4.6 Design of orthogonal pr-fir filterbanks via halfband spectral

Design of orthogonal pr-fir filterbanks via halfband spectral factorization

Recall that analysis-filter design for orthogonal PR-FIR filterbanks reduces to the design of a real-coefficient causalFIR prototype filter $H_0(z)$ that satisfies the power-symmetry condition

If, in addition to being real-valued,

First, realize $F(e^{i\omega })$ is easily modified to ensure non-negativity: construct $q(n)=f(n)+\epsilon \delta (n)$ for sufficiently large $\varepsilon$ , which will raise $F(e^{i\omega })$ by $\epsilon$ uniformly over $\omega$ (see ).

The resulting $Q(z)$ is non-negative and satisfies the amplitude-symmetry condition $Q(e^{i\omega })+Q(e^{i(\pi -\omega )})=1+2\epsilon$ . We will make up for the additional gain later. The procedureby which $H_0(z)$ can be calculated from the raised halfband $Q(z)$ , known as spectral factorization , is described next.

Since $q(n)$ is conjugate-symmetric around the origin, the roots of $Q(z)$ come in pairs $\{a_i, \frac{1}{\overline{a_i}}\}()$ . This can be seen by writing $Q(z)$ in the factored form below, which clearly corresponds to a polynomial with coefficientsconjugate-symmetric around the $0^{\mathrm{th}}$ -order coefficient.

Let us assume, without loss of generality, that $\left|a_i\right|< 1$ . If we form $H_0(z)$ from the roots of $Q(z)$ with magnitude less than one:

Actually, in order to make $\left|H_0(e^{i\omega })\right|^2=Q(e^{i\omega })$ , we are not required to choose all roots inside the unit circle; it is enough to choose one root from everyunit-circle-symmetric pair. However, we do want to ensure that $H_0(z)$ has real-valued coefficients. For this, we must ensure that roots come in conjugate-symmetric pairs, i.e. , pairs having symmetry with respect to the real axis in the complex plane ( ).

Because $Q(z)$ has real-valued coefficients, we know that its roots satisfythis conjugate-symmetry property. Then forming $H_0(z)$ from the roots of $Q(z)$ that are strictly inside (or strictly outside) the unitcircle, we ensure that $H_0(z)$ has real-valued coefficients.

Finally, we say a few words about the design of the halfband filter $F(z)$ . The window design method is one technique that could be used in this application. The window design methodstarts with an ideal lowpass filter, and windows its doubly-infinite impulse response using a window function withfinite time-support. The ideal real-valued zero-phase halfband filter has impulse response (where $n\in \mathbb{Z}$ ):

Design procedure summary

We now summarize the design procedure for a length- $N$ analysis lowpass filter for an orthogonal perfect-reconstruction FIRfilterbank:

• Design a zero-phase real-coefficient halfband lowpass filter $F(z)=\sum_{n=-(N-1)}^{N-1} f(n)z^{-n}$ where $N$ is a positive even integer (via, e.g. , window designs, LS, or equiripple).
• Calculate $\epsilon$ , the maximum negative value of $F(e^{i\omega })$ . (Recall that $F(e^{i\omega })$ is real-valued for all $\omega$ because it has a zero-phase response.) Then create "raised halfband" $Q(z)$ via $q(n)=f(n)+\epsilon \delta (n)$ , ensuring that $Q(e^{i\omega })\ge 0$ , forall $\omega$ .
• Compute the roots of $Q(z)$ , which should come in unit-circle-symmetric pairs $\{a_i, \frac{1}{\overline{a_i}}\}()$ . Then collect the roots with magnitude less than one into filter ${\widehat{H}}_0(z)$ .
• ${\widehat{H}}_0(z)$ is the desired prototype filter except for a scale factor. Recall that we desire $$\left|H_0(e^{i\omega })\right|^2+\left|H_0(e^{i(\pi -\omega )})\right|^2=1$$ Using Parseval's Theorem , we see that $\{{\widehat{h}}_0(n)\}()$ should be scaled to give $\{h_0(n)\}()$ for which $\sum_{n=0}^{N-1} h_0(n)^2=\frac{1}{2}$ .