy wrote at 23:47, right? Notice that this is exactly what I wanted to raise to your attention: You can of course write things recursively, but you gain nothing. – Marcus Müller Nov 4 '18 at 15:16

## 1 Answer

The frequency sampling representation of a length $$N$$ FIR filter's transfer function

$$H(z)=\frac{1-z^{-N}}{N}\sum_{k=0}^{N-1}\frac{\tilde{H}[k]}{1-e^{j2\pi k/N}z^{-1}}\tag{1}$$

is a systematic way of finding a recursive implementation for any FIR filter. The coefficients $$\tilde{H}[k]$$ in $$(1)$$ are the DFT coefficients of the impulse response $$h[n]$$. I do not know of any other such systematic way.

However, there are special cases for which a recursive implementation can be found by using standard finite sum identities. One such class of finite impulse response transfer functions is

$$H(z)=K\sum_{n=0}^{N-1}a^nz^{-n}=K\frac{1-a^Nz^{-N}}{1-az^{-1}}\tag{2}$$

A special case of $$(2)$$ is the moving average filter

$$H(z)=\frac{1}{N}\sum_{n=0}^{N-1}z^{-n}=\frac{1}{N}\frac{1-z^{-N}}{1-z^{-1}}\tag{3}$$

• So, I guess there are only these two ways to represent it in this form.But Kindly inform if you come across any other ways also.Thankyou – Kartikay Mani Tripathi Nov 4 '18 at 15:55
• @KartikayManiTripathi hey, that's only one thing! The moving average is just a special case! Quite obviously, since all this is linear, you could also represent any linear combinations of filters of type $(2)$. – Marcus Müller Nov 4 '18 at 16:09