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## MSc Data Communications

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**By R. A. Carrasco**Professor in Mobile Communications School of Electrical, Electronic and Computing Engineering University of Newcastle-upon-Tyne 2006 MSc Data Communications**Recommended Text Books**• “Essentials of Error-Control Coding”, Jorge Costineira, Patrick Guy Farrell • “Digital Communications”, John G. Proakis, Fourth Edition**Goals Of A Digital Communication System**• Deliver/Data from the source to the user in a: • FAST • INEXPENSIVE (Efficient) • RELIABLE WAY**Digital Modulation Schemes**• Task: to compare different modulation schemes with different values of M • Choice of modulation scheme involve trading off • Bandwidth • Power • Complexity • Define: • Bandwidth • Signal-to-noise ration • Error probability**Source Symbols**0 0 1 1 0 1 1 0 0 1 Ts A a t -A M=2 T=Ts S1(t) = A, 0t<T S2(t) = -A T 10 01 01 01 01 b t Sinusoids with 4 different phases M=4 T=2Ts 010 011 c t M=8 T = 3Ts 8 different amplitude levels T 011001 010101 d T Examples: Memoryless Modulation (Waveforms are chosen independently – each waveform depends only on mi) 101**A crucial question is raised**• what is the difference? • If T is kept constant, the waveforms of scheme C requires less bandwidth than those of 2, because the pulse duration is longer • In the presence of noise, and if the same average signal power is used, it is more difficult to distinguish among the waveforms of c. AM/AM = Amplitude Modulation – Amplitude Modulation conversion AM/PM = Amplitude Modulation – Phase Modulation conversion**Notice:**• Waveforms b have constant envelopes. • This choice is good for nonlinear radio channels A Output Envelope B Input Envelope A: Output envelop (AM/AM conversion) B: Output phase shift (AM/PM conversion)**TRADE-OFF BETWEENBANDWIDTH AND POWER**• In a Power-Limited Environment, use low values of M • In a Band-Limited Environment, use high values of M What if both Bandwidth and Power are Limited ? • Expand Complexity • DEFINE: BANDWIDTH SIGNAL-TO-NOISE RATIO ERROR PROBABILITY**DIGITAL MODULATION TRADE-OFFS**SHANNON CAPACITY LIMIT FOR AWGN C = W LOG (1 + S/N) • S = Signal Power = e/T • N = Noise Power = 2NoW • W = Bandwidth**dB**S(f) Different bandwidth definitions of the power density spectrum of (5.2). B1 is the half-power bandwidth: B2 is the equivalent noise bandwidth: B3 is the null-to-null bandwidth: B4 is the fractional power containment bandwidth at an arbitrary level: B5 is the bounded power spectral density at a level of about 18dB. Notice that the depicted bandwidths are those around the frequency f0. 0 -10 -20 -3 -1 0 1 2 3 -2 fT B1 • Half – power • Equivalent – noise • Null – to – null • Fractional power containment • Bounded power spectral density B2 B3 B4 B5 In general, W = /T Hz, depends on modulation scheme and on bandwidth definition Define Bandwidth W**Rate at which the source outputs binary symbols**Average signal energy • Average Signal Power b = average energy per bit • Signal-to-Noise Ratio Noise power spectral density. Bits/sec Hz (Bandwidth DEFINE SNR**Digital Modulation Trade – Offs(Comparison among different**schemes) 16 SHANNON CAPACITY BOUND 32 16 16 8 8 4 x 8 16 x 16 4 8 4 8 x 2 BANDWIDTH LIMITED REGION 4 2 2 x 2 1 1 2 4 8 POWER LIMITED REGION 0.5 2 4 16 8 32 0.25 16 0.125 64 32 -2 0 2 4 6 8 10 12 14 16 18 20 (dB) PAM (SSB) (COHERENT) PSK AM-PM DCPSK COHERENT FSK INCOHERENT FSK x**u1**k=1 Encoder for the (3, 1) repetition code x1= u1, x2=u1, x3=u3 0 000 1 111 x2 x3 x1 n=3 k=2 The whole word is defined by x1=u1, x2=u2 x3=u1+u2 Encoder for the (3,2) parity-check code u2 u1 00 000 01 011 10 101 11 110 x3 x2 x1 n=3 [1], pages 157 - 179**The codeword is defined by**xi = ui, i = 1,2,3,4 x5 = u1 + u2 + u3 x6 = u2 + u3 + u4 x7 = u1 + u2 + u4 (7, 4) Hamming Code Hamming Code (7,4) Source symbols encoded symbols Block Encoders Notice that only 16 of 128 sequences of length 7 are used for transmission**Convolutionally encoding the sequence 101000...