0.8 Magnitude l_p problem

In some applications, the effects of phase are not a necessary factor to consider when designing a filter. For these applications, control of the filter's magnitude response is a priority for the designer. In order to improve the magnitude response of a filter, one must not explicitly include a phase, so that the optimization algorithm can look for the best filter that approximates a specified magnitude, without being constrained about optimizing for a phase response too.

Power approximation formulation

The magnitude approximation problem can be formulated as follows:

Unfortunately, the second term inside the norm (namely the absolute value function) is not differentiable when its argument is zero. Although one could propose ways to work around this problem, I propose the use of a different design criterion, namely the approximation of a desired magnitude squared. The resulting problem is

The autocorrelation $r\left(n\right)$ of a causal length- $N$ FIR filter $h\left(n\right)$ is given by

The Fourier transform of the autocorrelation $r\left(n\right)$ is known as the Power Spectral Density function [link] $R\left(\omega \right)$ (or simply the SPD), and is defined as follows,

From the properties of the Fourier Transform [link] one can show that there exists a frequency domain relationship between $h\left(n\right)$ and $r\left(n\right)$ given by

This relationship suggests a way to design magnitude-squared filters, namely by using the filter's autocorrelation coefficients instead of the filter coefficients themselves. In this way, one can avoid the use of the non-differentiable magnitude response.

An important property to note at this point is the fact that since the filter coefficients are real, one can see from [link] that the autocorrelation function $r\left(n\right)$ is symmetric; thus it is sufficient to consider its last $N$ values. As a result, the PSD can be written as

in a similar way to the linear phase problem.

The symmetry property introduced above allows for the use of the $l_p$ linear phase algorithm of [link] to obtain the autocorrelation coefficients of $h\left(n\right)$ . However, there is an important step missing in this discussion: how to obtain the filter coefficients from its autocorrelation. To achieve this goal, one can follow a procedure known as Spectral Factorization . The objective is to use the autocorrelation coefficients $r\in {\mathbb{R}}^N$ instead of the filter coefficients $h\in {\mathbb{R}}^N$ as the optimization variables. The variable transformation is done using [link] , which is not a one-to-one transformation. Because of the last result, there is a necessary condition for a vector $r\in {\mathbb{R}}^N$ to be a valid autocorrelation vector of a filter. This is summarized [link] in the spectral factorization theorem , which states that $r\in {\mathbb{R}}^N$ is the autocorrelation function of a filter $h\left(n\right)$ if and only if $R\left(\omega \right)\ge 0$ for all $\omega \in [0,\pi ]$ . This turns out to be a necessary and sufficient condition [link] for the existence of $r\left(n\right)$ . Once the autocorrelation vector $r$ is found using existing robust interior-point algorithms, the filter coefficients can be calculated via spectral factorization techniques.

Assuming a valid vector $r\in {\mathbb{R}}^N$ can be found for a particular filter $h$ , the problem presented in [link] can be rewritten as

In [link] the existence condition $R\left(\omega \right)\ge 0$ is redundant since $0\le L{\left(\omega \right)}^2$ and, thus, is not included in the problem definition. For each $\omega$ , the constraints of [link] constitute a pair of linear inequalities in the vector $r$ ; therefore the constraint is convex in $r$ . Thus the change of variable transforms a nonconvex optimization problem in $h$ into a convex problem in $r$ .