LMIs, interior point methods, complexity theory, and robustness analysis

Studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean-squared error (MSE). When a symmetric source density with infinite support is sufficiently well behaved, the optimal step size /spl Delta//sub N/ for symmetric uniform scalar quantization decreases as 2/spl sigma/N/sup -1/V~/sup -1/(1/6N/sup 2/), where N is the number of quantization levels, /spl sigma//sup 2/ is the source variance and V~/sup -1/(/spl middot/) is the inverse of V~(y)=y/sup -1/ /spl int//sub y//sup /spl infin// P(/spl sigma//sup -1/X>x) dx. Equivalently, the optimal support length N/spl Delta//sub N/ increases as 2/spl sigma/V~/sup -1/(1/6N/sup 2/). Granular distortion is asymptotically well approximated by /spl Delta//sub N//sup 2//12, and the ratio of overload to granular distortion converges to a function of the limit /spl tau//spl equiv/lim/sub y/spl rarr//spl infin//y/sup -1/E[X|X>y], provided, as usually happens, that /spl tau/ exists. When it does, its value is related to the number of finite moments of the source density, an asymptotic formula for the overall distortion D/sub N/ is obtained, and /spl tau/=1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by /spl Delta//sub N//sup 2//12. Applying these results to the class of two-sided densities of the form b|x|/sup /spl beta//e(-/spl alpha/|x|/sup /spl alpha//), which includes Gaussian, Laplacian, Gamma, and generalized Gaussian, it is found that /spl tau/=1, that /spl Delta//sub N/ decreases as (ln N)/sup 1//spl alpha///N, that D/sub N/ is asymptotically well approximated by /spl Delta//sub N//sup 2//12 and decreases as (ln N)/sup 2//spl alpha///N/sup 2/, and that more accurate approximations to /spl Delta//sub N/ are possible. The results also apply to densities with one-sided infinite support, such as Rayleigh and Weibull, and to densities whose tails are asymptotically similar to those previously mentioned.

Given an integer /spl sigma/>1, a vector (/spl delta//sub 1/, /spl delta//sub 2/,..., /spl delta//sub /spl sigma/-1/), of nonnegative integers, and an undirected graph G=(V, E), an L(/spl delta//sub 1/, /spl delta//sub 2/,..., /spl delta//sub /spl sigma/-1/)-coloring of G is a function f from the vertex set V to a set of nonnegative integers, such that |f(u)-f(v)|/spl ges//spl delta//sub i/, if d(u,v)=i, for 1<i<(/spl sigma/-1), where d(u, v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(/spl delta//sub 1/, /spl delta//sub 2/,..., /spl delta//sub /spl sigma/-1/)-coloring for G is one using the smallest range /spl lambda/ of integers over all such colorings. This problem has relevant application in channel assignment for interference avoidance in wireless networks, where channels (i.e., colors) assigned to interfering stations (i.e., vertices) at distance i must be at least /spl delta//sub i/ apart, while the same channel can be reused in vertices whose distance is at least /spl sigma/. In particular, two versions of the coloring problem - L(2, 1, 1) and L(/spl delta//sub 1/, 1,..., 1) - are considered. Since these versions of the problem are NP-hard for general graphs, efficient algorithms for finding optimal colorings are provided for specific graphs modeling realistic wireless networks, including rings, bidimensional grids, and cellular grids.

Bandpass modulators sampling at high IFs (/spl sim/200 MHz) allows direct sampling of the RF signal, reducing analog hardware, and makes it easier to realize completely software programmable receivers. This paper presents the circuit design of and test results from a continuous time tunable IF sampling 4/sup th/ order bandpass delta-sigma modulator implemented in InP HBT IC technology for use in a multi-mode digital receiver application. The bandpass /spl Delta//spl Sigma/ modulator is fabricated in AlInAs/GaInAs heterojunction bipolar technology with a peak unity current gain cutoff frequency (f/sub T/) of 130 GHz and a maximum frequency of oscillation (f/sub MAX/) of 130 GHz. The 4/sup th/ order bandpass /spl Delta//spl Sigma/ modulator consists of two bandpass resonators that can be tuned to optimize both wideband and narrowband operation. The IF is tunable from 140 MHz to 210 MHz in this /spl Delta//spl Sigma/ modulator for use in multiple platform applications. Operating from /spl plusmn/5 V power supplies, the fabricated 4/sup th/ order /spl Delta//spl Sigma/ modulator sampling at 4 GSPS demonstrates stable behavior and achieves a signal to (noise+distortion) ratio (SNDR) of 78 dB @ 1 MHz BW and 50 dB @ 60 MHz BW. The average SNDR performance measured on over 250 parts is 72.5 dB @ 1 MHz BW and 47.7 dB @ 60 MHz BW.

