Delta-sigma modulation


Delta-sigma modulation is an oversampling method for encoding signals into low bit depth digital signals at a very high sample-frequency as part of the process of delta-sigma analog-to-digital converters and digital-to-analog converters. Delta-sigma modulation achieves high quality by utilizing a negative feedback loop during quantization to the lower bit depth that continuously corrects quantization errors and moves quantization noise to higher frequencies well above the original signal's bandwidth. Subsequent low-pass filtering for demodulation easily removes this high frequency noise and time averages to achieve high accuracy in amplitude, which can be ultimately encoded as pulse-code modulation.
Both ADCs and DACs can employ delta-sigma modulation. A delta-sigma ADC encodes an analog signal using high-frequency delta-sigma modulation and then applies a digital filter to demodulate it to a high-bit digital output at a lower sampling frequency. A delta-sigma DAC encodes a high-resolution digital input signal into a lower-resolution but higher sample-frequency signal that may then be mapped to voltages and smoothed with an analog filter for demodulation. In both cases, the temporary use of a low bit depth signal at a higher sampling frequency simplifies circuit design and takes advantage of the efficiency and high accuracy in time of digital electronics.
Primarily because of its cost efficiency and reduced circuit complexity, this technique has found increasing use in modern electronic components such as DACs, ADCs, frequency synthesizers, switched-mode power supplies and motor controllers. The coarsely-quantized output of a delta-sigma ADC is occasionally used directly in signal processing or as a representation for signal storage.
While this article focuses on synchronous modulation, which requires a precise clock for quantization, asynchronous delta-sigma modulation instead runs without a clock.

Motivation

When transmitting an analog signal directly, all noise in the system and transmission is added to the analog signal, reducing its quality. Digitizing it enables noise-free transmission, storage, and processing. There are many methods of digitization.
In Nyquist-rate ADCs, an analog signal is sampled at a relatively low sampling frequency just above its Nyquist rate and quantized by a multi-level quantizer to produce a multi-bit digital signal. Such higher-bit methods seek accuracy in amplitude directly, but require extremely precise components and so may suffer from poor linearity.

Advantages of oversampling

Oversampling converters instead produce a lower bit depth result at a much higher sampling frequency. This can achieve comparable quality by taking advantage of:
  • Higher accuracy in time.
  • Higher linearity afforded by low-bit ADCs and DACs.
  • Noise shaping: moving noise to higher frequencies above the signal of interest, so they can be easily removed with low-pass filtering.
  • Reduced steepness requirement for the analog low-pass anti-aliasing filters. High-order filters with a flat passband cost more to make in the analog domain than in the digital domain.

    Frequency/resolution tradeoff

Another key aspect given by oversampling is the frequency/resolution tradeoff. The decimation filter put after the modulator not only filters the whole sampled signal in the band of interest, but also reduces the sampling rate, and hence the representable frequency range, of the signal, while increasing the sample amplitude resolution. This improvement in amplitude resolution is obtained by a sort of averaging of the higher-data-rate bitstream.

Improvement over delta modulation

is an earlier related low-bit oversampling method that also uses negative feedback, but only encodes the derivative of the signal rather than its amplitude. The result is a stream of marks and spaces representing up or down of the signal's movement, which must be integrated to reconstruct the signal's amplitude. Delta modulation has several drawbacks. The differentiation alters the signal's spectrum by amplifying high-frequency noise, attenuating low-frequencies, and dropping the DC component. This makes its dynamic range and signal-to-noise ratio inversely proportional to signal frequency. Delta modulation suffers from slope overload if signals move too fast. And it is susceptible to transmission disturbances that result in cumulative error.
Delta-sigma modulation rearranges the integrator and quantizer of a delta modulator so that the output carries information corresponding to the amplitude of the input signal instead of just its derivative. This also has the benefit of incorporating desirable noise shaping into the conversion process, to deliberately move quantization noise to frequencies higher than the signal. Since the accumulated error signal is low-pass filtered by the delta-sigma modulator's integrator before being quantized, the subsequent negative feedback of its quantized result effectively subtracts the low-frequency components of the quantization noise while leaving the higher frequency components of the noise.

1-bit delta-sigma modulation is pulse-density modulation

In the specific case of a single-bit synchronous ΔΣ ADC, an analog voltage signal is effectively converted into a pulse frequency, or pulse density, which can be understood as pulse-density modulation. A sequence of positive and negative pulses, representing bits at a known fixed rate, is very easy to generate, transmit, and accurately regenerate at the receiver, given only that the timing and sign of the pulses can be recovered. Given such a sequence of pulses from a delta-sigma modulator, the original waveform can be reconstructed with adequate precision.
The use of PDM as a signal representation is an alternative to PCM. Alternatively, the high-frequency PDM can later be downsampled through decimation and requantized to convert it into a multi-bit PCM code at a lower sampling frequency closer to the Nyquist rate of the frequency band of interest.

