March 22, 2005Week 10 1 EE521 Analog and Digital Communications James K. Beard, Ph. D. Tuesday, March 22,

Week 102 March 22, 2005 Attendance

Week 103 March 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 104 March 22, 2005 Today’s Topics Quiz 1 Gray code MPSK Waveform Coding, Part 1  Waveform coding and structured sequences  Types of error control  Structured sequences Discussion (as time permits)

Week 105 March 22, 2005 Question 3 Computations

Week 106 March 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 107 March 22, 2005 Gray Code MPSK Defining Characteristic The Hamming distance between adjacent codes is 1 Result: less opportunity for bit errors gives lower BER See Sklar pp

Week 108 March 22, 2005 Sklar Chapter 6 Information source Format Source encode Encrypt Channel encode Multi- plex Pulse modulate Bandpass modulate Freq- uency spread Multiple access X M I T Format Source decode Decrypt Channel decode Demul- tiplex Detect Demod- ulate & Sample Freq- uency despread Multiple access RCVRCV Channel Information sink Bit stream Synch- ronization Digital baseband waveform Digital bandpass waveform Digital output Digital input Optional Essential Legend: Message symbols Channel symbols From other sources To other destinations Message symbols Channel impulse response

Week 109 March 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 1010 March 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 1011 March 22, 2005 M-ary Signaling MPSK or MFSK Number of waveforms is M=2 k Advantages of each  Signals can be orthogonal with MFSK  MPSK uses one frequency channel Additional requirements  MFSK requires more bandwidth  MPSK requires more E b /N 0

Week 1012 March 22, 2005 The Orthogonality Condition Normalized orthogonality Orthogonality can be  Time – signals are nonzero at different times  Functional – orthogonal functions  Codes – orthogonal codes  In frequency – see orthogonal functions

Week 1013 March 22, 2005 Antipodal and Orthogonal signals Antipodal  Two signals  One the negative of the other Orthogonal  M signals  A matched filter for any one produces a near- zero result with any other as input  Orthogonality can be in time, frequency, or code

Week 1014 March 22, 2005 Walsh-Hadamard Sequences A simple way to formulate orthogonal code sequences Based on recursive augmentation of Walsh-Hadamard matrices

Week 1015 March 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 2 k Cross-correlation properties  Excellent for zero lag  Poor for other lags

Week 1016 March 22, 2005 Bi-Orthogonal Codes Made up of rows or columns from half a Hadamard matrix Codes of order M/2=2 k-1 appended to their antipodal opposite Slightly improved symbol error performance Half the bandwidth of orthogonal codes

Week 1017 March 22, 2005 Bi-Orthogonality

Week 1018 March 22, 2005 Transformational Codes Also called Simplex codes Generated from orthogonal sets First digit of each code is deleted Minimum energy code Characterized by

Week 1019 March 22, 2005 Summary of Codes For large values of M  All three codes have similar BER performance  Biorthogonal codes have bandwidth advantage Bandwidth requirements  Grow exponentially with M  True of all three codes

Week 1020 March 22, 2005 Primitive Error Control Older schemes were based on terminal connectivity  Simplex – one-way communication  Half duplex – first one direction then the other  Full duplex – both directions simultaneously Duplex allows Acknowledgement/negative acknowledgement (ACK/NAK) handshake

Week 1021 March 22, 2005 Structured Sequences Three kinds  Block codes  Convolutional codes (later)  Turbo codes (next semester) Increasing M improves symbol error performance and bandwidth requrements

Week 1022 March 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 1023 March 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 bits  Generally, redundancy bits This is an (n,k) code  Redundancy is (n-k)/k  Code rate is k/n

Week 1024 March 22, 2005 Parity codes Parity check codes  Single parity bit can detect even number of errors  Useful in triggering NAK with low BER Rectangular codes  Double parity, second on p th bit of k words  Parity on bit p and word q allows correction of a single error

Week 1025 March 22, 2005 Parameters in the Trade Space Error performance Bandwidth vs. data rate Power Coding gain as defined by decrease in E b /N 0 required to obtain a specified BER when coding is used

Week 1026 March 22, 2005 Relationship Between Some Basic Trade Parameters

Week 1027 March 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 1028 March 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 1029 March 22, 2005 Desirable PRN Code Properties Maximal length – 2 m codes before repeating Balance – equal number of (+1) and (-1) pulses Closed on circular shifts Contain shorter subsequences Good autocorrelation properties

Week 1030 March 22, 2005 Galois Field Vector Extensions of Order 2 m Polynomials modulo 2 of order m-1 Arithmetic is done modulo a generating polynomial of the form Proper selection of generating polynomial  Sequence of powers produces all 2 m elements  Set is closed on multiplication

Week 1031 March 22, 2005 An Important Isomorphism 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