Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation Nyquist Lecture 10 Pulses Chapter 8 and 9 Detection of a Binary Signal Detection of a Binary Wave- Optics Signal 1 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Posterior probability distribution Lecture 10 Prior and Discrete memoryless information channel Posterior Distribu- depends or is conditioned by the corresponding channel input symbol and tions by only that symbol. Methods of Coded Modulation For a well-designed encoder output of the encoder, will appear to be Nyquist random and independent Pulses Detection of a Binary The probability of each codeword symbol, seen in isolation, at the input Signal to the channel from an ideal encoder is described by a prior probability Detection distribution p s ( s ) , of a Binary Wave- called simply a prior , w/components called prior probabilities Optics Signal In constrast, the probability of each channel output symbol, seen in isolation, is described by a posterior probability distribution p r ( r ) on the output, called simply the posterior The conditional probability distribution p s | r ( s | r ) on the input for a fixed r at the output is called the posterior probability distribution on the input symbol. 2 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Detection Statistic Lecture 10 Prior and Posterior Distribu- tions Methods of Detection process at Rx converts a discrete-time sample of the electrical Coded waveform, real or complex , into a senseword symbol that is sent to the Modulation decoder Nyquist Pulses Detection This sample value is called a detection statistic of a Binary Signal Detection A detection statistic can have many forms of a Binary Wave- Optics It is generated by transforming a received waveform into a sequence of Signal samples The decoder then determines the transmitted codeword or, equivalently, determines the user dataword represented by that codeword 3 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Different Viewpoints of an Information Channel Lecture 10 Prior and Posterior Distribu- The aspects of a communication system that constitute an information tions channel are described using a conditional probability distribution Methods of Coded Modulation This conditional distribution depends on the channel model Nyquist Pulses The information channel can be viewed from either the transmitter or Detection of a Binary from the receiver Signal Detection of a Binary When viewed from the transmitter, the memoryless information channel Wave- Optics is modeled as a conditional probability distribution p r | s ( r | s ) , abbreviated Signal p ( r | s ) , that the symbol r will be received when the symbol s is transmitted When viewed from the receiver, the information channel is modeled as a conditional probability distribution p s | r ( s | r ) , abbreviated p ( s | r ) , that when the symbol r is received, the symbol s was transmitted. 4 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Conditional, Joint and Prior Probalities Lecture 10 Using the Bayes rule, the combination of the information channel model Prior and and the prior distribution of symbols can be expressed as a joint Posterior Distribu- probability distribution p ( s , r ) tions Methods of p ( s , r ) = p s ( s ) p ( r | s ) = p r ( r ) p ( s | r ) , (1) Coded Modulation where p s ( s ) is the prior for the transmitted input symbol s , and p r ( r ) is Nyquist Pulses the posterior probability distribution for the received output symbol r Detection of a Binary Signal It follows immediately from (1) that (sum out one to get the other) Detection of a Binary � p r ( r ) = p s ( s ) p ( r | s ) (2) Wave- Optics Signal s and so p s ( s ) p ( r | s ) p ( s | r ) = s p s ( s ) p ( r | s ) . (3) � For the same physical channel, the expression for the joint probability p ( s , r ) based on continuous wave optics is different than the expression based on discrete photon optics because these are different models and use different methods of detection. 5 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Soft and Hard Decision Detection Lecture 10 The detected sequence of symbols form the senseword at the output of Prior and Posterior the information channel, which becomes the input to the decoder Distribu- tions In soft-decision detection , each component of the senseword is a sample Methods of Coded r k ( real or complex ) or a quantized form of that sample Modulation Nyquist Pulses In hard-decision detection , the detection process decides on a symbol Detection from a discrete output alphabet based on the received sample and on of a Binary Signal prior knowledge about the possible inputs Detection of a Binary The output of hard-decision detection is a sequence of logical symbols Wave- generated by a hypothesis-testing procedure that is used to form the Optics Signal senseword. Hypothesis testing is quantified by the probability of a detection error p e The probability of a detection error p e is not meaningful for soft-decision detection because the quantized samples are not generated by hypothesis testing Accordingly, the information channel defined using hard-decision detection is not the same as an information channel defined using soft-decision detection 6 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Methods of Coded Modulation Lecture 10 At the Tx, the information channel receives a sequence of logical symbols Prior and Posterior from the encoder and converts this sequence into a sequence of real or Distribu- tions complex electrical pulses for the electrical channel Methods of Coded At the Rx, the information channel receives a sequence of real or complex Modulation samples taken from the electrical waveform and converts this sequence Nyquist Pulses into a sequence of logical symbols for the decoder Detection of a Binary Signal This back and forth conversion is the task of modulation and Detection demodulation of a Binary Wave- Optics This sequence of real numbers can be described as a waveform w ( t ) on Signal continuous time using Dirac impulses as given by ∞ � s j δ � t − jT � w ( t ) = (4) , j = −∞ where T is the symbol interval, and the symbol s j in the j th interval is a point of the L -point signal constellation { s 0 , s 1 , ..., s L − 1 } that is specified by the user data. 7 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Signal Constellations Lecture 10 Prior and Posterior Distribu- tions d min Methods of Coded d min Modulation d min d min Nyquist φ Pulses Detection of a Binary Signal Detection (a) (b) (c) (d) of a Binary Wave- Optics Figure: Signal constellations. (a) A pulse amplitude modulation constellation. Signal (b) A square quadrature amplitude modulation (QAM) constellation. (c) A nonsquare slightly irregular QAM constellation. (d) A phase-shift keyed constellation. Also shown is the minimum euclidean distance d min . The minimum distance d min is the minimum euclidean distance between any two signal points in the constellation 8 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Signal Constellations -cont. Lecture 10 Prior and Posterior Distribu- tions Methods of Coded Modulation The elements s j of the encoded sequence must come from the chosen Nyquist signal constellation, but there may be constraints on the allowable Pulses sequence patterns to control or eliminate errors at the receiver Detection of a Binary Signal The abstract representation of the datastream given in (4) must appear Detection as a continuous waveform at the transmitter, and must then appear at of a Binary Wave- the receiver as a corresponding continuous waveform that is sampled Optics Signal An obvious way to form a continuous waveform is to replace each impulse by the transmit pulse shape x ( t ) 9 ECE243b Lightwave Communications - Spring 2019 Lecture 10
Pulse vs. Waveform Description of Channels Lecture 10 Prior and Posterior Distribu- tions (a) Pulse Description Methods of Coded Modulation δ ( t ) x ( t ) x ( t ) h ( t ) p ( t ) y ( t ) q ( t ) Nyquist Pulses Dirac Pulse Transmitted Physical Received Detection Filtered Impulse shaping pulse Channel pulse filter pulse Detection of a Binary Signal (b) Waveform Description Detection of a Binary Wave- Optics h ( t ) w ( t ) x ( t ) s ( t ) r ( t ) y ( t ) r k Signal Datastream Pulse Transmitted Physical Received Detection Sampler Sequence of shaping waveform Channel waveform filter Samples Figure: Channel response: (a) to an impulse (b) to a datastream. 10 ECE243b Lightwave Communications - Spring 2019 Lecture 10
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