Delta-sigma modulation

Delta-sigma (ΔΣ; or sigma-delta, ΣΔ) modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC). Delta-sigma (ΔΣ; or sigma-delta, ΣΔ) modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC). In a conventional ADC, an analog signal is sampled with a sampling frequency and subsequently quantized in a multi-level quantizer into a digital signal. This process introduces quantization error noise. The first step in a delta-sigma modulation is delta modulation. In delta modulation the change in the signal (its delta) is encoded, rather than the absolute value. The result is a stream of pulses, as opposed to a stream of numbers as is the case with pulse code modulation (PCM). In delta-sigma modulation, accuracy of the modulation is improved by passing the digital output through a 1-bit DAC and adding (sigma) the resulting analog signal to the input signal (the signal before delta modulation), thereby reducing the error introduced by the delta modulation. Both ADCs and DACs can employ delta-sigma modulation. A delta-sigma ADC first encodes an analog signal using high-frequency delta-sigma modulation, and then applies a digital filter to form a higher-resolution but lower sample-frequency digital output. A delta-sigma DAC encodes a high-resolution digital input signal into a lower-resolution but higher sample-frequency signal that is mapped to voltages, and then smoothed with an analog filter. In both cases, the temporary use of a lower-resolution signal simplifies circuit design and improves efficiency. 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 modulator is occasionally used directly in signal processing or as a representation for signal storage. For example, the Super Audio CD (SACD) stores the output of a delta-sigma modulator directly on a disk. Delta-sigma modulation converts an analog voltage signal into a pulse frequency, or pulse density, which can be understood as pulse-density modulation (PDM). 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. In contrast, without conversion to a pulse stream but simply transmitting the analog signal directly, all noise in the system would be added to the analog signal, reducing its quality. The use of PDM as a signal representation is an alternative to pulse-code modulation (PCM), sampling and quantizing to a multi-bit code at the Nyquist rate. A delta-sigma or other pulse-density or pulse-frequency modulator generates a pulse stream in which the frequency, f, of pulses in the stream is proportional to the analog voltage input, v, so that f = k · v, where k is a constant for the particular implementation. A feedback loop monitors the integral of v and when that integral has incremented by Δ, which is indicated by the integral waveform crossing a threshold, T, it subtracts Δ from the integral of v so that the combined waveform sawtooths between T and T − Δ. At each step a pulse is added to the pulse stream. A counter sums the number of pulses that occur in a predetermined period, P {displaystyle P} so that the sum, Σ {displaystyle Sigma } , is P ⋅ f = k ⋅ P ⋅ v {displaystyle Pcdot f=kcdot Pcdot v} . In a given implementation, k ⋅ P {displaystyle kcdot P} is chosen so that a digital display of the count, Σ {displaystyle Sigma } , is a display of v {displaystyle v} with a predetermined scaling factor. Because P {displaystyle P} may take any designed value, it may be made large enough to give any desired resolution or accuracy. For the purpose of introduction, Figure 1 illustrates the concept of voltage-to-frequency conversion, in an unclocked form that resembles delta-sigma modulation, and is called asynchronous modulation, asynchronous delta-sigma modulation, or free-running modulators. Shown below that are waveforms at points designated by numbers 1 to 5 for an input of 0.2 volts in the left column and 0.4 volts in the right column. The stream of delta impulses generated at each threshold crossing is shown at (2) and the difference between (1) and (2) is shown at (3). This difference is integrated to produce the waveform (4). The threshold detector generates a pulse (5) which starts as the waveform (4) crosses the threshold and is sustained until the waveform (4) falls below the threshold. The threshold (5) triggers the impulse generator to produce a fixed-strength impulse.