F = 0.8, only the most recent 25 or so data points contribute to the final result, since all
older data points are essentially weighted out of the solution.



                   1.0                                                                                                      F = ~ (equally weighted)

                   0.9                                                                                  !           F=0.J9                                            I

                   0.8                                                                                                --.:
                                                                                                                                       !                              I
                                                                                                                                                          ···········;···························


                   0.7                                                  --········-----···--                                                                                                   ....



        -
    ...i:!:        0.6 ····                                                         -• -1-                                         +
                                                                                                                                   ! -
        --
         u..
         .c
              C)   0.5       ........
                                                                                     =0.9! 5

                                                                          ............                  i ..........................   J ...........................+········--
         ·a5
         ~                                                                  0.99                                                       i                             I
         ca        0.4                  .....                                ...........................:.......                       ' ............................ ~............................


         ca
         Cl
                   0.3 .                    .....                       1                            /.-····---;                                                     i --
                   0.2                              -----i··········    -1..................
                                                       +o.s
                   0.1                                                                                                     .......... ,

                                                                                                                                       I
                   0.0
                         0                           50                100                        150                           200                             250                            300
                                                                       Data Index (older->)

                         Figure 35. Exponential Weights for Fading-Memory Filters

For the exponentially-weighted fading-memory filter, it can be shown that the
recursive filter factor used in Eq. (12) is
                                                                                          1-F
                                                                         a=--                                                                                                                         (20)
                                                                             n            1-Fn

Since OS F S 1, an in Eq. (20) does not approach zero as n approaches infinity (as the
other two filters do), but instead approaches the value (1 - F). If F = 0, then an= 1 for all
n, the filter has no memory at all, and the filtered value always equals the last
measurement. In the limit as F approaches one, L'Hospital' s rule can be applied to


9/10/96                                                                                  93                                                                                                           RTI