Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions
No edit summary |
Skew! |
||
Line 74: | Line 74: | ||
== Otherwise normed tuning == | == Otherwise normed tuning == | ||
If there is a weight, such as | If there is a weight W and/or a skew X, such as CTWE tuning, the weight-skew should be applied to the map and the constraint first: | ||
<math>\displaystyle | <math>\displaystyle | ||
\begin{align} | \begin{align} | ||
V &= | V &= AWX \\ | ||
M_{\rm C} &= | M_{\rm C} &= (WX)^+ B_{\rm C} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Working from here, we find the weighted projection map (P<sub>C</sub>)<sub> | Working from here, we find the weighted projection map (P<sub>C</sub>)<sub>WX</sub>: | ||
<math>\displaystyle | <math>\displaystyle | ||
(P_{\rm C}) | (P_{\rm C})_{WX} = | ||
\begin{bmatrix} | \begin{bmatrix} | ||
V^+VM_{\rm C} & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix} | V^+VM_{\rm C} & \begin{matrix} O \\ V_{\rm M}^+V_{\rm M} \end{matrix} | ||
Line 94: | Line 94: | ||
To reconstruct the original projection map, apply | To reconstruct the original projection map, apply | ||
<math>\displaystyle P_{\rm C} = | <math>\displaystyle P_{\rm C} = WX (P_{\rm C})_{WX} (WX)^+</math> | ||
== Nontrivially constrained tuning == | == Nontrivially constrained tuning == | ||
Line 128: | Line 128: | ||
<math>\displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1}</math> | <math>\displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1}</math> | ||
Similarly, if there is a weight, it should be applied to the map and the constraint first: | Similarly, if there is a weight and/or a skew X, it should be applied to the map and the constraint first: | ||
<math>\displaystyle | <math>\displaystyle | ||
\begin{align} | \begin{align} | ||
V &= | V &= AWX \\ | ||
M_{\rm C} &= | M_{\rm C} &= (WX)^+ B_{\rm C} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
and then the basis transformation matrix should be found out in this | and then the basis transformation matrix should be found out in this weight-skewed space: | ||
<math>\displaystyle | <math>\displaystyle | ||
S_{WX} = [\begin{matrix} M_{\rm C} & M_{\rm C}^\perp \end{matrix}] | |||
</math> | </math> | ||
We should apply this S<sub> | We should apply this S<sub>WX</sub> to the weight-skewed map and the weight-skewed constraint to convert them into the working basis: | ||
<math>\displaystyle | <math>\displaystyle | ||
\begin{align} | \begin{align} | ||
V_S &= | V_S &= VS_{WX} \\ | ||
(M_{\rm C})_S &= | (M_{\rm C})_S &= S_{WX}^+ M_{\rm C} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Proceed as before. The projection map found this way will be | Proceed as before. The projection map found this way will be weight-skewed and in the working basis. To reconstruct the original projection map, apply | ||
<math>\displaystyle P_{\rm C} = | <math>\displaystyle P_{\rm C} = WXS (P_{\rm C})_{WXS} S^{-1} (WX)^+</math> | ||
== Example == | == Example == |