1. Combined QPSK and MFSK Communication over an AWGN Channel Jennifer Christensen South Dakota School of Mines & Technology Advisor: Dr. Komo

2. Outline Background Material and Terminology Bandwidth Efficiency Plane Reed Solomon Coding Results

3. Background Multiple Frequency Shift Keying (MFSK) Multiple Phase Shift Keying (MPSK) Combination of BFSK, QPSK: 4 phases, 2 frequencies-16 signals

4. Background (cont’d) AWGN Channel –Gaussian distribution, μ=0, σ 2 = N o /2 E b /N o -bit energy divided by spectral noise density Probability of Bit Error for QPSK s i 2 =Es I Q

5. M=4 N Simulation s i (t) Decision Stage: r(t)=s i (t)+n(t) z(T)=a i (T)+n 0............ Transmitted signal plus noise Correlator Receiver Output - Gaussian random variable Determine closest match

6. At Pb=10 -5 Eb/No = 9.52 dB for M=4, 16, 64 Error Probability Plane

7. Bandwidth Efficiency R/W=Data rate/Bandwidth For coherently detected QPSK: Minimum tone spacing = 1/(2T s ) M=4 N - Minimum tone spacing = 1/T s

8. M=4 N, P b =10 -5 Bandwidth-Efficiency Plane

9. 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

10. Probability of Bit Error where and R c is the code rate (k/n) Coded Bandwidth: W c =W/R c Expand bit error calculations for M-ary signaling:

11. Probability Curves

12. Plot each point on Bandwidth Efficiency Plane n=256 Reed Solomon Codes

13. M=16, N=2 (no coding) k=192 k=224 k=240 k=160 n=256, P b =10 -5 Bandwidth-Efficiency Plane

14. 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

15. M=16, N=2 (uncoded) (256, 224) (256, 192) Bandwidth-Efficiency Plane

16. MFSK and QPSK combination has same bandwidth efficiency as QPSK Bandwidth efficiency decreases with Reed- Solomon coding Bit error probability also decreases Future work: Soft-decision decoding Evaluating across different channels Conclusions

17. Acknowledgements Dr. Komo Dr. Noneaker and Dr. Xu

18. 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.

19. Questions?

20. Errors and Erasures Add erasures for better performance