Chromatic Derivatives and Chromatic Approximations

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Last update 22/10/2010


What are the chromatic derivatives?


You can read about the theory of chromatic derivatives in the extended abstract or introduction of  this paper

or look at this practical, hands-on interactive Mathematica tutorial/implementation

or .pdf version


This page is intended to be a convenient “one stop” repository of papers, software, preprints, technical reports, patents, etc. on chromatic derivatives and chromatic expansions, especially for items that are hard to find elsewhere. The material is posted with permission from the authors.


Disclaimer: These documents are made available to ensure timely dissemination of scholarly work. Copyright and all rights therein are retained by the copyright holders. All parties copying this information are expected to adhere to the terms and constraints invoked by each copyright holder. In most cases, these documents may not be reposted without the explicit permission of the copyright holder. Other restrictions to copying individual documents may apply.







1.      Frequency estimation using chromatic derivatives,  Mathematica Tutorial/implementation, ideal for engineers who want to learn about the chromatic derivatives, accompanying papers 2 and 3 below. This is a NEW VERSION from July 8th


·       If you do not have Mathematica you can read the above tutorial by installing the free Mathematica Player from the Wolfram’s website, or, alternatively, you can look at the pdf of the above file.

·       Your comments are most welcome; please email me at


2.     Matlab version of the frequency estimation code, still very crude and poorly commented, soon to be cleaned up:


1.     Eigensystem Decomposition based method


2.     Singular Value Decompositionbased  method, also combining the  standard methods with the chromatic derivative method for a  large boost in accuracy



Some questions on chromatic derivatives can be found here.








3.     A. Ignjatovic and A. Zayed: Multidimensional chromatic derivatives and series expansions, to appear in the Proceedings of the American Mathematical Society.


4.     A. Ignjatovic: Frequency estimation using time domain methods based on robust differential operators, the 10th IEEE International Conference on Signal Processing (ICSP), 26 – 28 October 2010, Beijing, China. (preprint)  final version


5.     A. Ignjatovic: Signal interpolation using numerically robust differential operators, 14th WSEAS CSCC Multi-conference, July 22-25, 2010, Corfu Island, Greece.


6.     A. Zayed: Generalizations of Chromatic Derivatives and Series Expansions,  IEEE Transactions on Signal Processing, Volume 58 ,  Issue 3,  2010,  1638-1647  





7.     A. Ignjatovic:  Chromatic Derivatives, Chromatic Expansions and Associated Spaces, East Journal on Approximations, Volume 15, Number 3 (2009), 263-302.





8.     G. Walter: Chromatic Series With Prolate Spheroidal Wave Functions, Journal of Integral Equations and Applications, Volume 20, Number 2, 2008.


9.     A. Ignjatovic: Chromatic derivatives and local approximations,  IEEE Transactions on Signal Processing, Volume 57, Issue 8, 2009.



10.  A. Ignjatovic: Local Approximations Based on Orthogonal Differential Operators , Journal of Fourier Analysis and Applications, Vol. 13, Issue 3, 2007, pp. 309-330.



11.  G.  Walter and X. Shen: A sampling expansion for non bandlimited signals in chromatic derivatives, IEEE Transactions on Signal Processing 53, 2005, 1291–1298.



12.  M. J. Narasimha., A. Ignjatovic, and P. P. Vaidyanathan: Chromatic derivative filter banks, IEEE Signal Processing Letters, 9(7), 2002, 215–216.

13.  M. Cushman, M. J. Narasimha, and P.P. Vaidyanathan: Finite-channel chromatic derivative filter banks, IEEE Signal Processing Letters, 10(1), 2002, 5–17.




14.  M. Cushman: A Method for Approximate Reconstruction from Filter Banks, SIAM Conference on Linear Algebra in Signals, Systems and Control, Boston, 2001.

15.  T. Herron: Towards a New Transform Domain Adaptive Filtering Process Using Differential Operators and Associated Splines, International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS),  Nashville, 2001.

16.  J. Byrnes: Local Signal Reconstruction via chromatic differentiation Filter Banks, 35th Asilomar Conference on Signals, Systems and Computers, Monterey, 2001. 568–572.

17.  P. P.  Vaidyanathan,  A.  Ignjatovic, and M.J.  Narasimha: New sampling expansions of band limited signals based on chromatic derivatives, 35th Asilomar Conference on Signals, Systems and Computers, Monterey, 2001., 558–562.

18.  A. Ignjatovic: Numerical differentiation and signal processing, International Conference on Information, Communications and Signal Processing (ICICS), Singapore, 2001.




       Note: All of these patents are now in the public domain, free to use!

·      US Patent 6115726: Aleksandar Ignjatovic: Signal processor with local signal behavior.

This patent introduces the notion of chromatic derivatives, but the expansions use polynomials as interpolation functions.

Provisional Patent Disclosure 60/061,109 for this patent was filled October 3, 1997. Patent application 09/144,360 for this patent was filled May 28, 1998. The patent was issued September 5, 2000.

·      US Patent 6313778: Aleksandar Ignjatovic and Nicholas Carlin: Method and a system of acquiring local signal behavior parameters for representing and processing a signal.

This patent introduces chromatic expansions and describes basic signal processing methods based on chromatic expansions.

Provisional Patent Disclosure 60/143,074 for this patent was filled July 9, 1999.  Patent application 09/614,886 for this patent was filled July 9, 2000. The patent was issued November 6, 2001.

·      US Patent 6587064: M. Cushman and A. Ignjatovic: Signal Processor with Local Signal Behavior and Predictive Capability.

This patent describes  some prediction filters based on chromatic expansions

Patent application 09/897,325 for this patent was filled July 2, 2001. The patent was issued July 1, 2003.


SOFTWARE, ETC (if you need help, please feel free to email me!) more implementations to come soon!


1.      Filter coefficients for 129 tap transversal filters for twice oversampled signals for:

·         Legendre chromatic derivatives 

·      Chebyshev chromatic derivatives


2.      Mathematica scripts for the Remez exchange filter design algorithm used to produce the above filters.












This page is maintained by Aleksandar Ignjatovic