| dc.description.abstract | Adaptive Filtering (AF) [1], [2] is used in a variety of applications such as system identification, channel equalization, active noise cancellation, and echo cancellation. Over the years, significant research efforts have gone into the development of efficient algorithms for adaptive filtering. The efficiency of an adaptive algorithm is evaluated by performance indices such as convergence rate, computational complexity, accuracy of the solution, and robustness to round?off error accumulation. Most adaptive algorithms are designed using a squared?error–based cost function, achieving the best trade?off among various performance criteria. Computationally attractive LMS and fast RLS algorithms are the most celebrated examples of adaptive filtering.
Applications such as acoustic noise control typically require adaptive filters with hundreds of taps and the ability to process coloured input signals. In such cases, the performance of direct?form FIR?filter–based RLS and LMS cannot be guaranteed. The computational complexity of RLS is exorbitant, and the convergence of LMS becomes worse as filter length increases. Several improvements to these basic algorithms have therefore been explored, including Transform?Domain Adaptive Filtering (TDAF), Block Adaptive Filtering, Fast RLS, and the Affine Projection Algorithm.
In a system?identification scenario, a system can be alternatively modelled using a set of short adaptive filters operating in parallel on subband signals rather than a direct?form adaptive FIR filter. The subband signals are generated using a filterbank structure. One drawback of a filterbank?based adaptive filter is that it cannot model an arbitrary FIR filter exactly due to aliasing and the processing delay involved.
Filter?Bank Adaptive Filters (FBAF) overcome these limitations by employing the interpolated FIR (IFIR) approach. A sparse subfilter Ck(zD)C_k(z^D)Ck?(zD) (for k=0,1,…,M?1k = 0, 1, \ldots, M-1k=0,1,…,M?1) performs the interpolation, where DDD is the sparsity factor and MMM is the number of channels in the filterbank. FBAF includes TDAF as a special case when the sparse filter length is one. FBAF structures are capable of exactly modelling an arbitrary FIR transfer function. Usually, FBAFs employ interpolating filterbanks as the synthesis filterbank of a paraunitary PR filterbank. Such an FBAF can model any FIR system with a delay depending on the interpolator length. This case is referred to as the delaying IFIR structure.
Using a matrix formulation, the model of an FIR filter sss using the IFIR structure is given by
s=Fc,s = Fc,s=Fc,
where FFF is a matrix formed from the interpolators. For the delaying IFIR structure, the columns of FFF are orthogonal. Several delayless subband structures [4], [5] have been proposed. An FBAF for the case M=D+1M = D + 1M=D+1 can model a system without delay. In this case, the columns of FFF are generally not linearly independent. The properties of FFF depend on the interpolators used, and this is referred to as the delayless IFIR structure.
Details of the Work
In this thesis, using the matrix formulation, the generalized IFIR structure is analysed and its modelling capability is investigated. A delayless IFIR structure- a special case of the generalized structure- is discussed. This structure requires the design of interpolators. The design of these interpolators is addressed.
We propose Scheme A, an algorithm that adapts the interpolators along with the sparse filters using the same cost function. The step?size for interpolators is chosen to be very small compared to the step?size for sparse?filter adaptation, improving convergence properties. We also discuss an existing method (Scheme B) [7], which uses nonlinear optimisation to design interpolators assuming the input?signal statistics are known. While Scheme B involves an elaborate offline design and depends on known statistics, Scheme A is an online method and does not require prior statistical knowledge. However, Scheme A has higher online computational complexity.
An approximate delayless IFIR structure is also introduced, employing the synthesis filters of a paraunitary PR filterbank as the interpolating filterbank. This simpler structure provides insight into the trade?offs involved in designing delayless IFIR systems.
Conclusion
The delayless IFIR structure, a special case of the generalized IFIR framework, has been examined. The problem of choosing suitable interpolators has been addressed. An adaptive scheme that performs joint adaptation of interpolators and sparse filters has been proposed, analysed, and validated through simulations. A simplified approximate delayless IFIR structure has also been developed to enhance understanding of delayless IFIR design. | |
