Digital Communications I: Modulation and Coding Course. Term Catharina Logothetis Lecture 13

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1 Digital Communications I: Modulation and Coding Course Term 3-28 Catharina Logothetis Lecture 13

2 Last time, we talked aout: The properties of Convolutional codes. We introduced interleaving as a means to comat ursty errors y making the channel seem uncorrelated. We also studied Concatenated codes that simply consist of inner and outer codes. They can provide the required performance at a lower complexity. Lecture 13 2

3 Today, we are going to talk aout: Shannon limit Comparison of different modulation schemes Trade-off etween modulation and coding Lecture 13 3

4 Goals: Goals in designing a DCS Maximizing the transmission it rate Minimizing proaility of it error Minimizing the required power Minimizing required system andwidth Maximizing system utilization Minimize system complexity Lecture 13 4

5 rror proaility plane (example for coherent MPSK and MFSK) M-PSK M-FSK andwidth-efficient k5 power-efficient k4 Bit error proaility k3 k1 k2 k4 k5 k1,2 / [db] / [db] Lecture 13 5

6 Limitations in designing a DCS Limitations: The yquist theoretical minimum andwidth requirement The Shannon-Hartley capacity theorem (and the Shannon limit) Government regulations Technological limitations Other system requirements (e.g satellite orits) Lecture 13 6

7 yquist minimum andwidth requirement The theoretical minimum andwidth needed for aseand transmission of s symols per second is s /2 hertz. H ( f T ) h ( t) sinc( t / T ) 1 1 2T 1 2T f 2T T 2T Lecture 13 7 T t

8 Shannon limit Channel capacity: The maximum data rate at which error-free communication over the channel is performed. Channel capacity of AWGV channel (Shannon- Hartley capacity theorem): W S [Hz]: C W C W [ Watt]: [Watt]: log 2 Bandwidth 1+ S Average noise power [its/s] Average received signal power Lecture 13 8

9 Shannon limit The Shannon theorem puts a limit on the transmission data rate, not on the error proaility: Theoretically possile to transmit information at any rate C, with an aritrary small error proaility y using a sufficiently complicated coding scheme For an information rate > C, it is not possile to find a code that can achieve an aritrary small error proaility. Lecture 13 9

10 C/W [its/s/hz] Shannon limit Unattainale region Practical region S [its/s/hz] Lecture 13 1

11 Shannon limit C W log S C W S C W log C W As W 1 log 2 e or C W.693, we 1.6 get : [db] Shannon limit There exists a limiting value of / elow which there can e no error-free communication at any information rate. By increasing the andwidth alone, the capacity can not e increased to any desired value. Lecture 13 11

12 W/C [Hz/its/s] Shannon limit Practical region Unattainale region -1.6 [db] / [db] Lecture 13 12

13 /W [its/s/hz] Bandwidth efficiency plane >C Unattainale region C M8 M16 M64 M256 Bandwidth limited M4 M2 <C Practical region M4 M8 M2 Shannon limit M16 Power limited MPSK MQAM MFSK P B 5 1 / [db] Lecture 13 13

14 Power and andwidth limited systems Two major communication resources: Transmit power and channel andwidth In many communication systems, one of these resources is more precious than the other. Hence, systems can e classified as: Power-limited systems: save power at the expense of andwidth (for example y using coding schemes) Bandwidth-limited systems: save andwidth at the expense of power (for example y using spectrally efficient modulation schemes) Lecture 13 14

15 M-ary signaling Bandwidth efficiency: Assuming yquist (ideal rectangular) filtering at aseand, the required passand andwidth is: W / T s [Hz] M-PSK and M-QAM (andwidth-limited systems) / W log M 2 [its/s/hz] W log 2 M WT s 1 WT 1 s [its/s/hz] Bandwidth efficiency increases as M increases. MFSK (power-limited systems) / W log2 M / M [its/s/hz] Bandwidth efficiency decreases as M increases. Lecture 13 15

