Hongjia Mo

Spurious tones in digital delta-sigma modulators resulting from pseudorandom dither

Abstract Digital delta-sigma modulators (DDSMs) are finite state machines; their spectra are characterized by strong periodic tones (so-called spurs) when they cycle repeatedly in time through a small number of states. This happens when the input is constant or periodic. Pseudorandom dither generators are widely used to break up periodic cycles in DDSMs in order to eliminate spurs produced by underlying periodic behavior. Unfortunately, pseudorandom dither signals are themselves periodic and therefore can have limited effectiveness. This paper addresses the fundamental limitations of using pseudorandom dither signals that are inherently periodic. We clarify some common misunderstandings in the DDSM literature. We present rigorous mathematical analysis, case studies to illustrate the issues, and insights which can prove useful in design.

Experimental characterization of Arnold tongues in injection-locked CMOS LC frequency dividers with tail and direct injection

This paper describes an experimental technique for rapidly finding the boundaries of Arnold tongues. The method is suitable for characterizing the locking regions of injection-locked frequency dividers. The algorithm is described and examples are presented to demonstrate the value of the technique in comparing divider architectures with tail and direct injection.

Influence of Initial Conditions on the Fundamental Periods of LFSR-Dithered MASH Digital Delta–Sigma Modulators With Constant Inputs

A digital delta–sigma modulator (DDSM) with a constant input may produce a periodic output with a small fundamental period, resulting in strong tonal output behavior instead of the expected shaped white quantization noise. In practice, the problem is alleviated by dithering the DDSM. Pseudorandom dither generators based on linear feedback shift registers (LFSRs) are widely used to “break up” periodic cycles in DDSMs with constant inputs. Pseudorandom dither signals are themselves periodic and can lead to relatively short output sequences from dithered DDSMs. It is known that the fundamental period of the output signal depends not only on the input and the initial condition of the DDSM but also on the initial state of the LFSR. This brief shows that bad LFSR initial conditions can lead to ineffective dithering, producing short cycles and strong tonal behavior. Furthermore, it explains how to set the initial state of the DDSM as a function of the initial state of the LFSR in order to obtain a maximum-length dithered output.

Phenomenological study of an injection-locked CMOS LC frequency divider with direct injection

This paper reports experimental observations of injection-locking in a CMOS LC frequency divider with direct injection. Several factors are considered including the amplitude of the driving signal, the harmonic content of the driving signal, the switch size, and the forward body bias of the switch. The experiments are performed using a technique to determine rapidly the boundaries of the Arnold tongue locking regions. While some of the the observed phenomena are to be expected given the current theoretical understanding of injection-locking, others, such as the qualitatively different small- and large-signal locking regimes, have not been reported previously

The noise and spur delusion in fractional-N frequency synthesizer design

The standard design methodology for fractional-N frequency synthesizers assumes that the filtered shaped quantization noise from the requantizer is masked below the spectral envelope of the underlying integer-N synthesizer. Fractional-N frequency synthesizers are notorious for exhibiting an elevated noise floor and an unpredictable pattern of spurs. In this paper, we argue that designers should not be deluded by the overly conservative predictions of the simplified linear model but should instead consider nonlinearities as early as possible in the design process.

Comments on efficient dithering in digital delta-sigma modulator

The output spectra of digital delta sigma modulators (DDSMs) are characterized by unwanted spurious periodic tones (spurs). Additive pseudorandom dither is used to whiten the quantization noise of the DDSM in order to minimize these spurs. Unfortunately, pseudorandom dither signals are themselves periodic and may cause other spurs to appear. This paper describes a DDSM architecture with multiple periodic dither inputs and explains the limits of its performance.

0.3–4.3 GHz Frequency-Accurate Fractional-

If the modulus of the digital delta-sigma modulator (DΔΣM) in a fractional- N frequency synthesizer is a power of two, then the output frequency is constrained to be a rational multiple of the phase detector frequency (fPD), where the denominator of the rational multiplier is a power of two. If the required output frequency is not related to fPD in this way, one is forced to approximate the ratio by using a small programmable modulus DΔΣM or a very large power-of-two modulus. Both of these solutions involve additional hardware. Furthermore, the programmable modulus solution can suffer from spurs, while the large power of two lacks accuracy. This paper presents a new solution, based on mixed-radix algebra, where the required ratio is formed by combining two different moduli. The programmable modulus solves the accuracy problem, while the large power-of-two modulus minimizes the spur content. In addition, the phase detector can be clocked at high speed. This paper explains the theoretical foundations of the method, elaborates a design methodology, and presents measured results for an 0.18 μm SiGe BiCMOS prototype.

N

Static mismatch between the charge pump up and down currents in a fractional-N frequency synthesizer is known to elevate the in-band noise floor and to produce spurs. Offset current can be used to reduce nonlinear effects by moving the operating point of the charge-pump to a more linear region of its transfer characteristic. While both positive and negative offset current will work, one direction requires a smaller offset current and therefore adds less noise to the system. In this paper, a simplified feed-forward phase domain model is compared with a full time-domain behavioral model. We show that the feedforward model accurately predicts both the mismatch-related noise floor and the spurs. We also note a limitation of the feedforward model, namely that it fails to predict the 20 dB/decade slope due to non-uniform sampling.

Frequency Synthesizer With Integrated VCO and Nested Mixed-Radix Digital

Digital delta-sigma modulators (DDSMs) are finite state machines; their spectra are characterized by strong periodic tones (so-called spurs) when they cycle repeatedly in time through a small number of states. This can happen for a range of constant and periodic inputs. Pseudorandom dither generators are widely used to break up periodic cycles in DDSMs in order to eliminate spurs produced by underlying periodic behavior. Unfortunately, pseudorandom dither signals are themselves periodic and therefore can produce spurs in the output spectrum. We call pseudorandom dithering effective if it breaks up periodic cycles in the modulator without producing its own spurs in the output spectrum. This paper presents rigorous mathematical analysis and a design methodology to achieve effective dithering.

\Delta

Interaction between the requantizers periodic output and charge pump mismatch nonlinearity in a fractional-N frequency synthesizer causes it to exhibit an elevated inband noise floor and spurs. In this paper, we consider three leading analytical predictions from the literature. By comparing simulation results with analytical predictions, we show that the two most common approaches fail to deal correctly with offsets, while the method based on Prices theorem works well.