Using oversampled sigma-delta (/spl Sigma/-/spl Delta/) techniques a signal can be represented as a single bit binary data stream ({-1, 1}), with the quantization noise high pass filtered out of baseband. In this paper adaptive LMS digital filters are presented that directly use binary /spl Sigma/-/spl Delta/ input streams and update the PCM adaptive filter weights at the oversampled rate to realize architectures that are very suitable for mixed analog/digital VLSI implementation. Using the /spl Sigma/-/spl Delta/ adaptive architecture PCM values are multiplied by single bits, and can therefore be easily performed by a simple multiplexor. Also using the /spl Sigma/-/spl Delta/ approach, the decimation and interpolation stages required for converting from /spl Sigma/-/spl Delta/ to PCM (and vice-versa) suitable for Nyquist DSP algorithms are circumvented. The paper describes the sigma delta adaptive LMS filter architecture and presents preliminary results for adaptive echo cancellation using adaptive sigma delta and standard PCM architectures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The spectroscopic properties of Ho/sup 3+/ laser channels in KGd(WO/sub 4/)/sub 2/ crystals have been investigated using optical absorption, photoluminescence, and lifetime measurements. The radiative lifetimes of Ho/sup 3+/ have been calculated through a Judd-Ofelt (JO) formalism using 300-K optical absorption results. The JO parameters obtained were /spl Omega//sub 2/=15.35/spl times/10/sup -20/ cm/sup 2/, /spl Omega//sub 4/=3.79/spl times/10/sup -20/ cm/sup 2/, /spl Omega//sub 6/=1.69/spl times/10/sup -20/ cm/sup 2/. The 7-300-K lifetimes obtained in diluted (8/spl middot/10/sup 18/ cm/sup -3/) KGW:0.1% Ho samples are: /spl tau/(/sup 5/F/sub 3/)/spl ap/0.9 /spl mu/s, /spl tau/(/sup 5/S/sub 2/)=19-3.6 /spl mu/s, and /spl tau/(/sup 5/F/sub 5/)/spl ap/1.1 /spl mu/s. For Ho concentrations below 1.5/spl times/10/sup 20/ cm/sup -3/, multiphonon emission is the main source of non radiative losses, and the temperature independent multiphonon probability in KGW is found to follow the energy gap law /spl tau//sub ph//sup -1/(0)=/spl beta/exp(-/spl alpha//spl Delta/E), where /spl beta/=1.4/spl times/10/sup -7/ s/sup -1/, and /spl alpha/=1.4/spl times/10/sup 3/ cm. Above this holmium concentration, energy transfer between Ho impurities also contributes to the losses. The spectral distributions of the Ho/sup 3+/ emission cross section /spl sigma//sub EM/ for several laser channels are calculated in /spl sigma/- and /spl pi/-polarized configurations. The peak a /spl sigma//sub EM/ values achieved for transitions to the /sup 5/I/sub 8/ level are /spl ap/2/spl times/10/sup -20/ cm/sup 2/ in the /spl sigma/-polarized configuration, and three main lasing peaks at 2.02, 2.05, and 2.07 /spl mu/m are envisaged inside the /sup 5/I/sub 7//spl rarr//sup 5/I/sub 8/ channel.

