Transform Domain Adaptive Filters Research Articles

Performance analysis of the speech enhancement application with wavelet transform domain adaptive filters

Performance analysis of the speech enhancement application with wavelet transform domain adaptive filters

Delay Spoofing Reduction in GPS Navigation System based on Time and Transform Domain Adaptive Filtering

Delay Spoofing Reduction in GPS Navigation System based on Time and Transform Domain Adaptive Filtering

Algorithm and Architecture Design of Adaptive Filters With Error Nonlinearities

This paper presents a framework based on the logarithmic number system to implement adaptive filters with error nonlinearities in hardware. The framework is demonstrated through pipelined implementations of two recently proposed adaptive filtering algorithms based on logarithmic cost, namely, least mean logarithmic square (LMLS) and least logarithmic absolute difference (LLAD). To the best of our knowledge, the proposed architectures are the first attempts to implement both LMLS and LLAD algorithms in hardware. We derive error computing algorithms to realize the nonlinear error functions for LMLS and LLAD and map them onto hardware. We also propose a novel variable- $\alpha $ scheme to enhance the original LMLS algorithm and prove its robustness and suitability for VLSI implementations in practical applications. Detailed bit width and error analysis are carried out for the proposed VLSI fixed point implementations. Postlayout implementation results show that with an additional multiplier over conventional least mean square (LMS), 7-dB improvement in steady-state mean square deviation performance can be achieved and with the proposed variable- $\alpha $ scheme, 12-dB improvement can be achieved without compromising the convergence. We will show that LMLS can potentially replace LMS in practical applications, by demonstrating a proof-of-concept by extending the framework to transform domain adaptive filters.

Sliding Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform: Fast Algorithm and Applications

This paper presents a fast algorithm for the computation of sliding conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT). The algorithm calculates the values of window <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> /4 from those of window <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> , one length- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> /4 Walsh Hadamard transform (WHT) and one length- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> /4 Modified WHT (MWHT). The proposed algorithm requires O( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) arithmetic operations, which is more efficient than the block-based algorithms of various transforms and the sliding FFT algorithm, but less efficient than the sliding WHT algorithms. Compared to the recently proposed sliding inverse SCHT (ISCHT) algorithm, the proposed algorithm is more efficient for real input but less efficient for complex input. The applications of the sliding CS-SCHT in transform domain adaptive filtering (TDAF) to complex signal channel equalization and real speech signal acoustic echo cancellation are also provided.

Fast Algorithms for the Computation of Sliding Sequency-Ordered Complex Hadamard Transform

Fast algorithms for computing the forward and inverse sequency-ordered complex Hadamard transforms (SCHT) in a sliding window are presented. The first algorithm consists of decomposing a length-N inverse SCHT (ISCHT) into two length-N/2 ISCHTs. The second algorithm, calculating the values of window i+N/4 from those of window i and one length-N/4 ISCHT and one length-N/4 modified ISCHT (MISCHT), is implemented by two schemes to achieve a good compromise between the computation complexity and the implementation complexity. The forward SCHT algorithm can be obtained by transposing the signal flow graph of the ISCHT. The proposed algorithms require O(N) arithmetic operations and thus are more efficient than the block-based algorithms as well as those based on the sliding FFT or the sliding DFT. The application of the sliding ISCHT in transform domain adaptive filtering (TDAF) is also discussed with supporting simulation results.

Fault Tolerance in Transform-Domain Adaptive Filters Operating With Real-Valued Signals

Fault-tolerant adaptive filters (FTAFs) rely on inherent learning capabilities of the adaptive process to compensate for transient (soft) or permanent (hard) errors in the hardware implementation. In this paper, the use of the Walsh-Hadamard transform is first analyzed as a computationally efficient way of achieving adaptive fault tolerance, where a zero padding strategy is used to compensate for faulty filter coefficients. It is then shown that a fast-Fourier-transform (FFT)-based transform-domain FTAF operating on real-valued signals can provide a similar degree of fault tolerance without introducing redundancy in the form of zero padding. The complex arithmetic provides inherent fault-tolerant capabilities. This is achieved by using only the real part of the error to drive the coefficient updates. In the case of an N-tap filter, the FFT-based FTAF can protect up to N-2 filter coefficients.

On the application of a unified adaptive filter theory in the performance prediction of adaptive filter algorithms

Employing a recently introduced unified adaptive filter theory, we show how the performance of a large number of important adaptive filter algorithms can be predicted within a unified way. This approach is based on energy conservation arguments and does not need to assume the specific models for the regressors. This general performance analysis can be used to evaluate the mean square and tracking performance of the least mean square (LMS) algorithm, its normalized version (NLMS), the family of affine projection algorithms (APA), the recursive least squares (RLS), the data-reusing LMS (DR-LMS), its normalized version (NDR-LMS), and the transform domain adaptive filters (TDAF). Also, we establish the general expressions for the excess mean square in the stationary and nonstationary environments for all these adaptive algorithms. Finally, we demonstrate through simulations that these results are useful in predicting the adaptive filter performance.

