The Mallat algorithm for finite length signals is equivalent to a matrix transform on vector space, and the transform matrix is a finite 2-circular matrix. As a new concept, a minimal matrix, which is also a 2-circular matrix, is put forward. Through new developments of definitions and theorems for a minimal matrix, new algorithms for constructing wavelets filters are obtained. Employing these algorithms, we give several examples of orthonormal and biorthogonal wavelet filters. The methods avoid using a Z-transform or Fourier transform, so it is relatively simple to construct FIR wavelet filters. Because many filters constructed go beyond traditional wavelets, the new concept of discrete hyperwavelets transforms is put forward. Finally, an application in image compression is discussed briefly.