  In this paper we provide a complete and unifying characterization of
compactly supported univariate scalar orthogonal wavelets and vector-valued or
matrix-valued orthogonal multi-wavelets. This characterization is based on
classical results from system theory and basic linear algebra. In particular,
we show that the corresponding wavelet and multi-wavelet masks are identified
with a transfer function $$
  F(z)=A+B z (I-Dz)^{-1} \, C, \quad z \in \mathbb{D}=\{z \in \mathbb{C} \ : \
|z| < 1\}, $$ of a conservative linear system. The complex matrices $A,\ B, \
C, \ D$ define a block circulant unitary matrix. Our results show that there
are no intrinsic differences between the elegant wavelet construction by
Daubechies or any other construction of vector-valued or matrix-valued
multi-wavelets. The structure of the unitary matrix defined by $A,\ B, \ C, \
D$ allows us to parametrize in a systematic way all classes of possible wavelet
and multi-wavelet masks together with the masks of the corresponding refinable
functions.
