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Demodulation (BER analysis under fading) Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 1 Model for Channel and Noise For a narrowband channel, the received signal (in the baseband) is (Low-Pass (LP) representation) : complex-valued channel gain n(t): zero mean Gaussian noise i.e. n(t) has white spectrum. Autocorrelation of n(t) (and also the inphase and quadrature-phase components of n(t)) which becomes (as B → ∞) Inphase and quadrature-phase components of n(t) are uncorrelated. Pass-band (band-pass) representation of n(t) is Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 2 Signal-Space Representation Remember the signal-space representation of digitally modulated signals. Each modulation scheme can be represented by certain basis functions φn(t) and their weights: Valid for both LP and also BP representations. For BP φn(t) contains the carrier. The basis functions constitute the axes of the signal-space and the weights define the constellation points in that space. Each group of weight can be represented by a vector, sm, i.e. each constellation point is given by BPSK s2 Spring 2017 2-FSK φ2 s1 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS φ2 64-QAM φ1 s2 φ1 s1 s1 φ1 s64 3 Signal-Space Representation Received signal: where nn represent the noise terms in the signal space and n’(t) represents noise in the orthogonal space. We may simply ignore the noise in the orthogonal space. Does not affect the decision. Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 4 MAP/ML Detector All transmitted symbols are equally likely The modulation format does not have memory (e.g. ASK, PSK, FSK, QAM, but not CPM, GMSK etc.) The channel is AWGN. Both the channel gain α and phase rotation φ is known at the RX. Then, «if the signal r(t) was received, then which symbol sm(t) was most likely transmitted?» Which symbol maximizes Maximum A Posteriori (MAP) detector. With equiprobable symbols, the MAP detector becomes identical to Maximum Likelihood (ML) detector Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 5 MAP/ML Detector Using the signal space approach, ML rule becomes For Gaussian noise, the ML detector reduces to which is equivalent to where Spring 2017 Projection of rLP on to sLP,m . (For a constant modulus constellation (e.g. PSK, FSK) Em = E.) ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 6 Pairwise Error Probability (PEP) The probability that symbol sj is mistaken for symbol sk that has Euclidean distance djk from sj is φ2 djk φ1 Anti-podal, e.g. BPSK φ2 djk djk φ1 φ1 Orthogonal, e.g. FSK Bi-orthogonal, e.g. QPSK djk is determined by i. Signal energy εs,i ii. Channel gain α Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 7 Error Probability Binary Orthogonal Signals (2-FSK, 2-PPM) Signal (bit) energy: εb, SNR/symbol (bit): γs = γb = εb/N0, Symbol (bit) error probability = PEP: Antipodal Signals (BPSK) Signal (bit) energy: εb, SNR/symbol (bit): γs = γb = εb/N0, Symbol (bit) error probability = PEP: Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 8 Error Probability Bi-orthogonal Signals (QPSK, 4-QAM) Bit energy: εb, SNR/bit : γb = εb/N0, Signal energy: εs = 2εb, SNR/symbol : γs = εs/N0, Symbol (bit) error probability: can be complicated Upper-left figure: exact error region → difficult to calculate Calculate the PEP for each pair in error for sj Add the PEPs for each pair up → Pairwise error regions overlap → Sum-PEP gives an upper bound on symbol (bit) error probability. → This approach gives an approximation on exact error probability. (left & bottom points) (diagonal point) Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 9 Error Probability A simplification: pairwise error with nearest neighbours cover the whole error region, no need to calculate the last test. PEP: Symbol error probability (from union bound) If we use Gray coding to map pair-of-bits to symbols, bit error probability Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 10 Error Probability for Flat-Fading Channels In a flat-fading channel, the channel gain is NOT static but varies with time ! SNR is time varying ! BER is time varying ! Approach to calculate BER in flat-fading channels: 1. Determine the BER for any arbitrary SNR, 2. Determine the probability that a certain SNR occurs in the channel – in other words, determine the pdf of the power gain of the channel, 3. Average the BER over the distribution of SNRs. Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 11 Error Probability for Flat-Fading Channels In a fading channel, sometimes we are on a «mountain» (high SNR) and at other times in a «valley» (low SNR). BER-SNR relation is highly nonlinear (Q-function), Being on a «mountain» slightly improves BER as compared to the average conditions, Being in a «valley» drastically decreases BER as compared to the average conditions, Being in a «valley» dominates the behaviour → for the same average SNR, performance of a fading channel is significantly worse than that of an AWGN channel. Example: A fading channel has an average SNR of 10 dB. Fading causes the SNR to be - ∞ dB half of the time while it is 13 dB the rest of the time. Consider DBPSK. SNR = - ∞ dB → BER = 0.5 SNR = 13 dB → BER = 10-9. Average BER is For an AWGN channel with 10 dB SNR, Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 12 Error Probability for Flat-Fading Channels Signal strength: Rayleigh pdf To calculate SNR, we need signal power (its pdf). The mean power is Jacobian , to find the pdf of the received power: Then the pdf of the SNR (per bit) is ( . Use the : mean SNR/bit) For Rician-fading (Kr: Rice factor) Then, the average BER is calculated by Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 13 Error Probability for Flat-Fading Channels For a Rayleigh fading channel: Binary antipodal signals (BPSK): Binary orthogonal signals (2-FSK): Differential binary antipodal signals (DBPSK): Differential binary orthogonal signals: Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 14 Error Probability for Flat-Fading Channels For a Rician fading channel: Differential binary antipodal signals (DBPSK): Differential binary orthogonal signals: Example: Calculate the average BER of DBPSK with γB = 12 dB and Kr = -3 dB, 0 dB, 10 dB. Rician fading, use the above expression: For Kr = -3 dB: For Kr = 0 dB: BER = 2.3 x 10-2 For Kr = 10 dB: BER = 5.6 x 10-4 Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 15 Error Probability for Frequency Selective Channels That is another story !!!! Spring 2017 ELE 492 – FUNDAMENTALS OF WIRELESS COMMUNICATIONS 16