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__ANALYZE
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 s____tructure__

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 : |

or : | |

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 | answer to the causal signal | . One has so : |

or : | |

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 : |

__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 | . |