Transcript EE521 Analog and Digital Communications

EE521 Analog and
Digital
Communications
James K. Beard, Ph. D.
[email protected]
Tuesday, March 29, 2005
http://astro.temple.edu/~jkbeard/
March 29, 2005
Week 11
1
March 29, 2005
Week 11
4/19/2005
4/12/2005
4/5/2005
3/29/2005
3/22/2005
3/15/2005
3/8/2005
3/1/2005
2/22/2005
2/15/2005
2/8/2005
2/1/2005
1/25/2005
1/18/2005
Attendance
7
6
5
4
3
2
1
0
2
Essentials
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Text: Bernard Sklar, Digital Communications,
Second Edition
SystemView
Office
 E&A 349
 Tuesday afternoons 3:30 PM
 MWF 10:30 AM to 11:30 AM


to 4:30 PM & before class
Next quiz April 5
Final Exam Scheduled
 Tuesday, May 10, 6:00
 Here in this classroom
March 29, 2005
PM to 8:00 PM
Week 11
3
Today’s Topics
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Term Project
Waveform Coding, Part 1: Structured
Sequences and EDAC
 Linear
block codes
 Error-detection and correcting capability
 Cyclic codes
 Well-Known Block Codes
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Waveform Coding, Part 2: Convolutional Codes
Discussion (as time permits)
March 29, 2005
Week 11
4
The Term Project

Continue with the start that you turned in with
the first quiz backup
 Input
 Frequency sweep 1000 Hz to 3500 Hz
 Noise to obtain 20 dB SNR
 Sampling to obtain good performance
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Do NOT pitch your beginning and pick up the
ADC to bitstream modules as a template
Sample and encode/decode as instructed
Measure BER vs. Eb/N0 as instructed
Compare hard decoding with soft decoding
March 29, 2005
Week 11
5
Sklar Chapters 6 and 7
From
other
sources
Information
source
Message
symbols
Source
encode
Format
Essential
Legend:
Optional
Channel
symbols
Encrypt
Channel
encode
Multiplex
Pulse
modulate
gi  t 
ui
Digital
input
mi
Bit
stream
z T 
uˆi
Format
Information
sink
Message
symbols
March 29, 2005
Decrypt
Channel
decode
Demultiplex
si  t 
Digital
baseband
waveform
Synchronization
Digital
output
mˆ i
Source
decode
Bandpass
modulate
Detect
Frequency
spread
Digital
bandpass
waveform
X
M
IT
Multiple
access
hc  t 
Channel Channel
impulse
response
r t 
Demodulate &
Sample
Frequency
despread
R
C
V
Multiple
access
To other Channel
destinations symbols
Week 11
6
Linear block codes
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Begin with concepts of polynomials modulo 2
and m-vectors
Based on a closed set of vectors in m-space
A set of k-bit words maps to a this set of mvectors through a linear relationship
It’s a (k,m) code
Algorithm to define the m-vectors
 A complex
method that leads to the mapping
 Provides a basis for EDAC codes
March 29, 2005
Week 11
7
Reprise: Galois Field Vector
Extensions of Order 2m
Polynomials modulo 2 of order m-1
 Arithmetic is done modulo a generating
polynomial of the form
gg  x   x m  1 other powers of x
 Proper selection of generating polynomial
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 Sequence
of powers produces all 2m
elements
 Set is closed on multiplication
March 29, 2005
Week 11
8
Reprise: The Critical
Isomorphism
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Shift registers with feedback
 Bits
in shift register are isomorphic with
polynomial coefficients
 Shift is isomorphic with multiplication by x
 Modulo the generating polynomial is
isomorphic to multiple-tap feedback
Shift registers with feedback can produce
a Galois field in sequence of powers of x
 These codes are also called m-sequences
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March 29, 2005
Week 11
9
Error Detection and Correcting
(EDAC) Concepts

Concept of weight and distance
 Weight
is number of 1’s in a binary sequence
 Hamming distance between a pair of binary
sequences is number of 1’s in an XOR of them
 Any (k,k) mapping will have a minimum Hamming
distance of 1

The name of the game is to find a (n,k) code that
 Provides
a minimum Hamming distance dfree >2
 Can be easily implemented and decoded
 Can be decoded to the signal that codes to a
sequence with the nearest Hamming distance
March 29, 2005
Week 11
10
EDAC Practice
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Steps are
 Implement
(n,k) code from a k-bit signal
 Transmit the n-bit coded signal c through a channel
with less than dfree/2 bit errors
 Find the received code “c hat”
 Find the signal “s hat” that codes to the c nearest the
received c hat
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Can detect and correct up to dfree/2 bit errors
March 29, 2005
Week 11
11
Cyclic codes
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Definition
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The code space is a set of n-bit codes
The code space is closed
End-around shift of a code is still in the code space
Code as a modulo-2 polynomial: x.c mod (xn+1) is in the code
space
Properties
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Based on a generator polynomial like an m-sequence
Codes are signal polynomial times generator polynomial
Generating polynomials are factors of xn+1
Systematic codes possible
Decoding is done by dividing code by generating polynomial
March 29, 2005
Week 11
12
Error Correction With Cyclic
Codes
Divide received code by generating
polynomial
 Remainder represents bit error polynomial
divided by generating polynomial
 Multiply remainder by generating
polynomial to find bit errors
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March 29, 2005
Week 11
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Well-Known Block Codes
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Hamming codes
m
m
 n, k   2  1,2  1  m , m  2,3,


 The
dfree is 3
 Correct one, detect two bit errors
 A “perfect” code – all Hamming distances are dfree
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Extended Golay code
 Add
a parity bit to the perfect (24,12 ) Golay code
 Increases dfree from 7 to 8
 Produces a rate ½ code
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BCH Codes
March 29, 2005
Week 11
14
Assignment
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Review for quiz next week
 Baseband
signals
 Sources of corruption
 Quantization
 Modulation
 Demodulation and complex signals
 EDAC and convolutional codes
Read 7.1, 7.2, 7.3
 Quiz study guide to be posted on web site
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March 29, 2005
Week 11
15