On Exponential Digital Filters

ABSTRACT

This paper derives the weighting sequence of a linear digital filter whose output is an estimate of the predicted values of the derivatives of the input. The input functions considered are arbitrary linear combinations of n + 1 known functions, plus a random stationary signal and a random stationary noise component. The filter differs from previously considered minimum variance optimum filters in that the primary consideration here is the computational ease with which one can obtain the final solution. An optimization in the minimum variance sense is obtained as a secondary consideration in order to provide some control of the mean square output error. The exponential filter has its simplest form for the class of nonrandom input functions (hn) which are the complete solutions of a set of homogeneous linear difference equations of order n with constant coefficients. For this class the input and output are related by a time invariant recursion formula. The output contains a bias error which can be made to approach zero exponentially as the mean square error increases monotonically to a limit with increasing time. A modification of the exponential filter is considered such that the bias error is zero. The solution then involves a recursion formula with time varying coefficients.

REFERENCES

• 1: K.R. JOHNSON, Optimum linear, discrete filterings of signals containing a nonrandom eomponen$, I.R E. Trans. Vol. IT-2, Sept. 1956.
• 2: A. B. L~.s, Interpolation and_ extrapolation of sampled data, I.{d.E. Transactions or~ Information Theory, Vol. IT-2, Mar. 1956.
• 3: M. BLuM, An extension of the minimum mean square prediction theory for sampled input signals, I.R.E. Transactzons on Information Theory, Vol. IT-2, No. 3, Sept. 1956.
• 4: M. BLUM, P~ecurston formulas for gTowing memory digital fillers, I.R.E. Transactwns on Information Theory, Vol. IT-4, Mar. 1958

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
      G.1.1 Interpolation
          Subjects: Smoothing

General Terms:
Measurement, Performance, Reliability, Theory