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HOW 1s AND 0s RULE THE WORLD Utku Altunkaya

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Outline Introduction Basic Logic Operations Logic Circuits Base-2 (Binary) Number System Analog vs. Digital Signals and Systems Implementation Technologies for Digital Circuits

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Introduction All digital systems are built around the fundamentals of the base-2 (binary) number system, which uses only 1s and 0s to represent numbers. Anything that can be expressed as a binary number can be processed by a digital system.

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Introduction Here are a few concepts and acronyms related to digital systems that are widely used in daily life: Bits, bytes, kilo-, mega-, and gigabytes Microprocessors, RAM, ROM CD-ROM, CD- Audio, MP3 compression DVD, MPEG and DivX compression Internet, modem, ADSL, kbps, mbps…

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Logic Operations THE AND GATE The AND gate implements the Boolean AND function where the output is logic 1 only when all inputs to the AND gate are logic 1. The standard symbol and the truth table for a two-input AND gate is: The Boolean expression for the AND gate is Y = A · B

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Logic Operations THE OR GATE The OR gate implements the Boolean OR function where the output is logic 1 when any input to the OR gate is logic 1. The standard symbol and the truth table for a two-input OR gate is: The Boolean expression for the OR gate is Y = A + B

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Logic Operations THE NOT GATE The NOT gate (Inverter) implements the Boolean NOT function where the output is the inverse of the input. The standard symbol and the truth table for the NOT gate is: The Boolean expression for the NOT gate is Y = A. From these three basic logical gates it is possible to implement any Boolean expression in hardware. Some simple combinations of these functions have their own gate names and symbols; these are NAND, NOR, XOR, and XNOR gates.

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Logic Operations THE NAND GATE The NAND gate is an AND gate followed by a NOT gate. The output of a NAND gate is logic 0 when all inputs are logic 1. The standard symbol and the truth table for a two-input NAND gate is: The Boolean expression for the NAND gate is Y = (A · B)

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Logic Operations THE NOR GATE The NOR gate is a combination of an OR followed by a NOT gate. The output is logic 0 when any of the inputs is logic 1. The standard symbol and the truth table for a two-input NOR gate is: The Boolean expression for the OR gate is Y = (A + B)

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Logic Operations THE XOR GATE (EXCLUSIVE-OR) The output of this gate is logic 1 if an odd number of its inputs are logic 1; otherwise, the output of this gate is logic 0. The standard symbol and the truth table for a two-input XOR gate is: The Boolean expression for the XOR gate is Y = (A · B) + (A · B)

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Logic Operations THE XNOR GATE (EXCLUSIVE-NOR) The output of this gate is logic 1 if an even number of its inputs are logic 1; otherwise, the output of this gate is logic 0. The standard symbol and the truth table for a two-input XNOR gate is: The Boolean expression for the XOR gate is Y = [(A · B) + (A · B)]

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Logic Circuits A logic circuit is a compound circuit consisting of the basic logic gates AND, OR, NOT, NAND, NOR, XOR, and XNOR. A combinational logic circuit produces its output according to the logic values of its current inputs. The 'past' inputs do not influence the output of the circuit. Circuits that are able to 'remember' the past are called sequential circuits.

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Logic Circuits ABCL=(A · B)M=A+BN=(L · M)P=N+CQ=(N · C)R=(P · Q) 000101110 001101101 010110011 011110110 100110011 101110110 110011110 111011101

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Base-2 (Binary) Number System

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Analog vs. Digital A continuous valued signal is called an analog signal. A discrete time, quantized and binary coded signal is called a digital signal.

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Analog to Digital: Sampling The figure at the right shows a continuous-time, analog (continuous-valued) signal.

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Analog to Digital: Sampling The first step in digitizing an analog signal is sampling, which is done by taking samples of the original analog signal at equally spaced, fixed points in time.

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Analog to Digital: Sampling The time distance between these sampling points is called the sampling period, and the number of samples taken per second is denoted as the sampling frequency. Sampling period is the inverse of the sampling frequency: T s =1/f s

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Analog to Digital: Sampling As the sampling period gets smaller (and thus the sampling frequency gets higher) the sampled signal provides a better and better representation of the original signal.

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Analog to Digital: Sampling The samples are still analog in nature, but together they form a discrete-time signal. The sampled signal must now be quantized, and then binary codes must be assigned to each sample to obtain the digital signal.

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Analog to Digital: Quantization Quantization is achieved by rounding the value of each sample to a predetermined quantization level.

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Analog to Digital: Quantization The range between the minimum and maximum values of the signal is divided into a number of equally spaced levels, and the distance between two adjacent levels is called the quantization step.

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Analog to Digital: Quantization As the number of quantization levels increase, the quantization step gets smaller, enabling a better representation of the original signal.

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Analog to Digital: Quantization The result of the quantization operation is a discrete-valued, discrete-time signal. Finally, to obtain the digital signal, each quantization level will now be assigned a unique binary number code.

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Analog to Digital: Coding In this example, the sampled signal is quantized using 0.01V steps between the values 0V to 2.55V, giving 256 discrete quantization levels. These levels are then numbered using 8-bit binary codes ranging from 0 (0000 0000) to 255 (1111 1111).

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Analog to Digital: Coding The digital representation of the analog signal is thus a series of 1s and 0s forming 8-bit (1 byte) binary numbers for each sample. This information can now be stored in a file on a computer, or transmitted over the Internet.

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Digital Information The amount of data required to represent a digital signal is determined by the signals sampling frequency and the number of quantization levels. As these increase, so does the amount of data. In order to keep the amount of required data as small as possible, sampling frequency and quantization levels must be carefully chosen and these quantities should be no more than what is absolutely necessary to represent the analog signal to be digitized.

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Digital Information As an example, Audio CDs use 16-bit samples and the sampling frequency is 44.1 kHz. Since the audio recording is stereo, two channels (left and right) are digitized and stored on the CD. The amount of data required for 1 second of audio is thus: 2 channels x 16 bits/sample x 44100 samples/s = 1411200 bits/s = 1.35Mbits/s 1411200 bits/s = 176400 (1411200 / 8) bytes/s = 172.27 Kbytes/s

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Implementation Technologies for Digital Circuits Programmable Logic Devices Application Specific Integrated Circuits Microprocessors & Microcontrollers Digital Signal Processors FPGAs

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