This paper presents an investigation of the performance characteristics of a pair of bandpass Sigma-Delta (/spl Sigma/-/spl Delta/) analog-to-digital (A/D) converters. One of these /spl Sigma/-/spl Delta/ A/D converters has the conventional cascade-of-integrators structure while the other has the conventional cascade-of-resonators structure. These /spl Sigma/-/spl Delta/ A/D converters are subsequently compared in terms of their achievable signal-to-quantization noise ratios for corresponding (discrete) switched-capacitor implementations. Based on this comparison, a new bandpass /spl Sigma/-/spl Delta/ A/D converter structure is proposed which possesses the best practical features of both converters. These include, complementary signal and noise transfer functions and the fact that the zeros of the noise transfer function are constrained to lie on the unit-circle.

In this paper, the design of a continuous-time baseband sigma-delta (/spl Sigma//spl Delta/) modulator with an integrated mixer for intermediate-frequency (IF) analog-to-digital conversion is presented. This highly linear IF /spl Sigma//spl Delta/ modulator digitizes a GSM channel at intermediate frequencies up to 50 MHz. The sampling rate is not related to the input IF and is 13.0 MHz in this design. Power consumption is 1.8 mW from a 2.5-V supply. Measured dynamic range is 82 dB, and third-order intermodulation distortion is -84 dB for two -6-dBV IF input tones. Two modulators in quadrature configuration provide 200-kHz GSM bandwidth. Active area of a single IF /spl Sigma//spl Delta/ modulator is 0.2 mm/sup 2/ in 0.35-/spl mu/m CMOS.

It is well known that /spl Sigma/-/spl Delta/ A/D converters having complementary signal and noise transfer functions exhibit high signal-to-quantization-noise ratios (SQNRs) and high dynamic ranges (DRs), but that they tend to be overly sensitive to capacitor mismatches (potentially leading to unstable converter operation) in a corresponding switched-capacitor (SC) hardware implementation. Recently, a new class of /spl Sigma/-/spl Delta/ A/D converters was developed based on magnitude-squared or magnitude complementary signal and noise transfer functions to decrease the sensitivity to capacitor mismatches at the expense of slightly lower SQNRs and DRs. This paper is concerned with the development of a novel approach to the design of /spl Sigma/-/spl Delta/ A/D converters. This approach combines the transfer function complementarity (in the overall frequency band) with signal and noise transfer function magnitude complementarity (in the signal passband) to obtain a single /spl Sigma/-/spl Delta/ A/D converter with a cascade-of-resonators configuration. Consequently, the resulting A/D converters achieve not only high-stability (arising from magnitude complementarity), but also high DR and SQNR (arising from transfer function complementarity) in the SC hardware implementation. A practical application example is given to illustrate the results.

We present novel oversampled implementations of adaptive LMS FIR filters using sigma delta (/spl Sigma//spl Delta/) computation strategies. Using this approach in conjunction with /spl Sigma//spl Delta/ DACs and ADCs the need for expensive decimation and interpolation stages for the /spl Sigma//spl Delta/ codec has been circumvented. Hence the complete architecture can make considerable savings in total silicon area for a mixed signal ASIC implementation. The LMS algorithm performance of the /spl Sigma//spl Delta/ approach is shown to be comparable to the traditional Nyquist rate PCM LMS algorithm.

This paper presents a quadrature bandpass /spl Sigma//spl Delta/ modulator with continuous-time architecture. Due to the continuous-time architecture and the inherent anti-aliasing filter, the proposed /spl Sigma//spl Delta/ modulator needs no additional anti-aliasing filter in front of the modulator in contrast to quadrature bandpass /spl Sigma//spl Delta/ modulators with switched-capacitor architectures. The second-order /spl Sigma//spl Delta/ modulator digitizes complex analog I/Q input signals at 1-MHz intermediate frequency and operates within a clock frequency range of 25-100 MHz. The modulator chip achieves a peak signal-to-noise-distortion ratio (SNDR) of 56.7 dB and a dynamic range of 63.8 dB within a 1-MHz signal bandwidth and at a clock frequency of 100 MHz. Furthermore, it provides an image rejection of at least 40 dB. The 0.65-/spl mu/m BiCMOS chip consumes 21.8 mW at 2.7-V supply voltage.

