finite impulse response - Why are FIR filters still stable even though they contain poles?

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  How come FIR filters are always stable?

  
  


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  Since they contain poles, shouldn't they be more affected by stability issues than others?

  
  

Answer

FIR filters contain only zeros and no poles. If a filter contains poles, it is IIR. IIR filters are indeed afflicted with stability issues and must be handled with care.

EDIT:

After some further thought and some scribbling and google-ing, I think that I have an answer to this question of FIR poles that hopefully will be satisfactory to interested parties.

Beginning with the Z transform of a seemingly poleless FIR filter: $$H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + \cdots + b_N z^{-N}}{1}$$ As is shown in RBJ's answer, the FIR poles are revealed by multiplying the numerator and denominator of $H(z)$ by $z^{N}$: $$H(z) = \frac{b_0 z^{N} + b_1 z^{N-1} + b_2 z^{N-2} + \cdots + b_N }{z^{N}}$$ Thus yielding our $N$ poles at the origin of a general FIR filter.

However, in order to show this, the assumption of causality is placed on the filter. Indeed, if we consider a more general FIR filter where causality is not assumed: $$G(z) = \frac{b_0 z^{k} + b_1 z^{k-1} + b_2 z^{k-2} + \cdots + b_N z^{k-N}}{1}$$ A different number of poles $(N-k)$ appear at the origin: $$G(z) = \frac{b_0 z^{N} + b_1 z^{N-1} + b_2 z^{N-2} + \cdots + b_N }{z^{N-k}}$$

Thus, I conclude the following:

• (Answering the original question) In general, an FIR filter does have poles, though always at the origin of the Z-plane. Because they are never beyond the unit circle, they are no threat to the stability of an FIR system.


• The number of poles of the FIR signal corresponds to the filter order $N$ and the "degree" of acausality $k$. Thus, it is possible to construct FIR filters which have no poles but these filters are then acausal -- i.e. they aren't conceivable for realtime processing. For an $N^{th}$ order FIR filter which is exactly causal $(k=0)$, there are $N$ poles at the origin.


• Perhaps the simplest way to conceive of a pole at the origin is a simple delay element: $$ H(z) = z^{-1} = \frac{1}{z}$$ The typical FIR filters can then be viewed as acausal filters which are followed by enough delay elements to make them causal.