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Optimal Equiripple Design Technique 363
7.5.2 CONSTRAINT ON THE NUMBER OF EXTREMA
Before we give the solution to this above problem, we will first discuss
the issue: how many local maxima and minima exist in the error function
E(ω) for a given M-point filter? This information is used by the
Parks-McClellan algorithm to obtain the polynomial interpolation. The
answer is in the expression P (ω). From (7.41) P (ω) isatrigonometric
function in ω. Using trigonometric identities of the form
cos (2ω) =2cos 2 (ω) − 1
cos (3ω) =4cos 3 (ω) − 3 cos (ω)
. = . .
P (ω) can be converted to a trigonometric polynomial in cos (ω), which
we can write (7.41) as
P (ω) =
L∑
β(n) cos n ω (7.46)
□ EXAMPLE 7.21 Let h(n) = 1 [1, 2, 3, 4, 3, 2, 1] . Then M =7and h(n) issymmetric, which
15
means that we have a Type-1 linear-phase filter. Hence L =(M − 1)/2 =3.
Now from (7.7)
n=0
α(n) =a(n) =2h(3 − n), 1 ≤ n ≤ 2; and α(0) = a(0) = h(3)
or α(n) = 1 [4, 6, 4, 2]. Hence
15
or β(n) =
P (ω) =
3∑
α(n) cos ωn = 1 (4 + 6 cos ω +4cos 2ω +2cos 3ω)
15
0
= 1 15
{ 4+6cosω + 4(2 cos 2 ω − 1) + 2(4 cos 3 ω − 3 cos ω) }
=0+0+ 8 15 cos2 ω + 8 15 cos3 ω =
[
0, 0, 8
]
.
3∑
β(n) cos n ω
15 , 8 15
From (7.46) we note that P (ω) isanLth-order polynomial in cos(ω). Since
cos(ω) isamonotone function in the open interval 0