H. Howard Fan

Least squares parameter estimation of continuous-time ARX models from discrete-time data

When modeling a system from discrete-time data, a continuous-time parameterization is desirable in some situations, In a direct estimation approach, the derivatives are approximated by appropriate differences. For an ARX model this lead to a linear regression. The well-known least squares method would then be very desirable since it can have good numerical properties and low computational burden, in particular for fast or nonuniform sampling. It is examined under what conditions a least squares fit for this linear regression will give adequate results for an ARX model. The choice of derivative approximation is crucial for this approach to be useful. Standard approximations like Euler backward or Euler forward cannot be used directly. The precise conditions on the derivative approximation are derived and analyzed. It is shown that if the highest order derivative is selected with care, a least squares estimate will be accurate. The theoretical analysis is complemented by some numerical examples which provide further insight into the choice of derivative approximation.

Wavelet-based linear system modeling and adaptive filtering

It is shown how linear time-varying systems can be modeled in several different ways by discrete-time wavelets or, more generally, by some set of functions. Interpretation of physical meanings, possible efficiency, and other characteristics of the modeling are considered. System identification minimizing the mean square output error is studied. Optimal coefficients and the corresponding minimum mean square error are found, and they are, in general, time varying. Least-mean-square adaptive filtering algorithms are derived for on-line filtering and system Identification. Theoretically and by simulations, the advantages of using wavelet-based filtering are shown: separation of adaptation effects from unknown time-varying system behavior and fast convergence. Adaptive coefficients estimated by a recursive-least-square algorithm can tend toward constants, even in the case of time-varying systems. Time-invariant system identification and adaptive filtering is given as a special case of the general time-varying setting.

Estimation of continuous-time AR process parameters from discrete-time data

The problem of estimating continuous-time autoregressive process parameters from discrete-time data is considered. The basic approach used here is based on replacing the derivatives in the model by discrete-time differences, forming a linear regression, and using the least squares method. Such a procedure is simple to apply, computationally flexible and efficient, and may have good numerical properties. It is known, however, that all standard approximations of the highest order derivative, such as repeated use of the delta operator, gives a biased least squares estimate, even as the sampling interval tends to zero. Some of our previous approaches to overcome this problem are reviewed. Then. two new methods, which avoid the shift in our previous results, are presented. One of them, which is termed bias compensation, is computationally very efficient. Finally, the relationship of the above least squares approaches with an instrumental variable method is investigated. Comparative simulation results are also presented.

Direct blind multiuser detection for CDMA in multipath without channel estimation

We consider the blind multiuser detection problem for asynchronous DS-CDMA systems operating in a multipath environment. Using only the spreading code of the desired user, we first estimate the column vector subspace of the channel matrix by multiple linear prediction. Then, zero-forcing detectors and MMSE detectors with arbitrary delay can be obtained without explicit channel estimation. This avoids any channel estimation error, and the resulting methods are therefore more robust and more accurate. Corresponding batch algorithms and adaptive algorithms are developed. The new algorithms are extremely near-far resistant. Simulations demonstrate the effectiveness of these methods.

QR factorization based blind channel identification with second-order statistics

Most eigenstructure-based blind channel identification and equalization algorithms with second-order statistics need SVD or EVD of the correlation matrix of the received signal. In this paper, we address new algorithms based on QR factorization of the received signal directly without calculating the correlation matrix. This renders the QR factorization-based algorithms more robust against ill-conditioned channels, i.e., those channels with almost common zeros among the subchannels. First, we present a block algorithm that performs the QR factorization of the received data matrix as a whole. Then, a recursive algorithm is developed based on the QR factorization by updating a rank-revealing ULV decomposition. Compared with existing algorithms in the same category, our algorithms are computationally more efficient. The computation in each recursion of the recursive algorithm is on the order of O(m/sup 2/) if only equalization is required, where m is the dimension of the received signal vector. Our recursive algorithm preserves the fast convergence property of the subspace algorithms, thus converging faster than other adaptive algorithms such as the super-exponential algorithm with comparable computational complexities. Moreover, our proposed algorithms do not require noise variance estimation. Numerical simulations demonstrate the good performance of the proposed algorithms.

