مصفوفة تحويل

في نظرية النظم و علم الضبط و في تعاملنا مع نظم ذات عدة مداخل و عدة مخارج أي نظم من نوع ميمو (mimo = multiple input multiple output) فإن مفهوم مصفوفة التحويل transfer matrix يحل محل مفهوم دالة تحويل. وهي صياغة على شاكلة مصفوفة بلوك-توپلتس لمعادلة في متغيرين، تميّز refinable functions. Refinable functions تلعب دوراً هاماً في نظرية المويجات ونظرية العناصر المتناهية.

For the mask ${\displaystyle{\displaystyle h}}$, which is a vector with component indexes from ${\displaystyle{\displaystyle a}}$ to ${\displaystyle{\displaystyle b}}$, the transfer matrix of ${\displaystyle{\displaystyle h}}$, we call it ${\displaystyle{\displaystyle T_{h}}}$ here, تـُعرَّف كالتالي

${\displaystyle{\displaystyle(T_{h})_{j,k}=h_{2\cdot j-k}.}}$

وحرفياً

${\displaystyle{\displaystyle T_{h}={\begin{pmatrix}h_{a}&&&&&\\ h_{a+2}&h_{a+1}&h_{a}&&&\\ h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ &h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\ &&&h_{b}&h_{b-1}&h_{b-2}\\ &&&&&h_{b}\end{pmatrix}}.}}$

The effect of ${\displaystyle{\displaystyle T_{h}}}$ can be expressed in terms of the downsampling operator "${\displaystyle{\displaystyle\downarrow}}$":

${\displaystyle{\displaystyle T_{h}\cdot x=(h*x)\downarrow 2.}}$

 الخصائص

Actually not ${\displaystyle{\displaystyle n-2}}$ convolutions are necessary, but only ${\displaystyle{\displaystyle 2\cdot\log_{2}n}}$ ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the Fast Fourier transform.

• From the previous statement we can derive an estimate of the spectral radius of ${\displaystyle{\displaystyle\varrho(T_{h})}}$. It holds

${\displaystyle{\displaystyle\varrho(T_{h})\geq{\frac{a}{\sqrt{\#h}}}\geq{\frac{1}{\sqrt{3\cdot\#h}}}}}$

where ${\displaystyle{\displaystyle\#h}}$ is the size of the filter and if all eigenvalues are real, it is also true that

${\displaystyle{\displaystyle\varrho(T_{h})\leq a}}$,

where ${\displaystyle{\displaystyle a=\|C_{2}h\|_{2}}}$.

 انظر أيضاً

• محددة هورويتس Hurwitz determinant

 المراجع

• Strang, Gilbert (1996). "Eigenvalues of ${\displaystyle{\displaystyle(\downarrow 2){H}}}$ and convergence of the cascade algorithm". IEEE Transactions on Signal Processing. Vol. 44. pp. 233–238.
• Thielemann, Henning (2006). Optimally matched wavelets (PhD thesis). (contains proofs of the above properties)