Number systems
Computers and digital systems utilize several types of data systems that are essential for data manipulation. There are several types of these data systems: hexadecimal, decimal, binary, and octal number systems. The octal number system uses a base of 8 for conversion. All numbers are expressed to base 8. The decimal number system uses a base of 10, while the binary system uses a base of 2. In computer logic, binary number systems are used because any number can be expressed in terms of ones and zeroes in this system. This is useful for transistor logic since such numbers can be represented as on and off transistor signals. Other operations such as multiplication and division are done by shifting bits either to the left or to the right. Conversions between these number systems are straight forward as long as one central number system is adopted. It is customary to convert one number from either of those systems to decimal before converting to the other system.
Logic functions, devices and Boolean algebra
Digital systems are driven by logical operations. Digital logic is based on Boolean algebra, which converts statements into mathematical notation by assigning binary bits to them. When a statement is true, it is assigned binary bit 1, and when it is false, it is assigned binary bit 0. Combinations of logical statements can be used to form logical functions. Truth tables are used to carry out logic operations. A logic function takes in logical inputs and produces a single binary bit as the output. Logic gates are used to carry out logic manipulations. These are devices that are made using TTL (trasnsistos transistor logic). There are four types of logic gates: the AND, OR, NOR, NOT, NAND, and XOR logic gates. The OR gate produces an output if one or more inputs is true. The NOT gate inverts the input. The AND gate produces an output if all the inputs are true. The other gates are compliments of these three main gates. The XOR gate produces an output if one of the inputs is different.
Theorems, simplification and minimization
Truth tables are a great way of determining whether a logic function is generating the desired output. Kanaugh maps are used to minimize logic functions by eliminating redundant and dependent terms. Another useful but tedious method of function minimization is the use of Boolean algebra. De Morgan’s theorems come in handy in Boolean algebra as they help in minimization. De Morgan put forward two theorems that are used to minimize logical functions. This is similar to the normal math algebra except that the normal rules of addition do not apply (it is not numbers being added; rather, it is logical statements being added). K-maps simplify long expressions that are difficult to minimize and combine algebraically. The 1st is grouped in even groups of 2,4 extra in the K-maps. By checking which variable is not changing in each grouped set, then the term corresponding to that combination can be written down.
Combinational data processing logic systems
Combinational logic circuits are electrical systems that use logic gates combined in a certain way. Logical functions that we have discussed in the two sections above. In order to implement logic that uses many variables, combinational logic circuits are needed. For instance, a NOT gate can be connected at the output of an AND gate to invert the output. In this manner, the output of the combinational logic circuit is the inverse of the ANDED inputs. It is easy to develop these circuits once a function is known. One simply places that connects gates as per the terms in the function. To reduce cost and save hardware space, digital functions are first minimized so that the minimum number of terms is realized. This minimizes the number of logic gates required as well. The circuits are used to design digital controllers as well as devices such as calculators. Two circuits that are important in digital circuits are the full adder and subtractors. These are used in the arithmetic logic unit of a computer processor. They are composed of logic gates l.
Registers and counters
A register is a sequential device that can store a value. It is essentially a memory device. Infact, registers are the building blocks of all digital memory unlike combinational logic gates. Registers are driven by clocks. It is this clock signal that causes a register to “remember” it’s previous state. Registers are composed of circuit components known as flip flops. As the name suggests, these devices are able to hold their current state until the clock signal changes. If it is a level triggered flip flop, the state of the device will change when the clock signal moves from LOW to HIGH and vice versa. There are four main types of flip-flops: the SR, JK, T, and T flip-flops. In all flip flops, feedback is used to create the memory effect. When the flip flop generates a certain bit at the output, the value is fed back to the input, which causes it to hold its state. Flip flops are used to create counters. These shift bits sequentially, either upwards ( UP counter) or downwards (DOWN counter). The counter resets when the maximum count is reached.
Analog to digital and digital to analog conversion
An analog to digital (ADC) circuit conversions analog (continuously varying in time) signals to digital format (discrete). In an ADC, the analog signal is sampled at a rate determined by the frequency of the input signal. The minimum frequency of sampling is determined by the Shanon’s sampling theorem, which states that the sampling frequency must be at least twice the frequency of the input frequency. When a voltage sample is obtained at any one point in time, a digital value is assigned to it. In this manner, almost all points of the input analog signal are quantized. Some information is lost because not all points of the input waveform are represented. The R-2R circuit is the most commonly used quantization network since it ensures that the impedance is the same for all quantization levels. A DAC (digital to analog converter) conversions digital signals to analog form. A DAC uses a low pass filter made up of a resistor and a capacitor. When a stored digital voltage level is applied to the resistor-capacitor network, a discharge cycle starts. The voltage across the capacitor varies continuously in time. In this manner, the original analog signal is approximately recovered.
Microcontrollers
A microcontroller is a device that is used to control hardware devices using a programmable digital computer. It is essentially a stand-alone processor that can be programmed to perform specialized tasks. It has input and output pins for interfacing. In the input Port of the microcontroller, an analog to digital converter is used to convert analog signals to digital form for processing. The output Port contains a digital to analog converter. A microcontroller has a small memory that can be used to store sensor data, the program itself, and states such as flags. These devices are popular in the industry where they are used to control hardware devices such as pumps, heaters, and other machines. To program the device, special software is used to write the program code that is then compiled into binary form and uploaded into the device through a serial bussl connection to the programming computer. Once the program is stored in memory, it runs continuously until it is replaced by a new program. It can be said that the device cannot run more than one program in parallel. The devices are small in size and are essentially very small chips such the ATMEL series Microcontrollers..