A technique for shaping component mismatch error in the real and imaginary paths of a complex (quadrature) /spl Delta//spl Sigma/ are studied and simulated. Complex bandpass /spl Delta//spl Sigma/ modulators provide improved performance over a pair of real bandpass /spl Delta//spl Sigma/ modulators of the same order in the conversion of narrow-band quadrature intermediate frequency (IF) signals. This is possible due to elimination of redundant conjugate pairs of poles and zeros in its noise and signal transfer function. Mismatches in the modulators real and imaginary paths, however, results in spectral leakage of image band interference and image band quantization noise to the passband of the converted output. Strategic pole-zero placement, as well as adaptive techniques has been studied in earlier works. The work presented here takes advantage of the inherent dual paths of the complex bandpass /spl Delta//spl Sigma/ modulator to reduce component mismatch effects using a switched capacitor error shaping technique.

A quadrature bandpass /spl Delta//spl Sigma/ modulator IC facilitates monolithic digital-radio-receiver design by allowing straightforward "complex A/D conversion" of an image reject mixer's I and Q, outputs. Quadrature bandpass /spl Delta//spl Sigma/ modulators provide superior performance over pairs of real bandpass /spl Delta//spl Sigma/ modulators in the conversion of complex input signals, using complex filtering embedded in /spl Delta//spl Sigma/ loops to efficiently realize asymmetric noise-shaped spectra. The fourth-order prototype IC, clocked at 10 MHz, converts narrowband 3.75-MHz I and Q inputs and attains a dynamic range of 67 dB in 200-kHz (GSM) bandwidth, increasing to 71 and 77 dB in 100- and 30-kHz bandwidths, respectively. Maximum signal-to-noise plus distortion ratio (SNDR) in 200-kHz bandwidth is 62 dB. Power consumption is 130 mW at 5 V. Die size in a 0.8-/spl mu/m CMOS process is 2.4/spl times/1.8 mm/sup 2/.

One way to increase the conversion bandwidth of the Delta-Sigma (/spl Delta//spl Sigma/) A/D converter is to use parallelism. The three main types of parallel /spl Delta//spl Sigma/ A/D converter which are practical for implementation are the time-interleaved, the parallel /spl Delta//spl Sigma/ (/spl Pi//spl Delta//spl Sigma/), and the frequency band decomposition A/D converters. In this paper, it is shown that the operation of all three approaches can be explained using the same underlying theory. This common structure is then used to explore the design tradeoffs among these architectures.

An exact recursive formula is derived to describe the structure of an ideal first-order /spl Sigma/-/spl Delta/ output sequence as a function of its input. Specifically, it is shown that every /spl Sigma/-/spl Delta/ sequence generated by the constant input x/spl isin/[0, 1] can be decomposed into a shorter E-A subsequence whose input x'/spl isin/[0, 1) may be used to recover that of the original /spl Sigma/-/spl Delta/ sequence. This formula is applied to develop an O(N log N) algorithm for decoding an N-length sequence. Without knowledge of the modulator's initial state, it exhibits an average improvement, over all initial states, of 4.2 dB in output signal-to-noise ratio (SNR) compared with a near-optimal linear finite impulse response (FIR) filter. The regularity of the ideal first-order /spl Sigma/-/spl Delta/ structure with constant inputs permits the algorithm to be extended to bandlimited and noise-corrupted data. A simple error correction procedure is demonstrated, and it is shown that the recursive algorithm can outperform FIR filters on sequences of length N<64 having input SNRs as low as 30 dB.

We present an integrable solution for the detection of defective sensor elements using sigma-delta-(/spl Sigma//spl Delta/)-modulation and a matched filter. The sensor element is stimulated using a pseudo random binary sequence (PRBS). The sensor signal is read out and the analog output is digitized using a /spl Sigma//spl Delta/-modulator. The binary pulse density stream of the /spl Sigma//spl Delta/-modulator is the output of the sensor system and thus should ideally contain the PRBS. A matched filter has the task of detecting the pseudo random sequence in the pulse density stream and its sampled output is compared to a threshold thus making it possible to judge the functionality of the sensor element. By evaluating the magnitude of the matched filter output it is also possible to measure the sensor sensitivity. We present a discrete solution of this method, but an integrated chip using a standard 1.2 /spl mu/m CMOS process has been designed and is being fabricated.