07-02-2013, 04:54 PM
The Kalman Filter as the Optimal Linear Minimum Mean-Squared Error Multiuser CDMA Detector
1The Kalman Filter.pdf (Size: 180.78 KB / Downloads: 32)
Abstract
In this paper, it is shown that a first-order linear
state–space model applies to the asynchronous code-division
multiple-access (CDMA) channel, and thus the Kalman filter
produces symbol estimates with the minimum mean-squared
error (MMSE) among all linear filters, in long- or short-code
systems for a given detection delay. This result may be used as a
benchmark against which to compare the performance of other
linear detectors in asynchronous channels. It also reveals that a
time-varying recursive filter with a fixed and finite complexity
implements the fixed-lag linear MMSE (LMMSE) detector, which
hitherto has been assumed to require a processing window (and
hence complexity) that grows with time.
Bit-Error Rate Calculations
The recursive nature of the Kalman filter makes exact BER
calculations more complicated than with TDL filters. Nevertheless,
it is possible to obtain approximate BER expressions for
binary phase-shift keying (BPSK) in an AWGN channel that are
accurate to an arbitrary level, depending on the complexity that
can be tolerated. A much simpler BER computation using the
Gaussian assumption can also be derived.
SIMULATION RESULTS
To verify the BER expressions (23) and (24), we simulated the
Kalman smoother algorithm with four users, a processing gain
of , and a detection delay arbitrarily selected to be ,
and measured the BER. Randomly selected short codes and time
delays were used. The results are plotted in Fig. 3, where the
solid curve represents the BER calculated using both (23) and
(24) because they are indistinguishable. The simulated curve
falls almost exactly over the analytical curve, as expected.
CONCLUSION
In this paper, we have shown that the Kalman filter or
smoother is the exact or IIR linear MMSE detector for a given
detection delay. Although it effectively accounts for all past
information, the Kalman filter’s complexity is fixed for a given
detection delay, unlike in the case of the windowed LMMSE
detector, which for the same performance will require a window
whose length increases with time.
It was demonstrated that in time-invariant systems, the
Kalman filter is a time-invariant filter, which allows one to
use adaptive IIR filtering concepts in an implementation with
training sequences in place of channel side information [10]. In
addition, the BER of the Kalman detector in an AWGN channel
can be computed semianalytically either with or without the
Gaussian assumption on the MAI. The optimal MMSE BER
can thus be obtained easily and used as a benchmark against
which to compare new linear detector structures.