Analyse of nonrecursive filters or FIR

1. Recurrence equation

    FIR filters have a transfer function polynomial which contains a big number of coefficients.They are
unconditionally stable. Knowing that filter FIR is systems for which a output value [Image]
is obtained by a balanced sum a set finished by input values representing
the samples of the signal to be leaked out, one has:

[Image] With M the order of the filter. Remark :  a FIR filter of order M has M+1 coefficients.

2. Structure of realization

      2.1. Direct structure    

[Image]

Application requires for every output value M multiplications and M additions. It is necessary also M+1 memories for coefficients and M+1 data memories . The functioning of this structure is put rhythm in the time by the sample period . One realizes so this operation :

[Image]

       2.2 Transposed structure

               [Image]
Here, cells with delay memorize partial sums. One realizes in output filter the calculation of [Image] .

Remark : It is noticed that there is no loop of reaction , one does not so use the values of the previous output to calculate current output. It is for it that such a filter is called filter nonrecursive.

3. Impulse answer
  Impulse answer is the answer to the causal sequence [Image] . One has so :

[Image]

or : [Image]
[Image]
etc ...

       The coefficients of level-headedness are so the values of the impulse answer of the filter. This answer nullifies at the end of M+1 values ( more coefficients). Impulse answer being finished,one speaks about filter FIR.

4. Indexed answer  
     It is the signal [Image] answer to the causal signal [Image] . One has so :

           [Image]

or : [Image]
[Image]
etc...

      The final value of the indexed answer is equal to the sum the coefficients of the filter FIR and is reached at the end of M+1 output.

    5. Frequency answer 
 The transfer function in z spells : [Image]

Remark :There are no poles but only zéros.The filter FIR is so always stable.

 
    To have answer in frequency, one replaces z by [Image] , one has so :

[Image]
This answer is periodic of frequency [Image] .