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2. The Digital Abstraction x Computation Structures Part 1 Digital Circuits Copyright 2015 MIT EECS Computation Structures L2: The Digital Abstraction, Slide #1 Concrete Encoding of Information

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2. The Digital Abstraction x Computation Structures Part 1 Digital Circuits Copyright 2015 MIT EECS Computation Structures L2: The Digital Abstraction, Slide #1 Concrete Encoding of Information To this point we ve discussed encoding information using bits. But where do bits come from? If we re going to design a machine that manipulates information, how should that information be physically encoded? Rosetta Stone Captmondo (CC BY-SA 3.0) DNA What makes a good bit? - small, inexpensive (we want a lot of them) - stable (reliable, repeatable) - ease and speed of manipulation (access, transform, combine, transmit, store) Madeleine Price Ball (CC BY-SA 3.0) Computation Structures L2: The Digital Abstraction, Slide #2 Let s Use Electrical Phenomenon Consider using phenomenon associated with charged particles: voltages phase currents frequency In this course we ll use voltages to encode information. But the best choice depends on the intended application Voltage pros: easy generation, detection lots of engineering knowledge potentially low power in steady state zero Voltage cons: easily affected by environment DC connectivity required? R & C effects slow things down Computation Structures L2: The Digital Abstraction, Slide #3 Representing Information with Voltage Representation of each (x,y) point on a B&W image: 0 volts: BLACK 1 volt: WHITE 0.37 volts: 37% Gray How much information at each point? John Phelan (CC BY 3.0) Suppose we can reliably distinguish voltages that differ by 1/2 N volts. Then we can represent N bits of information by voltages in the range 0V to 1V. What are realistic values for N? Computation Structures L2: The Digital Abstraction, Slide #4 Using Voltages to Encode a Picture Representation of a picture: Scan points in some prescribed raster order Generate voltage waveform: John Phelan (CC BY 3.0) NTSC TV signal white black sync Computation Structures L2: The Digital Abstraction, Slide #5 Information Processing = Computation v Copy v v Inv 1-v Why have processing blocks? Pre-packaged functionality: rely on behavior without having to be an analog engineer Predictable composition of functions Tinker-toy assembly Guaranteed behavior: if components work, system will work! Wow, rules simple enough for a programmer to follow! Computation Structures L2: The Digital Abstraction, Slide #6 Let s Build a System! Copy Inv input Copy Inv Copy Copy Inv Inv (In(Reality) Theory)? output Computation Structures L2: The Digital Abstraction, Slide #7 Why Did Our System Fail? Why doesn t theory match reality? 1. COPY block doesn t work right 2. INV block doesn t work right 3. Theory is imperfect 4. Reality is imperfect 5. Our system architecture stinks ANSWER: all of the above! Noise and inaccuracy are inevitable; we can t reliably reproduce infinite information we must design our system to tolerate some amount of error if it is to process information reliably Computation Structures L2: The Digital Abstraction, Slide #8 The Digital Abstraction Real World Manufacturing Variations Noise Ideal Abstract World 0/1 Bits Volts or Electrons or Ergs or Gallons Keep in mind that the world is not digital, we would simply like to engineer it to behave that way. Furthermore, we must use real physical phenomena to implement digital designs! Computation Structures L2: The Digital Abstraction, Slide #9 Using Voltages Digitally Key idea: encode only one bit of information: 2 values 0, 1 Use the same uniform representation convention for every component and wire in our digital system Attempt #1: Hard to distinguish V TH -εfrom V TH +ε V V TH interpreted as 0 V V TH interpreted as 1 volts Attempt #2: V TH? V V L interpreted as 0 V L V V H Forbidden to ask V V H interpreted as 1 volts V L V H Computation Structures L2: The Digital Abstraction, Slide #10 A Digital Processing Element A combinational device is a circuit element that has one or more digital inputs one or more digital outputs Static discipline a functional specification that details the value of each output for every possible combination of valid input values a timing specification consisting (at minimum) of an upper bound t PD on the required time for the device to compute the specified output values from an arbitrary set of stable, valid input values input A input B input C Output a 1 if at least 2 out of 3 of my inputs are a 1. Otherwise, output 0. I will generate a