Resolution can also be described by electrical characteristics in volts. The difference between the minimum input voltages required to change the output discrete signal is called least significant bit (LSB) voltage. Therefore, the resolution q of the analog-to-digital converter is equal to the LSB voltage. The voltage resolution of an analog-to-digital converter is equal to its total voltage measurement range divided by the number of discrete voltage intervals:
Where n is the number of discrete voltage intervals, EFSR is the total voltage measurement range, and EFSR is given by the following formula.
Here, VRefHi and VRefLow are the upper and lower limits of allowable voltage during conversion.
Normally, the number of voltage intervals is equal to
Where m is the resolution of the analog-to-digital converter in bits. The response type of most analog-to-digital converters is linear, and "linearity" here means that the size of the output signal is linearly proportional to the size of the input signal.
The response types of some early converters are logarithmic, so the A-law algorithm or μ-law algorithm coding is realized. These encodings are now implemented by a high-resolution linear analog-to-digital converter (for example, 12 or 16 bits), and its 8 is plotted as the encoded output value. The error of analog-to-digital converter comes from several sources. Quantization error and nonlinear error (assuming that the ADC is nominally linear) are inherent errors in any ADC. There is also aperture error, which often occurs in the process of time domain signal digitization because of poor clock oscillation.
This error is measured by a parameter called least significant bit. Analog signals are continuous in time domain, so they can be converted into a series of digital signals that are continuous in time. Therefore, it is necessary to define a parameter to represent the new sampling rate of digital signals from analog signals. This rate is called the sampling rate or sampling frequency of the converter.
It is possible to collect signals with continuous changes and limited bandwidth (that is, measure and store a signal value every once in a while), and then restore the converted discrete signals to the original signals through interpolation. The accuracy of this process is limited by the quantization error. However, only when the sampling rate is higher than twice the signal frequency can the original signal be faithfully restored, which is embodied in the sampling theorem.
Because the actual analog-to-digital converter can't convert completely in real time, some additional methods must be used to keep the input signal constant during the first conversion. Sample-and-hold circuit is a common circuit. In most cases, the input analog voltage can be stored by using a capacitor, and the connection between the capacitor and the input signal can be closed or disconnected by a switch or a gate circuit. Many analog-to-digital conversion integrated circuits already contain such a sample-and-hold subsystem internally. All analog-to-digital converters work by periodic sampling. Therefore, their output signals are only an incomplete description of the behavior of input signals. During the period between a certain sampling and the next sampling, it is impossible to know the form of the input signal only based on the output signal. If the input signal changes at a rate lower than the sampling rate, it can be assumed that the signal between these two samples is between the signal values obtained from these two samples. However, if the input signal changes too fast, this assumption is wrong.
If the signal generated by the analog-to-digital converter passes through the digital-to-analog converter (DAC) in the later stage of the system, the output signal can faithfully reflect the original signal. If the change rate of the input signal is much greater than the sampling rate, it is another case, and the "false" signal output by the analog-to-digital converter is called "aliasing". The frequency of the aliased signal is the difference between the signal frequency and the sampling rate. For example, after the sampling rate is converted at 1.5 kHz, a 2 kHz sinusoidal signal will be reconstructed into a 500 Hz sinusoidal signal. Such a problem is called "aliasing".
In order to avoid aliasing, the input signal of analog-to-digital converter must be filtered by low-pass filter to filter out signals with frequency higher than half of the sampling rate. This kind of filter is also called anti-aliasing filter. It is very important in the actual analog-to-digital conversion system, and is often used in the conversion process of mixed analog signals and high-frequency signals.
Although aliasing is a bad phenomenon in most systems, it is worth noting that it can downmix high-frequency signals with limited bandwidth at the same time (see oversampling and mixer). In analog-to-digital converter, the working conditions can be improved by introducing jitter signal. Disturbance signal is a kind of random noise (white noise) mixed into the input signal before conversion. Its function is that when the input signal is extremely small, the LSB state oscillates randomly between 0 and 1 instead of at a fixed value. This can expand the effective range of the analog-to-digital converter, and will not completely cut off this signal at low input, but the price of doing so is that the noise will increase slightly and the quantization error will spread to a series of noise signal values. In the time range, it can still accurately reflect the change of signal with time. At the output, this small signal fluctuation can be recovered by using an appropriate electronic filters.
Low-amplitude audio signals without jitter signals sound very distorted and unpleasant. Because if there is no jitter signal, the low amplitude signal may cause the least significant bit to be fixed at 0 or 1. After introducing jitter signal, the actual amplitude of audio can be calculated by averaging the samples actually quantized over a period of time and a series of jitter signals. Jitter signal is also used in some integrated systems, such as watt-hour meters, which can make the signal value produce more accurate results than the least significant bit of analog-to-digital converter. Note that the introduction of jitter signal can only increase the resolution of the sampler, but not its linearity, so the accuracy may not be improved. Usually, for economic reasons, the signal is sampled at the lowest sampling rate allowed, resulting in white noise distributed in the whole passband of the converter. If the signal is sampled at a frequency higher than the Nyquist frequency and then digitally filtered, the bandwidth of the signal will be limited, which has the following advantages:
Digital filter has better characteristics (sharper roll-off and phase) than analog filter, so it can form a sharper anti-aliasing filter, which can down-sample the signal and provide better results.
A 20-bit ADC can be used as a 24-bit ADC with 256 times over-dense sampling.
Despite quantization noise, the signal-to-noise ratio will be higher than using the whole available bandwidth. After using this technology, it is possible to obtain higher resolution than using the converter alone;
The signal-to-noise ratio (SNR) improvement of over-dense sampling per frequency doubling (not enough in many applications) is 3 dB (equivalent to 0.5 bit). Therefore, over-dense sampling is usually accompanied by noise signal shaping. Through noise shaping, the improvement can reach 6L+3 dB per octave (where L is the order of the loop filter used for noise shaping, for example, the second-order loop filter can provide an improvement of 15 dB per octave). The speed of analog-to-digital converter varies greatly with different types. Wilkinson ADC is limited by its clock rate. At present, the frequency may exceed 300 MHz. The time required for conversion is proportional to the number of channels. For successive approximation ADC, its conversion time is proportional to the logarithm of the number of channels. In this way, a large number of channels can make the successive approximation converter faster than the Wilkinson converter. But the time of Wilkins converter trumpet is digital, while the successive approximation converter is analog. Because analog itself is slower than digital, when the number of channels increases, the time required also increases. In this way, they have a process of competing with each other in their work. Flash ADC is the fastest of the three, and the conversion is basically a single parallel process. For an 8-bit cell, the conversion can be completed in more than ten nanoseconds.
People expect to achieve the best balance between speed and accuracy. Flash ADC has drift and uncertainty at the comparator level, which will lead to uneven channel width. As a result, the linearity of Flash ADC is not good. For successive approximation ADC, the linearity difference is also obvious, but it is still better than Flash ADC. Here, nonlinearity is the accumulation of errors produced by the subtraction process. At this point, Wilkinson converter is the best. They have the best differential nonlinearity. Other types of converters need smooth channels to reach the level of Wilkinson converter. [3][4]