Numerical method for determining the quantization error PDF of single-bit /spl Sigma//spl Delta/ Modulators

Existing models for the quantizer of /spl Sigma//spl Delta/ modulators make assumptions on the probability density function (pdf) of the quantization error, or some other convenient signal of the modulator. In this paper, a method for the determination of this pdf for single-bit /spl Sigma//spl Delta/ modulators is presented. First, a numerical method is proposed in order to solve the simplified equation for the quantization error pdf for first-order systems considering noiseless and noisy dc input signals. Then, it is shown how most practical high-order (>2)/spl Sigma//spl Delta/ modulators, resulting from well-established design methods, can be modeled as first-order systems plus an additive noise source at the input. Hence, their quantization error pdf is analyzed using the proposed method. Simulation results are shown to be in considerable agreement with those of the proposed method.     

Noise-coupled /spl Delta//spl Sigma/ ADC's

The first-order noise coupling scheme proposed earlier is generalised to the realisation of higher-order enhancement. It is also extended to single-stage DeltaSigma loops, and to split structures with self-enhancement. The advantages and limitations of these new DeltaSigma architectures are compared with those of conventional single-stage and cascade DeltaSigma structures. As demonstrated by an example, they exhibit improved stability and robustness under practical fabrication conditions     

Quadrature /spl Sigma//spl Delta/ modulators with a dynamic element matching scheme

This paper presents a new topology of a multibit quadrature bandpass sigma-delta modulator which employs a simple dynamic element matching (DEM) technique in order to reduce the effects of path mismatch, namely aliasing in the signal band of the mirror images of the signal and of the quantization noise. The DEM scheme results in a reduction of the aliasing of the quantization noise mirror image while it reduces the input signal mirror image alias problem to a self-image problem. It is shown that the self-image can be completely removed in switched-capacitor implementations by using the same capacitors to sample the input and the reference of the feedback digital-analog converters (DACs). Moreover, a simple method for extending low-pass mismatch noise shaping techniques to the complex bandpass case is proposed for the case of multibit feedback DACs.     

Sturdy MASH /spl Delta//spl Sigma/ modulator

A new Delta-Sigma modulator is proposed. Its operation is similar to that of a multi-stage noise-shaping structure but requires no digital noise cancellation filters. Thus, the need for matching required between analogue and digital filters is eliminated. Simulation results and mathematical analysis demonstrate the effectiveness of this structure     

Nonlinearity correction for multibit /spl Delta//spl Sigma/ DACs

This paper presents a digital correction technique for wide-band multibit error-feedback (EF) digital-to-analog converters (DACs). The integral nonlinearity (INL) error of the multibit DAC is estimated (on line or off line) by a calibration analog-to-digital converter (CADC) and stored in a random-access memory table. The INL values are then used to compensate for the multibit DAC's distortion by a simple digital addition. The accuracy requirements for the error estimates are derived. These requirements can be significantly relaxed when the correction is combined with data-weighted averaging (DWA). Simulation and discrete-component measurement results are presented for a fourth-order 5-bit EF DAC. The results show a 14-bit DAC operating at an oversampling ratio of 8, which is suitable for digital subscriber line applications. The correction uses simple digital circuitry and a 3-bit CADC enhanced by DWA.     

A high-frequency double-sampling second-order /spl Delta//spl Sigma/ modulator

As the minimum feature size of VLSI technologies scales down, more of the signal processing tasks are performed in the digital domain. This results in increased speed, resolution, and dynamic range requirements for the analog-to-digital converter (ADC). High-speed and high-accuracy designs can be achieved by using oversampling ADC structures, which demand amplifiers with a high gain and a high unity-gain frequency. Due to the difficulty to meet both of these specifications, the ADC resolution at a frequency in the megahertz range appears to be limited by amplifier settling requirements. Design techniques to improve the ADC performance are presented. The proposed modulator structure uses the double-sampled technique, which increases by a factor of two the maximum speed of operation and correctly operates even with low dc gain amplifiers. Furthermore, the signal-to-noise ratio is significantly improved by a calibration stage, which dynamically estimates the offset errors to be removed by a simple subtraction from the output signal.     

Integrated stereo /spl Delta//spl Sigma/ class D amplifier

A 2/spl times/40 W class D amplifier chip is realized in 0.6-/spl mu/m BCDMOS technology, integrating two delta-sigma (/spl Delta//spl Sigma/) modulators and two full H-bridge switching output stages. Analog feedback from H-bridge outputs helps achieve 67-dB power supply rejection ratio, 0.001% total harmonic distortion, and 104-dB dynamic range. The modulator clock rate is 6 MHz, but dynamically adjusted quantizer hysteresis reduces output data rate to 450 kHz, helping achieve 88% power efficiency. At AM radio frequencies, the modulator output spectrum contains a single peak, but is otherwise tone-free, unlike conventional pulse-width modulation (PWM) modulators which contain energetic tones at harmonics of the PWM clock frequency.     

