Filterbank optimization with convex objectives and the optimality of principal component forms

This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs

Linear phase paraunitary filter banks: theory, factorizations and designs

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. In this paper, we study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. We begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, we develop a minimal factorizdion for a large class of such systems. This factorization will be proved to be complete for even M. Further, we structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. We then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory

Discrete multitone modulation with principal component filter banks

Discrete multitone (DMT) modulation is an attractive method for communication over a nonflat channel with possibly colored noise. The uniform discrete Fourier transform (DFT) filter bank and cosine modulated filter bank have in the past been used in this system because of low complexity. We show in this paper that principal component filter banks (PCFB) which are known to be optimal for data compression and denoising applications, are also optimal for a number of criteria in DMT modulation communication. For example, the PCFB of the effective channel noise power spectrum (noise psd weighted by the inverse of the channel gain) is optimal for DMT modulation in the sense of maximizing bit rate for fixed power and error probabilities. We also establish an optimality property of the PCFB when scalar prefilters and postfilters are used around the channel. The difference between the PCFB and a traditional filter bank such as the brickwall filter bank or DFT filter bank is significant for effective power spectra which depart considerably from monotonicity. The twisted pair channel with its bridged taps, next and fext noises, and AM interference, therefore appears to be a good candidate for the application of a PCFB. This is demonstrated with the help of numerical results for the case of the ADSL channel

A class of M-Channel linear-phase biorthogonal filter banks and their applications to subband coding

This correspondence presents a new factorization for linearphase biorthogonal perfect reconstruction (PR) FIR filter banks. Using this factorization, we propose a new family of lapped transform called the generalized lapped transform (GLT). Since the analysis and synthesis filters of the GLT are not restricted to be the time reverses of each other, they can offer more freedom to avoid blocking artifacts and improve coding gain in subband coding applications. The GLT is found to have higher coding gain and smoother synthesis basis functions than the lapped orthogonal transform (LOT). Simulation results also demonstrated that the GLT has significantly less blocking artifacts, higher peak signal-tonoise ratio (PSNR), and better visual quality than the LOT in image coding. Simplified GLT with different complexity/performance tradeoff is also studied. © 1999 IEEE.published_or_final_versio

Multiplier-less discrete sinusoidal and lapped transforms using sum-of-powers-of-two (SOPOT) coefficients

This paper proposes a new family of multiplier-less discrete cosine and sine transforms called the SOPOT DCTs and DSTs. The fast algorithm of Wang [10] is used to parameterize all the DCTs and DSTs in terms of certain (2×2) matrices, which are then converted to SOPOT representation using a method previously proposed by the authors [7]. The forward and inverse transforms can be implemented with the same set of SOPOT coefficients. A random search algorithm is also proposed to search for these SOPOT coefficients. Experimental results show that the (2×2) basic matrix can be implemented, on the average, in 6 to 12 additions. The proposed algorithms therefore require only O(N log2N) additions, which is very attractive for VLSI implementation. Using these SOPOT DCTs/DSTs, a family of SOPOT Lapped Transforms (LT) is also developed. They have similar coding gains but much lower complexity than their real-valued counterparts.published_or_final_versio

Perfect reconstruction modulated filter banks with sum of powers-of-two coefficients

IEEE International Symposium on Circuits and Systems, Geneva, Switzerland, 28-31 May 2000In this paper, a new family of multiplier-less modulated filter banks, called the SOPOT MFB, is presented. The coefficients of the proposed filter banks consist of sum of powers-of-two coefficients (SOPOT), which require only simple shifts and additions for implementation. The modulation matrix and the prototype filter are derived from a fast DCT-IV algorithm of Wang and the lattice structure in [1]. The design of the SOPOT MFB is performed using the genetic algorithm(GA). An 16-channel SOPOT MFB with 34 dB stopband attenuation is given as an example, and its average number of terms per SOPOT coefficient is only 2.6.published_or_final_versio

Signal Flow Graph Approach to Efficient DST I-IV Algorithms

In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having $(n-1)$ points signal flow graph for DST-I and $n$ points signal flow graphs for DST II-IV