March 29, 2005 Week 11 1

EE521 Analog and Digital CommunicationsJames K. Beard, Ph. D.

jkbeard@temple.edu

Tuesday, March 29, 2005

http://astro.temple.edu/~jkbeard/

Week 11 2March 29, 2005

Attendance

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1/18

/200

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1/25

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2/8/

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2/15

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2/22

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Week 11 3March 29, 2005

Essentials Text: Bernard Sklar, Digital Communications,

Second Edition SystemView Office

E&A 349 Tuesday afternoons 3:30 PM to 4:30 PM & before class MWF 10:30 AM to 11:30 AM

Next quiz April 5 Final Exam Scheduled

Tuesday, May 10, 6:00 PM to 8:00 PM Here in this classroom

Week 11 4March 29, 2005

Today’s Topics

Term Project Waveform Coding, Part 1: Structured

Sequences and EDAC Linear block codes Error-detection and correcting capability Cyclic codes Well-Known Block Codes

Waveform Coding, Part 2: Convolutional Codes Discussion (as time permits)

Week 11 5March 29, 2005

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 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

Week 11 6March 29, 2005

Sklar Chapters 6 and 7

Information

source

FormatSource encode

EncryptChannel encode

Channel encode

Multi-plex

Pulse modulate

Bandpass modulate

Freq-uency spread

Multiple access

X M I T

FormatSource decode

DecryptChannel decode

Channel decode

Demul-tiplex

DetectDemod-ulate & Sample

Freq-uency

despread

Multiple access

R C V

Channel

Information

sink

Bit stream

Synch-ronization

Digital baseband waveform

Digital bandpass waveformDigital

outputˆ im

Digital input

im

ˆiu z T r t

iu ig t is t

Optional

Essential

Legend:

Message symbols

Channel symbols

Channel symbols

From other

sources

To other destinations

Message symbols

Channel impulse

response

ch t

Week 11 7March 29, 2005

Linear block codes

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 m-

vectors 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

Week 11 8March 29, 2005

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

Proper selection of generating polynomialSequence of powers produces all 2m elementsSet is closed on multiplication

1 other powers of xmgg x x

Week 11 9March 29, 2005

Reprise: The Critical Isomorphism Shift registers with feedback

Bits in shift register are isomorphic with polynomial coefficients

Shift is isomorphic with multiplication by xModulo 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

Week 11 10March 29, 2005

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

Week 11 11March 29, 2005

EDAC Practice

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

Can detect and correct up to dfree/2 bit errors

Week 11 12March 29, 2005

Cyclic codes

Definition 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

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

Week 11 13March 29, 2005

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

Week 11 14March 29, 2005

Well-Known Block Codes

Hamming codes

The dfree is 3 Correct one, detect two bit errors A “perfect” code – all Hamming distances are dfree

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

BCH Codes

, 2 1,2 1 , 2,3,m mn k m m

Week 11 15March 29, 2005

Assignment

Review for quiz next weekBaseband signalsSources of corruptionQuantizationModulationDemodulation and complex signalsEDAC and convolutional codes

Read 7.1, 7.2, 7.3 Quiz study guide to be posted on web site