16 Design example of uncoded systems Design goals: 1. The it error proaility at the modulator output must meet the system error requirement. 2. The transmission andwidth must not exceed the availale channel andwidth. Input [its/s] M-ary modulator s log 2 M [symols/s] Output s P ( M ) f, PB g M-ary demodulator [ P ( M )] Pr s s Lecture 13 16

17 Design example of uncoded systems Choose a modulation scheme that meets the following system requirements: An Pr AWG channel with WC 4 [Hz] 53 [db.hz] 96 [its/s] M 8 P P B s > (log 2 2 ( M 8) W C P ( M ) log M 2Q 2 [ 2 / sin( π / M )] 7.3 / log (log P B Band - limited channel MPSK modulation s M ) s 5 5 M 96 / 3 32[sym/s] < < Pr M ) W C 4 [Hz] Lecture 13 17

18 Choose a modulation scheme that meets the following system requirements: An Pr ( M AWG channel with WC 45 [khz] 48 [db.hz] 96 [its/s] M 16 W P Design example of uncoded systems s < W Pr C (log 1 and relatively small 2 M 16) [db] M ) M s (log 2 M 1 exp /(log Pr M ) s 2 / 2 M ) P B 1 2 k 2 5 power - limited channel MFSK / [ksym/s] < P B k 1 P 1 W ( M ) 7.3 C 45 [khz] 1 6 < 1 5 Lecture 13 18

19 Design example of coded systems Design goals: 1. The it error proaility at the decoder output must meet the system error requirement. 2. The rate of the code must not expand the required transmission andwidth eyond the availale channel andwidth. 3. The code should e as simple as possile. Generally, the shorter the code, the simpler will e its implementation. Input [its/s] P B Output f ( p c ) ncoder Decoder c s P ( M ) f, pc n k g M-ary modulator [its/s] M-ary demodulator [ P ( M )] Pr s log 2 M Lecture c c [symols/s] s s

20 Design example of coded systems Choose a modulation/coding scheme that meets the following system requirements: An Pr AWG channel with 53 [db.hz] W C 96 4 [its/s] [Hz] P B 1 9 P M B > W C 8 P ( M ) log M 2 Band - limited channel s 7.3 / log M > 1 96 / 3 9 MPSK modulation 32 < 4 ot low enough : power - limited system The requirements are similar to the andwidth-limited uncoded system, except that the target it error proaility is much lower. Lecture 13 2

21 Design example of coded systems Using 8-PSK, satisfies the andwidth constraint, ut not the it error proaility constraint. Much higher power is required for uncoded 8-PSK. P B 1 9 uncoded 16 db The solution is to use channel coding (lock codes or convolutional codes) to save the power at the expense of andwidth while meeting the target it error proaility. Lecture 13 21

22 Design example of coded systems For simplicity, we use BCH codes. The required coding gain is: G(dB) uncoded (db) c coded (db) The maximum allowed andwidth expansion due to coding is: n n 96 n s WC log2 M k log2 M k 3 k The current andwidth of uncoded 8-PSK can e expanded y still 25% to remain elow the channel andwidth. Among the BCH codes, we choose the one which provides the required coding gain and andwidth expansion with minimum amount of redundancy. 2.8 db Lecture 13 22

23 Design example of coded systems Bandwidth compatile BCH codes Coding gain in db with MPSK n k t P B 1 PB Lecture 13 23

24 Lecture Design example of coded systems xamine that the comination of 8-PSK and (63,51) BCH codes meets the requirements: [Hz] 4 [sym/s] log 2 < C s W M k n ) ( log ) ( sin 2 2 ) ( < j n c j c n t j B c s s r s p p j n j n P M M P p M Q M P P π

25 ffects of error-correcting codes on error performance rror-correcting codes at fixed S influence the error performance in two ways: 1. Improving effect: The larger the redundancy, the greater the errorcorrection capaility 2. Degrading effect: nergy reduction per channel symol or coded its for real-time applications due to faster signaling. The degrading effect vanishes for non-real time applications when delay is tolerale, since the channel symol energy is not reduced. Lecture 13 25