History and variations

The seminal paper combining feedback with oversampling to achieve delta modulation was by F. de Jager of Philips Research Laboratories in 1952.
File:Patent-US3192371-Brahm-Feedback-integrating-system-fig-1.png|thumb|"Feedback Integrating System" by Charles B Brahm: The entire top half of its Fig 1 is a delta-sigma modulator. Box #10 is a two-input integrator. The 4-bit analog-to-digital quantizer uses designations "S", "1", "2", and "4" for each bit. Each "F" stands for flip-flop and each "G" is a gate, controlled by the 110 kHz oscillator.
The principle of improving the resolution of a coarse quantizer by use of feedback, which is the basic principle of delta-sigma conversion, was first described in a 1954-filed patent by C. Chapin Cutler of Bell Labs. It was not named as such until a 1962 paper by Inose et al. of University of Tokyo, which came up with the idea of adding a filter in the forward path of the delta modulator. However, Charles B Brahm of United Aircraft Corp in 1961 filed a patent "Feedback integrating system" with a feedback loop containing an integrator with multi-bit quantization shown in its Fig 1.
Wooley's "The Evolution of Oversampling Analog-to-Digital Converters" gives more history and references to relevant patents. Some avenues of variation are the modulator's order, the quantizer's bit depth, the manner of decimation, and the oversampling ratio.

Higher-order modulator

Noise of the quantizer can be further shaped by replacing the quantizer itself with another ΔΣ modulator. This creates a 2-order modulator, which can be rearranged in a cascaded fashion. This process can be repeated to increase the order even more.
While 1-order modulators are unconditionally stable, stability analysis must be performed for higher-order noise-feedback modulators. Alternatively, noise-feedforward configurations are always stable and have simpler analysis.

Multi-bit quantizer

The modulator can also be classified by the bit depth of its quantizer. A quantizer that distinguishes between N-levels is called a log2N bit quantizer. For example, a simple comparator has 2 levels and so is 1-bit quantizer; a 3-level quantizer is called a 1.5 bit quantizer; a 4-level quantizer is a 2-bit quantizer; a 5-level quantizer is called a 2.5-bit quantizer. Higher bit quantizers inherently produce less quantization noise.
One criticism of 1-bit quantization is that adequate amounts of dither cannot be used in the feedback loop, so distortion can be heard under some conditions. Many of the issues of 1-bit modulation can be treated by look-ahead sigma-delta modulation.

Subsequent decimation

Decimation is strongly associated with delta-sigma modulation, but is distinct and outside the scope of this article. The original 1962 paper didn't describe decimation. Oversampled data in the early days was sent as is. The proposal to decimate oversampled delta-sigma data using digital filtering before converting it into PCM audio was made by D. J. Goodman at Bell Labs in 1969, to reduce the ΔΣ signal from its high sampling rate while increasing its bit depth. Decimation may be done in a separate chip on the receiving end of the delta-sigma bit stream, sometimes by a dedicated module inside of a microcontroller, which is useful for interfacing with PDM MEMS microphones, though many ΔΣ ADC integrated circuits include decimation. Some microcontrollers even incorporate both the modulator and decimator.
Decimation filters most commonly used for ΔΣ ADCs, in order of increasing complexity and quality, are:
  1. Boxcar moving average filter : This is the easiest digital filter and retains a sharp step response, but is mediocre at separating frequency bands and suffers from intermodulation distortion. The filter can be implemented by simply counting how many samples during a larger sampling interval are high. The 1974 paper from another Bell Labs researcher, J. C. Candy, "A Use of Limit Cycle Oscillations to Obtain Robust Analog-to-Digital Converters" was one of the early examples of this.
  2. Cascaded integrator–comb filters: These are called sinc filters, equivalent to cascading the above sinc filter N times and rearranging the order of operations for computational efficiency. Lower N filters are simpler, settle faster, and have less attenuation in the baseband, while higher N filters are slightly more complex and settle slower and have more droop in the passband, but better attenuate undesired high-frequency noise. Compensation filters can, however, be applied to counteract undesired passband attenuation. Sinc filters are appropriate for decimating sigma delta modulation down to four times the Nyquist rate. The height of the first sideload is -13·N dB and the height of successive lobes fall off gradually, but only the areas around the nulls will alias into the low frequency band of interest; for instance when downsampling by 8, the largest aliased high frequency component may be -16 dB below the peak of the band of interest with a sinc filter but -40 dB below for a sinc filter, and if only interested in a narrower bandwidth, even fewer high frequency components will alias into it.
  3. Windowed sinc-in-time filters: Although the sinc function's infinite support prevents it from being realizable in finite time, the sinc function can instead be windowed to realize finite impulse response filters. This approximated filter design, while maintaining almost no attenuation of the lower-frequency band of interest, still removes almost all undesired high-frequency noise. The downside is poor performance in the time domain, higher delay, and higher computational requirements. They are the de facto standard for high fidelity digital audio converters.