Characterizations of families of rectangular, finite impulse response, para-unitary systems

Abstract

We here study finite impulse response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class $\mathcal {U}$). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in $\mathcal {U}$, as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs. Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in $\mathcal {U}$, so is the whole family. A key role is played by Hankel matrices.

Appendix: Construction of families of FIRs

For each of the items I  through  VI  from Subsect. 2.2 we here fill-in the following details:

• 1. Construct new polynomials from the original one.

• 2. Show that whenever the original polynomial was in $\mathcal {U}$, so are the newly constructed polynomials.

I.   Reverse polynomial

Using ${\mathbf T}$ from (2.18),

$$\begin{aligned} \begin{array}{lll} F_\mathrm{rev}(z)&{}:=&{} z^{-1}B_n+z^{-2}B_{n-1}+~\ldots ~+z^{n-1}B_2+z^nB_1\\ ~&{}=&{} {\mathbf Z}{\mathbf T}_{n, p}{\mathbf B}_0= \hat{\mathbf B}{\mathbf T}_{n, m}\hat{\mathbf Z}\end{array}. \end{aligned}$$

The realtion with the corresponding Hankel matrix is straightforward and thus omitted.

II.   Preserving the McMillan degree.

For simplicity of exposition we consider as an illustrative example the polynomial F(z) in (2.1) with $n=4$ and the parameter q attaining the values $-1$ and $-2$.

• a.   For $q=-1$ the corresponding Hankel matrix, ${\mathbf H_1}$ from (2.8) can be, without loss of generality, extended with a row and a column of zeros so it is $6p\times 6m$. Now the resulting Hankel matrix may be partitioned in two forms,

  

  which produce the two polynomials in item  II  a.

  Following Theorem 4.1 note that

  

  so indeed the polynoial is in $\mathcal {U}$.

• b.   Substituting in F(z) in (2.12) $q=-2$ yield a Hankel matrix ${\mathbf H}_2$ with three partitionings

  

  so that the three polynomials in item  II  b  are obtained.

These three polynomials are in $\mathcal {U}$, as following Theorem 4.1 one obtains,

To make the last construction more realistic take for example, $m=p=2$ and

In items III through VI we produce more elaborate structures out of a given Hankel matrix. To this end, we find it convenient to introduce the following notation. Let $\alpha ,\beta \ge 0$ and $\eta ,\delta >0$ be integers. One can construct the following $(\alpha +\beta +\eta )\delta \times \eta \delta$ isometry, i.e. $U_\mathrm{Iso}^*U_\mathrm{Iso}=I_{\eta \delta }$,

(5.1)

Similarly, $U_\mathrm{Coiso}$ is the following $\eta \delta \times (\alpha +\beta +\eta )\delta$ coisometry, i.e. $U_\mathrm{Coiso}U_\mathrm{Coiso}^*=I_{\eta \delta }$,

(5.2)

Let $\rho \in \{1,~2,~\ldots ~,~n\}$ be so that $\frac{n}{\rho }$ is natural. Substitute in $~U_\mathrm{Iso}$ and in $~U_\mathrm{Coiso}$ see (5.1), (5.2) respectively: $\alpha =a\rho {p}$, $\beta =b\rho {p}$, with $a, b\ge 0$ integers, $\eta =\frac{n}{\rho }$, $\delta =\rho {p}~$ and consider the pair of products,

(5.3)

and

(5.4)

Both cases yield the same $(a+b+1)np\times (a+b+1)nm$ Hankel matrix, denoted by ${\mathbf H}(a, b, \rho )$. For example,

(5.5)

or

(5.6)

or for $~n=6$

III.   Doubling the powers.

First note that ${\mathbf H}(a, b, 1)$ (i.e. when $\rho =1$) corresponds to $p\times m$-valued polynomial in (2.13) with $\gamma :=a+b+1$.

As another example, the above Hankel matrix ${\mathbf H}(0,~1,~3)$ is associated with the polynomial F(z) in (2.14).

