In this chapter we deal with the following nonlinear fractional programming problem: 

$$P:\mathop{{\max }}\limits_{{x \in s}} q(x) = (f(x) + \alpha )/((x) + \beta )$$

where f, g: R n → R, α, β ∈ R, S ⊆ R n . To simplify things, and without restricting the generality of the problem, it is usually assumed that, g(x) + β + 0 for all x ∈ S,S is non-empty and that the objective function has a finite optimal value.

Nonlinear Fractional Programming

Optical coherence tomography (OCT) and optical coherence microscopy (OCM) are novel techniques for noninvasive biomedical imaging based on low-coherence interferometry. OCT achieves high-spatial resolution ( 100 dB) in a fiber-optically integrated system which is suitable for application in minimally invasive diagnostics, including endoscopy. The technique of OCM combines the depth-ranging capability of OCT with the micron-scale resolution imaging capability of confocal microscopy to extend the available imaging depth of confocal microscopy up to several hundred micrometers deep in highly scattering tissues. The theoretical and technical bases for OCT and OCM imaging are described. Example OCT images are provided in gastrointestinal (GI) tissues to illustrate contrast between histological layers of the GI mucosa and differentiation of the mucosa from submucosa. Example OCM images revealing cellular-level microstructure up to several hundred micrometers deep in GI tissue are presented for the first time. The potential applications of OCT and OCM imaging in clinical diagnsotic medicine are discussed.

Optical Coherence Tomography and Microscopy in Gastrointestinal Tissues

Imaging systems that construct an image from phase information in received signals include synthetic aperture radar (SAR) and optical Doppler tomography (ODT) systems. A fundamental problem in the image formation is phase ambiguity, i.e., it is impossible to distinguish between phases that differ by 2/spl pi/. Phase unwrapping in two dimensions essentially consists of detecting the pixel locations of the phase discontinuities, finding an ordering among the pixel locations for unwrapping the phase, and adding offsets of multiples of 2/spl pi/. In this paper, we propose a new method for detecting phase discontinuities. The method is based on a supervised feedforward multilayer perceptron neural network. We train and test the neural network on simulated phase images formed in an ODT system. For the ODT phase images, the new method detects the correct unwrapping locations where some conventional methods fail. The key contribution of the paper is a one-pass pixel-parallel low-complexity method for detecting phase discontinuities.

/pdf/two-dimensional-phase-unwrapping-using-neural-networks-4yds8bufvl.pdf

Two-dimensional phase unwrapping using neural networks

To ease equalization in a multicarrier system, a cyclic prefix (CP) is typically inserted between successive symbols. When the channel order exceeds the CP length, equalization can be accomplished via a time-domain equalizer (TEQ), which is a finite impulse response (FIR) filter. The TEQ is placed in cascade with the channel to produce an effective shortened impulse response. Alternatively, a bank of equalizers can remove the interference tone-by-tone. This paper presents a unified treatment of equalizer designs for multicarrier receivers, with an emphasis on discrete multitone systems. It is shown that almost all equalizer designs share a common mathematical framework based on the maximization of a product of generalized Rayleigh quotients. This framework is used to give an overview of existing designs (including an extensive literature survey), to apply a unified notation, and to present various common strategies to obtain a solution. Moreover, the unification emphasizes the differences between the methods, enabling a comparison of their advantages and disadvantages. In addition, 16 different equalizer structures and design procedures are compared in terms of computational complexity and achievable bit rate using synthetic and measured data.

/pdf/unification-and-evaluation-of-equalization-structures-and-2tjr9xomnz.pdf

Unification and evaluation of equalization structures and design algorithms for discrete multitone modulation systems

Time domain synchronous OFDM (TDS-OFDM) has a higher spectrum and energy efficiency than standard cyclic prefix OFDM (CP-OFDM) by replacing the unknown CP with a known pseudorandom noise (PN) sequence. However, due to mutual interference between the PN sequence and the OFDM data block, TDS-OFDM cannot support high-order modulation schemes such as 256QAM in realistic static channels with large delay spread or high-definition television (HDTV) delivery in fast fading channels. To solve these problems, we propose the idea of using multiple inter-block-interference (IBI)-free regions of small size to realize simultaneous multi-channel reconstruction under the framework of structured compressive sensing (SCS). This is enabled by jointly exploiting the sparsity of wireless channels as well as the characteristic that path delays vary much slower than path gains. In this way, the mutually conditional time-domain channel estimation and frequency-domain data demodulation in TDS-OFDM can be decoupled without the use of iterative interference removal. The Cramer-Rao lower bound (CRLB) of the proposed estimation scheme is also derived. Moreover, the guard interval amplitude in TDS-OFDM can be reduced to improve the energy efficiency, which is infeasible for CP-OFDM. Simulation results demonstrate that the proposed SCS-aided TDS-OFDM scheme has a higher spectrum and energy efficiency than CP-OFDM by more than 10% and 20% respectively in typical applications.

