FIR (IIR) filters does not mean it is a Non Recursive ( Recursive) Filter in Digital signal processing

Sometime even published books state something wrong linking FIR filter and Recursiveness.

Though FIR filters are majorly non recursive and IIR filters are majorly realized recursively, the two are not related in any way.

FIR digital systems in DSP simply means the impulse response of the Finite number of Input samples x(n) is a finite number of out samples y(n).
IIR digital systems on the contrary have impulse response which is not a finite set of samples.

Recursive systems are those in which the output y(n) is fed back as a loop as an input. If no output is fed back then it is non recursive.

The two definitions is not correlated and remains separate. FIR any day does not mean non recursive!

Here is an example where a simple FIR system is defined using both in a recursive and non recursive way.

Say FIR system is defined by the difference equation:
y(n) = x(n) + x(n-1)
this is a non recursive implementation

Now the same difference equation can be represented in a recursive fashion as shown below

y(n) = x(n) + x(n-1)

now:
            y(n-1) = x(n-1) + x(n-2)
      => x(n-1) = y(n-1) - x(n-2)

Substituting above , we get

y(n) = x(n) - x(n-2) + y(n-1)

where y(n-1) represents feed back. So This is a recursive implementation.
Hence watch out all DSP buffs!