COMPARISON OF THEORETICAL PROBABILITY ERROR AND THE
BER SIMULATION OF QPSK AND QFSK MODULATION
MILAN MOSKOVLJEVIĆ
Technical Test Center, Belgrade, [email protected]
MIHAJLO STEFANOVIĆ
Electronic faculty, Niš, [email protected]
PREDRAG RAKONJAC
Technical Test Center, Belgrade, [email protected]
Abstract: This paper presents comparison the two modulation techniques QPSK and QFSK according to the theoretical
probability of bit errors and the simulation values of BER in communication systems with additive white Gaussian noise
(AWGN) and the optimal receiver that is modeled in Matlab Simulink.
Keywords: QPSK, QFSK, BER, bit error probability, Matlab Simulink.
1. INTRODUCTION
In this paper, after the introduction on theoretical error
probability, given the simulation models used to estimate
the BER of selected modulation techniques in a
communication system with additive white Gaussian
noise. At the end of the paper presents the results of
simulation and analysis.
The communication is very important to have accurate
information. Due to random factors such as different
atmospheric conditions, attenuation or malfunction of
equipment perfect transmission can not be provided. With
some parameters, such as coding, different modulation
techniques and filtering can affect the transmission
quality and accuracy of the received message [1]. Coding
and modulation means some kind of digital signal
processing in terms of optimizing the performance of
digital communication systems. Performance optimization
usually involves a compromise must be made between
certain system parameters such as signal strength,
bandwidth, or the complexity of signal processing needed
to errors in transmission of data maintained below set
limits [2].
2. THEORETICAL ASPECTS OF
PROBABILITY ERRORS QPSK AND QFSK
Error probability is different for different modulation
techniques. Common to all is that the modulation is
proportional to the relative probability of error signal
noise ratio (Eb/N0), where Eb is the energy of one bit and
N0 noise power in the range of 1 Hz [3].
The probability of symbol error in the coherent M-PSK
demodulation is given by the formula:
Because of imperfections in a digital communication
system during data leads to errors. The logic level 1 can
be received as a logic level 0 and vice versa. Usually, the
number of errors that are likely to occur in the system is
expressed as the bit error rate (BER).
( )
⎛
⎞
Ps=2Q ⎜ 2log 2 M* Eb *sin 2 π ⎟ , for M ≥ 4
No
M⎠
⎝
(1)
where Q-function can be expressed in terms of the
complementary error function
The bit error rate or bit error ratio (BER) is the number of
bit errors divided by the total number of transferred bits
during a studied time interval.
⎛
⎞
Q ( x ) = 1 *erfc ⎜ x ⎟ , for x ≥ 0 ,
2
⎝ 2⎠
BER = Errors/Total Number of Bits
(2)
The probability of bit errors is equal to
The bit error probability (Pb) is the expectation value of
the BER. The BER can be considered as an approximate
estimate of the bit error probability. This estimate is
accurate for a long time interval and a high number of bit
errors.
Pb=
507
1 Ps
log 2 M
(3)
And probability of bit error is equal to
Substituting equation (1) the equation for the probability
of bit error in the M-PSK, its arranging for M = 4 are
given equation to calculate the theoretical probability of
bit error for QPSK modulation technique.
⎛
⎞
Pb= 1 erfc ⎜ 2*Eb ⎟ ,
2
⎝ No ⎠
⎛
⎞
Pb= 3 Q ⎜ 2*Eb ⎟
2 ⎝ No ⎠
(4)
3. BER MODELING FOR QPSK AND QFSK
Modeling BER for selected modulation techniques in the
channel with additive white Gaussian noise was
conducted in Matlab Simulink.
