Analog-to-Digital Converters (ADCs): Decrypting Resolutions and Sampling Rates

Resolution and sample rate are perhaps the two most significant factors to consider when choosing an analog-to-digital converter  (ADC). Before making a decision, these two factors should be carefully weighed. They'll have an impact on everything from the price to the underlying architecture of the required analog-to-digital converter during the selection process. A reasonable comprehension of these features is required in order to appropriately identify the correct resolution and sampling rate for a given application.

Some mathematical definitions of terminology relevant to analog-to-digital conversion are provided here. Although math is vital, the concepts it depicts are far more so. You will be able to limit the number of  ADC s that fit your application and make a much easier selection if you can live with the arithmetic and comprehend the principles offered.

 Ⅰ. Quantisation

A continuous signal (voltage or current) is converted into a sequence of numbers represented by discrete logic levels via an analog-to-digital converter,  The process of transforming a large number of values into a smaller or discrete set of values is known as quantization. An  ADC can be represented mathematically as quantizing a large-domain function to yield a smaller-domain function.

Analog to Digital Conversion Process

The analog-to-digital conversion process is mathematically described by the equation above. The input voltage Vin is described as a series of bits b N-1...b 0. The number of quantization levels is represented by 2 N in this formula. More quantization levels, on the surface, appear to result in a more accurate digital reproduction of the original analog signal. Because each quantization level reflects a lower amplitude range, if we can represent the signal with 1024 quantization levels instead of 256, we can improve the ADC's accuracy.

The greatest input voltage that can be effectively translated to an accurate digital representation is represented by Vref. As a result, it's critical that V ref be more than or equal to V in's maximum value. However, take in mind that a value substantially higher than V will result in fewer quantization levels representing the original signal. If we knew our signal would never rise over 2.4 V, for example, having a voltage reference of 5 V would be inefficient because we would be utilizing more than half of the quantization level.

Quantization of a Continuous Signal

Ⅱ. Quantization Error

The disparity between the original signal and the discrete representation of the signal is referred to as quantization error.

Quantum description

A quantum can be defined as shown above, with A denoting amplitude and the signal spanning A to -A. The number N denotes the number of bits that the signal is quantized to.

It's time to look at what quantization means for ADCs now that we've looked at quantization. We'll need to do more math to do this. The quantization error is described by the equation below.

Quantization Error

As a result, the quantization error's power can be characterized as follows.

Definition of Power in Quantization Error

Quantization of a Shine Wave

Take a look at the signal in the illustration above. The following equation can be used to calculate the signal's power.

Power Definition of Signal

As a result, the  Signal Quantization Noise Ratio  (SQNR) can be expressed in decibels. This equation shows that having more quantization levels in an ADC enhances the SQNR ratio.

Definition of SQNR

The signal-to-noise ratio (SNR) of an ideal ADC will equal the SQNR value. Other sources of noise are unfortunately related to the analog-to-digital conversion process. Nonetheless, careful examination and consideration of the analog signal to identify the SQNR required for your application will help with the decision. The resolution of an analog-to-digital converter is the number of bits of quantization.

Feature 1: Resolution  - The number of bits of quantization in the ADC.

In most cases, it is preferable to have the highest resolution available. Other factors, such as resources and cost in the digital arena, often limit its resolution. As a result, determining the minimum resolution required for your application is critical.

Ⅲ. Sampling

Signals in the continuous-time domain must be quantized not only in amplitude but also in time. Consider the following series of pulses, where Ts is the sample period.

Ts is Defined as Sampling Time Period

Impulse Train and Analog Signal

The sampled signal y(t) can be formally characterized as stated in the equation below.

Sampled Signal Definition

This produces the pulse train seen in the image below for the pulse train and analog signal in the image above.

Sampled Data

When looking at signals in the frequency domain, the  Dirac delta function is useful for mathematically explaining the concept of sampling. However, it's important to note that these characteristics don't exist in real-world devices. Instead, virtually rectangular pulses take their place.

Ⅳ. The Nyquist Criterion and Shannon’s Theorem

The frequency-domain of the analog signal must be examined in order to identify the desired sampling rate. This, too, necessitates some math knowledge. The Fourier transform of w(t) can be calculated using the equation below.

Fourier Transform Definition of W(t)

This equation essentially states that the  Dirac delta function is repeated at each harmonic of its frequency Fs. Let's take a look at an analog signal with a spectrum like the one below. The spectrum of the sampled signal Y(f) turns out to be the convolution of X(f) and W. (f).

Two Sided Frequency Spectrum of X(f)

This signifies that the signal repeats for all multiples of the sampling frequency after sampling. If the sampling frequency is not high enough, the spectrum pictures of the signal will overlap, as seen in the diagram below. The Nyquist rate is defined as the minimum frequency equal to twice the bandwidth of the signal to be sampled.

Minimum Spectrum

Spectrum of Signal Sampled at Different Hz

As a result of the Nyquist criterion, it is evident that we must know the spectrum content of the analog signal in order to accurately specify the correct ADC for an application.

Filtering the analog signal before digitization is one approach to ensure that the Nyquist requirement is met.

Looking at the above image again, it's clear that the spectrum after filtering using a proper filter is identical to that of the original signal. There is no data loss, and the original signal can be recovered. Shannon's theorem is the name for this.

Feature 2: Sampling  Rate - The sampling frequency of the analog signal.

When determining the ADC necessary for an application, the sampling rate and resolution of the ADC must be carefully examined. In order to accurately digitize an analog signal, a compromise between sample rate and resolution is frequently required. It's crucial to determine the sample rate and resolution you want before choosing an ADC. In order to appropriately specify the needed resolution and sample rate, careful examination of analog signals and the digital resources required to process digital data is required.