1 Facuty of Electronics & Telecommunications, HCMUNS BÀI 3: Format (Channel coding) Facuty of Electronics & Telecommunications, HCMUNS Đặng Lê Khoa Email:danglekhoa@yahoo.com dlkhoa@fetel.hcmuns.edu.vn 2 Facuty of Electronics & Telecommunications, HCMUNS Facuty of Electronics & Telecommunications, HCMUNS Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 9 2006-02-16 Lecture 9 4 Facuty of Electronics & Telecommunications, HCMUNS Last time we talked about:  Evaluating the average probability of symbol error for different bandpass modulation schemes  Comparing different modulation schemes based on their error performances. 2006-02-16 Lecture 9 5 Facuty of Electronics & Telecommunications, HCMUNS Today, we are going to talk about: • Channel coding • Linear block codes – The error detection and correction capability – Encoding and decoding – Hamming codes – Cyclic codes 2006-02-16 Lecture 9 6 Facuty of Electronics & Telecommunications, HCMUNS Block diagram of a DCS Format Source encode Format Source decode Channel encode Pulse modulate Bandpass modulate Channel decode Demod. Sample Detect Channel Digital modulation Digital demodulation 2006-02-16 Lecture 9 7 Facuty of Electronics & Telecommunications, HCMUNS • Channel coding: Transforming signals to improve communications performance by increasing the robustness against channel impairments (noise, interference, fading, ) • Waveform coding: Transforming waveforms to better waveforms • Structured sequences: Transforming data sequences into better sequences, having structured redundancy. -“Better” in the sense of making the decision process less subject to errors. What is channel coding? 2006-02-16 Lecture 9 8 Facuty of Electronics & Telecommunications, HCMUNS Error control techniques • Automatic Repeat reQuest (ARQ) – Full-duplex connection, error detection codes – The receiver sends a feedback to the transmitter, saying that if any error is detected in the received packet or not (Not- Acknowledgement (NACK) and Acknowledgement (ACK), respectively). – The transmitter retransmits the previously sent packet if it receives NACK. • Forward Error Correction (FEC) – Simplex connection, error correction codes – The receiver tries to correct some errors •Hybrid ARQ (ARQ+FEC) – Full-duplex, error detection and correction codes 2006-02-16 Lecture 9 9 Facuty of Electronics & Telecommunications, HCMUNS Why using error correction coding? – Error performance vs. bandwidth – Power vs. bandwidth – Data rate vs. bandwidth – Capacity vs. bandwidth (dB) / 0 NE b B P A F B D C E Uncoded Coded Coding gain: For a given bit-error probability, the reduction in the Eb/N0 that can be realized through the use of code: [dB][dB] [dB] c 0 u 0 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = N E N E G bb 2006-02-16 Lecture 9 10 Facuty of Electronics & Telecommunications, HCMUNS Channel models • Discrete memory-less channels – Discrete input, discrete output • Binary Symmetric channels – Binary input, binary output • Gaussian channels – Discrete input, continuous output [...]... ⋅ u + a ⋅ v 2 ∀a, b ∈ F , ∀v ∈ V ⇒ (a ⋅ b) ⋅ v = a ⋅ (b ⋅ v ) ∀v ∈ V, 1⋅ v = v 2006-02-16 Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS 14 Some definitions… – Examples of vector spaces • The set of binary n-tuples, denoted by Vn V4 = {(000 0), (000 1), (001 0), (001 1), (010 0), (010 1), (011 1), (100 0), (100 1), (101 0), (101 1), (110 0), (110 1), (111 1)} • Vector subspace: Vn – A subset S of the... that has (001 1), cardinality is called – A spanning set011 0), (110 0), minimal (100 1)} spans V4 a basis for V • Cardinality of a set is the number of objects in the set • Example: {(100 0), (010 0), (001 0), (000 1)} is a basis for V4 2006-02-16 Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS 16 Linear block codes • Linear block code (n,k) – A set with cardinality is called k linear block 2 a... Error pattern Syndrome U = (10111 0) transmitted r = (00111 0) is received 000000 000 000001 000010 101 011 000100 001000 110 001 S = rHT = (001110)H T = (10 0) 010000 100000 010 100 010001 111 Error pattern corresponding to this syndrome is ˆ e = (10000 0) 2006-02-16 The syndrome of r is computed : The corrected vector is estimated ˆ ˆ U = r + e = (00111 0) + (10000 0) = (10111 0) Facuty of Electronics & Lecture... 