Principle of Communications, Fall 2017 Lecture 04 Reliable Communication I-Hsiang Wang ihwang@ntu.edu.tw National Taiwan University 2017/10/25,26
Channel Coding Binary Interface x ( t ) { c i } { u m } x b ( t ) ECC Symbol Pulse Up { b i } Encoder Mapper Shaper Converter Information coded discrete baseband passband Noisy bits bits sequence waveform waveform Channel Filter + y ( t ) { ˆ c i } { ˆ u m } y b ( t ) ECC Symbol Down { ˆ Sampler + b i } Decoder Demapper Converter Detection Previous lectures: Focusing on digital modulation, we can ensure that the coded bits { c i } can be reconstructed optimally (i.e., minimize avg. prob. of error) at the receiver 2
Averaged symbol probability of error is exponentially decaying with SNR P e . = exp( − c SNR ) For each symbol, P e = 10 –3 is already pretty good! However, this is not good enough … Consider a file mapped and converted into n = 250 symbols The file cannot be reconstructed if one symbol is wrong The “file” probability of error is 1 − (1 − P e ) n ≈ n P e = 250 / 1000 = 0 . 25 Pretty bad … But we cannot do much because noise is inevitable, while modulation only focus on the symbol level, not the the file level 3
Channel Coding Binary Interface x ( t ) { c i } { u m } x b ( t ) ECC Symbol Pulse Up { b i } Encoder Mapper Shaper Converter Information coded discrete baseband passband Noisy bits bits sequence waveform waveform Channel Filter + y ( t ) { ˆ c i } { ˆ u m } y b ( t ) ECC Symbol Down { ˆ Sampler + b i } Decoder Demapper Converter Detection This lecture: Reliable Communication! Introduce error correction coding, to add redundancy to the original file. � We are able to make the overall “file” probability of error arbitrarily small! Prices to pay : data rate and energy 4
Soft decision: jointly consider detection and decoding; directly work on the demodulated symbols b � [ b 1 b 2 ... b k ] c � [ c 1 c 2 ... c n ] u � [ u 1 u 2 ... u ˜ n ] ECC Digital u c Equivalent b Encoder Modulator Discrete-time We focus on n = n/ � ˜ Complex Rate: R = k / n Baseband soft decision Channel Detection ˆ V = u + Z b V first! + Decoder Hard decision: only consider decoding; directly work on the detected bit sequences b � [ b 1 b 2 ... b k ] c � [ c 1 c 2 ... c n ] u � [ u 1 u 2 ... u ˜ n ] ECC Digital u c Equivalent b Encoder Modulator Discrete-time n = n/ � ˜ Complex Baseband Channel V ECC d Demod. + ˆ V = u + Z b Decoder Detection 5
Outline • Prelude: repetition coding • Energy-e ffi cient reliable communication: orthogonal code • Rate-e ffi cient reliable communication: linear block code • Convolutional code 6
Part I. Prelude: Repetition Coding Repetition code, Rate and Energy e ffi ciency 7
Repetition: a simple way to enhance reliability • Idea: repeat each bit N times � data rate R = 1/ N . original bit seq. b 3 b 5 b 1 b 2 b 4 Many ways for repetition coded bit seq. 1 b 1 b 1 b 2 b 2 b 3 b 3 b 4 b 4 b 5 b 5 coded bit seq. 2 b 1 b 2 b 3 b 4 b 5 b 1 b 2 b 3 b 4 b 5 repeat N times • � � We focus on the architecture below: b � +1 ∼ b 2 � b 1 ∼ b � b 1 ∼ b � b 1 ∼ b � c = b � [ b 1 b 2 ... b k ] c � [ c 1 c 2 ... c n ] u � [ u 1 u 2 ... u ˜ n ] Digital u c Repetition Equivalent b Modulator Discrete-time n = kN n = n/ � ˜ Complex � : # of bits in a symbol Baseband Channel Detection ˆ V = u + Z b V + Decoder 8
repeat N times � � b � +1 ∼ b 2 � b 1 ∼ b � b 1 ∼ b � b 1 ∼ b � c = mod mod mod mod � � u 1 u 2 u N Equivalent vector symbol ∈ C N u � � � u 1 u 2 · · · u N Since the noises are i.i.d., it su ffi ces to use the N -dim. demodulated V = u + Z to optimally decode b 1 ∼ b � 9
BPSK + repetition coding V = u + Z ∈ C N Z 1 , ...Z N i.i.d. Equivalent channel model: ∼ CN (0 , N 0 ) Equivalent constellation set: u ∈ { a 0 , a 1 } � � � � d d · · · d d d · · · d a 1 = + a 0 = − Performance analysis: Repetition effectively � � �� �� � � P ( N ) ∥ a 1 − a 0 ∥ N · 4 d 2 N 2 d 2 = Q = Q = Q e 2 √ increase SNR by N -fold! 2 N 0 N 0 N 0 / 2 � . � √ = exp( − N SNR ) = Q N 2 SNR = d 2 SNR � average energy per uncoded symbol total noise variance per symbol N 0 10
Rate and energy efficiency Rate: R = 1 /N → 0 as N → ∞ Energy per bit: E b = Nd 2 → ∞ as N → ∞ Achieving arbitrarily small prob. of error at the price of zero rate and infinite energy per bit Question: can we resolve the issue with more general constellation sets? 11
