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Recent Articles on Digital Electronics and Logic Design. Topics: Number System and Representation; Programs; Boolean Algebra and Logic Gates; Gate Level.

**Table of contents**

- CSE 261 - Digital Logic Design
- Number System and Representation
- Digital logic circuits in yeast with CRISPR-dCas9 NOR gates
- CS Digital Logic Design

Theoretical and practical issues concerning design with combinational and sequential logic circuits, and programmable logic devices. Number systems, Boolean algebra, representation and simplification o fBoolean functions, universal logic families. Finite-state machines, state tables and state diagrams, flip-flops, counters, registers. Adders, decoders, comparators, multiplexers, memories and applications. Principles of analog circuits are presented in the context of real world problems, such as 'glitches,' power and ground bounce, contact bounce, tri-state logic and bus interfacing, timing circuits, asynchronous versus synchronous circuit components.

Manufacturing yields were also quite low by today's standards. The wide adoption of the MOSFET transistor by the early s led to the first large-scale integration ICs with more than 10, transistors on a single chip. As of , billions of MOS transistors are manufactured every day. An advantage of digital circuits when compared to analog circuits is that signals represented digitally can be transmitted without degradation caused by noise.

In a digital system, a more precise representation of a signal can be obtained by using more binary digits to represent it.

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While this requires more digital circuits to process the signals, each digit is handled by the same kind of hardware, resulting in an easily scalable system. In an analog system, additional resolution requires fundamental improvements in the linearity and noise characteristics of each step of the signal chain. With computer-controlled digital systems, new functions to be added through software revision and no hardware changes. Often this can be done outside of the factory by updating the product's software. So, the product's design errors can be corrected after the product is in a customer's hands.

Information storage can be easier in digital systems than in analog ones. The noise immunity of digital systems permits data to be stored and retrieved without degradation. In an analog system, noise from aging and wear degrade the information stored. In a digital system, as long as the total noise is below a certain level, the information can be recovered perfectly.

Even when more significant noise is present, the use of redundancy permits the recovery of the original data provided too many errors do not occur. In some cases, digital circuits use more energy than analog circuits to accomplish the same tasks, thus producing more heat which increases the complexity of the circuits such as the inclusion of heat sinks.

In portable or battery-powered systems this can limit use of digital systems. For example, battery-powered cellular telephones often use a low-power analog front-end to amplify and tune in the radio signals from the base station.

## CSE 261 - Digital Logic Design

However, a base station has grid power and can use power-hungry, but very flexible software radios. Such base stations can be easily reprogrammed to process the signals used in new cellular standards. Many useful digital systems must translate from continuous analog signals to discrete digital signals. This causes quantization errors.

Quantization error can be reduced if the system stores enough digital data to represent the signal to the desired degree of fidelity. The Nyquist-Shannon sampling theorem provides an important guideline as to how much digital data is needed to accurately portray a given analog signal. In some systems, if a single piece of digital data is lost or misinterpreted, the meaning of large blocks of related data can completely change.

For example, a single-bit error in audio data stored directly as linear pulse code modulation causes, at worst, a single click. Instead, many people use audio compression to save storage space and download time, even though a single bit error may cause a larger disruption. Because of the cliff effect , it can be difficult for users to tell if a particular system is right on the edge of failure, or if it can tolerate much more noise before failing.

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Digital fragility can be reduced by designing a digital system for robustness. For example, a parity bit or other error management method can be inserted into the signal path. These schemes help the system detect errors, and then either correct the errors , or request retransmission of the data.

### Number System and Representation

A digital circuit is typically constructed from small electronic circuits called logic gates that can be used to create combinational logic. Each logic gate is designed to perform a function of boolean logic when acting on logic signals. A logic gate is generally created from one or more electrically controlled switches, usually transistors but thermionic valves have seen historic use. The output of a logic gate can, in turn, control or feed into more logic gates. Another form of digital circuit is constructed from lookup tables, many sold as " programmable logic devices ", though other kinds of PLDs exist.

Lookup tables can perform the same functions as machines based on logic gates, but can be easily reprogrammed without changing the wiring. This means that a designer can often repair design errors without changing the arrangement of wires. Therefore, in small volume products, programmable logic devices are often the preferred solution. They are usually designed by engineers using electronic design automation software.