Continuous-time /spl Delta//spl Sigma/ modulators using distributed resonators

We derive a method for using distributed resonators in /spl Delta//spl Sigma/ modulators and demonstrate these /spl Delta//spl Sigma/ modulators have several advantages over existing /spl Delta//spl Sigma/ modulator architectures. Like continuous-time (CT) /spl Delta//spl Sigma/ modulators, the proposed /spl Delta//spl Sigma/ modulators do not require a high-precision track-and-hold, and additionally can take advantage of the high-Q of distributed resonators. Like discrete-time /spl Delta//spl Sigma/ modulators, the proposed /spl Delta//spl Sigma/ modulators are relatively insensitive to feedback loop delays and can subsample. We present simulations of several types of these /spl Delta//spl Sigma/ modulators and examine the challenges in their design.     

The tiling phenomenon in /spl Sigma//spl Delta/ modulation

The theoretical error signal analysis of a sigma-delta (/spl Sigma//spl Delta/) modulator is a difficult problem due to the presence of a nonlinear operation (the amplitude quantization) in a feedback loop. In this paper, new deterministic knowledge on the transfer function of a /spl Sigma//spl Delta/ modulator is established, thanks to some recently observed properties of its state variables. For a large class of typical /spl Sigma//spl Delta/ modulators with constant inputs, the state variables appear to remain in a tile. We show what characteristics in a /spl Sigma//spl Delta/ modulator are specifically responsible for this property and give some initial proof of it. Under a constant input, the tiling phenomenon has as fundamental consequence that the output is a fixed and memoryless modulo function of n successive integrated versions of the input. This gives the theoretical knowledge that the modulator has an equivalent feedforward circuit expression. We give some immediate theoretical consequences on error analysis including the case of time-varying inputs.     

Hexagonal /spl Sigma//spl Delta/ modulators in power electronics

Design techniques for /spl Sigma//spl Delta/ modulators from communications are applied and adapted to improve the spectral characteristics of high frequency power electronic applications. A high frequency power electronic circuit can be regarded as a quantizer in an interpolative /spl Sigma//spl Delta/ modulator. We review one dimensional /spl Sigma//spl Delta/ modulators and then generalize to the hexagonal sigma-delta modulators that are appropriate to three-phase converters. A range of interpolative modulator designs from communications can then be generalized and applied to power electronic circuits. White noise spectral analysis of sigma-delta modulators is generalized and applied to analyze the designs so that the noise can be shaped to design requirements. Simulation results for an inverter show significant improvements in spectral performance.     

Hybrid /spl Sigma//spl Delta/ modulators with adaptive calibration

This paper describes an architecture for stable high-order /spl Sigma//spl Delta/ modulation. The architecture is based on a hybrid /spl Sigma//spl Delta/ modulator, wherein hybrid integrators replace conventional analog integrators. The hybrid integrator, which is a combination of an analog integrator and a digital integrator, offers an increased dynamic range and helps make the resulting high-order /spl Sigma//spl Delta/ modulator stable. However, the hybrid /spl Sigma//spl Delta/ modulator relies on precise matching of analog and digital paths. In this paper, a calibration technique to alleviate possible mismatch between analog and digital paths is proposed. The calibration adaptively adjusts the digital integrators so that their transfer functions match the transfer functions of corresponding analog integrators. Through behavioral-level simulations of fourth-order /spl Sigma//spl Delta/ modulators, the calibration technique is verified.     

11 GHz CMOS /spl Sigma//spl Delta/ frequency synthesiser

A fully integrated 11 GHz fractional-N PLL with a three-stage MASH SigmaDelta modulator with DC dither in all stages is implemented in a standard 0.13 mum CMOS technology. The synthesiser generates no fractional spurs at frequency offsets outside the loop bandwidth and a small number of fractional spurs with the power not exceeding -44 dBc within the loop bandwidth at the carrier frequency of 11 GHz     

Self-tuning algorithms for high-performance bandpass switched-capacitor /spl Sigma//spl Delta/ modulators

Switched-capacitor high-frequency bandpass /spl Sigma//spl Delta/ modulators could suffer from capacitor mismatch, finite opamp dc gain, and finite opamp bandwidth. These problems make the notch frequency and the quality factor of the zeros of the noise transfer function to deviate from their nominal values, strongly affecting the modulator dynamic range (DR). In order to avoid this situation, two sampled-data algorithms have been developed which allow to self-calibrate the bandpass /spl Sigma//spl Delta/ modulators. They use 3500 gate and 0.043 mm/sup 2/ area and consume power only when they are active, while, when the system is on, they are off and do not interfere with standard operation. The validity of the proposal is demonstrated by a silicon prototype in which the proposed solution allows to guarantee a 75-dB DR performance also under worst case conditions. In the particular case, it allows for the recovery of 3 dB in the SNR for the 200-kHz FM band (from 73 to 76 dB).     

