Infinite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 238

Magnitude response of a Chebyshev II highpass filter

0

− 20

− 40

− 60

− 80

− 100

− 120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized frequency

Figure 4.28 Magnitude response (in decibels) of a Chebyshev II highpass filter.

4.10 YULE-WALKER APPROXIMATION

Now we introduce another function called yulewalk to find an IIR filter that
approximates an arbitrary magnitude response. The method minimizes the error
between the desired magnitude represented by a vector D and the magnitude of
the IIR filter H(e jω ) in the least-squares sense.
In addition to the maximally flat approximation and the minimax (Chebyshev
or equiripple) approximation we have discussed so far, there is the least- squares
approximation, which is used extensively in the design of filters as well as other
systems. The error that is minimized in a more general case is known as the
least- p th approximation. It is defined by

W(e jω ) H(e jω ) D(e jω )

p

J 3 (ω) = − dω

ω ∈ R

and when p = 2, it is known as the least-squares approximation . In the error

function shown above, D(e jω ) is the desired frequency response and H(e jω )
is the response of the filter designed, whereas W(e jω ) is a weighting function
chosen by the designer. It has been found that as p approaches ∞ , the error

is minimized in the minimax sense, and in practice, choosing p = 4 , 5 , 6gives

a good approximation to D(e jω ) in the least- p th sense [14]. It is best to avoid
sharp transitions in the specifications for the desired magnitude for the IIR filter
when we use the MATLAB function yulewalk. The function has the form

[num,den] = yulewalk(N,F,D)