A numeral system (or system of numeration) is a mathematical notation for representing a set of numbers by symbols in a consistent manner. It can be seen as the context that allows the numeral “11” to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in different bases. Ideally, a …
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Analog Signals An analog or analogue signal is any time continuous signal where some time varying feature of the signal is a representation of some other time varying quantity. It differs from a digital signal in that small fluctuations in the signal are meaningful. Analog is usually thought of in an electrical context, however mechanical, …
Representation of Digital Signals Digital systems do not store numbers the way humans do. A human can remember the number 4.5 and understand that it represents a quantity. The digital system does not have this capability. Instead, digital systems work with numbers using millions of tiny switches called transistors. Each transistor can remember only one …
The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Owing to its straightforward implementation in electronic circuitry, the binary system is used internally by virtually all modern …
Binary Conversion Unsigned Binary Counting The simplest form of numeric representation with bits is unsigned binary. When we count upward through the positive integers using decimal, we start with a 0 in the one’s place and increment that value until we reach the upper limit of a single digit, i.e., 9. At that point, we’ve …
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In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave (a continuous-time signal) to a sequence of samples (a discrete-time signal). A sample refers to a value or set of values at a point in time and/or space. A sampler …
Hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. Its primary purpose is to represent the binary code in a format easier for humans to read, and acts as a form of shorthand, in which one hexadecimal digit stands …
Binary-Coded Decimal (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence. Its main virtue is that it allows easy conversion to decimal digits for printing or display and faster decimal calculations. Its drawbacks are the increased complexity of circuits needed to implement mathematical operations and a relatively …
The use of binary counting sequences is common in digital applications. For example, an n-bit binary value can be used to identify the position of a rotating shaft as being within one of 2n different arcs. As the shaft turns, a sensor can detect which of the shaft’s arcs it is aligned with by reading a digital …
Basic Concepts behind the Binary System To understand binary numbers, begin by recalling elementary school math. When we first learned about numbers, we were taught that, in the decimal system, things are organized into columns: H | T | O 1 | 9 | 3 such that “H” is the hundreds column, “T” is the …
Binary Subtraction Just as with addition, we’re going to use the decimal numbering system to illustrate the process used in the binary numbering system for subtraction. There are four possible cases of single-bit binary subtraction: 0 – 0, 0 – 1, 1 – 0, and 1 – 1. As long as the value being subtracted from (the …
Floating Point Arithmetic Binary numbers can also have decimal points, and to show you how, we will once again begin with decimal numbers. For decimal numbers with decimal points, the standard way to represent the digits to the right of the decimal point is to continue the powers of ten in descending order starting with -1 …
Hexadecimal numbers (base 16) can be added using the same method. The difference is that there are more digits in hexadecimal than there are in decimal. For example, in decimal, adding 5 and 7 results in 2 with a carry to the next highest position. In hexadecimal, however, 5 added to 7 does not go beyond the range …
Unless you are an electrical engineer, an understanding of the operation of transistors is unnecessary. One level above the transistors, however, is a set of basic building blocks for digital circuitry. These building blocks are called logic gates, and it is at this level that we will begin our discussion. A logic gate has the basic format shown below in the …
The expression X = A * B reads as “X equals A AND B”. The multiplication sign stands for the ANDoperation, same for ordinary multiplication of 1s and 0s. The AND operation produces a result of 1 occurs only for the single case when all of the input variables are 1. The output is 0 for any case where one or more …
This logic gate, sometimes referred to as an inverter, is the only one in that has a single input. Its input goes into the left side of a triangle symbol while its output exits the gate through a small circle placed at the tip of the opposite corner. Note that it is the small circle that defines …
The OR gate is a digital logic gate that implements logical disjunction – it behaves according to the truth table to the right. A HIGH output (1) results if one or both the inputs to the gate are HIGH (1). If neither input is HIGH, a LOW output (0) results. The expression X = A …
XOR (Exclusive OR) Gate The XOR gate (sometimes EOR gate) is a digital logic gate that implements exclusive disjunction – it behaves according to the truth table to the right. A HIGH output (1) results if one, and only one, of the inputs to the gate is HIGH (1). If both inputs are LOW (0) …
A truth table is a mathematical table used in logic — specifically in connection with Boolean algebra, Boolean functions, and propositional calculus — to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables. In particular, truth tables can be used to …
Logic Gates Timing Diagram The operation of a logic gate can also be represented with a timing diagram. Images below shows the output that results from three binary input signals for an AND gate, OR gate, and XOR gate respectively. Remember that the AND gate outputs a one only if all its inputs equal one, the OR gate outputs a one if any input equals …