A bandpass mismatch noise-shaping technique for /spl Sigma/-/spl Delta/ Modulators

A simple bandpass mismatch noise-shaping technique for /spl Sigma/-/spl Delta/ modulators is presented. This technique uses the data-directed scrambler structure and achieves /spl sim/20 dB improvement in signal-to-noise ratio (SNR) for an oversampling ratio (OSR) of 32. Compared to the technique of bandpass mismatch noise shaping of Lin and Schreier, here, the SNR is improved by /spl sim/ 7dB resulting in a 1-bit improvement in resolution. The selection logic is half the size of the logic in Lin and Schreier and is fully compatible with the low-pass selection logic of Kwan et al.. This compatibility allows implementation of bandpass/low-pass modulator as a single product.     

Double-sampling single-loop /spl Sigma//spl Delta/ modulator topologies for broad-band applications

This paper presents novel double sampling high-order single loop sigma-delta modulator structures for wide-band applications. To alleviate the quantization noise folding into the inband frequency region, two previously reported techniques are used. The digital-to-analog converter's sampling paths are implemented with the single-capacitor approach and an additional zero is placed at the half of the sampling frequency of the modulator's noise transfer function (NTF). The detrimental effect of this additional zero on both the NTF and signal transfer function is also resolved through the proposed modulator architectures with a low additional circuit requirement.     

A fourth-order /spl Sigma//spl Delta/ interface for micromachined inertial sensors

This paper presents the design and implementation of a high-order /spl Sigma//spl Delta/ interface for micromachined inertial sensors, which employs an electronic filter in series with the mechanical sensor element to reject the excessive in-band quantization noise inherently present in state-of-the-art second-order solutions. A fourth-order prototype was fabricated in a standard 0.5-/spl mu/m CMOS process. The active circuit area measures 0.9 mm/sup 2/, and the interface consumes 13 mW from a 5-V supply and achieves resolution of 1/spl deg//s//spl radic/Hz with a gyroscope and 150/spl mu/g//spl radic/Hz with an accelerometer. Comparison between the measured and simulated behavior of the system shows that the contribution of the quantization error to the total noise is negligible.     

Theory and applications of incremental /spl Delta//spl Sigma/ converters

Analog-Digital (A/D) converters used in instrumentation and measurements often require high absolute accuracy, including very high linearity and negligible dc offset. The realization of high-resolution Nyquist-rate converters becomes very expensive when the resolution exceeds 16 bits. The conventional delta-sigma (/spl Delta//spl Sigma/) structures used in telecommunication and audio applications usually cannot satisfy the requirements of high absolute accuracy and very small offset. The incremental (or integrating) converter provides a solution for such measurement applications, as it has most advantages of the /spl Delta//spl Sigma/ converter, yet is capable of offset-free and accurate conversion. In this paper, theoretical and practical aspects of higher order incremental converters are discussed. The operating principles, topologies, specialized digital filter design methods, and circuit level issues are all addressed. It is shown how speed, resolution, and A/D complexity can be optimized for a given design, and how with some special digital filters improved speed/resolution ratio can be achieved. The theoretical results are verified by showing design examples and simulation results.     

Quasi-equivalent feedforward system in asymptotic /spl Sigma//spl Delta/ modulation

It was previously shown that sigma-delta (/spl Sigma//spl Delta/) modulators of "asymptotic" type theoretically yield an equivalent feedforward system where the recursive nonlinear mechanisms are extracted from the feedback loop and reduced to a memoryless function. With time-varying inputs, we show in this paper, partially by mathematical derivations and partially by experiment, that this system is quasi-equivalent to the original modulator in a sense that we explain. This reduction of the nonlinear mechanisms should permit more refined modeling of the /spl Sigma//spl Delta/ errors in future research, with a better account of the original nonlinearities of asymptotic /spl Sigma//spl Delta/ modulation.     

Improved design method for continuous-time quadrature bandpass /spl Sigma//spl Delta/ ADCs

In continuous-time quadrature bandpass /spl Sigma//spl Delta/ ADCs it is desirable to limit the number of cross-couplings. This can be achieved by implementing the loop as a cascade of complex integrators with only real coefficients. It is shown that this may result in a very poor approximation of the desired noise transfer function, because the effect of the DAC pulse is not taken into account correctly. A simple implementation that solves this problem is proposed.     

IF-sampling fourth-order bandpass /spl Delta//spl Sigma/ modulator for digital receiver applications

Bandpass modulators sampling at high IFs (/spl sim/200 MHz) allow direct sampling of an IF signal, reducing analog hardware, and make 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 fourth-order bandpass /spl Delta//spl Sigma/ modulator implemented in InP HBT IC technology for use in a multimode 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 fourth-order bandpass /spl Delta//spl Sigma/ modulator consists of two bandpass resonators that can be tuned to optimize both wide-band and narrow-band operation. The IF is tunable from 140 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 fourth-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 at 1 MHz BW and 50 dB at 60 MHz BW. The average SNDR performance measured on over 250 parts is 72.5 dB at 1 MHz BW and 47.7 dB at 60 MHz BW.