From
equation 2 in the module titled Statement of Optimal Linear Phase FIR Filter Design Problem,
the peaktopeak 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.