The probability of symbol error in the coherent M-FSK
demodulation is given by the formula:
( )
⎛
⎞
Ps= ( M-1) Q ⎜ log 2 M* Eb ⎟ , for M ≥ 4
No
⎝
⎠
(6)
Model of a coherent QPSK digital communication system
with BER analysis is presented in Figure 1
(5)
Figure 1: QPSK coherent digital communication system with BER analysis
The model used two subsystems (subsystem with Gray
coding and IQ correlation receiver). The specified random
binary sequence Random integer generator is encoded in
four-level Gray encoded symbols Figure 2
Figure 2: 4-level Gray coded bit to symbol converter
Four Gray code symbols are obtained in the Lookup
Table Block. Coded symbols come in AM modulator with
a carrier frequency fc = 20 kHz, phase φo = π/4 and phase
deviation factor kp = π/2. At the modulator output for the
four symbols have four different phases (π/4, 3π/4, 5π/4,
7π/4). Before entering the channel with additive white
Gaussian noise signal is amplified by a factor of 5th
When leaving the AWGN channel signal passes through a
QPSK coherent receiver with I-Q correlator, Figure 3 The
signal from the receiver to send the part for comparing the
input and the received bits.
Figure 3: QPSK coherent receiver uses an I-Q correlator
508
Every two-bit random sequence generator are grouped to
form dibit (symbols) that are not encoded. The signal still
goes through the FM modulator and has the same as for
QPSK, and then sends the corresponding gain in the
channel with the AWGN.
A simulation model of BER analysis QFSK
communication system with optimal receiver and additive
white Gaussian noise in the transmission channel is given
in Figure 4.
Figure 4: QFSK coherent digital communication system with BER analysis
After communication channel, signal comes at correlation correlator have symbol speed Ts, Figure 5.
receiver with four correlator with time integration. Four
Figure 5: QFSK correlation receiver
The results for the simulation of BER of coherent
modulation techniques QFSK Gray without coding and
optimal receiver in the communication system with
additive white Gaussian noise are given in Table 2
4. RESULTS
The results for the simulation of BER of QPSK coherent
modulation techniques with Gray coding and optimal
receiver in the communication system with additive white
Gaussian noise are compared with theoretical probability
Pb, and errors are given in Table 1
Table 2. The values of BER for coherent modulation
techniques QFSK
Eb/No (dB)
12
10
8
6
4
2
0
Table 1. The values of BER for QPSK coherent
modulation techniques
Eb/No (dB)
12
10
8
6
4
2
0
BER
0
0
2×10-4
2.3×10-3
1.20×10-2
3.62×10-2
7.65×10-2
Pb
9×10-9
3.87×10-6
1.9×10-4
2.3×10-3
1.25×10-2
3.75×10-2
7.86×10-2
BER
0
0
1×10-4
5.1×10-3
2.26×10-2
5.97×10-2
1.209×10-1
Pb
1.79×10-8
7.68×10-6
3.71×10-4
4.4×10-3
2.18×10-2
6.07×10-2
1.18×10-1
Figure 6 shows the comparison of simulated BER results
for QPSK and QFSK modulation technique with the
optimal receiver and the communication channel with the
AWGN.
509
Figure 6: Comparison of simulation BER results QFSK and QPSK modulation techniques
Figure 7 shows the Comparison of theoretical bit error
probability for QFSK and QPSK modulation techniques
with the optimal receiver and the communication channel
with the AWGN.
Figure 7: Comparison of theoretical bit error probability for QFSK and QPSK modulation techniques
5. CONCLUSION
References
[1] I.Stojanović: Fundamentals of Telecommunications,
Građevinska knjiga, Belgrade, 1977.
[2] M.Moskovljević: Digital modulation techniques,
doctoral studies, seminars, Niš, 2012.
[3] Mihajlo Č. Stefanović: Detection of signals in white
and colored Gaussian noise, the first edition of
monograph, Niš 1999.
[4] Harold P.E. Stern, Samy A. Mahmoud:
Communication Systems Analysis and Design,
Prentice hall, 2003.
[5] Haykin S, Michael M.: Introduction to Analog And
Digital Communications Second Edition, Hamilton,
2006.
Comparing and reviewing the results of these two
techniques can be concluded that the theoretical bit error
probability and simulation results of are the same order
and that there are minimal differences between them.
For lower values of the signal noise (below seven) QPSK
modulation technique has a better BER, while QFSK has
less BER for values about eight. For larger values of ten
for both modulation techniques BER values are zero.
By increasing the signal to noise ratio (SNR-Signal Noise
Ratio) for each modulation technique reduces the BER.
510