0.8 "notes on the design of optimal fir filters" appendix a

The formula for converting betweenAnd 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 indecibels, is given by

where ${\delta }_1$ is the peak amplitude deviation in the passband. Suppose now that

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

Now recall that $lo{g}_e(1+x)\approx x$ , when $x$ is small compared to unity, and that $lo{g}_{10}x\approx 0.434·lo{g}_{e}x$ . Combining these facts, leads to the equation

This formula holds as long as ${\delta }_1$ is small compared to unity. Using ${\delta }_1=0.1$ 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.