The Hilbert Transform Of B-spline Wavelets

The wavelets and the Hilbert transform are widely used in the field of signal processing.Because of this,it is of practical significance to study the relationship between these two mathematical concepts.Since B-spline wavelets have many good properties in vanishing moment,symmetry,compact support,etc.,this thesis focuses on the Hilbert transform of B-spline wavelets with orderm.We firstly describe the B-spline wavelets with ordermin the form of piecewise polynomials.Then this paper explores the important properties of Pascal's triangle transformation.Based on the discovered results,an explicit form of the Hilbert transform of the B-spline wavelets with ordermis established.Furthermore,this paper discusses in detail the vanishing moment,symmetry and asymptote behavior of the Hilbert transform of the B-spline wavelets with orderm.It is found that the vanishing moment and symmetry of the Hilbert transform of them-order B-spline wavelets are related to the parity of the orderm.When the ordermis even,the vanishing moment of the Hilbert transform of the B-spline wavelet is 2m+1,and its image is antisymmetric with respect to x=m-1/2;when the ordermis odd,the vanishing moment of the Hilbert transform of the B-spline wavelet is 2m,and its image is symmetric about x=m-1/2.Moreover,it is discovered that when the wavelet is symmetric,the Hilbert transform of the wavelet is anti-symmetric;when the wavelet is anti-symmetric,the Hilbert transform of the wavelet is symmetric.The asymptote behavior of them-order B-spline wavelet Hilbert transform is positively correlated with the order,and it decays to 0 at a speed of x^-(m+1) when??.In order to help the engineers to directly apply the results developed in this paper,two examples with m=3 and m=4 as well as the graphs of these two B-spline wavelets and their Hilbert transforms are given.