Integrated circuits consist of multiple transistors on one silicon chip, and are the least expensive way to make large number of interconnected logic gates. Integrated circuits are usually interconnected on a printed circuit board which is a board which holds electrical components, and connects them together with copper traces. Engineers use many methods to minimize logic redundancy in order to reduce the circuit's complexity. Reduced complexity reduces component count and potential errors and therefore typically reduces cost. Logic redundancy can be removed by several well-known techniques, such as binary decision diagrams , Boolean algebra , Karnaugh maps , the Quineâ€”McCluskey algorithm , and the heuristic computer method.

These operations are typically performed within a computer-aided design system. When the volumes are medium to large, and the logic can be slow, or involves complex algorithms or sequences, often a small microcontroller is programmed to make an embedded system. These are usually programmed by software engineers. When only one digital circuit is needed, and its design is totally customized, as for a factory production line controller, the conventional solution is a programmable logic controller , or PLC.

These are usually programmed by electricians, using ladder logic. Representations are crucial to an engineer's design of digital circuits.

Some analysis methods only work with particular representations. The classical way to represent a digital circuit is with an equivalent set of logic gates. Each logic symbol is represented by a different shape. Another way, often with the least electronics, is to construct an equivalent system of electronic switches usually transistors.

One of the easiest ways is to simply have a memory containing a truth table. The inputs are fed into the address of the memory, and the data outputs of the memory become the outputs. For automated analysis, these representations have digital file formats that can be processed by computer programs. Most digital engineers are very careful to select computer programs "tools" with compatible file formats. To choose representations, engineers consider types of digital systems. Most digital systems divide into " combinational systems " and " sequential systems.

## Digital logic circuits in yeast with CRISPR-dCas9 NOR gates

It is basically a representation of a set of logic functions, as already discussed. A sequential system is a combinational system with some of the outputs fed back as inputs. This makes the digital machine perform a "sequence" of operations. The simplest sequential system is probably a flip flop , a mechanism that represents a binary digit or " bit ". Sequential systems are often designed as state machines.

## CS Digital Logic Design

In this way, engineers can design a system's gross behavior, and even test it in a simulation, without considering all the details of the logic functions. Sequential systems divide into two further subcategories. Synchronous sequential systems are made of well-characterized asynchronous circuits such as flip-flops, that change only when the clock changes, and which have carefully designed timing margins.

The usual way to implement a synchronous sequential state machine is to divide it into a piece of combinational logic and a set of flip flops called a "state register. The fastest rate of the clock is set by the most time-consuming logic calculation in the combinational logic. The state register is just a representation of a binary number. If the states in the state machine are numbered easy to arrange , the logic function is some combinational logic that produces the number of the next state. As of , most digital logic is synchronous because it is easier to create and verify a synchronous design.

However, asynchronous logic is thought can be superior because its speed is not constrained by an arbitrary clock; instead, it runs at the maximum speed of its logic gates. Building an asynchronous system using faster parts makes the circuit faster. Nevertherless, most systems need circuits that allow external unsynchronized signals to enter synchronous logic circuits.

These are inherently asynchronous in their design and must be analyzed as such. Examples of widely used asynchronous circuits include synchronizer flip-flops, switch debouncers and arbiters. Asynchronous logic components can be hard to design because all possible states, in all possible timings must be considered.

The usual method is to construct a table of the minimum and maximum time that each such state can exist, and then adjust the circuit to minimize the number of such states. Then the designer must force the circuit to periodically wait for all of its parts to enter a compatible state this is called "self-resynchronization". Without such careful design, it is easy to accidentally produce asynchronous logic that is "unstable," that is, real electronics will have unpredictable results because of the cumulative delays caused by small variations in the values of the electronic components.

Many digital systems are data flow machines. These are usually designed using synchronous register transfer logic , using hardware description languages such as VHDL or Verilog. In register transfer logic, binary numbers are stored in groups of flip flops called registers. The outputs of each register are a bundle of wires called a " bus " that carries that number to other calculations. A calculation is simply a piece of combinational logic. Each calculation also has an output bus, and these may be connected to the inputs of several registers. Sometimes a register will have a multiplexer on its input, so that it can store a number from any one of several buses.

Alternatively, the outputs of several items may be connected to a bus through buffers that can turn off the output of all of the devices except one.