Institute for Experimental Mathematics Ellernstrasse Essen - Germany DATA COMMUNICATION 2-dimensional transmission A.J. Han Vinck May 1, 2003

University Duisburg-Essen digital communications group 2 Content we describe  orthogonal signaling  2-dimensional transmission model

University Duisburg-Essen digital communications group 3 „orthogonal“ binary signaling 2 signals S 1 (t) S 2 (t) in time T Example: Property:orthogonal energy E TT

University Duisburg-Essen digital communications group 4 Quadrature Amplitude Modulation: QAM S(t)

University Duisburg-Essen digital communications group 5 QAM receiver +/- 1/0 +/- 1/0 r(t) r(t) = S(t) + n(t) Note: sin(x)sin(x) = ½ (1 – cos (2x) ) sin(x)cos(x) = ½ sin (2x)

University Duisburg-Essen digital communications group 6 about the noise

University Duisburg-Essen digital communications group 7 about the noise Conclusion: n 1 and n 2 are Gaussian Random Variables zero mean uncorrelated (and thus statistically independent (f(x,y) =f(x)f(y) ) with variance  2.

University Duisburg-Essen digital communications group 8 Geometric presentation (1)

University Duisburg-Essen digital communications group 9 Geometric presentation (2) ML receiver: find maximum p(r|s)  min p(n)  decision regions

University Duisburg-Essen digital communications group 10 performance From Chapter 1: P(error) =

University Duisburg-Essen digital communications group 11 extension 4-QAM  2 bits 16-QAM  4 bits/s Channel 1 Channel 2

University Duisburg-Essen digital communications group 12 Geometric presentation (2) 11 22 transmitted received noise vector n The noise vector n has length |n| = (   2 2 ) ½ n has a spherically symmetric distribution! equal density

University Duisburg-Essen digital communications group 13 Geometric presentation (1) r‘ r d/2 Prob (error) = Prob(length noise vector > d/2)

University Duisburg-Essen digital communications group 14 Error probability for coded transmission The error probabiltiy is similar to the 1-dimensional situation: We have to determine the minimum d 2 Euclidean between any two codewords Example: C C‘ d 2 Euclidean =

University Duisburg-Essen digital communications group 15 Error probability The two-code word error probability is then given by:

University Duisburg-Essen digital communications group 16 modulation schemes On-offFSK 8-PSK  3 bits/s 16-QAM  4 bits/s 4-QAM  2 bits 1 bit/symbol

University Duisburg-Essen digital communications group 17 transmitted symbol energy energy: per information bit must be the same FSK

University Duisburg-Essen digital communications group 18 performance From Chapter 1: P(error) = d/2 FSK

University Duisburg-Essen digital communications group 19 Coding with same symbol speed In k symbol transmissions, we transmit k information bits. We use a rate ½ code In k symbol transmissions, we transmit k bits ML receiver:

University Duisburg-Essen digital communications group 20 Famous Ungerböck coding In k symbol transmissions transmit We can transmit 2k information bits and k redundant digits In k symbol transmissions transmit 2k digits Hence, we can use a code with rate 2/3 with the same energy per info bit!

University Duisburg-Essen digital communications group 21 modulator info encoder cici c i  {000,001,010,...111} Signal mapper 2323

University Duisburg-Essen digital communications group 22 example transmit or Parity even Parity odd Decoder: 1) first detect whether the parity is odd or even 2) do ML decoding given the parity from 1) Homework: estimate the coding gain

University Duisburg-Essen digital communications group 23 Example: Frequency Shift Keying-FSK Transmit:s(1):= s(0):= Note: FSK

University Duisburg-Essen digital communications group 24 Modulator/demodulator S(t) m m r(t) m Select largest demodulator modulator

University Duisburg-Essen digital communications group 25 Ex: Binary Phase Shift Keying-BPSK Transmit:s(1):= s(0):=  m  > or < 0? m‘

University Duisburg-Essen digital communications group 26 On-off BFSK BPSK Modulation formats

University Duisburg-Essen digital communications group On-off BPSK QPSK Eb/N0 dB Error rate PERFORMANCE