ECEN5533 Modern Commo Theory Dr. George Scheets. Lesson #34 ECEN5533 Modern Commo Theory Dr. George Scheets Lesson #34 6 April 2016 Read 6.4 – 6.5 Problems: 9.2, 4, & 7 Design #2 due Friday, 8 April Corrected tests due Friday, 15 April

ECEN5533 Modern Commo Theory Dr. George Scheets. Lesson #35 ECEN5533 Modern Commo Theory Dr. George Scheets Lesson #35 8 April 2016 Skim 6.6 – 6.9 Problems: 5.8, 6.2, & 6.3 Design #2 due Friday, 8 April Corrected tests due Friday, 15 April

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No Coding Transmitter Receiver Modulator 11 Data bits in 11 BPSK Data bits out Receiver Matched Filter Detector Data bits out P(Data BE) = 9.730*10-6 P(Data Word Error) = 107.4*10-6 BPSK 11 Data bits in

Detects and/or corrects FEC Coding Transmitter FEC Coder Adds extra parity bits. 11 Data bits in 15 Code bits out From Receiver Matched Filter Detector FEC Decoder Removes parity bits. Detects and/or corrects errors. 15 Code bits in P(Code BE) = 127.6*10-6 11 Data bits out P(Data Word Error) ≈ 1.707*10-6 P(Data BE) ≈ .2278*10-6 (was 9.720*10-6) Equation 6.46

FEC Example Uncoded (15,11) Code Bit Rate 4800 bps 6545 bps P(BE) out of 9.720(10-6) 127.6(10-6) Matched Filter detector P(11 bit Data 107.0(10-6) ≈ 1.708(10-6) Word Error) P(data BE) 9.720(10-6) ≈ 0.2278(10-6) (using Eqn 6.46)

FEC Examples Coding can make things worse MFD P(BE) gets worse as bit rate increases h(t) = 1; 0 < t < T, for an integrator H(f) = sinc with a phase shift Main lobe BW = 1/T Hz Integration time becomes shorter H(f) becomes wider, less of a low pass filter

FEC Examples In the limit, as bit interval T approaches zero # of independent samples approaches 1 MFD P(BE) approaches SSD P(BE) Suppose you have a system where P(BE) = 0.02 for MFD at bit rate R (no FEC) P(BE) = 0.03 for MFD at bit rate 2R (2:1 FEC) P(BE) = 0.04 for MFD at bit rate 3R (3:1 FEC)

Matched Filter Detector & No coding: Block Diagram Source Channel Coder Symbol Detector: Matched Filter Channel Suppose P(Data Bit Error) = .02

Suppose P(code bit error) = .03 MFD 2:1 FEC 2R code bps Source Coder: Input = 1 bit. Output = Input + Parity bit. Source Channel Coder R application bps R app. bps 2R code bps Source Decoder: Looks at blocks of 2 bits. Outputs 1 bit. Symbol Detector: Matched Filter Channel Suppose P(code bit error) = .03

Example) MFD 2 bit code words Suppose you transmit each bit twice, smaller bit width will cause P(Code Bit Error) to increase to, say 0.03 Legal Transmitted code words; 00, 11 Possible received code words 00, 11 (appears legal, 0 or 2 bits in error) 01, 10 (clearly illegal, 1 bit in error) P(No code bits in error) = .97*.97 = .9409 P(One code bit in error) = 2*.97*.03 = .0582 P(Both code bits in error) = .03*.03 = .0009 Decoder takes 2 code bits at a time and outputs 1 data bit If illegal code word received, it can guess 0 or 1. 94.09% + 5.82%(1/2) = 97% of time correct bit output .09% + 5.82%(1/2) = 3% of time the incorrect bit is output FEC makes it worse: 3% data bit error vs 2% No Coding

MFD 2:1 FEC 2R code bps Source Coder: Input = 1 bit. Source Output = Input + Parity bit. Source Channel Coder R application bps R app. bps 2R code bps Source Decoder: Looks at blocks of 2 bits. Outputs 1 bit. Symbol Detector: Matched Filter Channel P(data bit error) = .03 P(code bit error) = .03

Matched Filter Detector & No coding: Block Diagram Source Channel Coder Symbol Detector: Matched Filter Channel P(Data Bit Error) = .02 FEC coding can make things worse.

