Accelerated plane-wave destruction

Appendix A: Polynomial form of PWD

If all the coefficients of $B(Z_t)$ are polynomials of $p$, equation 4 is also a polynomial of $p$, and the plane-wave destruction equation becomes in turn a polynomial equation of $p$. The problem is to design a $2N+1$ points filter $B(Z_t)$ with polynomial coefficients such that the allpass system $H(Z_t)=\frac{B(1/Z_t)}{B(Z_t)}$ can approximate the phase-shift operator $Z_t^p=e^{j\omega p}$. Denoting the phase response of the system as $\theta(\omega)$, that is $H(e^{j\omega})=e^{j\theta(\omega)}$, the group delay of the system is

$$\tau(\omega)=\frac{\partial \theta(\omega)}{\partial \omega}.$$ (24)

The maximally flat criteria designs a filter with a smoothest phase response. There are $2N$ unknown coefficients in $H(Z_t)$, so we can add $2N$ flat constraints for the first $2N-$th order deviratives of the phase response. It becomes (Zhang, 2009, equation 7)

$$\left\{\begin{array}{l} \tau(\omega)=p \\ \displaystyle{\fr... ...partial \omega^n}}=0 \qquad n=1,2,\dots,2N \end{array}\right.,$$ (25)

which is equivalent to the following linear maximally flat conditions (Thiran, 1971):

$$\sum_{k=-N}^N (d-k)^{2n+1}b_k =0,$$ (26)

where $n=0,1,\dots,2N-1$ and $d=p/2$ is the fractional delay of $B(1/Z_t)$ or $1/B(Z_t)$.

In order to solve $b_k$ from the above equations, Thiran (1971) used an additional condition $b_0=1$, which leads $b_k(k\neq 0)$ to be a fractional function of $p$. Differently from that, we use the following condition,

$$\sum_{k=-N}^N b_k =1,$$ (27)

where $b_k$ can be proved to be polynomials of $p$.

Let vector $\mathbf b=[b_0,b_N,\dots,b_1,b_{-1},\dots,b_{-N}]^\textrm T$. Combining equations I-3 and I-4, we rewrite them into the following matrix form:

$$\left[\begin{array}{c\vert cccccc} 1 & 1 & \dots & 1 & 1 & \... ...begin{array}{c} 1 0 0 \vdots 0 \end{array}\right].$$

The matrix on the left side, denoted as $\tensor V$, can be split into four blocks $\left[\begin{array}{c\vert c} \tensor A & \tensor B \hline \tensor C & \tensor D \end{array}\right]$ as shown above. Following the lemma of matrix inversion,

$$\tensor V^{-1}=\left[\begin{array}{cc} (\tensor A-\tensor B\... ...\tensor C)^{-1}\tensor B\tensor D^{-1} \\ \end{array}\right],$$ (28)

therefore the coefficients

$$\mathbf b=\tensor V^{-1}[1,0,\dots,0]^\textrm T= \left[\begi... ... A-\tensor B\tensor D^{-1}\tensor C)^{-1} \end{array}\right].$$ (29)

Let subindex $i=-N,-N+1,\dots,-1,1,2,\dots,N$ and $x_i=d+i$. Submatrix $\tensor D$ can be expressed as

$$\begin{eqnarray*} \tensor D &=& \tensor E\tensor X \\ &=& \left[\begin{array}{c... ...} \vdots x_{-1} x_1 \vdots x_N \end{array}\right], \end{eqnarray*}$$

so $\tensor D^{-1}=\tensor X^{-1}\tensor E^{-1}$. Denoting $\tensor U=\tensor E^{-1}$ with elements $u_{ij}, j=1,2,\dots,2N$, as $\tensor E$ is a Vandermonde matrix, $u_{ij}$ and Lagrange intepolating polynomials have the following relationship:

$$\sum_{j=1}^{2N}u_{ij}x^{2j-2}=\ell_i(x),$$ (30)

where $i=-N,\dots,-1,1,\dots,N$, and $\ell_i(x)$ is the Lagrange polynomial related to the basis $d+i$,

$$\ell_i(x)= \prod_{-N\leq m\leq N}^{m\neq i,m\neq 0} \frac{x^2-(d+m)^2}{(d+i)^2-(d+m)^2}.$$ (31)

Substituting the above equation, $u_{ij}$ and $x$ into equation I-7, we can prove equation I-7. It follows that

$$[\tensor E^{-1}\tensor C]_i=d\ell_i(d),$$ (32)

$$[\tensor D^{-1}\tensor C]_i= [\tensor X^{-1}\tensor E^{-1}\tensor C]_i= \frac{d}{d+i}\ell_i(d),$$ (33)

with

$$\ell_i(d)=\frac{(-1)^{i+1}N!N!}{(N+i)!(N-i)!} \frac{d+i}{d} \prod_{m=-N}^N \frac{2d+m}{2d+m+i}.$$ (34)

Thus hence

$$\tensor A-\tensor B\tensor D^{-1}\tensor C = \sum_{i=-N}^N (-1)^{i}\frac{N!N!}{(N+i)!(N-i)!} \prod_{m=-N}^N \frac{p+m}{p+m+i}$$

$$= \frac{(4N)!N!N!} {(2N)!(2N)!} \frac{1}{ \displaystyle{\prod_{m=N+1}^{2N}(m^2-p^2)}}$$ (35)

and

$$[\tensor A-\tensor B\tensor D^{-1}\tensor C]^{-1} = \frac{(2N)!(2N)!}{(4N)!N!N!} \prod_{m=N+1}^{2N}(m^2-p^2).$$ (36)

It is the coefficient $b_0$, a $2N$-th degree polynomial of $p$. Substituting it into equation I-6, the coefficients at $k=\pm 1,\pm 2,\dots \pm N$ are expressed as

$$b_k = -[\tensor D^{-1}\tensor C]_k [\tensor A-\tensor B\tensor D^{-1}\tensor C]^{-1}$$

$$= \frac{(2N)!(2N)!}{(4N)!(N+k)!(N-k)!} \prod_{m=0}^{N-1-k}(m-2N+p) \prod_{m=0}^{N-1+k}(m-2N-p).$$ (37)

With the additional condition I-4 in $2N+1$ points approximation, all the coefficients are polynomials of $p$ of $2N$-th degree. Thus the plane-wave destruction equation 6 therefore is proved to be a polynomial equation of $2N$-th degree.

Accelerated plane-wave destruction