1. 1 INF244 Textbook: Lin and Costello Lectures (Tu+Th 12.15-14) covering roughly Chapter 1;Chapters 9-19? Weekly exercises: For your convenience Mandatory problem: Programming project (counts towards final mark) Oral exam some time in late November or early December

2. 2 Areas: Data transmission Modems, Mobile phones, satellite connections, deep space communication, microwave links, power line communication, optical fibers,… Data storage CD’s, DVD’s, hard disks, RAM,… Others: E.g. “Personnummer”, ISBN, credit cards and bank account numbers Advantages: Increased reliability Reduced effect for a given reliability level Increased data rate/storage density Coding theory : Applications Signal/Støy Feil

3. 3 Inf 244 focuses on this part! Sender Channel Receiver Channel model A sender transmits information across a noisy channel to a receiver ECC encoder ECC decoder Source encoder Source decoder Corrupted by noise!

4. 4 Code types Block codes ECC encoder ECC decoder Channel u:ku:k v:n, v = f(u) r = v + n u’:k Convolutional codes ECC encoder ECC decoder Channel ui:kui:k v i :n, v i = f(u i, u i-1,..., u i-m ) r i = v i + n i ui’:kui’:k

5. 5 Block codes vs. Convolutional codes Any code is limited by Code rate Error probability in a given channel Decoding complexity Block codes (classical approach): Algebraic structure Convolutional codes: Emphasis on decoding complexity Which is best? A matter of discussion, but: A block code is a convolutional code with m=0 A convolutional code with finite input length is a block code

6. 6 Error Detection and Correction Error-Detecting Codes Automatic Repeat Request (ARQ) If frame error: ask for retransmission Error-Correcting Codes If frame error: Instead of asking for retransmission, try to estimate what was originally sent

7. 7 Coding Theory: Error Detection SenderReceiver 1101 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 SenderReceiver 1101 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0

8. 8 Coding Theory: Error Detection Advantages Simple! Extensions/generalizations: Schemes that detect more than one error? Disadvantages and conditions Requires return channel Extra delay Return channel may not exist

9. 9 0 0g0g Error correction SenderReceiver 1 1 0 1 a b c d a:1 d:1 b:1c:0 Claim: Can correct one error or detect up to 2 errors 1 1e1e 0 0f0f

10. 10 a b c d e f g 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 1 ( ) Parity check matrix H Error correction : Hamming codes a d bc f g e 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 ( ) Generator matrix G: G·H T =0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 KODE

11. 11 Error Correction SenderReceiver Coding Decoding Rate: k/n Distance 2t+1 between codewords: Can correct t errors Problems: How to find good codes (high rate, large distance)? Construction; bounds for what can be achieved How to find efficient decoding algorithms? Exploit the mathematical structure of codes k n k

12. 12 Channel model, more detail Sometimes it is convenient to consider modulation as part of the coding process: ECC encoder ECC decoder ChannelDemodulator Modulator Analog channel Digital processes

13. 13 Modulation and coding Binary coding: 1 corresponds to s 1 (t)= K  cos (2  f 0 t), 0  t  T 0 corresponds to s 0 (t)= K  cos (2  f 0 t+  ) = -s 1 (t), 0  t  T Where T = duration of one signal; f 0 = 1/T K =  2E s /T E s is the energy dedicated to sending one signal Binary phase shift keying (BPSK) General M-ary PSK (QPSK, 8-PSK, etc.): M input symbols, Symbol j corresponds to s j (t)= K  cos (2  f 0 t+  j ), 0  t  T

14. 14 Sampling and noise Communication is a continuous-time and continuous-signal process: r(t) = s (t) + n(t) (Considering additive noise only) Noise n(t) is often a Gaussian random process (AWGN) r(t), s(t), n(t) are continuous-time functions, where pieces of duration T seconds correspond to individually transmitted signals A matched filter and a sampler produces discrete time outputs y =  0 T r(t) K cos (2  f 0 t)dt The received sequence that can be fed to the decoder is a sequence of real numbers …y i-1, y i, y i+1,… In general inconvenient to work with real numbers In practical implementations often quantized to a discrete number of levels

15. 15 More on quantization Assume M-ary input. Quantizing to Q-ary output where the output signal is independent of previously transmitted signals and the noise affecting them gives a Discrete Memoryless Channel (DMC) Q = 2 : Provided symmetric quantization, this gives a binary symmetric channel (BSC) Bit error transition probability Hard-decision decoding : Simple processing, suitable for fast, algebraic decoding algorithms (INF 243) Q > 2 : Soft-decision decoding Q = 3 : Provided symmetric quantization, this gives a binary symmetric erasure channel (BSEC) Larger Q: “Softer decisions”

16. 16 Some other channels Channels with memory E. g. intersymbol interference channels caused by multipath Fading channels Caused by multipath fading and other r(t) =  (t)  s (t) + n(t) Nongaussian noise Asymmetric channels Burst channels 1-D 2-D When designing codes, the channel model should be clear!

17. 17 Transmission rate Symbol transmission rate: 1/T k information bits fed into the encoder, producing n output bits R = k/n Bandwidth W Hz: 2W  1/T Data rate = Information transmission rate R/T = 2RW Coded system: R<1 Requires bandwidth expansion to maintain data rate If extra bandwidth is not available and data rate must be maintained, nonbinary coding must be used : Combined coding and modulation