Typical FEC Performance Coded Plot changes as type of symbol, type of detector, and type of FEC coder change. P(BE) Uncoded Plot changes as type of symbol, and type of detector change. Last example is operating here. There generally always is a cross-over point. The max possible P(BE) = 1/2. Eb/No

MFD 2:1 FEC 2R code bps Source Coder: Input = 1 bit. Source Output = Input + Parity bit. Source Channel Coder R application bps R app. bps 2R code bps Source Decoder: Looks at blocks of 2 bits. Outputs 1 bit. Symbol Detector: Matched Filter Channel P(app. bit error) = .03 P(code bit error) = .03

Suppose P(code bit error) = .04 MFD 3:1 FEC 3R code bps Source Coder: Input = 1 bit. Output = Input + two parity bits. Source Channel Coder R application bps R app. bps 3R code bps Source Decoder: Looks at blocks of 3 bits. Outputs 1 bit. Symbol Detector: Matched Filter Channel Suppose P(code bit error) = .04

Example) MFD 3 bit code words Transmit each bit thrice, P(Bit Error) again increases to, say 0.04, due to further increase in the bit rate. Legal Transmitted code words; 000, 111 Possible received code words 000, 111 (appears legal, 0 or 3 bits in error) 001, 010, 100 (clearly illegal, 1 or 2 code bits in error) 011, 101, 110 (clearly illegal, 1 or 2 code bits in error) P(No code bits in error) = .96*.96*.96 = .884736 P(One code bit in error) = 3*.962*.04 = .110592 P(Two code bits in error) = 3*.96*.042 = .004608 P(Three code bits in error) = .04*.04*.04 = .000064 Decoder takes 3 bits at a time & outputs 1 bit. Majority Rules. 88.4736% + 11.0592% = 99.5328% of time correct bit is output .0064% + .4608% = 0.4672% of time incorrect bit is output FEC makes Data BER better (.5% vs 2%) @ thrice the bit rate

MFD 3:1 FEC 3R code bps Source Coder: Input = 1 bit. Source Output = Input + two parity bits. Source Channel Coder R application bps R app. bps 3R code bps Source Decoder: Looks at blocks of 3 bits. Outputs 1 bit. Symbol Detector: Matched Filter Channel P(app. bit error) = .005 P(code bit error) = .04

Matched Filter Detector & No coding: Block Diagram Source Channel Coder Symbol Detector: Matched Filter Channel P(Data Bit Error) = .02

Trellis Decoding source: 1991 IEEE Communications Magazine

Coding Gain P(BE) Coded Uncoded Eb/No

Coding Gain P(BE) Coded Uncoded Target Data P(BE) Eb/No Required Eb/No

Coding Gain with using coder P(BE) Coded Uncoded Target Data P(BE) Eb/No Eb/No you can get by with using coder

Coding Gain Link Analysis using FEC: 1) Increase Bit Rate R 2) Include Coding Gain 3) Use Uncoded P(BE) equation. P(BE) Coded Uncoded Target Data P(BE) Coding Gain Eb/No

Rate 1/2 Turbo Coder uk (data) & vk (parity bits) are transmitted to far side vk = v1k 1/2 of the time & v2k other half Source: Figure 8.26 from Sklar's Digital Communications

Rate 1/2 Turbo Decoder ← Matched Filter ← Matched Filter xk (corrupted data) & yk (corrupted parity bits) yk = y1k 1/2 of the time & y2k other half Source: Figure 8.27 from Sklar's Digital Communications

Rate 1/2 Turbo Coding Performance P(data bit error) = 0.00001 when Eb/No = 0.7 dB & 18 reps P(bit error) = 0.06264 for BPSK when Eb/No = 0.7 dB Source: Figure 8.28 from Sklar's Digital Communications