1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2010 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring10/ecomms/ Lecture 10a March 30, 2010

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Plan Digital Baseband Communications Source Encoding Huffman code Error Control Coding Hamming Distance Error Detection Coding Parity Check Code Error Correction Coding Hamming Code Pulse Amplitude Modulation Sampling Quantization TDM Standards T1 (DS-1) Line North American TDMA Hierarchy

3. S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

4. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Recall: Huffman Coding 2-Step Process Reduction List symbols in descending order of probability Reduce the two least probable symbols into one symbol equal to their combined probability Reorder in descending order of probability at each stage Repeat until only two symbols remain Splitting Assign 0 and 1 to the final two symbols remaining and work backwards Expand code at each split by appending a 0 or 1 to each code word Example m(j)ABCDEFGH P(j)0.10.180.40.050.060.10.070.04

5. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Huffman Code C: 0.4 B: 0.18 A: 0.1 F: 0.1 G: 0.07 E: 0.06 D: 0.05 H: 0.04 0.4 0.18 0.1 0.09 0.07 0.06 0.4 0.18 0.13 0.1 0.09 0.4 0.19 0.18 0.13 0.1 0.4 0.23 0.19 0.18 0.4 0.37 0.23 0.6 0.4

6. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Digital Communications Transceiver Anti- aliasing Filter SamplingQuantization Data Encryption Encoder Source Encoder Error Control Encoder Channel/ Line Encoder Modulator MUX Audio Amp Source Decoder Data Encryption Decoder Error Control Decoder Equalization / Decision Circuits Demod- ulator DEMUX Reconstruction/ DAC ADC CODEC MODEM Analog o/p Multiple access channel Analog i/p

7. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Error Control Coding Hamming Distance The number of locations (bits) at which two code words differ Theorem 1 A code with a Hamming distance of d >= t+1 can detect t errors in the received code word Theorem 2 A code with a Hamming distance of 2t+1 <= d <= 2t+2 can detect and correct t errors in the received code word Error Detection (ARQ Technique) Error Correction (FEC Technique)

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Error Control Codes Principle Block Codes (memoryless) Convolutional Codes (with memory) Will not discuss! Block Coder k information bits n encoded bits Information bits Parity bits k n-k n-bit codeword (n, k) systematic block code Add Redundancy!!

9. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Parity Check Codes P I4I4 I1I1 I2I2 I3I3 I5I5 I6I6 I7I7 P is set such that the total no. of bits in the code word is even or odd

10. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Remediation for Detected Errors: ARQ 1 Rx 1 Tx 2 3 3 23 ACK NACK Error detected!!!

11. S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Single error detection and correction code Hamming distance, d = 3 Fits into a general category of coding techniques called BCH codes Employs a Code Generator Matrix Syndrome Decoding Technique I1I1 P1P1 P2P2 P3P3 I2I2 I3I3 I4I4

12. S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Code Generator s1s1 s2s2 s3s3 I1I1 I2I2 I3I3 I4I4 C4C4 C1C1 C2C2 C3C3 C5C5 C6C6 C7C7 Encoding Info bits Code word Decoding Parity Check R4R4 R1R1 R2R2 R3R3 R5R5 R6R6 R7R7 Received code word Error position indicator I1I1 P1P1 P2P2 P3P3 I2I2 I3I3 I4I4 = Parity bits

13. S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Code Generator Matrix

14. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Encoder P1P1 P2P2 P3P3 I1I1 I2I2 I3I3 I4I4 + + + P 3 = I 4  I 2  I 1 P 2 = I 4  I 3  I 1 P 1 = I 4  I 3  I 2 Info bits Parity bits

15. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Syndrome Decoding Parity Check Matrix

16. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Decoder + R4R4 R1R1 R2R2 R3R3 R5R5 R6R6 R7R7 s1s1 s2s2 s3s3 + + Received code word Error Position Indicator s 3 = R 7  R 5  R 4  R 3 s 2 = R 7  R 6  R 4  R 2 s 1 = R 7  R 6  R 5  R 1

17. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Decoder Error Position, e Error position indicator (syndrome) s No error

18. S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

19. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Digital Communications Transceiver Anti- aliasing Filter Data Encryption Encoder Source Encoder Error Control Encoder Channel/ Line Encoder Modulator MUX Audio Amp Source Decoder Data Encryption Decoder Error Control Decoder Equalization / Decision Circuits Demod- ulator DEMUX Reconstruction/ DAC ADC CODEC MODEM Analog o/p Multiple access channel Analog i/p Sampling Quantization

20. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Pulse Code Modulation (PCM) Standard for all digital communications Sample & Hold Quantize Encode Analog Signal Digital Bitstream PCM PAM

21. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Pulse Amplitude Modulation: Flat-top Sampling -0.500.511.522.533.54 -6 -4 -2 0 2 4 6 8 time Analog PAM Matlab Demo: pam.m

22. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Quantization UniformNon-Uniform Mid-rise Mid-tread  -Law A-Law Mid-rise Mid-tread Mid-rise Mid-tread Matlab Demos: quantization.m pcm.m Matlab Demos: companding.m pcm_mucompand.m pcm_mucompand2.m pcm_acompand.m pcm_acompand2.m

23. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Digital Communications Transceiver Anti- aliasing Filter Data Encryption Encoder Source Encoder Error Control Encoder Channel/ Line Encoder Modulator MUX Audio Amp Source Decoder Data Encryption Decoder Error Control Decoder Equalization / Decision Circuits Demod- ulator DEMUX Reconstruction/ DAC ADC CODEC MODEM Analog o/p Multiple access channel Analog i/p Sampling Quantization

24. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Time Division Multiplexing Message 1 Message 2 Message N Quantize & Encode Channel Message 1 Message 2 Message N Decode LPF

25. S. Mandayam/ ECOMMS/ECE Dept./Rowan University T1-Line TDM Format F Channel 1 18765432 Channel 2 18765432 Channel 24 18765432 1 Frame = (24 Channels + F) = 125  s 193 bits Frame Sync Bit Even Frames:0 0 1 1 1 0 Odd Frames:1 0 1 0 1 0 Every Sixth Frame LSB (8th bit) is a Signaling bit Bipolar R-Z Line Encoding

26. S. Mandayam/ ECOMMS/ECE Dept./Rowan University North American TDMA Hierarchy 1 st M U X 1 24 64 kbs 1 VF DS-0 1 2 DS-4 274.176 Mbs 4032 VF 5 th M U X 1 6 DS-3 44.736 Mbs 672 VF 4 th M U X 1 7 DS-2 6.312 Mbs 96 VF 3 rd M U X 1 4 DS-1 1.44 Mbs 24 VF 2 nd M U X DS-5 560.160 Mbs 8064 VF Figure 3-40, p. 207 in Couch Table 3-8, p. 208 in Couch Table 3-9, p. 209 in Couch

27. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary