IEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 1, JANUARY 2003 15 Finite-Channel Chromatic Derivative Filter Banks M. Cushman, Member, IEEE, M. J. Narasimha, Fellow, IEEE, and P. P. Vaidyanathan, Fellow, IEEE Abstract—Two recent contributions discussed the theory of perfect-reconstruction (PR) chromatic derivative filter banks comprising an infinite number of channels. This letter extends the theory to the case of finite channels. A novel time domain procedure is delineated for designing the synthesis filters that achieve PR in this case. Index Terms—Biorthogonal filter banks, chromatic derivatives, perfect reconstruction. I. INTRODUCTION I N THIS LETTER, we present the problem of multichannel sampling discussed in [3] and [4] from the view of a contin- uous-time biorthogonal filter bank, and we show that this view- point leads to new insights. A particular filter bank of this kind, called the chromatic derivative filter bank, is explored in greater detail. See [6] for a comprehensive survey of single and multi- channel sampling theory. Chromatic derivatives are linear combinations of the ordi- nary derivatives, where the coefficients of the combination are derived from orthogonal polynomial theory. Surprisingly, by taking combinations of a notoriously difficult operation such as differentiation, we obtain perfectly stable filters. This is because of the well-known orthogonal properties of the underlying poly- nomial set. The weighting function employed in the definition of these polynomials basically determines the characteristics of the filters. We believe that these derivatives are preferable to the ordinary derivatives in situations where wide-band channel splitting is needed. Multichannel analog-to-digital conversion of high-bandwidth signals is one such example. The theory of perfect reconstruction (PR) in chromatic derivative filter banks with an infinite number of channels is discussed in [1] and [2]. We extend that theory to the finite- channel case here. Although one can obtain an approximate reconstruction of the original signal by truncating the infinite series expansions, as explained in [1], we show that it is in fact possible to achieve PR by proper choice of the synthesis filters. We delineate a time domain procedure for designing such filters. Manuscript received May 28, 2002; revised August 1, 2002. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Markus Pueschel. M. Cushman was with Kromos Technology, Los Altos, CA 94022 USA. He is now with Deephaven Capital Management, Santa Clara, CA 95054 USA (e-mail: mcushman@alumni.uchicago.edu). M. J. Narasimha is with Stanford University, Stanford, CA 94305 USA (e-mail: sim@nova.stanford.edu). P. P. Vaidyanathan is with the California Institute of Technology, Pasadena, CA 91125 USA (e-mail: ppvnath@systems.caltech.edu). Digital Object Identifier 10.1109/LSP.2002.806706 II. BIORTHOGONALITY IN FINITE-CHANNEL ANALOG FILTER BANKS Fig. 1 shows an -channel filter bank, where the input signal is bandlimited to , and the sampling rate of the subbands is times smaller than the Nyquist rate of one sample per second. The output signal can be expressed as . We as- sume that the input belongs to the space spanned by the doubly indexed set of (linearly independent) functions , , i.e., for some set of con- stants . For such an input, following the procedure in [1] and [2], we can show that the PR property holds if the analysis and synthesis filters satisfy the biorthogonality condition (1) where is the Kronecker delta function. If denotes the cascaded impulse response of the th analysis filter and the th synthesis filter, then (1) can equivalently be written as . III. CHROMATIC DERIVATIVES BASED ON ORTHOGONAL POLYNOMIALS Consider a sequence of polynomials in the interval [ 1, 1], which are orthogonal with respect to a nonnegative even weighting function that is not identically zero over the defining interval. It is well known [5] that such polynomials can be obtained using the recursion (2) with and . The values of the constants , , , and depend on the weighting function that defines the polynomial family. As in [1], we identify the th analysis filter of the filter bank using the th orthogonal polynomial as follows: (3) The factor in the definition of the analysis filters guarantees that their impulse responses will be real for all values of . The output of the th analysis filter in Fig. 1 is known as the th chromatic derivative of evaluated at . Using (2), it can be shown that the analysis filters are poly- nomials in . Specifically, we can express them as (4) 1070-9908/03$17.00 © 2003 IEEE