Filter Bank Optimization with Applications in Noise Suppression and Communications

Citation

Akkarakaran, Sony John (2001) Filter Bank Optimization with Applications in Noise Suppression and Communications. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4712-9414. https://resolver.caltech.edu/CaltechTHESIS:10072010-095516188

Abstract

A filter bank (FB) is used to analyze or decompose a signal into several frequency bands, which are processed separately and then combined. This allows us to allocate processing resources in a manner tailored to the distribution of the relevant signal features among the bands. A judicious allocation leads to improved system performance over direct processing of the input signal (without using an FB). FBs have found applications in almost every area of modern digital signal processing, including audio, image and video compression and communications.

The main thrust of this thesis is towards the optimization of FBs based on the statistical properties of their input. We establish the optimality of a type of FB called the principal component filter bank (PCFB) for numerous signal processing problems. The PCFB depends on the input power spectrum and on the class of M channel orthonormal FBs over which we seek to optimize the FB. PCFB optimality for compression and progressive transmission has been observed to varying degrees in the past. Our work provides a unified framework for orthonormal FB optimization, that includes these earlier results as special cases. It also covers many other problems not observed earlier, notably in noise suppression and communications.

A central result that we establish is that the PCFB is the optimum orthonormal FB whenever the minimization objective is a concave function of the vector of subband variances of the FB. Many signal processing problems result in such objectives. The earlier results on PCFB optimality for compression can be explained by this framework. Another example not noticed earlier is FBbased white noise reduction using zeroth order Wiener filters or hard thresholds in the subbands. Yet another case involves the discrete multitone modulation (DMT) communication system, used in ADSL (asymmetric digital subscriber line) and wireless OFDM (orthogonal frequency division multiplexing) technologies. These systems use the transmultiplexer configuration of an FB, which is usually chosen as a DFT or cosine-modulated FB for efficiency of implementation. We show that at increased implementation cost, we can minimize the transmission power requirement (for a given bitrate and error probability) by using the PCFB associated with a certain normalized noise spectrum. We present simulation examples with realistic channel and noise models for the ADSL system to compare the performance of the PCFB against other types of FBs, such as the DFT.

We study various extensions of the basic PCFB optimality result. The noise suppression problem becomes more involved when the noise is colored, because the objective then depends on both signal and noise subband variances. For a specific FB class, namely the orthogonal transform coder class, we show that a simultaneous PCFB for the signal and noise is optimal (if it exists). For the class of unconstrained FBs, this does not hold in general; we develop an algorithm that computes the best FB for piecewise constant spectra. In some cases, PCFB optimality extends to classes of biorthogonal FBs too, although there are many open problems in this area, as we point out. We study the effect of nonexistence of a PCFB on the FB optimizations and show how they usually become analytically intractable. We show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs. We also study nonuniform FB optimization: We establish the definition of nonuniform PCFBs and study their existence and optimality, which are shown to be much more restricted when compared with uniform PCFBs.

Lastly, we study a related open problem on the parameterization of nonuniform perfect reconstruction (PR) FBs of various classes, such as the rational and FIR classes. Not all nonuniform PRFBs can be built by the common method of using tree structures of uniform PRFBs. Given a set of decimators, is there a rational PRFB using them? If so, what are all the PRFBs possible? When are they necessarily derivable from a tree structure? Very little is known about the answers to many such questions. For example, for existence of rational PRFBs with a given set of decimators, certain conditions on the decimators are known to be necessary, while certain others are sufficient. However, conditions that are both necessary and sufficient are unknown. One of our contributions is to strengthen considerably the known conditions. This is an important step towards a complete PR theory for nonuniform filter banks.

Thesis Files