The performance of an adaptive filter may be studied through the behaviourudof the optimal and adaptive coefficients in a given environment. This thesisudinvestigates the performance of finite impulse response adaptive lattice filters forudtwo classes of input signals: (a) frequency modulated signals with polynomialudphases of order p in complex Gaussian white noise (as nonstationary signals),udand (b) the impulsive autoregressive processes with alpha-stable distributions (asudnon-Gaussian signals).udInitially, an overview is given for linear prediction and adaptive filtering. Theudconvergence and tracking properties of the stochastic gradient algorithms are discussedudfor stationary and nonstationary input signals. It is explained that theudstochastic gradient lattice algorithm has many advantages over the least-meanudsquare algorithm. Some of these advantages are having a modular structure,udeasy-guaranteed stability, less sensitivity to the eigenvalue spread of the input autocorrelationudmatrix, and easy quantization of filter coefficients (normally calledudreflection coefficients).udWe then characterize the performance of the stochastic gradient lattice algorithmudfor the frequency modulated signals through the optimal and adaptiveudlattice reflection coefficients. This is a difficult task due to the nonlinear dependenceudof the adaptive reflection coefficients on the preceding stages and the inputudsignal. To ease the derivations, we assume that reflection coefficients of eachudstage are independent of the inputs to that stage. Then the optimal lattice filterudis derived for the frequency modulated signals. This is performed by computingudthe optimal values of residual errors, reflection coefficients, and recovery errors.udNext, we show the tracking behaviour of adaptive reflection coefficients forudfrequency modulated signals. This is carried out by computing the tracking modeludof these coefficients for the stochastic gradient lattice algorithm in average. Theudsecond-order convergence of the adaptive coefficients is investigated by modelingudthe theoretical asymptotic variance of the gradient noise at each stage. Theudaccuracy of the analytical results is verified by computer simulations.udUsing the previous analytical results, we show a new property, the polynomialudorder reducing property of adaptive lattice filters. This property may be used toudreduce the order of the polynomial phase of input frequency modulated signals.udConsidering two examples, we show how this property may be used in processingudfrequency modulated signals. In the first example, a detection procedure in carriedudout on a frequency modulated signal with a second-order polynomial phaseudin complex Gaussian white noise. We showed that using this technique a betterudprobability of detection is obtained for the reduced-order phase signals comparedudto that of the traditional energy detector. Also, it is empirically shown thatudthe distribution of the gradient noise in the first adaptive reflection coefficientsudapproximates the Gaussian law. In the second example, the instantaneous frequencyudof the same observed signal is estimated. We show that by using thisudtechnique a lower mean square error is achieved for the estimated frequencies atudhigh signal-to-noise ratios in comparison to that of the adaptive line enhancer.udThe performance of adaptive lattice filters is then investigated for the secondudtype of input signals, i.e., impulsive autoregressive processes with alpha-stableuddistributions . The concept of alpha-stable distributions is first introduced. Weuddiscuss that the stochastic gradient algorithm which performs desirable resultsudfor finite variance input signals (like frequency modulated signals in noise) doesudnot perform a fast convergence for infinite variance stable processes (due to usingudthe minimum mean-square error criterion). To deal with such problems, theudconcept of minimum dispersion criterion, fractional lower order moments, andudrecently-developed algorithms for stable processes are introduced.udWe then study the possibility of using the lattice structure for impulsive stableudprocesses. Accordingly, two new algorithms including the least-mean P-normudlattice algorithm and its normalized version are proposed for lattice filters basedudon the fractional lower order moments. Simulation results show that using theudproposed algorithms, faster convergence speeds are achieved for parameters estimationudof autoregressive stable processes with low to moderate degrees of impulsivenessudin comparison to many other algorithms. Also, we discuss the effect ofudimpulsiveness of stable processes on generating some misalignment between theudestimated parameters and the true values. Due to the infinite variance of stableudprocesses, the performance of the proposed algorithms is only investigated usingudextensive computer simulations.
展开▼
