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Linear-Phase FIR Filters: Amplitude Formulas

Module by: Ivan Selesnick. E-mail the author

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SUMMARY: AMPLITUDE FORMULAS

Figure 1: where M=N−12 M N 1 2
Type θ⁢ω θ ω A⁢ω A ω
I −(M⁢ω) M ω h⁢M+2⁢∑n=0M−1h⁢n⁢cos(M−n)⁢ω h M 2 n 0 M 1 h n M n ω
II −(M⁢ω) M ω 2⁢∑n=0N2−1h⁢n⁢cos(M−n)⁢ω 2 n 0 N 2 1 h n M n ω
III −(M⁢ω)+π2 M ω 2 2⁢∑n=0M−1h⁢n⁢sin(M−n)⁢ω 2 n 0 M 1 h n M n ω
IV −(M⁢ω)+π2 M ω 2 2⁢∑n=0N2−1h⁢n⁢sin(M−n)⁢ω 2 n 0 N 2 1 h n M n ω

AMPLITUDE RESPONSE CHARACTERISTICS

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

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

EVALUATING THE AMPLITUDE RESPONSE

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 N≤L N L . Samples of the frequency response of the filter can be written as H⁢2⁢πL⁢k=∑n=0N−1h⁢n⁢e(−i)⁢2⁢πL⁢n⁢k H 2 L k n 0 N 1 h n 2 L n k Define the LL-point signal g⁢n 0≤n≤L−1 g n 0 n L 1 as g⁢n={h⁢n  if  0≤n≤N−10  if  N≤n≤L−1 g n h n 0 n N 1 0 N n L 1 Then H⁢2⁢πL⁢k=G⁢k= DFT L ⁢g⁢n H 2 L k G k DFT L g n where G⁢k G k is the LL-point DFT of g⁢n g n .

Types I and II

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

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