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A m + 1 ( ω ) = λ A ^ m + 1 ( ω ) + ( 1 - λ ) A m ( ω ) .

The Matlab program listed in the appendix uses this form. This type of updating in the frequency domain allows different p to be used in different bands of A ( ω ) and different update parameters λ to be used in the appropriate bands. In addition, it allows a different constant K weighting to be used for the different bands. The error for this problem is changed from [link] to be

q = k = 0 k 0 | A ( ω k ) - A d ( ω k ) | 2 + K k = k 0 + 1 L - 1 | A ( ω k ) - A d ( ω k ) | p

[link] shows the frequency response of a filter designed with a passband p = 2 , a stopband p = 4 , and a stopband weight of K = 1 .

Figure six is identical to figure three, but is titled IRLS Designed Filter for p=2 in the passband, 4 in stopband.
Response of an IRLS Design with p = 2 in the Stopband and p = 4 in the Stopband

[link] gives the frequency response for the same specifications but with p = 100 and [link] adds a constant weight to the stopband.

Figure seven is identical to figure four, with the title IRLS Designed FIR Filter for p=2 in Passband, 100 in Stopband.
Response of an IRLS Design with p = 2 in the Stopband and p = 100 in the Stopband
Figure 8 is a graph titled IRLS Designed FIR Filter for p=2 in Passband, 100 + Weight in Stopband. The horizontal axis is labeled normalized frequency, and the vertical axis is labeled amplitude response, A. The horizontal axis ranges in value from 0 to 1 in increments of 0.2. The vertical axis ranges in value from 0 to 1.2 in increments of 0.2. The curve in the graph begins at (0, 0.95). The curve begins with three waves of increasing amplitude until (0.38, 1.2), when after a final peak it begins sharply decreasing to (0.45, 0). At this point, the curve wavers in a sinusoidal fashion above and below the horizontal axis with an amplitude of approximately 0.02, and completes 4 waves along the axis before it terminates at the right end of the figure.
Response of an IRLS Design with p = 2 in the Stopband and p = 100 plus a weight in the Stopband

The constrainedApproximation

In some design situations, neither a pure L 2 nor a L or Chebyshev approximation is appropriate. If one evaluates both the squarederror and the Chebyshev error of a particular filter, it is easily seen that for an optimal least squares solution, a considerable reduction ofthe Chebyshev error can be obtained by allowing a small increase in the squared error. For the optimal Chebyshev solution the opposite is true.A considerable reduction of the squared error can be obtained by allowing a small increase in the Chebyshev error. This suggests a better filtermight be obtained by some combination of L 2 and L approximation. This problem is stated and addressed by Adams [link] and by Lang [link] , [link] .

We have applied the IRLS method to the constrained least squares problem by adding an error based weighting function to unity in the stopband onlyin the frequency range where the response in the previous iteration exceeds the constraint. The frequency response of an example is the thatwas illustrated in [link] as obtained using the CLS algorithm. The IRLS approach to this problem is currently being evaluatedand compared to the approach used by Adams. The initial results are encouraging.

Application to the complex approximation and the 2d filter design problem

Although described above in terms of a one dimensional linear phase FIR filter, the method can just as easily be applied to the complexapproximation problem and to the multidimensional filter design problem. We have obtained encouraging initial results from applications of our newIRLS algorithm to the optimal design of FIR filters with a nonlinear phase response. By using a large p we are able to design essentially Chebyshev filters where the Remez algorithm is difficult toapply reliably.

Our new IRLS design algorithm was applied to the two examples considered by Chen and Parks [link] and by Schulist [link] , [link] and Preuss [link] , [link] . One is a lowpass filter and the other a bandpass filter, both approximating a constant group delay over theirpassbands. Examination of magnitude frequency response plots, imaginary vs. real part plots, and group delay frequency response plots for thefilters designed by the IRLS method showed close agreement with published results [link] . The use of an L p approximation may give more desirable results than a true Chebyshev approximation. Our results on thecomplex approximation problem are preliminary and we are doing further investigations on convergence properties of the algorithm and on thecharacteristics of L p approximations in this context.

Application of the new IRLS method to the design of 2D FIR filters has also given encouraging results. Here again, it is difficult to apply theRemez exchange algorithm directly to the multi-dimensional approximation problem. Application of the IRLS to this problem is currently beinginvestigated.

We designed 5 × 5 , 7 × 7 , 9 × 9 , 41 × 41 , and 71 × 71 filters to specifications used in [link] , [link] , [link] , [link] . Our preliminary observations from these examples indicate the new IRLS method is faster and/or gives lower Chebyshev errors than any of the other methods [link] . Values of K in the 1.1 to 1.2 range were required for convergence. As for the complex approximation problem, further research is being done on convergenceproperties of the algorithm and on the characteristics of L p approximations in this context.

Section conclusions

We have proposed applying the iterative reweighted least squared error approach to the FIR digital filter design problem. We have shown how alarge number of existing methods can be cast as variations on one basic successive approximation algorithm called Iterative Reweighted LeastSquares. From this formulation we were able to understand the convergence characteristics of all of them and see why Lawson's method hasexperimentally been found to have slow convergence.

We have created a new IRLS algorithm by combining an improved acceleration scheme with Fletcher's and Kahng's Newton type methods to give a very gooddesign method with good initial and final convergence properties. It is a significant improvement over the Rice-Usow-Lawson method.

The main contribution of the paper was showing how to use these algorithms with different p in different frequency bands to give a filter with different pass and stopband characteristics, how to solve the constrained L p problem, and how the approach is used in complex approximation and in 2D filter design.

Minimum phase design

Here we design optimal approximations that can be “lifted" to give a positive function that when viewed as a magnitude squared, can befactored to give a minimum phase optimal design. However, the factoring can be a problem for long filters.

Window function design of fir filters

One should not use Hamming, Hanning, Blackman, or Bartlet windows for the design of FIR filters. They are appropriate for segmenting long datastrings into shorter blocks to minimize the effects of blocking, but they do not design filters with any control over the transition band and do notdesign filters that are optimal in any meaningful sense.

The Kaiser window does have the ability to control the transition band. It also gives a fairly good approximation to a least squares approximationmodified to reduce the Gibbs effect. However, the design is also not optimal in any meaningful sense and does not allow individual control ofthe widths of multiple transition bands. The spline transition function method gives the same control as the Kaiser window but does have acriterion of optimality and does allow independent control over individual transition bands. No window method allows any separate weighting of theerror in different bands.

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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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