March 22, 2005 Week 10 1

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

jkbeard@temple.edu

Tuesday, March 22, 2005

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

Week 10 2March 22, 2005

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Week 10 3March 22, 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 March 22 Final Exam Scheduled

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

Week 10 4March 22, 2005

Today’s Topics

Quiz 1 Gray code MPSK Waveform Coding, Part 1

Waveform coding and structured sequencesTypes of error controlStructured sequences

Discussion (as time permits)

Week 10 5March 22, 2005

Question 3 Computations

Name BW W DC, %

Min fS LPF BW

k DPD Min fS

LPF Stop

Deci-mation

Booth 25 65 5 130 12.5 153 130.29 97.79 2.005 Dessino 35 85 6 170 17.5 116 171.67 129.17 2.020 Mountney 45 105 7 210 22.5 94 211.64 159.14 2.016 Mungo 55 125 8 250 27.5 79 251.57 189.07 2.013 Salaria 65 145 9 290 32.5 68 291.97 219.47 2.014 Kamanou 75 165 10 330 37.5 60 330.58 248.08 2.004 Yesminl 85 185 11 370 42.5 53 373.83 281.33 2.021

Week 10 6March 22, 2005

Gray Codes

Sometimes called reflected codes Defining property: only one bit changes

between sequential codes Conversion

Binary codes to Gray Work from LSB up XOR of bits j and j+1 to get bit j of Gray code Bit past MSB of binary code is 0

Gray to binary Work from MSB down XOR bits j+1 of binary code and bit j of Gray code to get bit j

of binary code Bit past MSB of binary code is 0

Week 10 7March 22, 2005

Gray Code MPSK

0,0,0

0,0,1 0,1,1

0,1,0

1,1,0

1,1,1 1,0,1

1,0,0

Defining Characteristic

The Hamming distance between adjacent codes is 1

Result: less opportunity for bit errors gives lower BER

See Sklar 4.9.4 pp. 234-235

Week 10 8March 22, 2005

Sklar Chapter 6

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 10 9March 22, 2005

Channel Coding Topic Areas

Overview: Waveform Coding and Structured Sequences

Modulation M-ary signaling Antipodal and orthogonal pulses Trellis-coded modulation

Codes as structured sequences Block codes Convolutional codes Turbo codes

Week 10 10March 22, 2005

Waveform Coding and Structured Sequences Channel coding

Structured sequences (EDAC) Waveform design

Structured sequences Coding digital sequences for transmission Increases the number of bits and provides EDAC

capability Waveform design

How to code a pulse for RF use A design point that selects containment in time and

frequency regions

Week 10 11March 22, 2005

M-ary Signaling

MPSK or MFSK Number of waveforms is M=2k

Advantages of eachSignals can be orthogonal with MFSKMPSK uses one frequency channel

Additional requirementsMFSK requires more bandwidthMPSK requires more Eb/N0

Week 10 12March 22, 2005

The Orthogonality Condition

Normalized orthogonality

Orthogonality can beTime – signals are nonzero at different timesFunctional – orthogonal functionsCodes – orthogonal codes In frequency – see orthogonal functions

0

1,1

0,

T

ij i j

i jz s t s t dt

E i j

Week 10 13March 22, 2005

Antipodal and Orthogonal signals Antipodal

Two signalsOne the negative of the other

OrthogonalM signalsA matched filter for any one produces a near-

zero result with any other as inputOrthogonality can be in time, frequency, or

code

Week 10 14March 22, 2005

Walsh-Hadamard Sequences

A simple way to formulate orthogonal code sequences

Based on recursive augmentation of Walsh-Hadamard matrices

1

1

1 1

1 1

i ii

i i

H

H HH

H H

Week 10 15March 22, 2005

Properties of Walsh-Hadamard Sequences Matrices are symmetrical Matrices are self-orthogonal Each matrix has rows or columns are a

sequence of orthogonal sequences of length 2k

Cross-correlation propertiesExcellent for zero lagPoor for other lags

Week 10 16March 22, 2005

Bi-Orthogonal Codes

Made up of rows or columns from half a Hadamard matrix

Codes of order M/2=2k-1 appended to their antipodal opposite

Slightly improved symbol error performance

Half the bandwidth of orthogonal codes

Week 10 17March 22, 2005

Bi-Orthogonality

,

1, ,2

0, ,2

ij

i j

Mz i j i j

Mi j i j

Week 10 18March 22, 2005

Transformational Codes

Also called Simplex codes Generated from orthogonal sets First digit of each code is deleted Minimum energy code Characterized by

,

1,

1ij

i jz

i jM

Week 10 19March 22, 2005

Summary of Codes

For large values of MAll three codes have similar BER performanceBiorthogonal codes have bandwidth

advantage Bandwidth requirements

Grow exponentially with MTrue of all three codes

Week 10 20March 22, 2005

Primitive Error Control

Older schemes were based on terminal connectivitySimplex – one-way communicationHalf duplex – first one direction then the otherFull duplex – both directions simultaneously

Duplex allows Acknowledgement/negative acknowledgement (ACK/NAK) handshake

Week 10 21March 22, 2005

Structured Sequences

Three kindsBlock codesConvolutional codes (later)Turbo codes (next semester)

Increasing M improves symbol error performance and bandwidth requrements

Week 10 22March 22, 2005

Channel Models

Discrete memoryless channel (DMC) Discrete input and output alphabets BER depends only on signal at current epoch BER equations are as studied before

Gaussian channel DMC with binary input, continuous output Gaussian noise is added to symbols

Binary symmetric channel A DMC with a binary alphabet: only 1, 0 A Gaussian channel with hard decoding on output

Week 10 23March 22, 2005

Code Rate and Redundancy

Begin with k data bits per symbol Add EDAC bits to form a symbol of n bits

Parity bits or check bitsGenerally, redundancy bits

This is an (n,k) codeRedundancy is (n-k)/kCode rate is k/n

Week 10 24March 22, 2005

Parity codes

Parity check codesSingle parity bit can detect even number of

errorsUseful in triggering NAK with low BER

Rectangular codesDouble parity, second on pth bit of k wordsParity on bit p and word q allows correction of

a single error

Week 10 25March 22, 2005

Parameters in the Trade Space

Error performance Bandwidth vs. data rate Power Coding gain as defined by decrease in

Eb/N0 required to obtain a specified BER when coding is used

Week 10 26March 22, 2005

Relationship Between Some Basic Trade Parameters

0

0

b

b

SignalPowerE

BitRate

NoisePower N Bandwidth

E SignalPower Bandwidth

N NoisePower BitRate

Week 10 27March 22, 2005

Linear Block Codes

These are (n,k) codes based on polynomials in binary arithmetic

Polynomials are added and subtracted Arithmetic is modulo 2 Polynomial coefficients considered as vectors Sets closed on addition are called Vector

subspaces

Week 10 28March 22, 2005

Maximal-Length Sequences

Bit sequence is essentially random Pseudo-random noise (PRN) code Codes Construction

Shift registers with feedback Recursive modulo-2 polynomial arithmetic

PRN codes are then selected for good cross-correlation properties

Week 10 29March 22, 2005

Desirable PRN Code Properties

Maximal length – 2m codes before repeating

Balance – equal number of (+1) and (-1) pulses

Closed on circular shifts Contain shorter subsequences Good autocorrelation properties

Week 10 30March 22, 2005

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 10 31March 22, 2005

An Important Isomorphism

Shift registers with feedbackBits in shift register are isomorphic with

polynomial coefficientsShift 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