Spectrum- and Energy-Efficient OFDM Based on Simultaneous Multi-Channel Reconstruction

In a multicarrier modulation system, a time domain equalizer (TEQ) traditionally shortens the transmission channel impulse response (CIR) to mitigate intersymbol interference (ISI). In this paper, we propose a data-rate optimal TEQ filter bank whose data rates at the equalizer output of this filter bank are significantly better than those of the Maximum Bit Rate and Minimum ISI methods and similar to those of the least-squares per-tone equalizer method for ADSL CIRs, transmit filters, and receive filters. The contributions of this study are: (1) a new model for multicarrier demodulator output SNR that includes ISI, near-end crosstalk, white Gaussian noise, and the digital noise floor, (2) data rate optimal time domain per-tone TEQ filter bank, and (3) a new achievable upper bound on achievable bit rate.

https://users.ece.utexas.edu/~bevans/papers/2002/optimalTEQ/optimalTEQAsil2002paper.pdf

DMT bit rate maximization with optimal time domain equalizer filter bank architecture

Several high-speed communication standards modulate encoded data on multiple-carrier frequencies using the fast Fourier transform (FFT). The real part of the quantized inverse FFT samples form a symbol. The symbol is periodically extended by prepending a copy of its last few samples, also known as a cyclic prefix. When the cyclic prefix is longer than the channel order, amplitude and phase distortion can be equalized entirely in the frequency domain. In the receiver, prior to the FFT, a time-domain equalizer, in the form of a finite-impulse response filter, shortens the effective channel impulse response. Alternately, a bank of equalizers tuned to each carrier frequency can be used. In earlier literature, we unified optimal multicarrier equalizer design algorithms as a product of generalized Rayleigh quotients. In this paper, we convert the unified theoretical framework into a framework for fast design algorithms. The relevant literature is reviewed and classified according to this framework. We analyze the achieved bit rate versus implementation complexity (in terms of multiply-and-accumulate operations) tradeoffs in the original and fast design algorithms. The comparison includes multiple implementations of each of 16 different equalizer structures and design algorithms using synthetic and measured discrete multitone modulated data

https://users.ece.utexas.edu/~bevans/papers/2005/equalizationII/multicarrierEqualizerDesignDraft.pdf

Implementation Complexity and Communication Performance Tradeoffs in Discrete Multitone Modulation Equalizers

A time domain equalizer (TEQ) is a finite impulse response filter that shortens the channel impulse response (CIR) to mitigate intersymbol interference (ISI). P. Melsa et al. (see IEEE Trans. on Commun., vol.44, p.1662-72, 1996) minimized ISI in the time domain by maximizing the shortening signal-to-noise ratio (SSNR) of the energy in a target window to the energy outside the target window in the shortened CIR. Infinite SSNR means zero ISI. Melsa et al. also developed a joint channel shortening method to design a single TEQ to shorten a channel and a near-end echo impulse response. We extend the joint SSNR method to design a single TEQ to shorten multiple channels by maximizing the composite SSNR. The composite SSNR is a weighted sum of normalized channel SSNRs. The normalized SSNR is the ratio of the energy in the target window samples to the energy of all samples in the shortened CIR and has a range of [0, 1] so that it is better suited for numerical stability and fixed-point implementation. Our proposed method outperforms the joint channel shortening method. because it achieves higher weighted sum of SSNRs of the used channels.

/pdf/simultaneous-multichannel-time-domain-equalizer-design-based-4xj3qc4s0g.pdf

Simultaneous multichannel time domain equalizer design based on the maximum composite shortening SNR

The traditional discrete multi-tone equalizer is a cascade of a time domain equalizer (TEQ) as a single finite impulse response filter, a multicarrier demodulator as a fast Fourier transform (FFT), and a frequency domain equalizer (FEQ) as a one-tap filter bank. The TEQ shortens the transmission channel impulse response (CIR) to mitigate inter-symbol interference (ISI). Maximum Bit Rate (MBR) and Minimum ISI (Min-ISI) methods achieve higher data rates at the TEQ output than previously published methods. As an alternative to the traditional equalizer, the per-tone equalizer (PTE) moves the TEQ into the FEQ and customizes a multi-tap FEQ for each tone. In this paper, we propose atime domainTEQ filter bank (TEQFB) and single TEQ that demonstrate better data rates at the FEQ output than MBR, Min-ISI, and least-squares PTE methods with standard CIRs, transmit filters, and receive filters. The contributions of this paper are: (1) a model for the signal-to-noise ratio (SNR) at the FFT output that includes ISI, near-end crosstalk, white Gaussian noise, analog-to-digital converter quantization noise and the digital noise floor; (2) data rate optimal time domain per-tone TEQ filter bank and the upper bound on bit rate performance it achieves; and (3) data rate maximization single TEQ design algorithm. SP EDICS: SP 3-TDSL: Telephone Networks and Digital Subscriber Loops. Milo s̆ Milo s̆evíc *, B. L. Evans and R. Baldick are with the Dept. of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712-1084, L. F. C. Pessoa is with Motorola, Inc.,7700 West Parmer Lane, MD: TX32-PL30, Austin, TX 78729. E-mail: {milos,bevans,baldick }@ece.utexas.edu andLucio.Pessoa@motorola.com . B. L. Evans was supported by a gift from the Motorola Semiconductor Products Sector, Austin, Texas, and The State of Texas Advanced Technology Program under grant 003658-0614-2001.

/pdf/optimal-time-domain-equalization-design-for-maximizing-data-4079gl1cmr.pdf

Milos Milosevic

Papers

Unification and evaluation of equalization structures and design algorithms for discrete multitone modulation systems

DMT bit rate maximization with optimal time domain equalizer filter bank architecture

Implementation Complexity and Communication Performance Tradeoffs in Discrete Multitone Modulation Equalizers

Simultaneous multichannel time domain equalizer design based on the maximum composite shortening SNR

Optimal Time Domain Equalization Design for Maximizing Data Rate of Discrete Multi-Tone Systems