Polynomials Polynomials P N x have an interesting scaling property that if we scale the domain variable x by a real number M, the resulting function is still a polynomial with the same degree. A polynomial of degree N over real numbers is defined as follows: P N x k 0 N a k x k The coefficients a k of the monomial x k are real for real polynomials. In general they could be complex numbers and P N x can be defined over the complex domain. Consider P N x M f x , we find f x k 0 N a k x M k k 0 N a k M k x k k 0 N b k x k So f x is also a polynomial over x with degree N.

Splines We know that splines are smoothly connected pieces of polynomials. Let φ x be a spline. φ x is a piecewise polynomial over integer intervals. Dilate φ x by M. φ x M is also a piecewise polynomial over integer intervals with same degree. So it can be represented by using B-Spline Bases. In particular integer dilates of B-splines can be represented by the unscaled B-spline bases themselves which gives us a scaling equation for B-Splines.

M-Scale Relation Consider shifted causal B-splines φ x n β n x n 1 2 . These are obtained by the ( n 1 )-fold convolution of φ 0 x , the Haar scaling function. φ 0 x M k 0 M 1 φ 0 x k k 0 M 1 h M 0 k φ 0 x k where h M 0 k is the filter with z-transform H M 0 z k 0 M 1 z k (discrete pulse of length M). Convolve φ 0 x M with itself n 1 times. This gives us the following result: φ n x M k 0 M 1 k h M n k φ n x k where H M n z 1 M n H M 0 z n 1 1 M n k 0 M 1 z k Thus we have an M-scale relation for all integers M. If M 2 , H 2 n z is a binomial filter whose coefficients form the Pascal's triangle.

Spline Wavelets We saw that shifted B-splines φ n x β n x n 1 2 are scaling functions. We can form a nested set of subspaces V j , such that j V j V j + 1 V j Span k φ 2 j t k And they form a multiresolution analysis system that is dense in L 2 . How do we find the spline wavelets? φ n x k are not orthogonal so we need to find either biorthogonal or semi-orthogonal wavelets. The biorthogonal spline wavelets are compact and give FIR filters for the wavelet filter bank. The semi-orthogonal spline wavelets generate IIR filter bank structures. There are also other spline wavelets such as Stromberg's one sided orthogonal splines and the Battle-Lemarie Wavelets. BIORTHOGONAL SPLINE WAVELETS These are the wavelets obtained using biorthogonality of the filters on synthesis and analysis sides of the filter bank. For further details refer to BIORTHOGONAL WAVELETS. φ 1 x β 1 x 1 , the Hat function gives the 5/3 filter bank. H z 1 z 1 2 2 is the scaling filter for φ 1 x . The biorthogonal scaling filter is H ^ z 1 z 1 2 2 1 4 z 1 z 2 2 . The scaling filters and wavelets generated are plotted in .

Spline 5/3 Biorthogonal Wavelets

IIR and Semiorthogonal Basis φ t k for V 0 are non-orthogonal. We impose wavelets w t k to be orthogonal to φ t . ⊥ V 0 W 0 and V 0 W 0 as in orthogonal scaling functions. w t k need not be an orthogonal set. The analysis filters are IIR. The high pass synthesis filter G z is given by G z z 1 2 N H z A z H ~ z and G ~ z are the low pass and high pass analysis filters respectively. H ~ z 2 z H z A z A z 2 G ~ z 2 H z A z 2 The A z 2 in the denominator makes the analysis filters IIR, where A z is the z-transform of the sequence, a k φ t φ t k .