| dc.description.abstract |
In practical applications, the statistics of the signal are unknown and time-varying.
Hence, the coefficients of an adaptive filter converge towards the optimal values and
track the time-varying statistics of the input signal during the steady state time. In
order to analyse the performance of adaptive filters. the behaviour of optimal and
adaptive coefficients as well as the resulting output error signals need to be carefully
studied. Frequency modulated (FM) input signals in noise are a well-known class of
non stationary random processes with time-varying spectra. They possess a structure
which facilitates the theoretical analysis and are also encountered in many real applications.
In investigating the behaviour of adaptive filters tracking linear FM signals, a
great effort has been devoted to transversal filters [9), [I). However, an alternative
structure is the lattice filter with its attractive properties. Some of these properties
include high convergence speed, easy computations, less sensitivity to eigenvalues
spread, ease of implementations, and providing a Gram-Schmidt type of orthogonalisation
of the input signal [8).
The intent of this paper is to investigate new properties of FIR lattice filters in
presence of quadratic and linear FM signals [10). It will theoretically be shown that
the reflection coefficients for such signals produce linear FM and sinusoidal signals.
respectively_ We have called this property as the polynomial order reducing (POR)
property of the lattice filter. Additionally, the relationship between the forward and
backward error signals (residuals) with the filter order and the input signal to noise
ratio is presented. To illustrate the tracking behaviour of the adaptive fi Iter, in the
first experiment, the instantaneous frequency (IF) [2] of a quadratic FM signal is estimated
under the assumption that the input signal has an AR structure. To update the
adaptive coefficients, the unnormalised stochastic gradient (SG) algorithm is
applied. While other on-line IF estimation techniques such as zero-crossing and central
finite difference (CFO) methods produce a high variance even al high SNRs [3),
adaptive FIR lattice filters perform very well, even at low SNRs. In the second experiment,
a new technique is presented based on the POR property to estimate the IF of a linear FM signal. The results compared to those of the adaptive line enhancer (ALE)
[51 show a lower mean-square error (MSE) and bias in the IF estimates. |
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