# Vertices of a Hypercube: Combining QPSK and MFSK Communication ```Combined QPSK and MFSK
Communication over an
AWGN Channel
Jennifer Christensen
South Dakota School of Mines &amp; Technology
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
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