PowerPoint PresentationTopicsAnalysisArithmeticSlide 5Discrete vs continuousHow close is good enough?The advantage of discrete techniquesGeorge Boole 1815-1864Claude ShannonDigital representationsIntegersDigital Signal RepresentationsSo Why Digital?Logic FunctionsLogical ExpressionsLogic Function ExampleLogic Function Example 2Evaluation of Logical Expressions with “Truth Tables”Slide 20The Important Logical FunctionsAnd & OrSome Important Logical FunctionsLogic GatesLogic Circuits10/19/2004 EE 42 fall 2004 lecture 21 1Lecture #21 Intro to Digital ElectronicsThis week: Circuits for digital devices10/19/2004 EE 42 fall 2004 lecture 21 2TopicsToday:•Digital electronics•Boolean logic•And, Or, and Not gates10/19/2004 EE 42 fall 2004 lecture 21 3Analysis•Analysis is the systematic study of real and complex-valued continuous functions. Important subfields of analysis include calculus, differential equations, and functional analysis. Derivatives, integrals, and infinite series are examples of the tools of analysis.10/19/2004 EE 42 fall 2004 lecture 21 4Arithmetic•Arithmetic is a branch of mathematics which studies the properties of the natural numbers (the integers) 1, 2, 3, … and operations on those numbers such as addition, subtraction, multiplication and division.•From Geometry problems, it was realized by the ancients that there was a relationship between the natural numbers and continuous measures like distances.•Of course, there are always quantities which don’t come out to an even multiple, so they invented the concept of a fraction.10/19/2004 EE 42 fall 2004 lecture 21 5•It came as quite a surprise when it turned out that some measurements could not be expressed exactly as a fraction (ratio of two integers)•So much so that they are still called “irrational” numbers!2 e10/19/2004 EE 42 fall 2004 lecture 21 6Discrete vs continuous•For thousands of years, mathematics has been split between:–Arithmetic, the study of the discrete–Analysis, the study of the continuous. •Until the 1930’s analysis was considered the pinnacle of mathematics. •Computer Science and digital circuits are a triumph of discrete mathematics•Most of Electrical engineering is analysis!10/19/2004 EE 42 fall 2004 lecture 21 7How close is good enough?•A Fraction can be constructed to be arbitrarily close to any real number, but can not perfectly represent all numbers.•Decimals are just an extension of the fraction concept, as are floating point numbers.•A continuous quantity can not be expressed exactly with these, but it can be represented with any desired degree of accuracy.10/19/2004 EE 42 fall 2004 lecture 21 8The advantage of discrete techniques•The great advantage of discrete techniques such as the integers, fractions, etc, are that quantities can be transmitted and stored without degradation.•If you convey “This far” to your neighbor, it is not possible for him to transmit the information exactly to his neighbor, but “4/5” can be recorded and accurately transmitted for three thousand years! •(4/5ths was what was left after the Pharaoh took his share)10/19/2004 EE 42 fall 2004 lecture 21 9George Boole 1815-1864A schoolteacher and sonof a cobbler, George Boole was a largely self-taught mathematician. He came up with a way to treat logic (which was regarded as a part of philosophy) with the tools of mathematic expressions.His 1848 paper is on the web site, in case you are interested.His work was not at all well known for the next 90 years, when Claude Shannon applied it to electronic circuits.10/19/2004 EE 42 fall 2004 lecture 21 10Claude ShannonClaude Shannon was a student working on a mechanical calculating machine at MIT. He considered ways of improving it, perhaps by using electrical circuits instead of mechanical parts. It occurred to Shannon that the discrete mathematics of Boolean algebra he had learned as an undergraduate could be implemented in hardware, in electronic circuits. Shannon's 1937 thesis was a key piece in the development of digital electronics and modern computers.10/19/2004 EE 42 fall 2004 lecture 21 11Digital representations•We have been looking mostly at circuits which do what we want by making them as linear as possible, and transferring information as a voltage, or as a current.•In a digital system, we do pretty much the opposite, using highly nonlinear responses (like the comparator) to force a signal to be one of a few discrete values (often just 2: high or low, true or false, 0 or 1)10/19/2004 EE 42 fall 2004 lecture 21 12Integers•All operations on integers can be reduced to operations on multiple variables each of which takes on only one of two values, 0 and 1, and the operations and, or and not.•For example, a specific integer can be represented by its binary equivalent•5=1012•Implemented as electronic circuits, this would correspond to three wires, and each having a high or low voltage•These three wires could represent integers 0 to 710/19/2004 EE 42 fall 2004 lecture 21 13Digital Signal RepresentationsExample: Possible digital representation for a pure sine wave of known frequency. We must choose maximum value and “resolution” or “error,” then we can encode the numbers. Suppose we want 1V accuracy of amplitude with maximum amplitude of 50V, We could use a simple pure binary code with 6 bits of information. ( Why 6 bits…. What if we only use 5?)By using binary numbers we can represent any quantity. For example a binary two (10) could represent a 2 Volt signal. But we generally have to agree on some sort of “code” and the dynamic range of the signal in order to know the form and the minimum number of bits. Example: We want to encode to an accuracy of one part in 64 (i.e. 1.5% precision). Answer: with 5 binary digits we can represent only 32 valuesIt takes 6 binary digits (or “bits”) to represent any number 0 to 63.10/19/2004 EE 42 fall 2004 lecture 21 14So Why Digital?(For example, why CDROM audio vs vinyl recordings?)•Digital signals can be transmitted, received, amplified, and re-transmitted with no degradation.•Binary numbers are a natural method of expressing logical variables.•Complex logical functions are easily expressed as binary functions (e.g., in control applications … see next page).•Digital signals are easy to manipulate (as we shall see).•With digital representation, we can achieve arbitrary levels of “dynamic range,” that
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