**Rate , K = 3 From B Sklar, Digital Communiations. Prentice-Hall, 1988**Convolutionally encoding the sequence 101000...**Output sequence: 11 10 00 10 11**1**(7, 5) Convolutional Encoder c1 d1 d3 d2 1 Input Data d3 d2 d1 Constraint length K = 3 Code Rate = 1/2 Output 2 d1 d3 c2**Example message:**Input 0 1 1 0 1 0 0 1 0 1 Output 00 11 01 01 00 10 11 11 10 00 The Finite State Machine a=00 b=01 c=10 d=11 1/10 d 11 0/01 1/00 1/01 c 01 b 10 0/10 0/11 1/11 output bit a 00 0/00 input bit The 0 or 1 The coder in state a= 00 A 1 appearing at the input produces 11 at the output, the system moves to state b = 01**If in state b, a 1 at the input produces 01 as the output**bits. The system then moves to state d (11). • If a 0 appears at the input while the system is in state b, the bit sequence 10 will appear at the output, and the system will move to state c (10).**Tree Representation**00 a b c d 00 11 00 10 11 00 01 11 a b 10 00 11 01 Time c d 0 01 10 00 11 a b 1 11 K=3 Rate= 10 Input data bits 10 c d 00 01 11 11 a b 01 00 01 01 c d 10 10 1 2 3 4 0 bit 1 bit Upward transition Downward transition**D2**Signal-flow Graph**Transfer Function T(D)**Therefore,**Transfer Function T(D)**Performing long division gives: This gives the number of paths in the state diagram with their corresponding distances. In this case, the minimum distance of the code is 5**s1=(00)**s2=(01) s3=(10) s4=(11) 1 Block Encoders by Professor R.A Carrasco STATE 000 U Ui Ui-1 Ui-2 S1 00 001 111 110 S3 10 S2 01 011 010 100 X S4 11 0 Source Symbol 101 “State Diagram” of Code u= (11011.....) corresponds to the paths s1 s3 s4 s2 s3 s4 through the state diagram and the output sequence is x=(111100010110100)**Tree Diagram**The path corresponding to the input sequence 11011 is shown as an example.**Signal Flow Graph**Xa = S1 Xb = S3 Xc = S4 Xd = S2**Transfer Function of the State Diagram**We have dfree = 6, for error events: S1 to S3 to S2 to S1 and S1 to S3 to S4 to S2 to S1**Trellis Diagram for Code (Periodic from time 2 on)**Legend Input 0 Input 1 The minimum distance of the convolutional code l = N dmin = dc (N), the column distance. The free distance dfree of the convolutional code**Trellis Diagram for the computation of dfree**Trellis labels are the Hamming distances of encoder outputs and the all-zero sequence.**We want to compute**Functions whose arguments l can take on a finite number of values Simplest situation arises when T2, T1…….are “independent” (The value taken on by each one of them does not influence the other variables) 1 Then D C 0 B A Viterbi Algorithm [1], pages 181 – 185**2. We have, for a binary symmetric channel:**1-P 0 0 P Rx Tx Irrelevant multiplicative constant Irrelevant additive constant P 1 1 1-P Hamming distance Between xl and yl Viterbi Decoding of Convolutional Codes is a maximum received sequence transmitted symbols Observe (memoryless channel) received no-tuple n0-tuple of coded digits**Brute force approach:**• Compute all the values of the function and choose the smallest one. • We want a sequential algorithm**What if 0, 1, … are not independent?