Digital Communications I: Modulation and Coding Course Period - 2007 Catharina Logothetis Lecture 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 Lecture 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 Lecture Block diagram of a DCS Format Source encode Channel encode Pulse modulate Bandpass modulate Channel Digital modulation Digital demodulation Format Source decode Channel decode Lecture Detect Demod Sample What is channel coding? „ 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 Lecture Error control techniques „ Automatic Repeat reQuest (ARQ) „ „ „ „ Forward Error Correction (FEC) „ „ „ 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 Simplex connection, error correction codes The receiver tries to correct some errors Hybrid ARQ (ARQ+FEC) „ Full-duplex, error detection and correction codes Lecture Why using error correction coding? „ „ „ „ Error performance vs bandwidth Power vs bandwidth P Data rate vs bandwidth Capacity vs bandwidth B Coded A F Coding gain: For a given bit-error probability, the reduction in the Eb/N0 that can be realized through the use of code: ⎛ Eb ⎞ ⎛ Eb ⎞ ⎟⎟ [dB] − ⎜⎜ ⎟⎟ [dB] G [dB] = ⎜⎜ ⎝ N0 ⎠u ⎝ N ⎠c Lecture C B D E Uncoded Eb / N (dB) Channel models „ Discrete memory-less channels „ „ Binary Symmetric channels „ „ Discrete input, discrete output Binary input, binary output Gaussian channels „ Discrete input, continuous output Lecture Linear block codes „ Let us review some basic definitions first which are useful in understanding Linear block codes Lecture 9 Some definitions „ Binary field : „ The set {0,1}, under modulo binary addition and multiplication forms a field Addition 0⊕0 = 0 ⊕1 = 1⊕ = 1⊕1 = „ Multiplication 0⋅0 = 0 ⋅1 = 1⋅ = ⋅1 = Binary field is also called Galois field, GF(2) Lecture 10 ... detection and correction codes Lecture Why using error correction coding? „ „ „ „ Error performance vs bandwidth Power vs bandwidth P Data rate vs bandwidth Capacity vs bandwidth B Coded A F Coding. .. error detection and correction capability Encoding and decoding Hamming codes Cyclic codes Lecture Block diagram of a DCS Format Source encode Channel encode Pulse modulate Bandpass modulate... Bandpass modulate Channel Digital modulation Digital demodulation Format Source decode Channel decode Lecture Detect Demod Sample What is channel coding? „ Channel coding: Transforming signals to