24 Linear block codes – cont’d • Encoding in (n,k) block code U = mG ⎡ V1 ⎤ ⎢V ⎥ (u1 , u2 ,K , un ) = (m1 , m2 , K , mk ) ⋅ ⎢ 2 ⎥ ⎢M ⎥ ⎢ ⎥ ⎣Vk ⎦ – The rows of G, are linearly independent (u1 , u2 , K , un ) = m1 ⋅ V1 + m2 ⋅ V2 + K + m2 ⋅ Vk 2006-02-16 Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS 25 Linear block codes – cont’d • Example: Block code (6, 3) Message vector Codeword 2006-02-16... Lecture 9 Telecommunications, HCMUNS 26 Linear block codes – cont’d • Systematic block code (n,k) – For a systematic code, the first (or last) k elements in the codeword are information bits G = [P I k ] I k = k × k identity matrix Pk = k × (n − k ) matrix U = (u1 , u2 , , un ) = ( p1 , p2 , , pn − k , m1 , m2 , , mk ) 14 244 14243 4 3 4 4 parity bits 2006-02-16 Facuty of Electronics & Lecture 9 Telecommunications,... Example: {(000 0), (010 1), (101 0), (111 1)} is a subspace of V4 2006-02-16 Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS 15 Some definitions… • Spanning set: – A collection of vectors , G =include 2 , K , v n in a vector v1 , v all vectors the linear combinations of which space V, is said to be a spanning set for V or to span V • Example: { } • Bases: {(100 0), ( for V that has (001 1), cardinality... 19 Linear block codes – cont’d • The Hamming weight of vector U, denoted by w(U), is the number of non-zero elements in U • The Hamming distance between two vectors U and V, is the number of elements in which they differ • The minimum distance of a block code is d (U, V ) = w(U ⊕ V ) d min = min d (U i , U j ) = min w(U i ) i≠ j 2006-02-16 i Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS... orthogonal to the rows of H ( n − k ) n , : G • H is called the parity check matrix and its rows are T linearly independent • For systematic linear block codes: GH = 0 H = [I n − k 2006-02-16 PT ] Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS 28 Linear block codes – cont’d Data source Format m Channel encoding U Modulation channel Data sink Format ˆ m Channel decoding r Demodulation Detection... (b + c) = (a ⋅ b) + (a ⋅ c) 2006-02-16 Facuty of Electronics & Lecture 9 Telecommunications, HCMUNS 13 Some definitions… • Vector space: – Let V be a set of vectors and F a fields of elements called scalars V forms a vector space over F if: 1 Commutative: 2 ∀u, v ∈ V ⇒ u + v = v + u ∈ F 3 Distributive: V ⇒ a ⋅ v = u ∈ V ∀a ∈ F , ∀v ∈ 1 Associative: (a + b) ⋅ v = a ⋅ v + b ⋅ v and a ⋅ (u + v ) = a ⋅... coset 30 Linear block codes – cont’d • Standard array and syndrome table decoding S = rHT 1 Calculate 2 Find the coset leader, 3 Calculate – Note that • • 2006-02-16 If If ˆ , e = ei corresponding to ˆ andˆ U = r + ecorresponding S ˆ m ˆ ˆ ˆ ˆ U = r + e = (U + e) + e = U + (e + e) , error is corrected ˆ e=e , undetectable decoding error occurs ˆ e≠e Facuty of Electronics & Lecture 9 Telecommunications, . subspace a is )} 1111 (), 1010 (), 0101 (), 0000{( 4 V n V n V )} 1111 (), 1101 (), 1100 (), 1011 (), 1010 (), 1001 (), 1000( ), 0111 (), 0101 (), 0100 (), 0011 (), 0010 (), 0001 (), 0000{( 4 =V 2006-02-16. in the set. •Example: {} .for basis a is )0 001 (), 0010 (), 0100 (), 1000( 4 V {} . spans )1 001 (), 0011 (), 1100 (), 0110 (), 1000( 4 V { } n G vvv ,,, 21 K = 2006-02-16