simplicity, we only discuss the real valued wavelet tight frame systems, but note that it is not difficult to extend the idea to the complex case.

Denote by $\msH$ a Hilbert space and let $\la\cdot,\cdot\ra$ be the inner product defined on $\msH$. A countable set $\{\bvphi_n:n\in\Z\}\subseteq\msH$ is called a tight frame on $\msH$ if $$\begin{aligned}
\label{TightFrame}
\|\bu\|^2=\sum_{n\in\Z}|\la\bu,\bvphi_n\ra|^2~~~~~\text{for all}~~~\bu\in\msH.\end{aligned}$$ Given $\{\bvphi_n:n\in\Z\}\subseteq\msH$, we define the analysis operator $\bsW:\msH\to\ell_2(\Z)$ as $$\begin{aligned}
\bu\in\msH\mapsto \bsW\bu=\{\la\bu,\bvphi_n\ra:n\in\Z\}\in\ell_2(\Z).\end{aligned}$$ The synthesis operator $\bsW^T:\ell_2(\Z)\to\msH$ is defined as the adjoint of $\bsW$: $$\begin{aligned}
\bc\in\ell_2(\Z)\mapsto\bsW^T\bc=\sum_{n\in\Z}\bc[n]\bvphi_n\in\msH.\end{aligned}$$ Then $\{\bvphi_n:n\in\Z\}$ is a tight frame on $\msH$ if and only if $\bsW^T\bsW=\bsI$ where $\bsI$ is the identity on $\msH$. It follows that, for a given tight frame $\{\bvphi_n:n\in\Z\}$, we have the following canonical expression: $$\begin{aligned}
\bu=\sum_{n\in\Z}\la\bu,\bvphi_n\ra\bvphi_n,\end{aligned}$$ with $\bsW\bu=\{\la\bu,\bvphi_n\ra:n\in\Z\}$ being called the canonical tight frame coefficients. Hence, the tight frames are extensions of orthonormal bases to the redundant systems. In fact, a tight frame is an orthonormal basis if and only if $\|\bvphi_n\|=1$ for all $n\in\Z$.

One of the most widely used class of tight frames is the discrete wavelet frame generated by a set of finitely supported filters $\{\bq_1,\cdots,\bq_m\}$. In this paper, we only discuss the undecimated wavelet frames, which are also known as the translation 