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Filters can be used in any signal processing application. An example would be a FIR filter used in a camera. If the image has noise or if you cannot any the image properly, you apply an FIR filter to correct so that the image is filtered and clear to your eyes. Similarly you can apply filter to improve the signal quality or even remove unwanted noise. So basically filter (FIR or IIR) is the basic step in improving or processing the signal quality.
FIR filters: They do not have poles (actually all poles lies on the center of the unit circle in z transform). FIR filters only have zeros. Order of FIR filter decide the number of zeros. FIR filter is always stable because they do not have poles. There is no feedback in FIR filter.
IIR filters: They have poles and also zeros. If poles lies outside the unit circle then IIR filter is unstable otherwise it is stable if all poles lies inside the unit circle. IIR filters have feedback because of poles.
Practically IIR filters are easy to implement because of feedback with less filter order compared with FIR. In mobile radio communication the wireless channel is also model as FIR / IIR filer between transmitter and receiver. At receiver end the inverse channel response could be implemented using these two filters.