0.14 L_p approximation  (Page 2/2)

The frequency-dependent FIR problem was first introduced in [link] . Following the FIR approach, one can design IIR frequency-dependent filters by merely replacing the linear weighted least squares step by a nonlinear approach, such as the quasilinearization method presented in [link] (as in the complex $l_p$ IIR case). This problem illustrates the flexibility in design for $l_p$ IRLS-based methods.

Magnitude $l_p$ Iir design

The previous sections present algorithms that are based on complex specifications; that is, the user must specify both desired magnitude and phase responses. In some cases it might be better to specify a desired magnitude response only, while allowing an algorihm to select the phase that optimally minimizes the magnitude error. Note that if an algorithm is given a phase in addition to a magnitude function, it must then make a compromise between approximating both functions. The magnitude $l_p$ IIR approximation problem overcomes this dilemma by posing the problem only in terms of a desired magnitude function. The algorithm would then find the optimal phase that provides the optimal magnitude approximation. A mathematical formulation follows,

A critical idea behind the magnitude approach is to allow the algorithm to find the optimum phase for a magnitude approximation. It is important to recognize that the optimal magnitude filter indeed has a complex frequency response. Atmadji Soewito [link] published in 1990 a theorem in the context of $l_2$ IIR design that demonstrated that the phase corresponding to an optimal magnitude approximation could be found iteratively by updating the desired phase in a complex approximation scenario. In other words, given a desired complex response $D_0$ one can solve a complex $l_2$ problem and take the resulting phase to form a new desired response $D^{+}$ from the original desired magnitude response with the new phase. That is,

where $D_0$ represents the original desired magnitude response and $e^{j{\phi }_i}$ is the resulting phase from the previous iteration. This approach was independently suggested [link] by Leland Jackson and Stephen Kay in 2008.

This work introduces an algorithm to solve the magnitude $l_p$ IIR problem by combining the IRLS-based complex $l_p$ IIR algorithm from [link] with the phase updating ideas from Soewito, Jackson and Kay. The resulting algorithm is robust, efficient and flexible, allowing for different orders in the numerator and denominator as well as even or uneven sampling in frequency space, plus the optional use of specified transition bands. A block diagram for this method is presented in [link] .

The overall $l_p$ IIR magnitude procedure can be summarized as follows,

1. Experimental analysis demonstrated that a reasonable initial solution for each of the three main stages would allow for faster convergence. It was found that the frequency domain Prony method by Burrus [link] (presented in [link] ) offered a good initial guess . In [link] this method is iterated to update the specified phase. The outcome of this step would be an equation error $l_2$ magnitude design.
2. The equation error $l_2$ magnitude solution from the previous step initializes a second stage where one uses quasilinearization to update the desired phase. Quasilinearization solves the true solution error complex approximation. Therefore by iterating on the phase one finds at convergence a solution error $l_2$ magnitude design.
3. The rest of the algorithm follows the same idea as in the previous step, except that the least squared step becomes a weighted one (to account for the necessary $l_p$ homotopy weighting). It is also crucial to include the partial updating introduced in [link] . By iterating on the weights one would find a solution error $l_p$ magnitude design.

Figures [link] through [link] illustrate the effectiveness of this algorithm at each of the three different stages for length-5 filters $a$ and $b$ , with transition edge frequencies of 0.2 and 0.24 (in normalized frequency) and $p=30$ . A linear transition band was specified. Figures [link] , [link] and [link] show the equation error $l_2$ , solution error $l_2$ and solution error $l_p$ . [link] shows a comparison of the magnitude error functions for the solution error $l_2$ and $l_p$ designs. [link] shows the phase responses for the three designs.

From Figures [link] and [link] one can see that the algorithm has changed the phase response in a way that makes the maximum magnitude error (located in the stopband edge frequency) to be reduced by approximately half its value. Furthermore, [link] demonstrates that one can reach quasiequiripple behavior with relatively low values of $p$ (for the examples shown, $p$ was set to 30).