Cascade lattice IIR adaptive filters

The feedback lattice filter forms, including the two-multiplier form and the normalized form, are examined with respect to their relationships to the feedback direct form filter. Specifically, the transformation matrix between the lattice forms and the direct form is derived; parameter and state relationships between the lattice forms and the direct form are therefore obtained. An IIR filter structure-the cascade lattice IIR structure-is constructed. Based on this structure, three IIR adaptive filtering algorithms in the two-multiplier form can then be developed following the gradient approach, the Steiglitz-McBride approach and the hyperstability approach. Convergence of these algorithms is theoretically analyzed using either the ODE approach or the hyperstability theorem. These algorithms are then simplified into forms computationally as efficient as their corresponding direct form algorithms. Relationships of the simplified algorithms to the direct form algorithms are also studied, which disclose a consistency in algorithm structure regardless of the filter form. Three normalized lattice algorithms can also be derived from the two-multiplier lattice algorithms. Experimental results show much improved performance of the normalized lattice algorithms over the two-multiplier lattice algorithms and the direct form algorithms. >

A Newton-like algorithm for complex variables with applications in blind equalization

Although the Newton algorithm has been extended to the complex domain in different forms, none of them seems to be directly applicable to blind equalization. Therefore, the objective of this correspondence is to develop an algorithm for blind equalization in the complex domain. We propose a Newton-like algorithm based on a complex Taylor series. Stochastic Newton-like algorithms (SNLA) for two blind equalization cost functions are developed. Simulations show that the new algorithms perform slightly better than the self-orthogonalizing algorithm.

Linear prediction methods for blind fractionally spaced equalization

We describe adaptive methods for estimating FIR zero-forcing blind equalizers with arbitrary delay directly from the linear predictions of the observations. While most current methods require inversion or singular value decomposition (SVD) of the correlation matrix, our methods need only to solve two linear prediction problems. They can be implemented as RLS or LMS algorithms to recursively update the equalizer estimation. they are computationally efficient. The computational complexity in each recursion can be less than 15(LN)/sup 2/ in the RLS case, where LN equals the equalizer length, and 3L(LN) in the LMS case, where L is the number of subchannels. The performance of the proposed methods and comparisons with existing approaches are shown by simulation to demonstrate their effectiveness.

Direct estimation of blind zero-forcing equalizers based on second-order statistics

Most existing zero-forcing equalization algorithms rely either on higher than second-order statistics or on partial or complete channel identification. We describe methods for computing fractionally spaced zero-forcing blind equalizers with arbitrary delay directly from second-order statistics of the observations without channel identification. We first develop a batch-type algorithm; then, adaptive algorithms are obtained by linear prediction and gradient descent optimization. Our adaptive algorithms do not require channel order estimation, nor rank estimation. Compared with other second-order statistics-based approaches, ours do not require channel identification at all. On the other hand, compared with the CMA-type algorithms, ours use only second-order statistics; thus, no local convergence problem exists, and faster convergence can be achieved. Simulations show that our algorithms outperform most typical existing algorithms.

On-line identification of echo-path impulse responses by Haar-wavelet-based adaptive filter

Long-distance telephone communications of good quality require absence of echoes. The physical characteristics of an echo path are time varying and the impulse responses measured at different times can differ from each other. This means that an echo canceller must track the changes. Using a small number of coefficients in Haar-wavelet-based models we can efficiently identify echo paths which have certain typical impulse response shapes. The modeling error energy obtained is low (less than 2%). A simple wavelet-based LMS adaptive filter can be used for on-line estimation of the coefficients. A low number of time-consuming computations is obtained per input sample due to the usage of Haar wavelets. This number is less than the ones obtained by the FIR of DFT domain based modeling.