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"Notes on the Design of Optimal FIR Filters" Appendix A

Module by: John Treichler. E-mail the author

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The Formula for Converting between and Passband Ripple

From equation 2 in the module titled Statement of Optimal Linear Phase FIR Filter Design Problem, the peak-to-peak passband ripple, measured in decibels, is given by

P B R = 10 l o g 10 ( 1 + δ 1 ) 2 ( 1 - δ 1 ) 2 , P B R = 10 l o g 10 ( 1 + δ 1 ) 2 ( 1 - δ 1 ) 2 , (1)

where δ1δ1 is the peak amplitude deviation in the passband. Suppose now that

0 < δ 1 ≪ 1 . 0 < δ 1 ≪ 1 . (2)

If so, then the passband ripple PBR is closely approximated by

P B R ≈ 10 l o g 10 ( 1 + 4 δ 1 ) . P B R ≈ 10 l o g 10 ( 1 + 4 δ 1 ) . (3)

Now recall that loge(1+x)≈xloge(1+x)≈x, when xx is small compared to unity, and that log10x≈0.434·logexlog10x≈0.434·logex. Combining these facts, leads to the equation

P B R ≈ 10 l o g 10 ( 1 + 4 δ 1 ) ≈ 4 . 34 · l o g e ( 1 + 4 δ 1 ) ≈ 17 . 36 · δ 1 . P B R ≈ 10 l o g 10 ( 1 + 4 δ 1 ) ≈ 4 . 34 · l o g e ( 1 + 4 δ 1 ) ≈ 17 . 36 · δ 1 . (4)

This formula holds as long as δ1δ1 is small compared to unity. Using δ1=0.1δ1=0.1 as a benchmark, the formula holds for values of passband ripple less than 1.5 to 2 dB, the range in which most filter design falls.

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