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Digital first-order IIR filter impulse response

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Sandi2000

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The first formula :
A causal first-order IIR filter is characterized by the following difference equation

01.PNG (01)

The impulse response of that system can be computed via the Z-transform or otherwise, and it turns out to be
02.PNG (02)

The second formula :
the general causal IIR filter is

G.8 G.9.PNG (03)

has transfer function
G.10.PNG (04)

one step of long division yields

G.11.PNG (05)

The question is :

The common thing between the first and second formula is (b1−b0*a1) ,i want to understand more why using (b1−b0*a1) in first and second formula? what is the difference ?



Thank you in advance.
 

The question is :
The common thing between the first and second formula is (b1−b0*a1) ,
i want to understand more why using (b1−b0*a1) in first and second formula?
what is the difference ?
Nb >= Na.

In your case, Nb=Na, N=Na.
 
when can I apply the equation (2)
and when can i apply the equation (5)?
Describe sentences gramatically correctly.

Numbers of Pole are different for (2) and (5).
Can you understand ?
 

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