The Synthesis of Recursive Digital Filters

ABSTRACT

The circumstances and methods of the synthesis of linear digital recursive filters which are both stable and physically realizable are described. It is shown that any amplitude requency transfer function expressible as an even trigonometric rational polynomial can be synthesized by a real stable linear digital recursive filter. The degree of the corresponding difference equation is twice the degree of the denominator of the rational trigonometric polynomial. A class of even rational trigonometric functions which exhibit pointwise convergence to the deal rectangular low or high-pass amplitude frequency transfer function is chosen. A member of this class is shown to approximate more closely the ideal rectangular filter than does the corresponding classical continuous Butterworth filter. This class of filters is then mechanized. The phase and unit-impulse response functions are calculated for the corresponding difference equations of degrees 2 and 4.

REFERENCES

• 1: LANE, D.W. Digital shock spectrum analysis by reeursive filtering. Shock, Vibration and Associated Environment Bulletin, Off. Dir. Defense, Res. Eng., US Dept. Def., Feb. 1964.
• 2: GRANGER, C. W. J., AND HARANAKA, M. Spectral Analysis of Economic Time Series. Princeton U. Press, Princeton, N. J., 1964.
• 3: OTNES, R. K. Phase distortion correction by comptlter methods. Interoff. Corr. N0. 1775-3-24, Aerospace Corp., March, 1964.
• 4: Joseph F. A. Ormsby, Design of Numerical Filters with Applications to Missile Data Processing, Journal of the ACM (JACM), v.8 n.3, p.440-466, July 1961 [doi>10.1145/321075.321087]
• 5: TIMAN, A .F . Theory of Approximation of Functions of a Real Variable. Tr. by Berry, J., The Macmillan Co., New York, 1963, p. 240-246.
• 6: CHENEY, E, W., AND LOEB, H.L. Generalized rational approximatiom J. SIAM, Num. Anal. {8}, 1, 11-25.
• 7: NATANSON, L .P . Theory of Functions of a Real Variable, Vol. I, II. Ungar, New York, 1955, 1960.
• 8: FLECK, J. T., ANn FRYER, W.D. An exploration of numerical filtering techniques. ASTIA No. 18191, May, 1953.
• 9: MARTIN, M.A. Digital filters for data processing. Tech. Inf. Ser. No. 62SD484, Valley Forge, Pa., Oct. 1962.
• 10: GOODMAN, N. P. Some comments on spectral analysis of time aeries. Technometrics 3, 2 (May, 1961), 221-228.
• 11: BLACKMAN, R. B., AND TUKEY, J.W. The Measurement of Power Spectra. Dover Publ, New York, 1959.
• 12: John R. B. Whittlesey, A rapid method for digital filtering, Communications of the ACM, v.7 n.9, p.552-556, Sep. 1964 [doi>10.1145/364847.364878]
• 13: Kuo, B.C. Analysis and Synthesis of Sampled Data Control Systems. Prentice Hall, Inc., Englewood Cliffs, N. J., 1963.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.3 Enhancement
          Subjects: Filtering

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

General Terms:
Design, Measurement, Performance, Theory