A New Fast Bit-Reversal Permutation Algorithm Based on a Symmetry

This correspondence describes a new bit-reversal permutation algorithm based on a trivial symmetry that has not been exploited until now. According to timing experiments, this algorithm outperforms the fastest algorithms known to the author. This is of interest for applications using intensive fast Fourier transforms (or fast Hartley transforms) of constant length, such as transform domain adaptive filtering.

Complexity considerations for transform-domain adaptive filters

This paper is concerned with the computational complexity and convergence performance of transform-domain adaptive filtering algorithms. In particular, the transform-domain least-mean-square algorithm and the generalized subband decomposition LMS algorithm are considered. Reduced complexity variants of these algorithms are developed based on the concept of selective partial updating. The effect of power normalization on the computational complexity is analyzed, and an alternative implementation is proposed to reduce the number of divisions to one. This implementation is also shown to be the only realization that lends itself to complexity reduction by selective partial updating. The complexity and performance of the algorithms are illustrated in acoustic echo cancellation and channel equalization applications by way of computer simulations.

A new method for low-rank transform domain adaptive filtering

This paper introduces a least squares, matrix-based framework for adaptive filtering that includes normalized least mean squares (NLMS), affine projection (AP) and recursive least squares (RLS) as special cases. We then introduce a method for extracting a low-rank underdetermined solution from an overdetermined or a high-rank underdetermined least squares problem using a part of a unitary transformation. We show how to create optimal, low-rank transformations within this framework. For obtaining computationally competitive versions of our approach, we use the discrete Fourier transform (DFT). We convert the complex-valued DFT-based solution into a real solution. The most significant bottleneck in the optimal version of the algorithm lies in having to calculate the full-length transform domain error vector. We overcome this difficulty by using a statistical approach involving the transform of the signal rather than that of the error to estimate the best low-rank transform at each iteration. We also employ an innovative mixed domain approach, in which we jointly solve time and frequency domain equations. This allows us to achieve very good performance using a transform order that is lower than the length of the filter. Thus, we are able to achieve very fast convergence at low complexity. Using the acoustic echo cancellation problem, we show that our algorithm performs better than NLMS and AP and competes well with FTF-RLS for low SNR conditions. The algorithm lies in between affine projection and FTF-RLS, both in terms of its complexity and its performance.

Narrow-band interference excision in spread-spectrum systems using self-orthogonalizing transform-domain adaptive filters

Among many transform-domain interference excision techniques, transform-domain adaptive filtering has many advantages. It is based on a true optimization of some particular performance parameters such as the bit-error rate (BER). Moreover, it is insensitive to jammer frequency. However, transform-domain adaptive filtering also has the drawback of being incapable of tracking a rapidly changing interference because most adaptive algorithms require time to converge to the optimal solution. In this paper, a self-orthogonalizing transform-domain least mean square (SO-TRLMS) algorithm is used to speed up the convergence. Compared to a traditional transform-domain least mean square (TRLMS) algorithm, the SO-TRLMS algorithm can significantly improve the convergence rate of the LMS algorithm, thus making the transform-domain adaptive filtering technique more suitable for real-time processing. In order to show how the system performance is affected by various factors such as interference power and the transform used, this paper presents an analytical result for the BER performance that is applicable for arbitrary orthogonal linear transforms. Simulation results are also presented to demonstrate the validity of the analysis.

A VLSI Design for Implementation of Transform Domain Adaptive Filters

A VLSI implementation of a dedicated digital signal processor is presented. The processor is tailored for efficient implementation of transform domain adaptive filters. It incorporates a butterfly processor which performs butterfly operation to implement the required transformation. It is also able to perform complex addition, subtraction and multiplication. The butterfly processor makes use of a redundant binary tree multiplier with a recently proposed coding of signed-digit numbers which reduces the number of levels in the tree by one. An on-chip read only memory holds the transformation coefficients. The contents of the ROM determine the type of transform. The processor incorporates an ALU to perform integer arithmetic, address calculations and implementation of circular memory scheme. For fastest accessibility, the essential variables of the algorithm are implemented in a register fire.

The use of the modified escalator algorithm to improve the performance of transform-domain LMS adaptive filters

This paper describes a new algorithm that improves the convergence performance of the transform-domain least mean-square (TRLMS) algorithm. The algorithm exploits the sparse structure of the correlation matrix of the transformed input process to derive a data dependent Gram-Schmidt orthogonalization type transform of the process. We show its faster convergence compared with the time-domain least mean-square (LMS) algorithm and the DCT or the DWT-based TRLMS algorithm. The Gram-Schmidt orthogonalization is realized using a modified adaptive escalator algorithm. The modification significantly reduces the computational complexity of the adaptive escalator algorithm and determines the computational complexity of the proposed algorithm.

