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Error probability of digital signaling 15, Jan. 2001 jxu CONTENTS ■ Error probability of digital signaling Pairwise error probability Bounds on error probability Bit and symbol error probabilities ■ Error probability of digital signaling for an AWGN channel BPSK, M-PSK, M-PAM, M-QAM... ■ Error probability of digital signaling in the fading channel error probability in a flat fading channel error probability over frequency selective fading channel jxu Vector space representation of received signals ■ In a flat fading channel with AWGN, the received complex envelope can be defined as: ~ r (t ) = g (t )~ si (t ) + n~ (t ) If fmT<<1 ~ r (t ) = g~ si (t ) + n~ (t ) ~ Si (t) is M complex low-pass waveforms ~ { S k (t)} (k: from 0 to M-1) g(t): a complex Gaussian random variable. n~ (t ) : zero mean complex AWGN with a power spectral density (psd) of N0 watts/Hz. The vector can be expressed as: ~ ~ ~ r = gs i + n r~ = (~ r0 , ~ r1 ,..., r~N −1 ) ~ si = ( ~ si 0 , ~ si 1 ,..., ~ si N −1 ) n~ = (n~ , n~ ,..., n~ ). 0 jxu 1 N −1 Probability of error ■ ■ { } the set M signal vectors S~ m The decision regions R = {~r : m ■ P (e ~ sm ) = 1 − ∫ p ( ~ r g~ s m ) d~ r M −1 m =0 ~ r − g~ sm 2 2 ≤ ~ r − g~ smˆ , ∀mˆ ≠ m 1 P ( e) = M Rm jxu M −1 ~ P ( e S ∑ m) m =0 } Upper bounds and lower bounds on error probability ■ Upper bound can be obtained by computing the minimum squared Euclidean distance between any 2 2 two-signal points d~min = min ~ sn − ~ sm n,m the pair wise error probability , ~2 α 2 d min P (e) ≤ ( M − 1)Q ( ) 4N 0 ~2 α d min ~ ~ ) P(S j , S k ) ≤ Q( 4N 0 2 ■ lower bound on the error probability ~2 α d min 2 P ( e) ≥ Q( ) M 4N 0 2 jxu Pairwise error probability It can be defined for each pair of signal vectors in the signal constellation decision boundary ~ gS j the squared Euclidean distance ~ gS k ~2 ~ ~ d jk = S i − S k 2 So the pairwise error probability between the message vectors 2 ~2 d jk α ~ ~ ~ ~ ~ and ~ P S S P e S P e S Q ( , ) = ( ) = ( ) = ( ) j k j k Sj Sk 4N 0 jxu Bit and symbol error probabilities The symbol error probability is PM and the bit error probability is Pb ■ PM ≤ Pb ≤ PM log 2 M ■ Gray codes :symbol errors correspond to single bit errors where k= log2M. PM Pb ≈ k jxu Error probability of BPSK ■ two signal waveforms are s1(t)=g(t) and s2(t)= -g(t). ■ s1 = Eb , s 2 = − Eb When noise n is present, The received signal from the demodulator is r = s1+n p ( r s1 ) = 0 1 −( r − e πN 0 P (e s1 ) = ∫ p ( r s1 ) dr = Q ( −∞ P (e ) = Eb ) 2 / N 0 _ AND _ p ( r s2 ) = 1 −( r + e πN 0 Eb ) 2 / N 0 2 Eb ) N0 2 Eb 1 1 P(e s1 ) + P(e s 2 ) = Q( ) 2 2 N0 jxu Fig. Conditional pdfs of two signals Error probability of M-PSK The probability of M-ary symbol error, is just the probability that the received angle outsider the region [-´/M, ´ /M] ■ PM (γ s ) = 1 − ∫ π /M −π / M p(θ )dθ jxu Error probability of M-PAM ■ the Gray coded 8PAM system signal constellation 2α 2 E h Pi = 2 Q N0 2α 2 E h Po = Q N0 M −2 2 1 2α 2 E h Pm = Pi + P0 = 2(1 − )Q M M M N0 jxu Error probability of M-QAM ■ ■ Rectangular QAM signal constellations in practice two -PAM systems in quadrature most frequently used e.g.16-QAM system can be treated as two independent Gray coded 4PAM systems in quadrature, each operating with half the power of the 16-QAM system The symbol error probability for each-PAM system is 1 6 γs P M (γ s ) = 2(1 − )Q M M −1 2 the probability of error symbol reception in the M-QAM system is PM (γ s ) = 1 − (1 − P 2 ) M jxu Error probability in a flat fading channel (1/2) ■ When the channel experiences fading, the error probability must be averaged over the fading statistics. ■ The bit error probability of a fading channel is given by ∞ Pb = ∫ Pe (γ ) p(γ )dγ 0 and the average symbol error probability is the symbol error ∞ probability over the p (γ ) PM = ∫ PM (γ ) p (γ )dγ 0 ■ If the channel is Rayleigh faded. p (γ ) = 1 exp( − γ ) Eb 2σ 2 Eb 2 Ea = γ0 = , N0 N0 [ ] γ0 γ0 is the mean value of the SNR. jxu Error probability in a flat fading channel (2/2) ■ Rayleigh fading converts an exponential dependency of the bit error probability on the bit energy-to-noise ratio into an inverse linear one. Bit error probability for BPSK and QPSK on an AWGN channel and a Rayleigh fading channel with AWGN Bit error probability for M-QAM on an AWGN channel and a Rayleigh fading channel with AWGN jxu Doppler spread impacts For OFDM, if the channel is time-varying, inter-channel interference (ICI) is introduced due to a loss of sub-channel orthogonality ■ At low ~ , additive noise dominates the γb performance so that the extra noise due to ICI has little effect. However, ICI dominates the performance and causes an error floor at large . ■ γ~b Figure: BER for 16-QAM OFDM on a Rayleigh fading channel with various Doppler frequencies jxu Error probability over frequency selective fading channel ■ The frequency selective channel will lead to ISI in the receiver, which will cause signal distortion. The result is irreducible bit error probability floor, which means increasing the signal energy does not have any effect on the bit error probability. ■ This irreducible error floor can also due to the random FM caused by the time-varying Doppler spread, which in turn is due to the motion of the mobile. ■ The computer simulations are the main tool used for analyzing frequency selective fading effects jxu Reference [1] Gordon L. Stuber, "Principles of Mobile Communication (Second Edition)", Kluwer Academic Publishers [2] John G. Proakis, “Digital communications (Third Edition)”, by McGraw-Hill, Inc [3] L.Ahlin & J. Zander, Principle of Wireless Communications , Studentlitteratur, 1998 jxu