Unless otherwise specified, all SVM-based receivers use the RBF kernel with σ = 0.5 and the following system parameters are assumed in all the examples given in this section: M = 4, L = 1, U = 1, and η = 1. The reason to choice the RBF kernel is due to the fact that it has been proved that this kernel is universal, which means that linear combinations of this kernel computed at the data points can approximate any function[11].
Example 1 4FSK in AWGN channels.
0 1 2 3 4 5 6 7 8 9 10
10−5 10−4 10−3 10−2 10−1
100 4FSK in AWGN channel
Eb/N0
SER
Linear kernel Theorical result RBF kernel
Figure 4.5: Symbol error rate performance of optimum and SVM-based 4FSK receivers in AWGN channel.
Fig. 4.5 reveals that a receiver based on the SVM technique can provide performance very close to that achieved by a conventional optimal MFSK receiver. More specifically,
the probability of symbol error for the optimal noncoherent M -ary orthogonal signal detector is given by [16]
PM(e) = The receiver whose performance is given by the dashed curve uses RBF kernel, 600 training samples and C = 1. The solid one with linear kernel, 600 training samples and C = 6 is plotted as well. The circle points are calculated according to (3.19). Both receivers yield almost the same performance as that of the optimum receiver.
Example 2 4FSK in a single-user AWGN channel with diversity order L = 4.
0 2 4 6 8 10 12
4FSK with L=4 in AWGN channel
Eb/N0
SER
Union bound Receiver based on SVM
Figure 4.6: Symbol error rate performance of the optimum and SVM-based 4FSK re-ceivers in AWGN channel and diversity order L = 4.
In Fig. 4.6, we plot the performance curves for the SVM-based receiver and the optimal noncoherent 4FSK receiver. The performance of the latter is computed by using
the union bound [16] for general M -ary orthogonal signal with diversity L.
The solid line corresponds to the performance obtained with 600 training samples and C = 0.9. It is obvious that the performance is almost identical to that of the ideal noncoherent 4FSK receiver except in low SNRs. It is known that [16], for relatively small values of M , the union bound is sufficiently tight for most practical applications.
We conclude that the MFSK receiver based on SVM can offer performance very close or identical to the optimal receiver when diversity reception is involved.
Example 3 4FSK in Rayleigh fading channels.
As shown in Fig. 4.7, the performance of the proposed receiver is similar to that of the conventional optimal receiver (matched filter). The solid curve is the SER performance of the SVM-based receiver with 1800 training samples and C = 0.9. Again, we can see that the proposed SVM-based receiver provides performance identical to that of the conventional optimal receiver.
Example 4 4FSK in frequency nonselective Rayleigh fading channels with diversity or-der L = 4.
The SVM-based receiver is obtained with 1800 training samples and C = 0.1. The threshold b is chosen to be q
2 ∗ (1 + SN R1 )ln(1 + SN R) ∗ N0 according to [14].
The solid curve in Fig.4.8 represents the performance of the receiver proposed by Goodman et al. [9] and the dashed curve is the performance of our SVM-based receiver.
In this case our receiver outperforms Goodman’s, which is only a suboptimal receiver.
0 2 4 6 8 10 12 14 16 18 20 10−3
10−2 10−1
100 4FSK in Rayleigh channel
Eb/N0
SER
Receiver based on SVM Conventional receiver
Figure 4.7: Performance comparison of optimum and SVM-based 4FSK receivers in Rayleigh fading channels.
Example 5 4FSK in a Rayleigh fading channel with diversity order L = 4, B = 2 and two system users.
The dashed curve in Fig.4.9 is the performance of our SVM-based receiver. The proposed receiver is achieved with 1800 training samples and C = 0.1. The solid line is the performance of the receiver proposed by [17] with q = 8. It is clear from Fig. 4.9 that our proposed receiver outperform that in [17] too.
Example 6 4FSK in Rayleigh fading channels with diversity order L = 4, B = 2, η = 1.2 and two system users (one interferer).
The dashed line in Fig.4.10 is the performance of our SVM-based receiver. The proposed receiver is obtained with 1800 training samples and C = 0.1. The solid line is
0 2 4 6 8 10 12 14 16 10−4
10−3 10−2 10−1
100 4FSK with L=4 in Rayleigh channel
Eb/N0
SER
Receiver proposed by Goodman Receiver based on SVM
Figure 4.8: Performance comparison of Goodman’s and SVM-based 4FSK receivers in Rayleigh fading channels.
the performance of receiver proposed in [17] with q = 8. Again, Fig. 4.10 shows that our proposed receiver outperforms that in [17].
Example 7 Robustness of SVM-based detectors
In this example, it is illustrated by Fig. 4.11 that SVM may be insensitive to the param-eter: Eb/N0. The solid line is the performance under the SVM receiver whose support vectors are obtained with the assumption Eb/N0 = 20 dB but are used for different Eb/N0s. The dashed line, on the other hand, represents the performance of the SVM receiver using Eb/N0-dependent (optimized) support vectors.
The fact that both receivers yield almost identical performance indicates that the proposed SVM-based receiver is relatively robust against changing Eb/N0. Such robust
0 5 10 15 20 25 10−2
10−1
100 4FSK B=2 with one interferer in Rayleigh channel
Eb/N0
SER
Nonlinear receiver Receiver based on SVM
Figure 4.9: Performance comparison of the 4FSK receiver of [17] and SVM-based receiver in a Rayleigh fading channel with B = 2 and one interferer.
behavior also simplifies our receiver design for there is no need to store different sets of SVs for different Eb/N0s.
0 5 10 15 20 25 10−2
10−1
100 4FSK B=2 eta=1.2 with one interferer in Rayleigh channel
Eb/N0
SER
Receiver based on SVM Nonlinear receiver
Figure 4.10: Performance comparison of the 4FSK receiver of [17] and SVM-based re-ceiver in Rayleigh fading channels with B = 2, η = 1.2 and one interferer.
0 5 10 15 20 25 30 35 40 10−2
10−1
100 4FSK with B=2, one interferer in Rayleigh channel
Eb/N0
SER
SVM trainned in Eb/N0=20dB SVM trainned with different Eb/N0
Figure 4.11: Performance comparison of SVM-based receivers with different training ways.
Chapter 5
Conclusion and Future Works
5.1 Conclusion
This thesis documents our investigation results on the design of receiver based on SVM for MFSK/FHMA system. From the simulation results, we may infer that the receiver based on SVM can approximate to the optimal receiver or outperform proposed receivers in chapter 2. The main reason is due to the fact that the conditional probability density is intractable. The SVM-based receiver, however, does not require the knowledge and usually the performances can approximate to the ML results. Also, the SVM-based receiver is insensitive to the parameter: Eb/N0. With this property, we can save the memory and implement the receiver without sorting other results based on different Eb/N0.