26 Bandwidth efficient modulation schemes Offset QPSK (OQPSK) and Minimum shift keying Bandwidth efficient and constant envelope modulations, suitale for non-linear amplifier M-QAM Bandwidth efficient modulation Trellis coded modulation (TCM) Bandwidth efficient modulation which improves the performance without andwidth expansion Lecture 13 26

27 Course summary In a ig picture, we studied: Fundamentals issues in designing a digital communication system (DSC) Basic techniques: formatting, coding, modulation Design goals: Proaility of error and delay constraints Trade-off etween parameters: Bandwidth and power limited systems Trading power with andwidth and vise versa Lecture 13 27

modulation Digital demodulation Channel Format

28 Block diagram of a DCS Format Source encode Channel encode Pulse modulate Bandpass modulate Digital modulation Digital demodulation Channel Format Source decode Channel decode Detect Demod. Sample Lecture 13 28

29 Course summary cont d In details, we studies: 1. Basic definitions and concepts Signals classification and linear systems andom processes and their statistics WSS, cyclostationary and ergodic processes Autocorrelation and power spectral density Power and energy spectral density oise in communication systems (AWG) Bandwidth of signal 2. Formatting Continuous sources yquist sampling theorem and aliasing Uniform and non-uniform quantization Lecture 13 29

30 Course summary cont d 1. Channel coding Linear lock codes (cyclic codes and Hamming codes) ncoding and decoding structure Generator and parity-check matrices (or polynomials), syndrome, standard array Codes properties: Linear property of the code, Hamming distance, minimum distance, error-correction capaility, coding gain, andwidth expansion due to redundant its, systematic codes Lecture 13 3

31 Course summary cont d Convolutional codes ncoder and decoder structure ncoder as a finite state machine, state diagram, trellis, transfer function Minimum free distance, catastrophic codes, systematic codes Maximum likelihood decoding: Viteri decoding algorithm with soft and hard decisions Coding gain, Hamming distance, uclidean distance, affects of free distance, code rate and encoder memory on the performance (proaility of error and andwidth) Lecture 13 31

32 Course summary cont d 1. Modulation Baseand modulation Signal space, uclidean distance Orthogonal asic function Matched filter to reduce ISI qualization to reduce channel induced ISI Pulse shaping to reduce ISI due to filtering at the transmitter and receiver Minimum yquist andwidth, ideal yquist pulse shapes, raise cosine pulse shape Lecture 13 32

33 Course summary cont d Baseand detection Structure of optimum receiver Optimum receiver structure Optimum detection (MAP) Maximum likelihood detection for equally likely symols Average it error proaility Union ound on error proaility Upper ound on error proaility ased on minimum distance Lecture 13 33

34 Course summary cont d Passand modulation Modulation schemes One dimensional waveforms (ASK, M-PAM) Two dimensional waveforms (M-PSK, M-QAM) Multidimensional waveforms (M-FSK) Coherent and non-coherent detection Average symol and it error proailities Average symol energy, symol rate, andwidth Comparison of modulation schemes in terms of error performance and andwidth occupation (power and andwidth) Lecture 13 34

35 Course summary cont d 1. Trade-off etween modulation and coding Channel models Discrete inputs, discrete outputs Memoryless channels : BSC Channels with memory Discrete input, continuous output AWG channels Shannon limits for information transmission rate Comparison etween different modulation and coding schemes Proaility of error, required andwidth, delay Trade-offs etween power and andwidth Uncoded and coded systems Lecture 13 35

36 Information aout the exam: xam date: 8 th of March 28 (Saturday) Allowed material: Any calculator (no computers) Mathematics handook Swedish-nglish dictionary A list of formulae that will e availale with the exam. Lecture 13 36

Modulation and Coding Tradeoffs

0 Modulation and Coding Tradeoffs Contents 1 1. Design Goals 2. Error Probability Plane 3. Nyquist Minimum Bandwidth 4. Shannon Hartley Capacity Theorem 5. Bandwidth Efficiency Plane 6. Modulation and