Transfer function in the z domain

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Sandi2000

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A generic causal FIR or IIR filter can be represented by the following transfer function in the z domain as
shown in
(01)

in other case, A finite-dimensional LTI discrete-time causal filter can be represented by the following transfer function in the z domain:
(02)


Why there is two equations with the same form but with different order of coefficient in the numerator and denominator ?
 

Order of coefficients is just a matter of convention, both transfer functions are basically equivalent, except for the characteristic a0 = 1 in the IIR z transfer function. Of course not all coefficients must be populated, IR has e.g. a1 - aN = 0. In other words, 01 is a subset of 02, IR can be seen as subset of IIR (feedback path eliminated).
 

Thank you for the clarification , but what do you mean by ' IR can be seen as subset of IIR '? (the IR : impulse response and it represent (1)).
 

Sorry, a typo. Means FIR. If you zero out all ax terms (denominator) in the IIR transfer function, you get a FIR function.
 
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