FIR filter design - Continuous and Discrete Time Signals and Systems - page 697

specified by the impulse response h [ k ]. Let ε max max denote the maximum value
of the error ε ( Ω ) . The polynomial G ( Ω ), which best approximates H d ( Ω )
(i.e. minimizes ε max ), produces the error function ε ( Ω ) that must satisfy the
following property. There should be at least L + 2 discrete frequencies

Ω 1 <
Ω 2 < < Ω L + 2 ∈ S at which the maximum and minimum peak values of
the error alternate, i.e. ε ( Ω m + 1 ) =−ε ( Ω m ) = ε max .
Before presenting some examples of the application of the alternation theo-
rem, we briefly comment on the degree L of the error function ε ( Ω ) in the FIR
filters. The value of L is determined by evaluating the highest power of cos( Ω )
in the G ( Ω ) function of the filters. For the four types of FIR filters with length
N , the value of L is specified as follows:

N − 1
2

type I FIR filters L = ;

N − 2
2

type II FIR filters L = ;

N − 3
2

L =

type III FIR filters ;

N − 2
2

type IV FIR filters L = .

The alternation theorem states that the minimum number of alternations for the
optimal FIR filter should be at least L + 2. The actual number of alternations
in an optimal FIR may, however, exceed the minimum number specified by the
alternation theorem. An optimally designed lowpass or highpass filter can have
up to L + 3 alternations, while an optimal bandpass or bandstop filter can have
up to L + 5 alternations.

Example 15.8
The magnitude spectra of two lowpass FIR filters with lengths N = 13 and
20 are, respectively, shown in Figs. 15.18(a) and (b), where the filter gain
within the pass and stop bands is enclosed within a frame box. Determine if the
two filters satisfy the alternation theorem.

Solution
Figure 15.18(a) shows the frequency response of a type I FIR filter with
length N = 13. The degree L of cos( Ω ) in the polynomial ε ( Ω )isgiven
by L = (13 − 1) / 2 = 6. Based on the alternation theorem, there should be at
least L + 2 = 8 alternations in polynomial ε ( Ω ). Note that the absolute value
of error ε ( Ω ) is the difference H ( Ω ) − H d ( Ω ) , where H d ( Ω ) has a unity gain
within the pass band and zero gain within the stop band. Therefore, counting the
number of alternations in ε ( Ω ) is the same as counting the number of alterna-
tions in H ( Ω ) with respect to the pass- and stop-band ripples. From Fig. 15.18
we observe that there are indeed eight alternations (shown by symbols) in
H ( Ω ). One of these alternations occurs at the pass-band edge frequency Ω p ,