IIR Filters: Questions & Answers

[mnb96]

Hello,
I know that Finite Impulse Response (FIR) filters can be equivalently expressed as a convolution. The effect of convolution in frequency domain is well known. In conclusion it is easy to make sense of FIR filters.

My questions are:

- can also Infinite Impulse Response (IIR) filters be given a similar interpretation?
Are they convolution of some sort?

- There exists some digital IIR filters which manipulate the frequencies of the signal (e.g. low-pass, hi-pass, etc...).
How can one design an IIR filter which does specific operations to frequencies if we don't know how to handle it mathematically?
Alternatively, how can you prove that a low-pass IIR filter in time-domain, does indeed filter frequencies?

 

[marcusl]

Yes, IIR filters also operate as convolutions, with the difference that their impulse responses are infinite in time instead of finite. Hence the name IIR: "infinite impulse response". Your other questions seem to arise from the misconception that IIR filters cannot be described mathematically. This is false. These filters can be described mathematically, and can be designed to produce all of the filter responses you describe. They can be implemented digitally or with analog circuitry. You can read about analog implementations in books on systems and signals, for instance,

Wainstein and Zubakov, Extraction of Signals From Noise

Digital filters (including IIR) are covered everywhere. See. e.g.,

Oppenheim and Schafer, Digital Signal Processing

Even Wikipedia has some relevant information
http://en.wikipedia.org/wiki/Transfer_function"
http://en.wikipedia.org/wiki/Infinite_impulse_response"
http://en.wikipedia.org/wiki/LTI_system_theory#Impulse_response_and_convolution"