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Infinite Impulse Response (IIR) filters are important tools in signal processing. The flexibility they offer with the use of poles and zeros allows for relatively small filters meeting specifications that would require somewhat larger FIR filters. Therefore designing IIR filters in an efficient and robust manner is an inportant problem.

This section covers the design of a number of important l p IIR problems. The methods proposed are consistent with the methods presented for FIR filters, allowing one to build up on the lessons learned from FIR design problems. The complex l p IIR problem is first presented in [link] , being an essential tool for other relevant problems. The l p frequency-dependent IIR problem is also introduced in [link] . While the frequency-dependent formulation might not be practical in itself as a filter design formulation, it is fundamental for the more relevant magnitude l p IIR filter design problem, presented in [link] .

Some complications appear when designing IIR filters, among which the intrinsic least squares solving step clearly arises from the rest. Being a nonlinear problem, special handling of this step is required. It was detemined after thorough experimentation that the quasilinearization method of Soewito presented in [link] can be employed successfully to handle this issue.

A block diagram of complex l_p IIR algorithm. Starting from the top there is an oval containing the phrase 'Given a^_0,b^_0'. An arrow points down from this oval to a rectangular box containing 'w_k(ω)=...' To the right of this box is an pointing left to the box labeling it 'L_p Weighting'. An arrow points down from the box to another box containing 'Find a^,b^'. Another arrow points down to a diamond shaped box containing 'Stop?'. An arrow with the letter 'N' next to it points up to the arrow pointing to the third box containing 'Find a^,b^'. A dashed square surrounds the blocks 'Find a^,b^' and 'Stop?'. The box is labeled to the right with a bracket and the phrase 'Solve L_p complex problem via L_2 quasilinearization'. Below the last block, the one containing 'Stop?' is an arrow with a 'Y' to the left of it points down to another diamond shaped block containing the phrase 'Stop?'. To the left of this block there is an arrow with labeled 'N' that points up all the way to the first arrow pointing to 'w_k(ω)=...'. Below the diamond shaped box there is another arrow with a 'Y' to the left pointing down to an oval shaped block containing the word 'END'.
Block diagram for complex l p IIR algorithm.

Complex and frequency-dependent l p Approximation

Chapter [link] introduced the problem of designing l p complex FIR filters. The complex l p IIR algorithm builds up on its FIR counterpart by introducing a nested structure that internally solves for an l 2 complex IIR problem. [link] illustrates this procedure in more detail. This method was first presented in [link] .

A graph of the Results for complex l_100 IIR design. There are two wave forms. One is identified by a solid red line and labeled L_p solutions. The other wave is identified by a blue dashed line and labeled L_2 solution. The desired function is indicated with a blue dotted line. Both wave forms start at (0,1) and then at about (0.2,0) both waves have there largest peak before falling drastically to about (0.25,0). The waves then continue along the x axis till (.5,0) where the graph ends. The dotted blue line runs along y=1 to (.2,1). Then the dotted blue line continues at y=0 from (0.25,0) to (o.5,0).
Results for complex l 100 IIR design.
A graph containing two wave forms. One is identified with a solid red line and labeled L_p solution. The other is identified with a blue dashed line and labeled L_2 solution. The desired function is indicated by a horizontal blue dotted line at y=0. The wave of L_p solution begins are (0,.07) drops to about (0.25,0.02) and then climes to (0.28,0.07) and then falls to (0.35,0.04). The wave of L_2 starts at (0,0.15) drops to (0.26,0.01). Then it climbs to about (0.3,0.045) then drops to (0.36,0.03).
Maximum error for l 2 and l 100 complex IIR designs.
A graph labeled L_p error. There is one wave form present in the graph, and it is identified by a solid red line. The wave starts at (0.01,0) rises to (0.075,0.07) falls to (0.14,0.005). The wave continues by rising to (0.18,0.07) drops to (0.2,0.01) and rises to (0.2,0.07). The wave stops here and then the wave begins again at (0.24,0.07) and then drops to (0.25,0.015) and then rises to (0.27,0.07). Finally it drops to (0.35,0.037) and then exits the graphs at (0.5,0.05).
Error curve for l 100 complex IIR design.

Compared to its FIR counterpart, the IIR method only replaces the weighted linear least squares problem for Soewito's quasilinearization algorithm. While this nesting approach might suggest an increase in computational expense, it was found in practice that after the initial l 2 iteration, in general the l p iterations only require from one to only a few internal weighted l 2 quasilinearization iterations, thus maintaining the algorithm efficiency. Figures [link] through [link] present results for a design example using a length-5 IIR filter with p = 100 and transition edge frequencies of 0.2 and 0.24 (in normalized frequency).

[link] compares the l 2 and l p results and includes the desired frequency samples. Note that no transition band was specified. [link] illustrates the effect of increasing p . The largest error for the l 2 solution is located at the transition band edges. As p increases the algorithm weights the larger errors heavier; as a result the largest errors tend to decrease. In this case the magnitude of the frequency response went from 0.155 at the stopband edge (in the l 2 case) to 0.07 (for the l p design). [link] shows the error function for the l p design, illustrating the quasiequiripple behavior for large values of p .

Another fact worth noting from [link] is the increase in the peak in the right hand side of the passband edge (around f = 0 . 22 ). The l p solution increased the amplitude of this peak with respect to the corresponding l 2 solution. This is to be expected, since this peak occurs at frequencies not included in the specifications, and since the l p algorithm will move poles and zeros around in order to meet find the optimal l p solution (based on the frequencies included for the filter derivation). The addition of a specified transition band function (such as a spline) would allow for control of this effect, depending on the user's preferences.

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Source:  OpenStax, Iterative design of l_p digital filters. OpenStax CNX. Dec 07, 2011 Download for free at http://cnx.org/content/col11383/1.1
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