S. Mandayam/ECE Dept./Rowan University Digital Communications / Fall 2002 Shreekanth Mandayam ECE Department Rowan University Lecture 3 September 23, 2002

S. Mandayam/ECE Dept./Rowan University Plan Discussion: ECOMMS Review QUIZ Baseband Data Transmission Recall: Digital Signaling Orthogonal Vector Representation Binary and Multilevel Signals Baud Rate, Bit Rate and Bandwidth Binary Symmetric Channel Joint and Conditional Probabilities Baye’s Theorem BER Optimum Receiver Lab Project 1

S. Mandayam/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

S. Mandayam/ECE Dept./Rowan University Waveform Synthesis using Orthogonal Functions Clock Function Generator 1 Function Generator 2 Function Generator N W1W1 W2W2 WNWN  w(t)

S. Mandayam/ECE Dept./Rowan University Vector Representation of a 3-bit Binary Signal s(t) T T 0 =3T t 5V p 1 (t) t 5V t p 2 (t) t 5V p 3 (t) 0 T 2T 3T p(t) T t 5V 11 s 22 33 Digital signal Bit shape pulse Orthogonal set Signal vector

S. Mandayam/ECE Dept./Rowan University Multilevel Signaling Binary Source l-bit DAC Binary signal D symbols/second = R bits/second L=2 l Multilevel Signal D symbols/second = R/l R bits/second TbTb TsTs T0T0 t (ms)

S. Mandayam/ECE Dept./Rowan University Recall Waveforms Deterministic Stochastic Signal (desired) Noise (undesired) Probability

S. Mandayam/ECE Dept./Rowan University More Probability Joint Probability Probability of events A and B ocurring Conditional Probability Probability that A will occur, given that B has occured

S. Mandayam/ECE Dept./Rowan University Baye’s Theorem

S. Mandayam/ECE Dept./Rowan University Binary Channel Digital Modulation System Digital Demodulation System Discrete Memoryless Channel 0/1 r 0 /r 1 0/1 Noise Channel Transition Diagram m1m1 m0m0 r1r1 r0r0 m 0 /m 1

S. Mandayam/ECE Dept./Rowan University Binary Channel What can you do? Cannot change channel - limited by physics/current technology Objective Design an optimum receiver by developing a suitable decoding algorithm How? Develop a probabilistic model for the channel Assume that the probability of error is dependent on the message that was sent

S. Mandayam/ECE Dept./Rowan University Channel Transition Probability Diagram m1m1 m0m0 r1r1 r0r0 P(m 0 ) P(m 1 ) P(r 0 \m 0 ) P(r 1 \m 1 ) P(r 0 \m 1 ) P(r 1 \m 0 )

S. Mandayam/ECE Dept./Rowan University Summary