Report, 20150423, On Distance Rate

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Section 1. On "Distance Rate"

Definition 1. The Riemannian distance of two positive-definite symmetric matrices $P_1$ and $P_2$ is \[d\left( {{P_1},{P_2}} \right) = {\left\| {\log \left( {{P_1}^{ - 1}{P_2}} \right)} \right\|_F} = {\left( {\sum\limits_{i = 1}^n {{{\ln }^2}{\lambda _i}} } \right)^{\frac{1}{2}}}\] where $\lambda_i,~i=1,\ldots,n$ are the eigenvalues of $P_1^{-1}P_2$.

Lemma 1. If a function $H\left( z \right) = \sum\nolimits_{n = 0}^\infty  {{h_n}{z^{ - n}}} $ is minimum-phase and $h_0 \ne 0$, then \begin{equation} \nonumber \ln h_0^2 = \frac{1}{{2\pi }}\int_{ - \pi }^\pi  {\ln {{\left| {H\left( {{e^{j\omega }}} \right)} \right|}^2}d\omega }. \end{equation}

For a minimum-phase system, we have shown that \[\left[ {\begin{array}{*{20}{c}} {{y_0}}\\ {{y_1}}\\  \vdots \\ {{y_{n - 1}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{h_0}}&0& \cdots &0\\ {{h_1}}&{{h_0}}& \cdots &0\\  \vdots & \vdots &{}& \vdots \\ {{h_{n - 1}}}&{{h_{n - 2}}}& \cdots &{{h_0}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_0}}\\ {{u_1}}\\  \vdots \\ {{u_{n - 1}}} \end{array}} \right] \buildrel \Delta \over = A_n\left[ {\begin{array}{*{20}{c}} {{u_0}}\\ {{u_1}}\\  \vdots \\ {{u_{n - 1}}} \end{array}} \right].\]

Suppose that $h_0 \ne 0$ which implies that $A_n$ is nonsingular and let the system input be Gaussian white with zero mean and $\sigma_u^2$ as covariance. Let ${\tilde u_k} = {[ {\begin{array}[b]{*{20}{c}} {{u_0}}&{{u_1}}& \cdots &{{u_{k-1}}} \end{array}}]^T}$ and ${\tilde y_k} = {[ {\begin{array}[b]{*{20}{c}} {{y_0}}&{{y_1}}& \cdots &{{y_{k-1}}} \end{array}}]^T}$, it‘s easy to see that \begin{equation} \label{def_Uk_Yk} \begin{aligned} {U_k} &= E\left\{ {{{\tilde u}_k}{{\tilde u}_k}^T} \right\} = \sigma_u^2 I_k\\ {Y_k} &= E\left\{ {{{\tilde y}_k}{{\tilde y}_k}^T} \right\} = \sigma_u^2 A_k A_k^T \end{aligned} \end{equation} which implies that ${U_k}^{ - 1}{Y_k} = A_k A_k^T$.

Now we define the distance rate as

\begin{equation} \label{def_dist_rate}
\begin{aligned}
\bar d\left( {U,Y} \right) &= \mathop {\lim \sup }\limits_{k \to \infty } \frac{{d\left( {{U_k},{Y_k}} \right)}}{k^{1/2}} \\
 &= \mathop {\lim \sup }\limits_{k \to \infty }{\left( {\frac{1}{k}\sum\limits_{i = 1}^k {{{\ln }^2}{\lambda _i}\left( {U_k^{ - 1}{Y_k}} \right)} } \right)^{\frac{1}{2}}}.
\end{aligned}
\end{equation}

Directly substitute (\ref{def_Uk_Yk}) into (\ref{def_dist_rate}), we have \begin{equation}\nonumber \begin{aligned} \frac{{d\left( {{U_k},{Y_k}} \right)}}{{{k^{1/2}}}} &= {\left( {\frac{1}{k}\sum\limits_{i = 1}^k {{{\ln }^2}{\lambda _i}\left( {{A_k}A_k^T} \right)} } \right)^{\frac{1}{2}}} \\ &\ge \frac{1}{k}\sum\limits_{i = 1}^k {\ln {\lambda _i}\left( {{A_k}A_k^T} \right)}  \\ &= \frac{1}{k}\ln \left| {{A_k}A_k^T} \right| \\ &= \frac{1}{k}\ln {\left| {{h_0}} \right|^{2k}} \\ &= \ln {\left| {{h_0}} \right|^2} \end{aligned} \end{equation} or, more clearly, $\bar d\left( {U,Y} \right) \ge \ln {\left| {{h_0}} \right|^2}$. Furthermore, by lemma 1 and Papoulis‘s theorem, we know that \[\bar d\left( {U,Y} \right) \ge \frac{1}{{2\pi }}\int_{ - \pi }^\pi  {\ln {{\left| {H\left( {{e^{j\omega }}} \right)} \right|}^2}d\omega }  = \bar h\left( y \right) - \bar h\left( u \right).\] On the other hand, the distance rate is bounded by the absolute value of the logarithm of the maximum singular value of $A_k$‘s, i.e., \begin{equation} \begin{aligned} \bar d\left( {U,Y} \right) &= \mathop {\lim \sup }\limits_{k \to \infty } {\left( {\frac{1}{k}\sum\limits_{i = 1}^k {{{\ln }^2}{\lambda _i}\left( {{A_k}A_k^T} \right)} } \right)^{\frac{1}{2}}} \\ &\le \mathop {\lim \sup }\limits_{k \to \infty } {\left( {\frac{1}{k}\sum\limits_{i = 1}^k {\max\left\{ {{{\ln }^2}{\lambda _i}\left( {{A_k}A_k^T} \right)} \right\}} } \right)^{\frac{1}{2}}} \\ &\le \mathop {\sup }\limits_k \left( {\left| {\ln {\lambda _i}\left( {{A_k}A_k^T} \right)} \right|} \right). \end{aligned} \end{equation} Generally, we have \[\mathop {\lim \sup }\limits_{k \to \infty } \frac{1}{k}\ln \left| {U_k^{ - 1}{Y_k}} \right| \le \bar d\left( {U,Y} \right) \le \mathop {\sup }\limits_k \left( {\left| {\ln {\lambda _i}\left( {U_k^{ - 1}{Y_k}} \right)} \right|} \right).\] However, the upper bound $\mathop {\sup }\limits_k \left( {\left| {\ln {\lambda _i}\left( {U_k^{ - 1}{Y_k}} \right)} \right|} \right)$ may be conservative or even divergent.

Section 2. A Theorem

Let $f(x)$ be a real-valued function of the class $L$, \[{c_n} = \frac{1}{{2\pi }}\int_{ - \pi }^\pi  {{e^{-inx}}f\left( x \right)dx} ,~~~n = 0, \pm 1, \pm 2, \ldots ,\] its Fourier coefficients. We consider the finite Toeplitz forms \begin{equation} \begin{aligned} {T_n}\left( f \right) &= \sum\limits_{i,j = 0,1,2, \ldots ,n} {{c_{j - i}}{u_i}{{\bar u}_j}}  \\ &= \frac{1}{{2\pi }}\int_{ - \pi }^\pi  {{{\left| {{u_0} + {u_1}{e^{ix}} +  \cdots  + {u_n}{e^{inx}}} \right|}^2}f\left( x \right)dx} ,\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n = 0,1,2, \ldots. \end{aligned} \end{equation} The eigenvalues of the Hermitian form $T_n(f)$ are defined as the roots of the characteristic equation $\det T_n(f-\lambda)=0$; we denote them by $${\lambda _1^{\left( n \right)}},{\lambda _2^{\left( n \right)}},\ldots,{\lambda _n^{\left( n \right)}}.$$ As well known, these values are all real.

Theorem 1. [1] We denote by $m$ and $M$ the ‘essential‘ lower bound and upper bound of $f(x)$, respectively, and assume that $m$ and $M$ are finite. If $G(\lambda)$ is any continuous function defined in the finite interval $m \le \lambda \le M$, we have \[\mathop {\lim }\limits_{n \to \infty } \frac{{G\left( {\lambda _1^{\left( n \right)}} \right) + G\left( {\lambda _2^{\left( n \right)}} \right) +  \cdots  + G\left( {\lambda _{n + 1}^{\left( n \right)}} \right)}}{{n + 1}} = \frac{1}{{2\pi }}\int_{ - \pi }^\pi  {G\left[ {f\left( x \right)} \right]dx}.\]

Corollary. [2] \[\mathop {\lim }\limits_{n \to \infty } \frac{{\sum\nolimits_{i = 1}^n {G\left( {{\lambda _i}\left( {\Sigma \left( {e_1^k} \right)} \right)} \right)} }}{n} = \frac{1}{{2\pi }}\int_{ - \pi }^\pi  {G\left[ {{{\hat F}_e}\left( \omega  \right)} \right]d\omega } .\]

References

[1] Ulf Grenander and Gabor Szego. Toeplitz Forms and Their Applications. Univ. Calif. Press, 2001.

[2] N.C. Martins and M.A. Dahleh. Fundamental limitations of performance in the presence of finite capacity feedback. In American Control Conference, 2005. Proceedings of the 2005, pages 79–86 vol. 1, June 2005.

Report, 20150423, On Distance Rate

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原文地址：http://www.cnblogs.com/aujun/p/4451487.html