General modulation + repetition coding V = u + Z ∈ C N Z 1 , ...Z N i.i.d. Equivalent channel model: ∼ CN (0 , N 0 ) Equivalent constellation set: u ∈ { a 1 , ..., a M } M = 2 � Rate: R = � /N Energy per bit: E b ℓ SNR = SNR N 0 = N R → ∞ as N → ∞ → 0 as N → ∞ Probability of error (take M -ary PAM as an example) : �� � �� � P ( N ) 6 = 2(1 − 2 − ℓ )Q N = 2(1 − 2 − NR )Q 4 NR − 1 6 SNR 4 ℓ − 1 SNR N e lim N →∞ P ( N ) ⇒ lim N →∞ 4 NR − 1 = 0 ⇐ = 0 e N it is necessary that lim N →∞ R = 0 12
Why repetition coding is not very good • Repetition coding: high reliability at the price of asymptotically zero rate and infinite energy per bit • Repetition is too naive and does not utilize the available degrees of freedom in the N -dimensional space e ffi ciently • Is it possible to design better coding schemes with the following? ‣ Vanishing probability of error ‣ Positive rate ‣ Finite energy per bit YES! 13
Part II. Energy-Efficient Reliable Communication Orthogonal code, Optimal energy e ffi ciency 14
Orthogonal coding b � [ b 1 b 2 ... b k ] u � [ u 1 u 2 ... u ˜ n ] Encoder + Equivalent b u Modulation Discrete-time here we jointly consider Complex coding and modulation Baseband Channel Detection ˆ V = u + Z b V + Decoder • With N dimensions ( N time slots), use N equal-norm orthogonal vectors to encode log 2 N bits • Since the noises are i.i.d. circularly symmetric complex Gaussian, we can WLOG assume that these N vectors are simply scaled standard unit vectors: { d e i | i = 1 , ..., N } , e i ( j ) = { i = j } 15
Example: N = 8 info. bits symbol vector 000 [ d 0 0 0 0 0 0 0] 001 [0 d 0 0 0 0 0 0] Equivalent to encoding messages 010 [0 0 d 0 0 0 0 0] using the location of a pulse 011 [0 0 0 d 0 0 0 0] 100 [0 0 0 0 d 0 0 0] � Pulse Position Modulation (PPM) 101 [0 0 0 0 0 d 0 0] 110 [0 0 0 0 0 0 d 0] 111 [0 0 0 0 0 0 0 d ] 16
Performance analysis of orthogonal coding V = u + Z ∈ C N Z 1 , ...Z N i.i.d. Equivalent channel model: ∼ CN (0 , N 0 ) Equivalent constellation set: u ∈ { d e i | i = 1 , ..., N } Rate: Energy per bit: E b = d 2 / log 2 N R = log 2 N/N → 0 as N → ∞ Finite energy per bit su ffi ces! E b > (2 ln 2) N 0 Probability of error: √ d min = 2 d �� � �� �� � � d 2 P ( N ) d 2 log 2 N E b ≤ ( N − 1) Q = ( N − 1) Q ≤ N Q min e 2 N 0 N 0 N 0 � � �� → 0 as N → ∞ as long as E b ≤ 1 E b N 0 > 2 ln 2 2 exp − ln N (2 ln 2) N 0 − 1 17
Minimum energy per bit • Does orthogonal coding achieve the minimum E b / N 0 ? • Let us use Shannon’s capacity formula to derive the minimum E b / N 0 of all possible coding+modulation schemes: ‣ For the additive Gaussian noise channel with energy per channel use P , the best achievable rate follows (bits per channel use) R < C � log 2 (1 + P N 0 ) ‣ Energy per bit is hence ⇒ R < log 2 (1 + R E b E b = P/R = N 0 ) ‣ N 0 > E ∗ b ( R ) � 2 R − 1 The minimum energy per bit when the rate is R can be found: E b N 0 R ‣ Taking infimum over all R , we see: E ∗ E ∗ 2 R − 1 b ( R ) b � inf = lim = ln 2 N 0 N 0 R R> 0 R ↓ 0 • In fact, orthogonal code can achieve any E b / N 0 > ln 2 ! ‣ but union bound fails; new techniques required (see Gallager Ch. 8.5.3 for more details) 18
Part III. Rate-Efficient Reliable Communication Linear block code, Existence of rate-e ffi cient codes with vanishing error probability 19
Linear block code + BPSK modulation • Orthogonal code achieves vanishing probability of error with zero rate but finite energy per bit (energy-efficient reliable communication) . • Is it possible to achieve vanishing probability of error with positive rate and finite energy per bit (rate-efficient reliable communication) ? • We focus on the following architecture: linear block code + BPSK modulation ‣ It turns out this simple architecture can achieve rate-e ffi cient reliable communication! b � [ b 1 b 2 ... b k ] c � [ c 1 c 2 ... c n ] u � [ u 1 u 2 ... u ˜ n ] ECC Digital Linear Block Binary PAM u c Equivalent b Encoder Modulator Code Modulator Discrete-time n = n/ � ˜ R = k/n � = 1 Complex Baseband Channel Detection ˆ V = u + Z b V + Decoder 20
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