Two previous contributions discussed the theory of perfect-reconstruction (PR) chromatic derivative filter banks comprising an infinite number of channels. This paper 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

Cushman, M.

Narasimha, M. J.

Vaidyanathan, P. P.

Caltech Authors - Main

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 multichannelsampling 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
16 IEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 1, JANUARY 2003
Fig. 1. M -channel filter bank for a signal x(t) bandlimited to j!j < .
where are real constants. If these constants are arranged
as an matrix whose th element is , then it
can be verified that the matrix is lower triangular with nonzero
diagonal elements. This follows from the fact that the analysis
filters satisfy a recursive relation similar to (2). Since the factor
in the frequency domain corresponds to the th derivative
in the time domain, the matrix also defines the relationship
between the ordinary derivatives and the chromatic derivatives.
(Note that the row and column indexing of the matrix starts at
zero.)
IV. TIME DOMAIN DESIGN OF SYNTHESIS FILTERS FOR PR
With the analysis filters expressed as in (4), the biorthogo-
nality or PR condition (1) becomes
(5)
Noting that corresponds to differentiation in time, we can
write this equivalently as
(6)
for all , with superscript denoting the th derivative. We
now choose the synthesis filters as
(7)
where is a -bandlimited signal defined as the th power
of the function and is a polynomial in of degree
(8)
and
(9)
The values , , represent the coefficients of
the th polynomial. We will choose them so that the filter bank
satisfies the PR conditions.
Using the Leibnitz rule for the th derivative of the product in
(7), we get
(10)
Since and its first derivatives are zero at for
all , it follows that and its first derivatives
are also zero at these points. Thus, the above choice of synthesis
filters satisfies the PR condition (6) for all . Next, we pick
the polynomial coefficients to take care of the case .
Substituting (10) in (6) with , and noting that
, we get
(11)
Using matrix notation, we can rewrite (11) as , where
is the matrix that relates the ordinary derivatives to the
chromatic derivatives as explained in Section III; is an
matrix of the polynomial coefficients with th element ;
and is an lower triangular matrix with elements
.
(12)
The diagonal elements of are nonzero, ensuring nonsingu-
larity. The synthesis filters can, therefore, be uniquely identified
from the elements of the matrix . Since and
are both lower triangular, is also a lower triangular matrix.
Its th column denotes the coefficients of the th polynomial.
Specifically, we can write
.
CUSHMAN et al.: FINITE-CHANNEL CHROMATIC DERIVATIVE FILTER BANKS 17
Example
Let , and assume that the analysis filters are based on
the Chebyshev family of chromatic derivatives [1]. In this case,
the matrices are given by
and
The corresponding polynomials are ,
, , and
.
V. CONCLUSION
The interpretation of generalized sampling as a contin-
uous-time biorthogonal filter bank leads to many novel
sampling expansions. One such expansion based on chromatic
derivatives is discussed in this letter, along with an elegant time
domain design procedure for the synthesis filters to achieve PR.
Our procedure is simpler than the frequency domain approach
proposed in [3] for the case of ordinary derivative-based
sampling. The realization of a stable filter bank in which the
channel filters are all wideband is possible by employing
chromatic derivative operators for the analysis filters. Po-
tential applications of this filter bank include multichannel
analog-to-digital conversion, where sub-Nyquist samples of
the individual channels can be used to represent a broadband
signal, and transient analysis of signals.
REFERENCES
[1] M. J. Narasimha, A. Ignjatovic, and P. P. Vaidyanathan, “Chromatic
derivative filter banks,” IEEE Signal Processing Lett., vol. 9, pp.
215–216, July 2002.
[2] P. P. Vaidyanathan, A. Ignjatovic, and M. J. Narasimha, “New sampling
expansions for bandlimited signals based on chromatic derivatives,” in
Proc. 35th Asilomar Conf. Signals, Systems, and Computers, Monterey,
CA, Nov. 2001.
[3] A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits
Syst., vol. CAS-24, no. 11, pp. 652–654, Nov. 1977.
[4] J. L. Brown, Jr., “Multi-channel sampling of lowpass signals,” IEEE
Trans. Circuits Syst., vol. CAS-28, pp. 101–106, Feb. 1981.
[5] G. Szegö, Orthogonal Polynomials, 4th ed, ser. AMS Colloquium Pub-
lications. Providence, RI: Amer. Math. Soc., 1981, vol. 23, ch. II.
[6] M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE, vol. 88,
pp. 569–587, Apr. 2000.


Finite-channel chromatic derivative filter banks

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Finite-channel chromatic derivative filter banks

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