Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole. It resembles the algebra of real numbers as taught in high school, but with the numeric operations of multiplication xy, addition x + y, and negation -x replaced, respectively, by the logical operations of conjunction x?y, disjunction x?y, and …
Boolean algebra Symbols Analogous behavior can be shown between Boolean algebra and mathematical algebra, and as a result, similar symbols and syntax can be used. For example, the following expressions hold true in math. This looks like the AND function allowing an analogy to be drawn between the mathematical multiply and the Boolean AND functions. …
Laws & Rules of Boolean algebra The manipulation of algebraic expressions is based on fundamental laws. Some of these laws extend to the manipulation of Boolean expressions. For example, the commutative law of algebra which states th at the result of an operation is the same regardless of the order of operands holds true for Boolean algebra too. This is shown …
Adders Most mathematical operations can be handled with addition. For example, subtraction can be performed by taking the two’s complement of a binary value, and then adding it to the binary value from which it was to be subtracted. Two numbers can be multiplied using multiple additions. Counting either up or down (incrementing or decrementing) can be …
Multiplexers A multiplexer, sometimes referred to as a MUX, is a device that uses a set of control inputs to select which of several data inputs is to be connected to a single data output. With n binary “select lines,” one of 2n data inputs can be connected to the output. Image below presents a block diagram …
De-Multiplexers The previous section described how multiplexers select one channel from a group of input channels to be sent to a single output. Demultiplexers take a single input and select one channel out of a group of output channels to which it will route the input. It’s like having multiple printers connected to a computer. A document can only …
Decoders & Encoders Encoders An encoder is a device used to change a signal (such as a bitstream) or data into a code. The code may serve any of a number of purposes such as compressing information for transmission or storage, encrypting or adding redundancies to the input code, or translating from one code to another. This …
IC (Integrated Circuit) It may appear that much of our discussion up to this point has been theoretical, but in reality, each of the circuits we’ve presented can easily be implemented given the right tools. Prototypes used to test or verify circuit designs can be made by wiring together small plastic chips that offer access …
Bitwise Operations Most software performs data manipulation using mathematical operations such as multiplication or addition. Some applications, however, may require the examination or manipulation of data at the bit level. For example, what might be the fastest way to determine whether an integer is odd or even? The method most of us are usually taught to distinguish …
Parity One of the most primitive forms of error detection is to add a single bit called a parity bit to each piece of data to indicate whether the data has an odd or even number of ones. It is considered a poor method of error detection as it sometimes doesn’t detect multiple errors. When …
Checksum For digital systems that store or transfer multiple pieces of data in blocks, an additional data element is typically added to each block to provide error detection for the block. This method of error detection is common, especially for the transmission of data across networks. One of the simplest implementations of this error detection scheme is …
A cyclic redundancy check (CRC) is a type of function that takes as input a data stream of any length and produces as output a value of a certain fixed size. The term CRC is often used to denote either the function or the function’s output. A CRC can be used as a checksum to detect accidental alteration of data during …
Organization of Memory Device Modern memory has the same basic configuration as magnetic core memory although the rings have been replaced with electronic memory cells such as the D-Latch. The cells are arranged so that each row represents a memory location where a binary value would be stored and the columns represent different bits of those …
Memory-Processor Interfacing The previous topic presented the input and output lines for a memory device. These lines are shared across all of the devices that communicate with the processor. If you look at the electrical traces across the surface of a motherboard, you should see collections of traces running together in parallel from the processor …
Memory Terminology There are many different purposes for memory in the operation of a computer. Some memory is meant to store data and programs only while the computer is turned on while other memory is meant to be permanent. Some memory contains application code while other memory is meant to store the low-level driver code …
Characteristic of Hard Drive At the bottom of the hierarchy is long-term, high-capacity storage. This type of storage is slow making a poor choice for the processor to use for execution of programs and data access. It is, however, necessary to provide computer systems with high capacity, non-volatile storage. Hard drives are the most cost-effective …
Cache RAM Even with increases in hard drive performance, it will never be practical to execute programs or access data directly from these mechanical devices. They are far too slow. Therefore, when the processor needs to access information, it is first loaded from the hard drive into main memory where the higher performance RAM allows fast access …
Registers At the top of the memory, the hierarchy is a set of memory cells called registers. A register is a group of latches that have been combined in order to perform a special purpose. This group of latches may be used to store an integer, store an address pointing to memory, configure an I/O …
BUS, Registers, and Flags Bus A bus is a bundle of wires grouped together to serve a single-purpose. The main application of the bus is to transfer data from one device to another. The processor’s interface to the bus includes connections used to pass data, connections to represent the address with which the processor interested, and …
Buffers Rarely does a processor operate in isolation. Typically there are multiple processors supporting the operation of the main processor. These include video processors, the keyboard and mouse interface processor, and the processors providing data from hard drives and CROMs. There are also processors to control communication interfaces such as USB, Firewire, and Ethernet networks. These processors …