Sliding transforms for efficient implementation of transform domain adaptive filters

An efficient technique for the implementation of the widely used orthogonal transforms when their input samples are originating from a tap delay line (TDL), and the transformation has to be performed after each new sample enters the TDL, is addressed. The redundancy in the input data samples is used to reduce the computational complexity of these transforms from order of N log N to order of N, where N is the length of transform. The term sliding is used to refer to such transforms. We use the Bruun's (1978) technique, to propose a number of sliding implementations for the most commonly used transforms. Our emphasis is on the transform domain adaptive filters which employ such transforms for decorrelating their input data.

Adaptive filtering using filter banks

This paper derives a theory for adaptive filters which operate on filter bank outputs, here called filter bank adaptive filters (FBAFs). It is shown how the FBAFs are a generalization of transform domain adaptive filters and adaptive filters based on structural subband decompositions. The minimum mean-square error performance and convergence properties of FBAFs are determined as a function of filter bank used. A parametrization for a class of FIR perfect reconstruction filter banks is derived which is used to design FBAF's having optimal error performance given prior knowledge of the application. Simulations are performed to illustrate the derived theory and demonstrate the improved error performance of the FBAFs relative to the LMS algorithm, when prior knowledge is incorporated.

New Hadamard basis

A new basis that is similar to the Hadamard basis in sign sequency of the basis vectors is derived. The new basis may be useful for telecommunications in signalling across frequency selective fading channels and for signal processing in transform domain adaptive filtering.

The discrete Laguerre transform: derivation and applications

The discrete Laguerre transform (DLT) belongs to the family of unitary transforms known as Gauss-Jacobi transforms. Using classical methodology, the DLT is derived from the orthonormal set of Laguerre functions. By examining the basis vectors of the transform matrix, the types of signals that can be best represented by the DLT are determined. Simulation results are used to compare the DLT's effectiveness in representing such signals to that of other available transforms in applications such as data compression and transform-domain adaptive filters.

Time-recursive computation and real-time parallel architectures: a framework

The time-recursive computation has been proven a particularly useful tool in real-time data compression, in transform domain adaptive filtering, and in spectrum analysis. Unlike the FFT-based ones, the time-recursive architectures require only local communication. Also, they are modular and regular, thus they are very appropriate for VLSI implementation and they allow a high degree of parallelism. In this two-part paper, we establish an architectural framework for parallel time-recursive computation. We consider a class of linear operators that consists of the discrete time, time invariant, compactly supported, but otherwise arbitrary kernel functions. We show that the structure of the realization of a given linear operator is dictated by the decomposition of the latter with respect to proper basis functions. An optimal way for carrying out this decomposition is demonstrated. The parametric forms of the basis functions are identified and their properties pertinent to the architecture design are studied. A library of architectural building modules capable of realizing these functions is developed. An analysis of the implementation complexity for the aforementioned modules is conducted. Based on this framework, the time-recursive architecture of a given linear operator can be derived in a systematic routine way.

Order of N complexity transform domain adaptive filters

Implementation of the transform domain adaptive filters is addressed. Recent results have shown that if the input data to a radix-2 fast Fourier transform (FFT) structure is sliding one sample at a time, only N-1 butterflies need to be calculated for updating the FFT structure. This is opposed to most of the previous reports that assume order of NlogN complexity for such implementation. In this correspondence, a generalization of the sliding FFT, which introduces a wide class of orthogonal transforms that can be implemented with the order of N complexity is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transform-domain adaptive filters: an analytical approach

Transform-domain adaptive filters refer to LMS filters whose inputs are preprocessed with a unitary data-independent transformation followed by a power normalization stage. The transformation is typically chosen to be the discrete Fourier transform (DFT), although other transformations, such as the cosine transform (DCT), the Hartley transform (DHT), or the Walsh-Hadamard transform, have also been proposed in the literature. The resulting algorithms are generally called DFT-LMS, DCT-LMS, etc. This preprocessing improves the eigenvalue distribution of the input autocorrelation matrix of the LMS filter and, as a consequence, ameliorates its convergence speed. In this paper, we start with a brief intuitive explanation of transform-domain algorithms. We then analyze the effects of the preprocessing performed in DFT-LMS and DCT-LMS for first-order Markov inputs. In particular, we show that for Markov-1 inputs of correlation parameter /spl rho//spl isin/[0,1], the eigenvalue spread after DFT and power normalization tends to (1+/spl rho/)l(1-/spl rho/) as the size of the filter gets large, whereas after DCT and power normalization, it reduces to (1+/spl rho/). For comparison, the eigenvalue spread before transformation is asymptotically equal to (1+/spl rho/)/sup 2//(1-/spl rho/)/sup 2/. The analytical method used in the paper provides additional insight into how the algorithms work and is expected to extend to other input signal classes and other transformations. >