0.3 Finite impulse response (fir) l_p design

A Finite Impulse Response ( FIR ) filter is an ordered vector $h\in {\mathbb{R}}^N$ (where $0<N<\infty$ ), with a complex polynomial form in the frequency domain given by

The filter $H\left(\omega \right)$ contains amplitude and phase components $\left\{A_H,\left(\omega \right),,,{\phi }_H,\left(\omega \right)\right\}$ that can be designed to suit the user's purpose.

Given a desired frequency response $D\left(\omega \right)$ , the general $l_p$ approximation problem is given by

In the most basic scenario $D\left(\omega \right)$ would be a complex valued function, and the optimization algorithm would minimize the $l_p$ norm of the complex error function $ϵ\left(\omega \right)=D\left(\omega \right)-H\left(\omega \right)$ ; we refer to this case as the complex $l_p$ design problem (refer to [link] ).

One of the caveats of solving complex approximation problems is that the user must provide desired magnitude and phase specifications. In many applications one is interested in removing or altering a range of frequencies from a signal; in such instances it might be more convenient to only provide the algorithm with a desired magnitude function while allowing the algorithm to find a phase that corresponds to the optimal magnitude design. The magnitude $l_p$ design problem is given by

where $D\left(\omega \right)$ is a real, positive function. This problem is discussed in [link] .

Another problem that uses no phase information is the linear phase $l_p$ problem. It will be shown in [link] that this problem can be formulated so that only real functions are involved in the optimization problem (since the phase component of $H\left(\omega \right)$ has a specific linear form).

An interesting case results from the idea of combining different norms in different frequency bands of a desired function $D\left(\omega \right)$ . One could assign different $p$ -values for different bands (for example, minimizing the error energy ( ${\epsilon }_2$ ) in the passband while using a minimax error ( ${\epsilon }_{\infty }$ ) approach in the stopband to keep control of noise). The frequency-varying $l_p$ problem is formulated as follows,

where $\left\{{\omega }_{pb},,,{\omega }_{pb}\right\}$ are the passband and stopband frequency ranges respectively (and $2<p,q<\infty$ ).

Perhaps the most relevant problem addressed in this work is the Constrained Least Squares ( CLS ) problem. In a continuous sense, a CLS problem is defined by

The idea is to minimize the error energy across all frequencies, but ensuring first that the error at each frequency does not exceed a given tolerance $\tau$ . [link] explains the details for this problem and shows that this type of formulation makes good sense in filter design and can efficiently be solved via IRLS methods.

The irls algorithm and fir literature review

A common approach to dealing with highly structured approximation problems consists in breaking a complex problem into a series of simpler, smaller problems. Often, one can even prove important mathematical properties in this way. Consider the $l_p$ approximation problem introduced in [link] ,

For simplicity at this point we can assume that $f(·):{\mathbb{R}}^N\mapsto {\mathbb{R}}^M$ is linear. It is relevant to mention that [link] is equivalent to

In its most basic form the $l_p$ IRLS algorithm works by rewriting [link] into a weighted least squares problem of the form