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Linear-Phase FIR Filters: Amplitude Formulas

Module by: Ivan Selesnick. E-mail the author

Summary: (Blank Abstract)

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Figure 1: where M=N12 M N 1 2
Type θω θ ω Aω A ω
I (Mω) M ω hM+2n=0M1hncos(Mn)ω h M 2 n 0 M 1 h n M n ω
II (Mω) M ω 2n=0N21hncos(Mn)ω 2 n 0 N 2 1 h n M n ω
III (Mω)+π2 M ω 2 2n=0M1hnsin(Mn)ω 2 n 0 M 1 h n M n ω
IV (Mω)+π2 M ω 2 2n=0N21hnsin(Mn)ω 2 n 0 N 2 1 h n M n ω


To analyze or design linear-phase FIR filters, we need to know the characteristics of the amplitude response Aω A ω .

Table 1
Type Properties  
Figure 2
Figure 2 (fourAmps.png)


The frequency response H f ω H f ω of an FIR filter can be evaluated at LL equally spaced frequencies between 0 and π using the DFT. Consider a causal FIR filter with an impulse response $h(n)$ of length-NN, with NL N L . Samples of the frequency response of the filter can be written as H2πLk=n=0N1hne(i)2πLnk H 2 L k n 0 N 1 h n 2 L n k Define the LL-point signal gn 0nL1 g n 0 n L 1 as gn={hn  if  0nN10  if  NnL1 g n h n 0 n N 1 0 N n L 1 Then H2πLk=Gk= DFT L gn H 2 L k G k DFT L g n where Gk G k is the LL-point DFT of gn g n .

Types I and II

Suppose the FIR filter hn h n is either a Type I or a Type II FIR filter. Then we have from above

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