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360 Chapter 7 FIR FILTER DESIGN
Using simple trigonometric identities, each expression for H r (ω) can
be written as a product of a fixed function of ω (call this Q(ω)) and a
function that is a sum of cosines (call this P (ω)). For details see Proakis
and Manolakis [23] and Problems P7.2–P7.5. Thus
where P (ω) isofthe form
H r (ω) =Q(ω)P (ω) (7.40)
P (ω) =
L∑
α(n) cos ωn (7.41)
n=0
and Q(ω), L, P (ω) for the four cases are given in Table 7.3.
TABLE 7.3
Q(ω), L, and P (ω) for linear-phase FIR filters
LP FIR Filter Type Q(ω) L P(ω)
Type-1 1
M − 1
2
L∑
a(n) cos ωn
0
Type-2 cos ω 2
M
L∑
2 − 1 ˜b(n) cos ωn
0
Type-3
sin ω
M − 3
2
L∑
˜c(n) cos ωn
0
Type-4 sin ω 2
M
2 − 1 L∑
0
˜d(n) cos ωn
The purpose of the previous analysis was to have a common form
for H r (ω) across all four cases. It makes the problem formulation much
easier. To formulate our problem as a Chebyshev approximation problem,
we have to define the desired amplitude response H dr (ω) and a weighting
function W (ω), both defined over passbands and stopbands. The weighting
function is necessary so that we can have an independent control over
δ 1 and δ 2 . The weighted error is defined as
E (ω) △ = W (ω)[H dr (ω) − H r (ω)] , ω ∈S △ =[0,ω p ] ∪ [ω s ,π] (7.42)
These concepts are made clear in the following set of figures. It shows a
typical equiripple filter response along with its ideal response.
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