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|
|is obtained by a balanced sum a set finished by input values representing|
|the samples of the signal to be leaked out, one has:|
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
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 :
2.2 Transposed structure
|Here, cells with delay memorize partial sums. One realizes in output filter the calculation of||.|
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||. One has so :|
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||answer to the causal signal||. One has so :|
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 :|
Remark :There are no poles but only zéros.The filter FIR is so always stable.
|To have answer in frequency, one replaces z by||, one has so :|
|This answer is periodic of frequency||.|