Chebyshev nonuniform sampling cascaded with the discrete cosine transform for optimum interpolation

This correspondence presents a new method for discrete representation of signals { g ( t ) , IE [ O , L ] , g e C*(O, L ) } consisting of a cascade having two stages: a) nonuniform sampling according to Chebyshev polynomial roots; and b) discrete cosine transform applied on the nonuniformly taken samples. We have proved that the considered signal samples and the coefficients of the corresponding Chehyshev polynomial finite series are essentially a discrete cosine transform pair. It provides a method for fast computation of the coefficients of the optimum interpolation formula (which minimizes the maximum instantaneous error). If the signal g ( t ) is band limited and has a finite energy, we deduce the condition of convergence for interpolation. INTRODUCTION The Shannon sampling theorem and its variants [ 2 ] , [8], [I41 are well known as performing the reconstruction of a band-limited signal from the knowledge of its uniformly taken samples. There are also a few approaches to signal reconstruction from nonuniformly spaced samples [6], [8], [10] [12] . The classical results of function interpolation theory in computational mathematics [4], [7] show that the best choice of interpolation points to minimize the maximum modulus of the instantaneous error corresponds to the roots of the Chebyshev polynomials of the first kind. Recently, considerable attention has been paid to the use of orthogonal transforms applied to the uniformly spaced samples. These Manuscript received June 29. 1987; revised September 27. 1989. The author is with the Faculty of Electronics and Telecommunications, IEEE Log Number 9037810. Polytechnic Institute of Bucharest, Bucharest 16. Romania 77206. for h = 0 the values x, are given by ( 2 ) and T / , ( x ) is the hth degree normalized Chebyshev polynomial 1 T o ( x ) = -; T 1 2 ( x ) = ~ ~ ( x ) = cos ( h arc c o s x ) ; d5 h = 1, . . . , N 1. ( 4 ) In (3), h is the row index and; is the column index. Denote by gN = ( g(t,)),”=o’ the vector of nonuniformly spaced samples according to the Chebyshev sampling grid vector t N . Denote by CN = ( CO C, . . . CN_ , ) T the direct discrete cosine transformation (DDCT) of g,, defined as C N = WEDCT . g N ( 5 ) wEDCT = ( J ~ / N ) w E ‘ ~ . ( 6 ) g, = C N ( 7 ) where The inverse discrete cosine transform (IDCT) is expressed by .