valid output in no more than 2 minutes after seeing valid inputs output Y Computation Structures L2: The Digital Abstraction, Slide #11 A Combinational Digital System A set of interconnected elements is a combinational device if each circuit element is combinational every input is connected to exactly one output or to some vast supply of constant 0 s and 1 s the circuit contains no directed cycles Why is this true? Computation Structures L2: The Digital Abstraction, Slide #12 Is This a Combinational Device? A, B and C are combinational devices. Is the following circuit a combinational device? A C B Does it have digital inputs? Does it have digital outputs? Can you derive a functional description? Can you derive a t PD? Computation Structures L2: The Digital Abstraction, Slide #13 Will This System Work? Valid 0 : V L -ε Combinational device Noise V L +ε: not a valid signal Combinational device Upstream device transmits a signal at V L -ε, a valid 0. Noise on the connecting wire causes the downstream device to receive V L +ε, a signal in the forbidden zone. Hmm. Looks like the output voltage needs to be adjusted so that a signal will still be valid when it reaches an input even if there s noise Computation Structures L2: The Digital Abstraction, Slide #14 Where Does Noise Come From? Parasitic resistance, inductance, capacitance IR drop, L(dI/dt) drop, LC ringing from current steps Power supply Integrated circuit + - L s from chip leads R s and C s from Aluminum wiring layers Current loads from onchip devices Imprecision of component values Manufacturing variations, allowable tolerances Environmental effects External EM fields, temperature variations, etc Computation Structures L2: The Digital Abstraction, Slide #15 Needed: Noise Margins! Proposed fix: separate specifications for inputs and outputs digital output: 0 V OL, 1 V OH digital input: 0 V IL, 1 V IH V OL V IL V IH V OH VALID INPUT REPRESENTATIONS Valid 0 V OL V IL Forbidden Zone V IH V OH Valid 1 volts NOISE MARGINS VALID OUTPUT REPRESENTATIONS A combinational device accepts marginal inputs and provides unquestionable outputs (to leave room for noise) Computation Structures L2: The Digital Abstraction, Slide #16 A Buffer A simple combinational device: V out V OH Voltage Transfer Characteristic (VTC): Plot of V out vs. V in where each measurement is taken after any transients have died out. V OL V in Note: VTC does not tell you anything about how fast a device is it measures static behavior not dynamic behavior V IL V IH Static Discipline requires that the VTC avoid the shaded regions (aka forbidden zones ), which correspond to valid inputs but invalid outputs Computation Structures L2: The Digital Abstraction, Slide #17 Voltage Transfer Characteristic V out V OH V OL V in V IL V IH 1) Note the VTC can do anything when V IL V IN V IH. 2) Note that the center white region is taller than it is wide (V OH -V OL V IH -V IL ). Net result: combinational devices must have GAIN 1 and be NONLINEAR Computation Structures L2: The Digital Abstraction, Slide #18 Can This Be a Combinational Inverter? Suppose that you measured the voltage transfer curve of the device shown below. Can we find a signaling specification that would allow this device to be a combinational inverter? V OH V OUT (0,5) (1,4) The device must be able to actually produce the desired output level. Thus, V OL can be no lower than 0.5 V. VIH Try V OL = 0.5 V must be high enough to produce VOL Try V IH = 3 V V OL (3,0.5) (2.5,1) V IL V IH V IN Now, find noise margin N and compute V OH = V IH + N V IL = V OL + N Then verify that V OUT V OH when V IN V IL. Try N = 0.5 V Device is a combinational inverter when V OL =0.5, V IL =1, V IH =3, V OH = Computation Structures L2: The Digital Abstraction, Slide #19 Summary Use voltages to encode information Digital encoding valid voltage levels for representing 0 and 1 forbidden zone avoids mistaking 0 for 1 and vice versa gives rise to notion of signal VALIDITY. Noise Want to tolerate real-world conditions: NOISE. Key: tougher standards for output than for input devices must have gain and have a non-linear VTC Combinational devices Each logic family has Tinkertoy-set simplicity, modularity predictable composition: parts work whole thing works static discipline digital inputs, outputs; restore marginal input voltages complete functional spec valid inputs lead to valid outputs in bounded time Computation Structures L2: The Digital Abstraction, Slide #20 Next Time: Building Logic with Transistors It s about time! I d have preferred the DNA Computation Structures L2: The Digital Abstraction, Slide #21

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