Linear Predictive Methods
This technique for narrowband interference suppression has been
explored in detail through the use of fixed and adaptive linear predictors
(e.g., [17, 19, 20, 37, 199, 209, 210, 221, 228, 235, 253, 255, 310, 312, 327, 346, 366, 451, 480]; see [6, 244, 327, 332] for reviews). Two basic
architectures for fixed linear predictors are Kalman–Bucy predictors, based on a
state-space model for the interference, and finite-impulse-response (FIR) linear
predictors, based on a tapped-delay-line structure.
Kalman–Bucy Predictors
To use Kalman–Bucy prediction (cf. [235]) in this application, it is
useful to model the narrowband interference as a pth order Gaussian autoregressive [AR(p)] process:
Equation 7.9
where {en} is a white
Gaussian sequence, 
, and where the AR parameters F1, F2, ..., Fp are assumed to be constant or slowly
varying.
Under this model, the received discrete-time signal (7.5) has a state-space representation as
follows (assuming one spread-spectrum user, i.e., K = 1):
Equation 7.10
Equation 7.11
where
Equation 7.12
with
Given this state-space formalism, the linear minimum
mean-square-error (MMSE) prediction of the received signal (and hence of the
interference) can be computed recursively via the Kalman–Bucy filtering
equations (e.g., [377]), which predicts the nth observation rn, as 
, where
denotes the state prediction in (7.10)–(7.11), given
recursively through the update equations
Equation 7.13
Equation 7.14
with 
, denoting the variance of the
prediction residual, and where the matrix Mn
(which is the covariance of the state prediction error 
is
computed via the recursion
Equation 7.15
Equation 7.16
with
Equation 7.17
where e1 denotes a
p-vector with all entries being zeros, except for
the first entry, which is 1. The Kalman–Bucy prediction-based NBI suppression
algorithm based on the state-space model (7.10)–(7.11) is
summarized as follows. (Note that it is assumed that the model parameters are
known.)
Algorithm 7.1: [Kalman–Bucy
prediction-based NBI suppression] At time i, N received
samples {riN, riN+1, ..., riN+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 7.18
Equation 7.19
Equation 7.20
Equation 7.21
Equation 7.22
Equation 7.23
-
Detect the ith bit
b1[i] according to
Equation 7.24
Linear FIR Predictor
The Kalman–Bucy filter is, of course, an
infinite-impulse-response (IIR) filter. A simpler linear structure is a
tapped-delay-line (TDL) configuration, which makes one-step predictions via the
FIR filter
Equation 7.25
where L is the data length used
by the predictor, and a1, a2, ..., aL, are tap
weights. In the stationary case, the tap weights can be chosen optimally via the
Levinson algorithm (see, e.g., [377]). More important, though, the
FIR structure (7.25) can easily be
adapted using, for example, the least-mean-squares (LMS) algorithm (e.g., [463]). Denote 
. Let a[n] denote the tap-weight vector to be applied at the
nth chip sample (i.e., to predict rn+1). Also denote 
. Then the
predictor coefficients can be updated according to
Equation 7.26
where m
is a tuning constant. Although the Kalman–Bucy filter can also be adapted, the
ease and stability with which the FIR structure can be adapted makes it a useful
choice for this application. To make the choice of tuning constant invariant to
changes in the input signal levels, the LMS algorithm (7.26) can be normalized as follows:
Equation 7.27
where pn is an
estimate of the input power obtained by
Equation 7.28
The estimate of the signal power pn is an exponentially weighted estimate.
The constant m0 is chosen small enough to ensure
convergence, and the initial condition p0 should be large enough so that the
denominator never shrinks so small as to make the step size large enough for the
adaptation to become unstable.
A block diagram of a TDL-based linear predictor is shown in Fig. 7.4. The LMS linear prediction-based
NBI suppression algorithm is summarized as follows.
Algorithm 7.2: [LMS linear
prediction-based NBI suppression] At time i, N received
samples {riN, riN+1, ..., riN+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 7.29
Equation 7.30
Equation 7.31
-
Detect the ith bit
b1[i] according to
Equation 7.32
Performance and convergence analyses of these types of linear
predictor–subtractor systems have shown that considerable
signal-to-interference-plus-noise ratio (SINR) improvement can be obtained by
these methods. (See the above-cited references and the results in Section 7.4.)
Linear interpolation filters can also be used in this context, leading to
further improvements in SINR and to better phase characteristics compared with
linear prediction filters (e.g., [311]). For example, a simple linear
interpolator of order L1 + L2 for estimating rn is given by
Equation 7.33
where a-L1, ..., aL2, are tap weights. Such an
interpolator can be adapted similarly via the LMS algorithm.
Sections
