2.5 Constrained approximation and mixed criteria  (Page 9/10)

Each iteration of our new variable $p$ method is implemented by the basic algorithm described as Karlovitz's method but using the Newton's methodbased value of $\lambda$ from Fletcher or Kahng in [link] . Both Ekblom and Kahng only used $K=2$ which is too large in almost all cases.

We also tried the generalized acceleration scheme with the basic Karlovitz method and the RUL algorithm. Although it improved the initialperformance of the Karlovitz method, the slowness of each iteration still made this method unattractive. Use with the RUL algorithm gave only aminor improvement of initial performance and no improvement of the poor final convergence.

Our new algorithm uses three distinct concepts:

• The basic IRLS which is a straight forward algorithm with linear convergence [link] when it converges.
• The second order or Newton's modification which increases the number of cases where initial convergence occurs and gives quadratic asymptoticconvergence [link] , [link] .
• The controlled increasing of $p$ from one iteration to the next is a modification which gives excellent initial convergence and allowsadaptation for “difficult" cases.

The best total algorithm, therefore, combines the increasing of $p$ given in [link] the updating the filter coefficients using [link] , and the Newton's choice of $\lambda$ in [link] . By slowly increasing $p$ , the error surface slowly changes from the parabolic shape of $L_2$ which Newton's method is based on, to the more complicated surface of $L_p$ . The question is how fast to change and, from experience with manyexamples, we have learned that this depends on the filter design specifications.

A Matlab program that implements this basic IRLS algorithm is given in the appendix of this paper. It uses an updating of $A\left({\omega }_k\right)$ in the frequency domain rather than of $a\left(n\right)$ in the time domain to allow modifications necessary for using different $p$ in different bands as will be developed later in this paper.

An example design for a length $L=31$ , passband edge $fp=0.4$ , stopband edge $fs=0.44$ , and $p=2$ the program does not have to iterate and give the response in [link] .

For the same specifications except $p=4$ we get [link]

and for $p=100$ we get [link]

Different error criteria in different bands

Probably the most important use of the $L_p$ approximation problem posed here is its use for designing filters with different error criteria indifferent frequency bands. This is possible because the IRLS algorithm allows an error power that is a function of frequency $p\left(\omega \right)$ which can allow an $L_2$ measure in the passband and a Chebyshev error measure in the stopband or any other form. This is important if an $L_2$ approximation is needed in the passband because Parseval's theorem shows that the time domain properties of the filtered signal will be wellpreserved but, because of unknown properties of the noise or interference, the stopband attenuation must be less than some specified valued.

The new algorithm described in "A New Robust IRLS Method" was modified so that the iterative updating is done to $A\left(\omega \right)$ rather than to $a\left(n\right)$ . Because the Fourier transform is linear, the updating of [link] can also be achieved by