IV.   Rectangular polynomials.

Another sample of a Hankel matrix associated with $U_\mathrm{Iso}{\mathbf B_o}~$ in (5.3) (or $\hat{{\mathbf {B}}}_{o} U_\mathrm{Coiso}$ in (5.4)) is obtained when the parameters are $~a=2$, $b=2$ $\rho =2$, i.e. Now, multiplying ${\mathbf H}(a,~b,~\rho )$ from the  right  by $U_\mathrm{Iso}$ in (5.1) with the parameters $\alpha =(\rho -1)m$, $\beta =0$, $\eta =\frac{n}{\rho }(a+b+1)$ and $\delta =m$ yields the following $(a+b+1)np\times (a+b+1)\frac{n}{\rho }m$ Hankel matrix (here $a=2, b=2, \rho =2$)

(5.7)

which corresponds to the $p\rho \times m$-valued polynomial in IV a.

Similarly, multiplying ${\mathbf H}(a,~b,~\rho )$ from the  left  by $U_\mathrm{Coiso}$ in (5.2) with the parameters $~\alpha =(\rho -1)p$, $\beta =0$, $\eta =\frac{n}{\rho }(a+b+1)$ and $\delta =p~$ yields the following $(a+b+1)\frac{n}{\rho }p\times (a+b+1)nm$ Hankel matrix (here $a=2, b=2, \rho =2$)

(5.8)

which corresponds to the $p\times \rho {m}$-valued polynomial in IV b.

It is easy to verify that if ${\mathbf H_o}$ satisfies the first line in (4.5) then so do all Hankel matrices of the form ${\mathbf H}_0U_\mathrm{Iso}$ (5.7).

Similarly, if ${\mathbf H}_0$ satisfies the second line in (4.5) then so do all Hankel matrices of the form $U_\mathrm{Coiso}{\mathbf H}_0$ (5.8).

In the sequel, we shall adjust our previous notation in (2.8) of the Hankel matrix associated with

$$\begin{aligned} F(z)=z^{-(1+\eta )}B_1+~\ldots ~+z^{-(n+\eta )}B_n \quad \quad \quad \quad \eta \ge 0, \end{aligned}$$

to $~ {\mathbf H}_{\mathbf {B}, n, \eta }$ (in (2.8) the subscricts B and n were omitted, as so far they were evident from the context). For example, with the polynomial

$$\begin{aligned} z^{-(1+\eta )}C_1+~\ldots ~+z^{-(l+\eta )}C_l \quad \quad \quad \quad \eta =0,~1,~2\ldots \end{aligned}$$

one can associate the $(l+\eta )p_c\times (l+\eta )m_c$ Hankel matrix,

V.   Composition of polynomials.

With the pair of poynomilas in (2.16) one can associated the Hankel matrices ${\mathbf H}_{\mathbf {B}, n, 0}$ and ${\mathbf H}_{\mathbf {C}, l, 0}$, which are of dimensions $np_b\times nm_b$ and $lp_c\times nm_c$, respectively.

Out of this pair, one can construct (at least) the  three  following Hankel matrices, all of the form ${\mathbf H}_{\mathbf {D}, n, 0}~$:

• a.   A $~n(p_b+p_c)\times n(m_b+m_c)$ Hankel matrix

  

  or another $~n(p_b+p_c)\times n(m_b+m_c)$ Hankel matrix

  

• b.   For $m_c\ge m_b~$ a $~n(p_b+p_c)\times nm_c$ Hankel matrix

  

• c.   For $p_b\ge p_c$ a ${ np}_{ b}\times (m_b+m_c)n$ Hankel matrix

  

VI.   Product of polynomials

Recall that out of

$$\begin{aligned} \begin{matrix} F_b(z)&{}=&{}z^{-1}B_1+\ldots ~z^{-n}B_n&{}&{}p_b\times \rho \\ ~\\ F_c(z)&{}=&{}z^{-1}C_1+\ldots ~z^{-l}C_l&{}&{}\rho \times m_c \end{matrix} \end{aligned}$$

the following $~p_b\times m_c$-valued polynomial was obtained

$$\begin{aligned} F_d(z):=F_b(z)F_c(z)=z^{-1}\left( z^{-1}D_1+~\ldots ~+z^{-(n+l-1)}D_{n+l-1}\right) ~. \end{aligned}$$

(5.9)

where the coefficients $D_1~,~\ldots ~,~D_{n+l-1}~$ were explicitely given in (2.17).

