ACM Filter

ACM Filter

In [309], Masreliez proposed an approximate conditional mean (ACM) filter for estimating the state of a linear system with Gaussian state noise and non-Gaussian measurement noise. In particular, Masreliez proposed that some, but not all, of the Gaussian assumptions used in derivation of the Kalman filter be retained in defining a nonlinearly recursively updated filter. He retained a Gaussian distribution for the conditional mean, although it is not a consequence of the probability densities of the system (as is the case for Gaussian observation noise), hence the name approximate conditional mean that is applied to this filter. In [133, 387, 522] this ACM filter was developed for the model (7.10)–(7.11). To describe this filter, first denote the prediction residual by

Equation 7.34

This filter operates just as that of (7.13)–(7.17), except that the measurement update equation (7.14) is replaced with

Equation 7.35

and the update equation (7.16) is replaced with

Equation 7.36

The terms g_n and G_n are nonlinearities arising from the non-Gaussian distribution of the observation noise and are given by

Equation 7.37

Equation 7.38

where we have used the notation , and p() denotes the measurement prediction density. The measurement updates reduce to the standard equations for the Kalman–Bucy filter when the observation noise is Gaussian.

For the single-user system, the density of the observation noise in (7.12) is given by the following Gaussian mixture:

Equation 7.39

Let be the variance of the innovation (or residual) signal in (7.34):

Equation 7.40

we can then write the functions g_n and G_n in this case as

Equation 7.41

Equation 7.42

The ACM filter is thus seen to have a structure similar to that of the standard Kalman–Bucy filter. The time updates (7.13) and (7.15) are identical to those in the Kalman–Bucy filter. The measurement updates (7.35) and (7.36) involve correcting the predicted value by a nonlinear function of the prediction residual _n. This correction essentially acts like a soft-decision feedback to suppress the spread-spectrum signal from the measurements. That is, it corrects the measurement by a factor in the range [] that estimates the spread-spectrum signal. When the filter is performing well, the variance term in the denominator of tanh(·) is low. This means that the argument of tanh(·) is larger, driving tanh(·) into a region where it behaves like the sign(·) function, and thus estimates the spread-spectrum signal to be if the residual signal _n is positive and if the residual is negative. On the other hand, when the filter is not making good estimates, the variance is high and tanh(·) is in a linear region of operation. In this region, the filter hedges its bet on the accuracy of sign(_n) as an estimate of the spread-spectrum signal. Here the filter behaves essentially like the (linear) Kalman filter. The ACM-filter-based NBI suppression algorithm based on the state-space model (7.10)–(7.11) is summarized as follows.

Algorithm 7.3: [ACM-filter-based NBI suppression] At time i, N received samples {r_iN, r_iN+1, ..., r_iN+N-1} are obtained at the chip-matched filter output (7.5).

• For n = iN, iN + 1, ..., iN + N - 1 perform the following steps:

  Equation 7.43

  

  
  

  Equation 7.44

  

  
  

  Equation 7.45

  

  
  

  Equation 7.46

  

  
  

  Equation 7.47

  

  
  

  Equation 7.48

  

  
  

  where g_n and G_n are defined in (7.41) and (7.42), respectively.

• Detect the ith bit b₁[i] according to

  Equation 7.49

  

  
  

Simulation Examples

When the interference is modeled as a first-order autoregressive process, which does not have a very sharply peaked spectrum, the performance of the ACM filter does not seem to be appreciably better than that of the Kalman–Bucy filter. However, when the spectrum of the interference is made to be more sharply peaked by increasing the order of the autoregression, the ACM filter is found to give significant performance gains over the Kalman filter. Simulations were run for a second-order AR interferer with both poles at 0.99:

where {e_n} is an i.i.d. Gaussian sequence. The ambient noise power is held constant at s² = 0.01, while the total of noise plus interference power varies from 5 to 20 dB (all relative to a unity power spread-spectrum signal). In comparing filtering methods, the figure of merit is the ratio of SINR at the output of filtering to the SINR at the input, which reduces to

where _n is defined as in (7.34). The results from the Kalman and ACM predictors are shown in Fig. 7.5. The filters were run for 1500 points. The results reflect the last 500 points, and the values given represent averages over 4000 independent simulations.

Figure 7.5. Performance of the Kalman filter– and ACM filter–based NBI suppression methods.

To stress the effectiveness against the narrowband interferer (versus the background noise), the solid line in Fig. 7.5 gives an upper bound on SNR improvement, assuming that the narrowband interference is predicted with noiseless accuracy. This is calculated by setting equal to the power of the AWGN driving the AR process (i.e., the unpredictable portion of the interference). Note that the SINR improvement due to using the ACM filter is quite substantial and is very near the theoretical bound.