**0 = A 1 = C or 1 = D 1 D C 0 0 = B 1 = D B A What is the shortest route from Los Angeles to Denver? Viterbi Algorithm**l=5**l=0 l=4 l=1 l=2 2 1 3 1 0 3 2 2 2 1 1 1 0 3 4 1 2 4 1 1 2 2 l=1 l=2 2 l=3 2 4 2 1 3 3 1 1 2 2 3 1 1 0 4 2 1 6 4 (a) 1 2 2 (c) (b) 4 2 l=5 l=4 4 0 4 1 4 5 3 1 6 5 (d) (e) 4 Viterbi Algorithm l=3**Conclusion**• We maximise P(y|x) by minimising , the Hamming distance between the received sequence and coded sequence. • Brute-Force Decoding • Compute all the distances between y and all the possible x’s. Choose x that gives the minimum distance. • Problems with Brute-Force Decoding • - Complexity • - Delay • Viterbi algorithm solves the complexity problem (complexity increases only linearly with sequence length) • Truncated Viterbi algorithm also solves delay problem**Band-limited environment**(Terrestrial radio communications) • Power-limited environment (Satellite radio communications) • Use higher-order modulation schemes. (8 PSK instead of 4 PSK) Same BW, More bit/s per hz, more power • Use coding: Less power, less bit/s per hz, BW expanded Trellis-Coded Modulation A.K.A - Ungerboeck Codes - Amplitude-Redundant Codes - Modulation How to increase transmission efficiency? Reliability or Speed [2], pages 522-532 G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inform. Theory, vol. IT-28, pp. 55-67, 1982.**Construction of TCM**• Constellation is divided into smaller constellations with larger euclidean distances between constellation points. • Construction of Trellis (Ungerboeck’s rules) • Parallel transitions are assigned members of the same partition • Adjacent transitions are assigned members of the next larger transitions • Signals are used equally often**Model for TCM**Memory Part n Select Constellation an Select Signal From Constellation**2**3 1 4 0 d’ 5 7 6 Some examples of TCM schemes Consider transmission of 2 bits/signal We examine TCM schemes using 8-PSK With uncoded 4-PSK we have We use the octogonary constellation**The Free Distance of a convolutional code is the Hamming**distance between two encoded signals • dfree is a measure of the separation among encoded sequences: the larger dfree, the better the code (at least for large enough SNR). • Fact: To compute dfree for a linear convolutional code we may consider the distances with respect to the all-zero sequence.**Remerge**Split (00) (00) An “error event” • A simple algorithm to compute dfree is • Compute dc(l) for l = 1,2,… • If the sequence giving dc(l) merges into the all-zero sequence, store its weight as dfree**First Key Point**Constellation size must be increased to get the same rate of information Minimum distance between sequences • We gain Minimum distance for uncoded transmission Energy with coding • We lose Energy without coding Gain is**Second Key Point**How to introduce the dependence among signals Transmitted symbol at time nT Source symbol at time nT Previous source Symbols = “state” n (Describes output as a function of input symbol + encoder state) We write (Describes transitions between states)**TCM Example 1**Consider the 8-PSK TCM scheme, which involves the transmission of 2 bits/symbol using an uncoded 4-PSK constellation and the coded 8-PSK constellation for the TCM scheme as shown below Show that and**TCM Example 1: Solution**We have from the uncoded 4-PSK constellation We have from the coded 8-PSK constellation Using**0**0 0 4 5 6 1 2 3 1 7 Hence, we get the coding gain a0 a1 I 4PSK 8PSK Can this performance gain Trellis coded QPSK be improved? The answer is yes by going to more trellis states. TCM Scheme Based on 2-State Trellis**TCM Example 2**• Draw the set partition of 8-PSK with maximum Euclidean distance between two points. • By how much is the distance between adjacent signal points increased as a result of partitioning?