Expressing, (2.17) in terms of corresponding Hankel matrices yields

$$\begin{aligned} {\mathbf H}_{\mathbf {D}, n+l-1, 1}= {\mathbf H}_{\mathbf {B}, n, l}{\mathbf T}_{m+l, \rho } {\mathbf H}_{\mathbf {C}, l, n}~, \end{aligned}$$

(5.10)

where the Hankel matrices ${\mathbf H}_{\mathbf {B}, n, l}$, ${\mathbf H}_{\mathbf {C}, l, n}~$ and $~{\mathbf H}_{\mathbf {D}, n+l-1, 1}~$ are $(n+l)p_b\times (n+l)\rho$, $(n+l)\rho \times (n+l)m_c~$ and $~(n+l)p_b\times (n+l)m_c~$ respectively, while $~{\mathbf T}_{m+l, \rho }$ is the permutation matrix as in (2.18).

Next, to establish the fact that $F_d(z)$ is in $~\mathcal {U}$, we go through the following steps.

First, from (5.10) note that

$$\begin{aligned} {\mathbf H}_{\mathbf {D}, n+l-1, 1}^*{\mathbf H}_{\mathbf {D}, n+l-1 , 1} ={\mathbf H}_{\mathbf {C}, l, n}^*{\mathbf T}_{m+l, \rho } {\mathbf H}_{\mathbf {B}, n, l}^*{\mathbf H}_{\mathbf {B}, n, l} {\mathbf T}_{m+l, \rho }{\mathbf H}_{\mathbf {C}, l, n}~. \end{aligned}$$

Assuming now that $~F_b\in \mathcal {U}$, it follows from (4.5) that

$$\begin{aligned} {\mathbf H}_{\mathbf {D}, n+l-1, 1}^* {\mathbf H}_{\mathbf {D}, n+l-1, 1}= & {} {\mathbf H}_{\mathbf {C}, l , n}^*{\mathbf T}_{m+l, \rho }\cdot \mathrm{diag}\{I_{(l+1)\rho }\quad \Delta _{(n-1)\rho }\}\cdot {\mathbf T}_{m+l, \rho }{\mathbf H}_{\mathbf {C}, l,n}\\= & {} {\mathbf H}_{\mathbf {C}, l, n}^*\cdot \mathrm{diag}\{\Delta _{(n-1)\rho }\quad I_{(l+1)\rho }\}\cdot {\mathbf H}_{\mathbf {C}, l, n}~, \end{aligned}$$

where $\Delta _{(n-1)\rho }$ is $~(n-1)\rho \times (n-1)\rho ~$ positive semi-definite (weak) contraction.

Assuming now that also $~F_c\in \mathcal {U}$, (carefully following the dimensions) it follows from (4.5) that

$$\begin{aligned} {\mathbf H}_{\mathbf {D}, n+l-1, 1}^* {\mathbf H}_{\mathbf {D}, n+l-1, 1}= \mathrm{diag}\{I_{2p_b}\quad \hat{\Delta }_{(n+l-2)p_b}\}, \end{aligned}$$

where $\hat{\Delta }_{(n+l-2)p_b}$ is a $~(n+l-1)p_b\times (n+l-1)p_b~$ positive semi-definite (weak) contraction, this part is estabilished and indeed $F_d\in \mathcal {U}$.

Showing the relation for ${\mathbf H}_{\mathbf {D}, n+l-1, 1} {\mathbf H}_{\mathbf {D}, n+l-1, 1}^*$ is quite similar and thus omitted. $\square$