The two important pillars - Digital-Analog-Converter and Analog-Digital-Converter
What we haven’t checked yet in our journey around (analog) electronics and digital electronics where the converter circuits from analog to digital and vice versa - for this we first want to explain the different terms around.
The theory about analog signals, digital signals, sampling and quantisation
Analog signals are characterized by being continuous in time and continuous in value. The conversion process of analog signals is often referred to as quantization.
Digital signals, on the other hand, are signals that have undergone a form of quantization process, which means that instead of being continuous signals, they are discrete signals. Digital signals are both time-discrete and value-discrete.
Analog Signals
Analog signals are continuous in time and continuous in value. This means that the signal exists at every moment in time and can take any value within a certain range. There are no jumps or gaps in the signal. Many physical quantities, such as sound, temperature, or light intensity, can naturally be described by analog signals.
Because analog signals change continuously, they provide a direct representation of real-world processes.
Digital Signals
Digital signals, on the other hand, are not continuous. They are signals that have undergone a quantisation process. As a result, digital signals are discrete in time and discrete in value. Instead of being defined at every point in time, a digital signal is only defined at certain time instants. In addition, the signal values are restricted to a finite set of possible values. Digital signals therefore consist of individual values rather than a continuous waveform.
Sampling and Quantisation
Sampling
Sampling describes the process of measuring a continuous-time signal at discrete points in time. Quantisation describes the process of mapping these sampled values to a finite set of discrete amplitude levels.
Quantisation
Quantisation describes the process of converting an analog signal into a digital signal. During this process, the continuous signal is represented by a sequence of discrete values. This process makes it possible to process, store, and transmit signals using digital systems.
The Analog-Digital-Converter (ADC)
The other way around is described as Analog-Digital-Converter. The fastest one, is a flash converter, a single IC which contains (2^n) -1 comparators where n stands for the bit width.
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The Sigma-Delta Converter (ΣΔ-Converter)
In addition to fast converter architectures such as the flash ADC or simple DAC structures like the R-2R ladder, there exists another very important class of converters that focuses not on speed, but on precision and resolution: the sigma-delta converter.
Sigma-delta converters are widely used in modern electronics, especially in audio systems, sensor interfaces, and precision measurement equipment. Their strength lies in achieving very high resolution with comparatively simple analog circuitry.
Basic idea behind the Sigma-Delta Converter
The sigma-delta converter follows a fundamentally different approach compared to classical ADC architectures. Instead of converting an analog signal directly into a multi-bit digital value in a single step, it:
samples the input signal at a very high rate
uses feedback to measure the error between input and output
generates a 1-bit (or few-bit) digital data stream
reconstructs a high-resolution result using digital filtering
In essence, the converter measures the average value of the input signal over time.
Origin of the name "Sigma-Delta"
The name sigma-delta originates from two mathematical operations used in the converter loop:
Sigma (Σ) – integration (accumulation over time)
Delta (Δ) – difference (error between input and feedback signal)
The converter continuously integrates the difference between the input signal and a feedback signal derived from its own output.
Structure of a Sigma-Delta ADC
A basic first-order sigma-delta ADC consists of the following blocks:
Integrator (Σ)
The integrator receives the difference between the analog input signal and the feedback signal. By integrating this error over time, low-frequency components are emphasized while high-frequency noise is averaged out.
Quantizer (Δ)
The quantizer is usually extremely simple, often only a 1-bit comparator. It decides whether the integrator output is positive or negative and produces a corresponding digital output (0 or 1).
Feedback DAC
The digital output of the quantizer is converted back into an analog signal using a simple DAC. This feedback signal is subtracted from the input, closing the control loop.
Digital Filter and Decimator
The raw output of the quantizer is a high-speed bitstream. A digital low-pass filter removes unwanted high-frequency noise, and a decimator reduces the data rate to obtain a high-resolution digital output word.
Bitstream and Pulse-Density Modulation
The direct output of a sigma-delta modulator is not a conventional binary number, but a stream of bits:
111011101111010111…
The information is encoded in the density of ones:
many ones → higher input voltage
many zeros → lower input voltage
This representation is called pulse-density modulation (PDM).
Oversampling
Sigma-delta converters operate with a sampling frequency that is much higher than the Nyquist rate of the input signal. This is known as oversampling.
The oversampling ratio (OSR) is defined as:
\[ OSR = \frac{f_s}{2 \cdot f_{signal}} \]
Oversampling has several advantages:
reduced quantization noise
relaxed requirements for analog anti-aliasing filters
improved effective resolution
Noise Shaping
One of the most important concepts behind sigma-delta converters is noise shaping. While quantization noise cannot be eliminated, it can be redistributed.
Sigma-delta modulators push most of the quantization noise toward high frequencies, leaving the low-frequency signal band relatively clean. The digital filter then removes this high-frequency noise.
Advantages and Disadvantages
Advantages
Very high resolution (16–24 bits are common)
Excellent linearity
Simple and robust analog circuitry
Cost-effective for precision applications
Disadvantages
Limited conversion speed
Latency caused by digital filtering
Not suitable for very high-frequency signals
Typical Applications
Sigma-delta converters are commonly found in:
Audio ADCs and DACs
Precision sensor interfaces
Temperature and pressure measurement
Weighing scales and industrial instrumentation
Summary
The sigma-delta converter is a powerful conversion technique that achieves high resolution by combining oversampling, feedback, and noise shaping. By trading speed for accuracy, it has become one of the most important converter architectures in modern electronics.
The Digital-Analog-Converter (DAC)
One extraordinary simple example of a DAC is a resistor network R-2R-Ladder… It just consists of a simple resistor network and a simple single op-amp.
The R-2R-DAC
The most simple DAC ist the r-2r-ladder DAC. It is basically a resistor network, with a amplifier (op-amp) at the output.
