Abstract
Digital differentiators (DD) are extensively used in various computational systems. An ideal DD has the frequency response given by \( \tilde{H}_{d} (\omega ) = j\omega , - {\kern 1pt} \pi \le \omega \le \pi \). The most popular FIR design for the differentiators is the minimax relative error (MRE) approximation, proposed by Rabiner and Schafer [1]. The MRE-DDs are, in general, suitable for wideband frequency ranges with moderate accuracy, (say, 100 ± 1%) but are not flexible enough to be efficiently adopted for limited (low, mid or high) frequency bands. Moreover, the required weighting coefficients for these DDs are computed by using an optimization algorithm [2]. In this chapter, we discuss a new class of digital differentiators, investigated by us, which have maximally linear (ML) frequency response at ω = 0 or ω = π/2 or ω = π. Correspondingly, this design yields efficient approximations for low, mid or high frequency ranges, giving extremely low relative errors (RE). A universal digital differentiator has also been obtained, which covers the frequency range 0 ≤ ω ≤ 0.90π. Mathematical formulas for computing the exact values of the weighting coefficients have also been given.
Source: Balbir Kumar & S.C. Dutta Roy, “Maximally Linear FIR Digital Differentiators: A Review,” JIETE, vol 34, pp 347–357, September–October 1988.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L.R. Rabiner, R.W. Schafer, On the behaviour of minimax relative error digital differentiators. Bell Syst. Tech. J. 53, 333–362 (1974)
J.H. McClellan, T.W. Parks, L.R. Rabiner, A computer program for designing optimum FIR linear phase digital filters. IEEE Trans. Audio Electroacoust. AU-21, 506–526 (1973)
M.I. Skolnik, Introduction to Radar Systems, 2nd edn. (McGraw-Hill International Book Co., 1980)
M.I. Skolnik, Radar Handbook (McGraw-Hill Book Co., New York, 1970)
R.J. Urick, Principles of Underwater Sound (Mc-Graw-Hill Inc., 1975)
R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, New Jersey, 1983)
L.R. Rabiner, B. Gold, Theory and Applications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, New Jersey, 1975)
A.V. Oppenheim, R.W. Schafer, Digital Signal Processing (Prentice Hall, Englewood Cliffs, New Jersey, 1975)
A. Antoniou, Digital Filter Analysis and Design (Tata McGraw-Hill Publishing Co, New Delhi, 1983)
L.R. Rabiner, K. Steiglitz, The design of wideband recursive and non-recursive digital differentiators. IEEE Trans. Audio Electroacoust. AU-18, 204–209 (1970)
R.W. Hamming, Digital Filters (Prentice-Hall, Englewood Cliffs, New Jersey, 1977)
B.C. Kuo, Digital Control Systems (Holt, Rinehart and Winston Inc., New York, 1980)
B. Kumar, S.C. Dutta Roy, Design of digital differentiators for low frequencies. Proc. IEEE 76, 287–289 (1988)
C. Froberg, Introduction to Numerical Analysis (Addison Wesley, Reading, Massachusetts, 1969)
B. Kumar, S.C. Dutta Roy, Coefficients of maximally linear, FIR digital differentiators for low frequencies. Electron. Lett. 24, 563–565 (1988)
B. Kumar, S.C. Dutta Roy, Maximally linear FIR digital differentiators for midband frequencies. Int. J. Circuit Theory Appl. 17, 21–27 (1989)
B. Kumar, S.C. Dutta Roy, Design of maximally linear FIR digital differentiators and maximally flat FIR digital Hilbert transformers for midband frequency ranges. Int. J. Circuit Theory Appl. 17, 483–486 (1989)
B. Kumar, S.C Dutta. Roy, Optimal, FIR digital differentiators for high frequencies. IEEE Trans. Circuits Syst. CAS-36, 314–318 (1989)
L.R. Rabiner, C.M. Rader, Digital Signal Processing, Selected Reprint Series under the sponsorship of the IEEE Audio and Electroacoustic Group (IEEE Press, 1972)
A.V. Oppenheim, W.F.G. Mecklenbrauker, R.M. Mersereau, Variable cut-off linear phase digital filters. IEEE Trans. Circuits Syst. CAS-23, 199–203 (1976)
R.E. Crochiere, L.R. Rabiner, On the properties of frequency transformation for variable cut-off linear phase digital filters. IEEE Trans. Circuits Syst. CAS-23, 684–686 (1976)
S.C. Dutta Roy, S.S. Ahuja, Frequency transformation for linear-phase variable-cut off digital filters. IEEE Trans. Circuits Syst. CAS-26, 73–75 (1979)
S.N. Hazra, Linear phase bandpass digital filters with variable cut-off frequencies. IEEE Trans. Circuits Syst. CAS-31, 661–663 (1984)
B. Kumar, S.C. Dutta Roy, Design of universal variable frequency range FlR digital differentiators. Circuit Syst. Signal Process. 11, 431–439 (1992)
A.V. Oppenheim, et al., Selected Papers in Digital Processing, II, ed. by Digital Signal Processing Committee, IEEE Acoustics, Speech, and Signal Processing Society (IEEE Press, 1976)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Dutta Roy, S.C. (2020). Maximally Linear FIR Digital Differentiators: A Review. In: Topics in Signal Processing. Springer, Singapore. https://doi.org/10.1007/978-981-13-9532-1_21
Download citation
DOI: https://doi.org/10.1007/978-981-13-9532-1_21
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-9531-4
Online ISBN: 978-981-13-9532-1
eBook Packages: EngineeringEngineering (R0)