Transcript HOW 1s AND 0s RULE THE WORLD

HOW 1s AND 0s RULE
THE WORLD
Utku Altunkaya
Outline
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Introduction
Basic Logic Operations
Logic Circuits
Base-2 (Binary) Number System
Analog vs. Digital Signals and Systems
Implementation Technologies for Digital
Circuits
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.
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…
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
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
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.
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)’
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)’
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)
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)]’
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.
Logic Circuits
A
B
C
L=(A · B)’
M=A+B
N=(L · M)’
P=N+C
Q=(N · C)’ R=(P · Q)’
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Base-2 (Binary)
Number System
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.
Analog to Digital: Sampling
The figure at the right
shows a continuous-time,
analog (continuous-valued)
signal.
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.
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: Ts=1/fs
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.
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.
Analog to Digital: Quantization
Quantization is achieved by
rounding the value of each
sample to a predetermined
quantization level.
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.
Analog to Digital: Quantization
As the number of
quantization levels increase,
the quantization step gets
smaller, enabling a better
representation of the
original signal.
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.
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).
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.
Digital Information
The amount of data required to represent a digital
signal is determined by the signal’s 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.
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
Implementation Technologies
for Digital Circuits
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Programmable Logic Devices
Application Specific Integrated Circuits
Microprocessors & Microcontrollers
Digital Signal Processors
FPGAs
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