FIR Filter Design

Let $h_0,\cdots,h_{n-1}\in{\bf R}$ be the coefficients of a finite impulse response (FIR) filter of length $n$. The filter output signal $y$ is defined by the input signal $u$, by

$$y(k)=\sum^{n-1}_{j=0} h_ju(k-j).$$

The function of the frequency response is $H:[0,2\pi]\rightarrow{\bf C}$ defined by

$$H(\omega)=\sum^{n-1}_{k=0}h_ke^{-ik\omega },$$

where $i=\sqrt{-1}$ and $\omega$ is the frequency parameter. There are several different FIR filter designs, see, for example [5].

In this section, we consider the minimax dB linear phase lowpass FIR filter design,

$$\begin{array}{rll} \mbox{min}&t&\\ \mbox{s.t.}&\frac{1}{t}... ...eq\beta,&\omega_s\leq\omega\leq\pi,\\ &t\geq 1,& \end{array}$$

where $\beta\geq 0$ and $0<\omega_p<\omega_s<\pi.$ For simplicity we assume $n$ is even.

Note that this problem has infinite constraints. Subsequently, we discretize the frequency parameter $\omega$ to obtain a finite-constraint approximation. Moreover, there is a nonlinear term $\frac{1}{t}$ in the first part of the constraints. To convert it to a second order cone constraint, we introduce $u$ such that

$$\frac{1}{t}\leq u.$$

We take the following approach,

$$1\leq ut \Longleftrightarrow v^2\leq 2ut,$$   for $$u\geq 0, t\geq1,$$   and $$v=\sqrt{2}.$$

Now, we obtain the problem in SOCP form,

$$\begin{array}{rll} \mbox{min}&t&\\ \mbox{s.t.}&u\leq2\sum^{n... ... &t\geq 1, v=\sqrt{2},&\\ &v^2\leq 2ut, u\geq 0,& \end{array}$$ (3-9)

where $0=\omega_0<\cdots<\omega_{N_1-1}=\omega_p$, $\omega_s=\omega_{N_1}<\cdots <\omega_N=\omega.$ We use the following data in our test cases.

Example 5. Given $\beta=0.01$, $\omega_p=\pi/2$, and $\omega_s=2\pi /3.$ We descretize the frequency parameter $\omega$ by the uniform step $\frac{\pi}{180}.$ We choose $n=10,20,40,80$ as our test samples.

More details about this problem and the interior point approach can be found in [19].