1. March 29, 2005Week 11 1 EE521 Analog and Digital Communications James K. Beard, Ph. D. jkbeard@temple.edu Tuesday, March 29, 2005 http://astro.temple.edu/~jkbeard/

2. Week 112 March 29, 2005 Attendance

3. Week 113 March 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

4. Week 114 March 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)

5. Week 115 March 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

6. Week 116 March 29, 2005 Sklar Chapters 6 and 7 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

7. Week 117 March 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

8. Week 118 March 29, 2005 Reprise: 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

9. Week 119 March 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 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

10. Week 1110 March 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 d free >2  Can be easily implemented and decoded  Can be decoded to the signal that codes to a sequence with the nearest Hamming distance

11. Week 1111 March 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 d free /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 d free /2 bit errors

12. Week 1112 March 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 (x n +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 x n +1  Systematic codes possible  Decoding is done by dividing code by generating polynomial

13. Week 1113 March 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

14. Week 1114 March 29, 2005 Well-Known Block Codes Hamming codes  The d free is 3  Correct one, detect two bit errors  A “perfect” code – all Hamming distances are d free Extended Golay code  Add a parity bit to the perfect (24,12 ) Golay code  Increases d free from 7 to 8  Produces a rate ½ code BCH Codes

15. Week 1115 March 29, 2005 Assignment 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