Subtractive Dither Research Articles

Effect of reconstruction error in subtractive dither structure

This paper studies the causes and distribution of reconstruction errors in subtractive dither structure and provides solution to minimize these errors. We present the generation of reconstruction errors in two types of quantization, namely mid-riser and mid-tread, with their corresponding distributions. We then construct a 12-bit pipeline analog to digital converters (ADC) simulation model and use a 16-bit digital to analog converter (DAC) to convert the dither sequence into analog signals. We perform statistical analysis on the resulting data, verify the accuracy of the distribution, and find that reconstruction error only occurs when the dither signal accuracy surpasses that of the ADC. Additionally, we find that mid-tread quantization has the lowest probability of reconstruction error. Finally, we analyze the impact of reconstruction error on the signal to noise ratio (SNR) and spurious free dynamic range (SFDR) at various input frequencies. Our findings show that the reconstruction errors' effect on the SFDR and SNR of the input signal is less than 1 dB.

Dithered Depth Imaging for Single-Photon Lidar at Kilometer Distances

Depth imaging using single-photon lidar (SPL) is crucial for long-range imaging and target recognition. Subtractive-dithered SPL breaks through the range limitation of the coarse timing resolution of the detector. Considering the weak signals at kilometer distances, we present a novel imaging method based on blending subtractive dither with a total variation image restoration algorithm. The spatial correlation is well-considered to obtain more accurate depth profile images with fewer signal photons. Subsequently, we demonstrate the subtractive dither measurement at ranges up to 1.8 km using an array of avalanche photodiodes (APDs) operating in the Geiger mode. Compared with the pixel-wise maximum-likelihood estimation, the proposed method reduces the depth error, which has great promise for high-depth resolution imaging at long-range imaging.

Dithering Concepts for Spur-Free Nonlinear DTC-Based Frequency Synthesizers

Digital-to-time converters (DTCs) are a promising technology for radio frequency (RF) transceivers but are prone to spur generation. A common approach to change the spurious emissions to a spur-free shape is a method called dithering. The power added due to dithering is an important aspect of this approach and gives raise to investigations on additive dither as well as methods for subtractive dithering. This work presents a mathematical model for dithering DTC-based local oscillator (LO) generators. It proposes concepts for the application of subtractive dither and it introduces a novel generalization of quantization-dither to allow for optimal dithering of nonlinear quantizers.

A 6 $\mu$ W 95 dB SNDR Inverter Based $\Sigma\Delta$ Modulator With Subtractive Dithering and SAR Quantizer

This brief presents a low-power $ {\Sigma \Delta }$ modulator using the subtractive dither technique. The proposed technique eliminates tones at the $ {\Sigma \Delta }$ modulator’s output by injecting a dither into the quantizer. The injected dither is cancelled at the first integrator’s output, which results in minimal impact on the operational transconductance amplifiers’ (OTAs) output swings and enables the use of energy-efficient inverter-based OTAs. Dither is also subtracted from the digital output and has no impact on the signal-to-quantization-noise ratio. The modulator features a successive approximation register quantizer and a modified DAC which eliminates signal-dependent kick-back to the input and reference buffers. Implemented in 180-nm CMOS, the proposed modulator achieves 95.1 dB signal-to-noise-and-distortion ratio from dc to 1 kHz, and consumes only 6 $ {\mu }\text{W}$ from a 1.2-V supply according to transient noise simulation, leading to a state-of-the-art Schreier figure-of-merit of 177.3 dB. Simulation verifies the effectiveness of subtractive dither.

Estimation From Quantized Gaussian Measurements: When and How to Use Dither

Subtractive dither is a powerful method for removing the signal dependence of quantization noise for coarsely-quantized signals. However, estimation from dithered measurements often naively applies the sample mean or midrange, even when the total noise is not well described with a Gaussian or uniform distribution. We show that the generalized Gaussian distribution approximately describes subtractively-dithered, quantized samples of a Gaussian signal. Furthermore, a generalized Gaussian fit leads to simple estimators based on order statistics that match the performance of more complicated maximum likelihood estimators requiring iterative solvers. The order statistics-based estimators outperform both the sample mean and midrange for nontrivial sums of Gaussian and uniform noise. Additional analysis of the generalized Gaussian approximation yields rules of thumb for determining when and how to apply dither to quantized measurements. Specifically, we find subtractive dither to be beneficial when the ratio between the Gaussian standard deviation and quantization interval length is roughly less than 1/3. If that ratio is also greater than 0.822/$K^{0.930}$ for the number of measurements $K>20$, we present estimators more efficient than the midrange.

Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources

We deal with zero-delay source coding of a vector-valued Gauss-Markov source subject to a mean-squared error (MSE) fidelity criterion characterized by the operational zero-delay vector-valued Gaussian rate distortion function (RDF). We address this problem by considering the nonanticipative RDF (NRDF) which is a lower bound to the causal optimal performance theoretically attainable (OPTA) function and operational zero-delay RDF. We recall the realization that corresponds to the optimal "test-channel" of the Gaussian NRDF, when considering a vector Gauss-Markov source subject to a MSE distortion in the finite time horizon. Then, we introduce sufficient conditions to show existence of solution for this problem in the infinite time horizon. For the asymptotic regime, we use the asymptotic characterization of the Gaussian NRDF to provide a new equivalent realization scheme with feedback which is characterized by a resource allocation (reverse-waterfilling) problem across the dimension of the vector source. We leverage the new realization to derive a predictive coding scheme via lattice quantization with subtractive dither and joint memoryless entropy coding. This coding scheme offers an upper bound to the operational zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then for "r" active dimensions of the vector Gauss-Markov source the gap between the obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1 bits/vector. We further show that it is possible when we use vector quantization, and assume infinite dimensional Gauss-Markov sources to make the previous gap to be negligible, i.e., Gaussian NRDF approximates the operational zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian sources of any finite memory under mild conditions. Our theoretical framework is demonstrated with illustrative numerical experiments.

Subtractive Dither を用いた量子化器のフィードバック制御系における効果

Subtractive Dither を用いた量子化器のフィードバック制御系における効果

Networked LQG control over lossy channels with computational/packet-transmission delays and coarsely quantised packets

This article addresses the linear quadratic Gaussian (LQG) control problem of networked multi-input, multi-output systems where computational delay exists and the measurement and control signals are packetised and transmitted through a network within which random delay and packet loss may occur during transmissions. A transmission control protocol (TCP)-like protocol for the communication network is considered in which acknowledgement is sent from the actuator to the controller if and only if the control packet is received, assuming these acknowledgements always reach the estimator in time and without fail. To minimise the data word-length for transmissions over the network and to maximise control system performance, it is proposed that different quantisation resolutions be used for transmission data encapsulation, and control and output signals A/D-D/A conversions at sensor/actuator. To circumvent the problem of disparity between encapsulation and A/D-D/A quantisation resolutions, a pseudo-stochastic approach via subtractive dither is applied to quantise the transmission packets. This also enables us to model the quantisation errors as uncorrelated independent zero-mean additive white noises and apply standard LQG methodology and separation principle to design the estimator and the controller separately. An example is included to demonstrate the effectiveness of the approach.

A Study of Subtractive Digital Dither in Single-Stage and Multi-Stage Quantizers

The effect of subtractive digital dither on single-stage and multi-stage quantizers is examined. In the single stage case, it is proved that the conditional density of the quantization error becomes asymptotically independent of the input and uniformly distributed as the number of digital dither levels increases. For a fixed number of levels, the conditional expectation of a class of piecewise constant functions is shown to be independent of the input. A Fourier series representation of quantization error densities is used to construct a linear periodically time varying model of multistage quantization and dithering systems. For a homogeneous multi-stage quantization system, it is shown that when the least common multiple of the numbers of digital dither levels at successive stages is large, the statistical dependence of the quantization noise on the input disappears almost entirely. Computer simulations confirm this analysis.

Amplitude Scale Estimation for Quantization-Based Watermarking

In this paper we propose a maximum likelihood technique to combat amplitude scaling attacks within a quantization-based watermarking context. We concentrate on operations that are common in many applications and at the same time devastating to this class of watermarking schemes, namely, amplitude scaling in combination with additive noise. First we derive the probability density function of the watermarked and attacked data in the absence of subtractive dither. Next we extend these models to incorporate subtractive dither in the encoder. The dither sequence is primarily used for security purposes, and the dither is assumed to be known also to the decoder. We design the dither signal statistics such that an attacker having no knowledge of the dither cannot decode the watermark. Using an approximation of the probability density function in the presence of subtractive dither, we derive a maximum likelihood procedure for estimating amplitude scaling factors. Experiments are performed with synthetic and real audio signals, showing the feasibility of the proposed approach under realistic conditions.

Comment on “Noise averaging and measurement resolution” [Rev. Sci. Instrum. 70, 2038–2040 (1999)

The paper presents a superficial analysis, based on computer simulation, of a subject which has already been discussed at some length and for which analytic results have been presented. The authors appears to use “uniform dither” where several authors recommend “triangular dither” which gives 50% more quantization noise, but makes the quantization noise independent of signal level. The author also fails to mention the possibility of using “subtractive dither” which is technically difficult, but justifiable in expensive systems.

Rate-distortion performance in coding bandlimited sources by sampling and dithered quantization

The rate-distortion characteristics of a scheme for encoding continuous-time band limited stationary sources, with a prescribed band, is considered. In this coding procedure the input is sampled at Nyquist's rate or faster, the samples undergo dithered uniform or lattice quantization, using subtractive dither, and the quantizer output is entropy-coded, The rate-distortion performance, and the tradeoff between the sampling rate and the quantization accuracy is investigated, utilizing the observation that the coding scheme is equivalent to an additive noise channel. It is shown that the mean-square error of the scheme is fixed as long as the product of the sampling period and the quantizer second moment is kept constant, while for a fixed distortion the coding rate generally increases when the sampling rate exceeds the Nyquist rate. Finally, as the lattice quantizer dimension becomes large, the equivalent additive noise channel of the scheme tends to be white Gaussian, and both the rate and the distortion performance become invariant to the sampling rate. >