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Combined QPSK and MFSK Communication over an AWGN Channel Jennifer Christensen South Dakota School of Mines & Technology Advisor: Dr. Komo Outline Background Material and Terminology Bandwidth Efficiency Plane Reed Solomon Coding Results Background Multiple Frequency Shift Keying (MFSK) Multiple Phase Shift Keying (MPSK) Combination of BFSK, QPSK: 4 phases, 2 frequencies-16 signals si (t ) Es Ts (2i1 1) (2i2 1) cos 2 t cos 1t 4 4 0 t Ts , 0 i1 , i2 3, i 4i2 i Background (cont’d) AWGN Channel –Gaussian distribution, μ=0, σ2 = No/2 Eb/No-bit energy divided by spectral noise density Probability of Bit Error for QPSK Q Es 2 si2=Es I P(correct ) (1 Pb ) 2 Es Pb Q N 0 N M=4 Transmitted signal plus noise Simulation Correlator Receiver Output - Gaussian random variable 1 (t ) T dt r(t)=si(t)+n(t) n (t ) . . . 0 T dt 0 z(T)=ai(T)+n0 Decision Stage: . . . Determine closest match si(t) Error Probability Plane At Pb=10-5 Eb/No = 9.52 dB for M=4, 16, 64 Bandwidth Efficiency R/W=Data rate/Bandwidth For coherently detected QPSK: Minimum tone spacing = 1/(2Ts) R log 2 M W M=4N - Minimum tone spacing = 1/Ts M -1 1 M 1 R W Ts Ts 2MT 2T 2 Bandwidth-Efficiency Plane Eb W C W 2 1 NO C M=4N, Pb=10-5 Reed-Solomon Coding Error correction code adds redundancies to data (n,k) notation n = total number of code symbols k = number of data symbols encoded Corrects up to (n-k)/2 errors Probability of Bit Error Expand bit error calculations for M-ary signaling: 1 (1 Pb ) 2N 1 d Ps n where P 1 (1 P ) c s d b n j( j t 1 2N n j cj s c n j s )P (1 P ) 2 Rc E b and P Q No • Rc is the code rate (k/n) • Coded Bandwidth: Wc=W/Rc d b Probability Curves n=256 Reed Solomon Codes Plot each point on Bandwidth Efficiency Plane Bandwidth-Efficiency Plane n=256, Pb=10-5 k=240 k=224 k=192 k=160 M=16, N=2 (no coding) Qualitative Results M=16 for N=2 with Reed Solomon Coding (256, 192) Code: R/W=2*(3/4 ) ≈1.5 bps/Hz Coding gain = 9.52-5.71 = 3.81 dB (256, 224) Code: R/W=2*(7/8) ≈1.75 bps/Hz Coding gain = 9.52-6.16 = 3.36 dB Bandwidth-Efficiency Plane (256, 224) M=16, N=2 (uncoded) (256, 192) Conclusions MFSK and QPSK combination has same bandwidth efficiency as QPSK Bandwidth efficiency decreases with ReedSolomon coding Bit error probability also decreases Future work: Soft-decision decoding Evaluating across different channels Acknowledgements Dr. Komo Dr. Noneaker and Dr. Xu References Sklar, Bernard. “Digital Communications: Fundamentals and Applications.” Prentice Hall, PTR, 2nd Edition. 2001. Wicker, Stephen. “Error Control Systems for Digital Communication and Storage.” Prentice-Hall, 1995. Questions? Errors and Erasures •Add erasures for better performance QPSK Detection Decision Regions for RS Coding Original Signal γ γ Correct Signal Detected Erasure Region Error Region