The following is *an article* based on *a short presentation (PowerPoint)* on **Understanding Digital Systems**. The article may serve as a guide on how to understand digital systems better, walking you through the building blocks of digital systems for you to understand them properly.

*the link*to

**download the complete presentation**at the

*end*of this article.

## 1. What are Digital Systems

What are the systems that can be considered as Digital Systems? Personal Computers? Mobile Phones?

Yes, Personal computers and mobile phones can be thought as digital systems. But what these digital systems have in common? What characterizes a digital system? Let’s discuss some concepts related to digital systems which may serve as the building blocks to extend our understanding about digital systems.

## 2. Measurements

We measure an attribute of some object. We get a value and the corresponding unit.

We use **measurement tools** to measure **physical attributes** of objects. An *electronic scale* can be used to measure the weight of a box. A *ruler* can be used to measure the length of a pen. By measuring, we get a value and a unit of the measurement. Let’s consider three different cases of measuring things and our expectations of the measurement.

**Measurement** *(from Old French, mesurement)* is **the assignment of numbers to objects or events**. It is a cornerstone of most natural sciences, technology, economics, and quantitative research in other social sciences. The science of measurement is called **metrology**.

Read more about Measurements

### 2.1. Measuring the Weight of a Box

**The first case**, measuring the weight of a box, is done with an electronic scale. We get the weight of the box, a value and a unit. It could be 2.1kg, 2.11kg, 2.136kg. Now, we have the value and the unit of the measurement. What had been our expectation about the value before the measurement was done? We expected it to be **within a possible continuous range**. We did not expect it to be a discrete value.

### 2.2. Frequencies after exciting a Hydrogen Atom

**The emission spectrum** of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the **Rydberg formula**. These observed spectral lines are due to electrons moving between energy levels in the atom.

Read more about Hydrogen Spectral Series

**The second case**, we are now exciting a hydrogen atom. We measure the output frequencies when the electrons return to their ground states. Now, we have the the values*(frequencies)* and the unit*(hz)*. What had been our expectation about the values before the measurement was done? We expected to them to be **discrete values**. We did not expect the frequencies to be some arbitrary values.

### 2.3. Measuring IQ

**The third case**, measuring IQ of a group of students, is done using an IQ MCQ paper. We evaluate their choices in MCQs and assign marks based on whether their choice being right or wrong. Then we classify the students according to their total marks. At the end of the evaluation, we have one of the **discrete levels** assigned to each of the students for their IQ.

In the first case, our expectation was to have some arbitrary value as the measurement. But in the second case, we knew the quantum nature of the case and we expected the frequencies to be discrete values. As we can now see, our expectation varied depending on the nature of the system that we were measuring. Both the first and the second systems were natural. We measured something natural out of a system that was natural.

The third case was quite different from the first two cases for it was an artificial system that we implemented ourselves for evaluating IQ of the students. But our expectation of the final result was, finally, a discrete value. It can now be understood that, no matter whether the case is natural or artificial, our expectation of the measurement values can only be either continuous or discrete.

## 3. Classification of Systems

In teaching **Signals and Systems**, systems are often classified according to *the output they produce in response to an input*. i.e. We learn about **discrete time systems** having *discrete time input signals* and **continuous time systems** having *continuous time input signals*.

We use **Laplace transform** and **Z transform** to ease the analysis of continuous and discrete systems respectively. Let’s not dive that deeper into classifying the systems in this article. Let’s continue our classification based on the measurement that we made in the previous paragraphs.

When measuring, *continuous values* are expected from **continuous systems** and *discrete values* are expected in **discrete systems**. Yes, quite a simple classification, but it serves its purpose.

### Laplace Transform

The Laplace transform is a widely used integral transform in mathematics with many applications in physics and engineering. It is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms f(t) to a function F(s) with complex argument s, given by the above integral.

### Z Transform

The basic idea now known as the **Z-transform** was known to **Laplace**, and re-introduced in 1947 by W. Hurewicz as a tractable way to solve linear, constant-coefficient difference equations. It was later dubbed “the z-transform” by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952.

## 4. Numeral Systems

We use **decimal system** for our day to day expression of numerical values. In the decimal system, we have **10 symbols** to represent our values. *[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]*

Let’s take 200 symbol as an example. While it can be thought as a symbol itself in its entirely i.e. 200, it can also be thought as having composed of 2 different symbols that are 2 and 0. The value expressed in the symbol 200 depends on which numeral system it has been written. As a general rule, the expressed value of a symbol depends on the numeral system being used and the position of the symbol in the presentation.

A **numeral system** (or system of numeration) is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols “11” to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.

Read more about Numeral Systems

## 5. Binary Digital Systems

In a binary digital system, we only have 2 digits to represent values; 0 and 1.

## 6. Layered Model for Digital Systems

## 7. Presentation Summary

## 8. Questions and Answers

Q: We were made aware of the possibility of realizing ternary digital systems. Are you aware of any physical layer device available to implement such a system; or is there any research currently being done on ternary digital devices?

A: Please go thorough the following resources to find out more about ternary digital systems.

1. Design of Parallel Analog to Digital Converters for Ternary CMOS Digital Systems

2. Formal Veriﬁcation of Digital Circuits Using Symbolic Ternary System Models

You can download the complete **PowerPoint(.pptx)** presentation *(Understanding Digital Systems)* using the below button. The download will contain all the above slides and a few more which was not discussed in the article.