During the last few years, neural networks have received the attention of many
scientists. Due to their potential of solving difficult problems with no pre-knowledge
of the methodology of the resolution (Ghwanmeh et al.,
2006), they represent a powerful tool (Misra and Dehuri,
2007). Signal processing is one of the fields where the solution is not
all the time evident (Zaknich, 2003) and becomes even
more difficult to find when the background noise is taken into account. In signal
processing, neural networks present the advantage of real time parallel processing
(Samet and Miri, 2009), training with real data and
adaptability. Probabilistic Neural Networks (PNN) perform well in signal processing
(Specht, 1988), they have been used effectively to solve
many problems like image classification, hand digit recognition and alphabet
classification (Burcu and Tulay, 2006), but they suffer
from one major drawback: the generalization of the network is poor due to the
similar kernel function for all units (Galleske and Castellanos,
2002). Rotated Kernel Neural Networks can overcome this limitation by adopting
the same architecture as probabilistic neural networks and using different kernels
(instead of one) according to the shape of the different classes presented to
the network. In this work, we applied RKNN to the problem of Radar target detection
in non-Gaussian noise. Experiments show that RKNN outperforms the standard PNN
in detecting targets hidden in a background noise.
PROBABILISTIC NEURAL NETWORKS
The probabilistic neural network was developed by Donald Specht. His network
architecture was first presented in two papers (Specht,
1988). This network provides a general solution to pattern classification
problems by following an approach developed in statistics, called Bayesian classifiers.
Bayes theory, developed in the 1950's, takes into account the relative likelihood
of events and uses a priori information to improve prediction. The network paradigm
also uses Parzen Estimators, which were developed to construct the probability
density functions required by Bayes theory (Shen and Yan,
The architecture of a probabilistic neural network as defined by Specht
(1988) is shown in Fig. 1. This network is a special case
of a multi-layer perceptron that has three layers:
||An input layer which has as many elements as there are separable
parameters needed to describe the objects to be classified
||A pattern layer, which organizes the training set such that each input
vector is represented by an individual processing element
||An output layer, called the summation layer, which has as
many processing elements as there are classes to be recognized. Each element
in this layer combines via processing elements within the pattern layer
which relate to the same class and prepares that category for output
Sometimes a fourth layer is added to normalize the input vector, if the inputs
are not already normalized before they enter the network.
The radial units are copied directly from the training data, one per case.
Each one models a Gaussian function entered at the training case (Bolat
and Yildirim, 2003). There is one output unit per class (Wahab
et al., 2007). Each unit is connected to all the radial units belonging
to its class, with zero connections from all other radial units. Hence, the
output units simply add up the responses of the units belonging to their own
class. The outputs are each proportional to the kernel-based estimates of the
pdfs of the various classes and by normalizing these to sum to 1. 0 estimates
of class probability are produced.
The pattern units use the following Gaussian activation function:
where, wi s represent the weights, xi s represent the input vector elements and σ is a smoothing parameter (Fig. 2).
The training function may include a global smoothing factor to better generalize
classification results (Zhong et al., 2002).
||The pattern unit in a PNN
In any case, the training vectors do not have to be in any special order in
the training set, since the category of a particular vector is specified by
the desired output of the input. The learning function simply selects the first
untrained processing element in the correct output class and modifies its weights
to match the training vector.
Rotated kernel probabilistic neural networks: The major problem in standard
probabilistic neural networks is generalization which is poor because of Gaussian
functions similar for all units. This gives classes with the same shape which
is not the case in many real world data (as for Radar target data). The Rotated
Kernel Probabilistic Neural Network (RKPNN) keeps the architecture of the original
PNN and uses different Gaussian kernel functions for each unit and with different
kernel parameters (Galleske and Castellanos, 2002).
In this case, the classes represented by the network have different shapes and
can easily fit the shape of the input data classes.
The general idea of this method is to divide the covariance matrix Σi into two other matrices Si and Si to obtain the parameters of the ith training pattern with the formula:
where, Ri is the rotation matrix and Si is a diagonal matrix.
The kernel parameters of the network are estimated automatically in the training process and have not to be chosen to suit a specific classification problem like original probabilistic neural networks do.
The Radar target detection problem can be seen as testing two hypotheses H0 and H1:
where, xs are the samples of the waveform, sis are samples
of the signal and nis are samples of the background noise (McDonough
and Whalen, 1995).
We can then calculate the maximum likelihood ratio L(r) and compare it to a threshold as follows:
||The conditional density function of x given H1
||The conditional density function of x given H0 is true
To simplify, we take a threshold = 1 in this document. So, the decision rule
||H1 is true ⇒ f(x/H1)>f(x/H0)
||H0 is true ⇒ f(x/H0)>f(x/H1)
The probability of detection Pd is:
The probability of false alarm is given by:
From Eq. 4, we can say that if we want to make a decision,
we have to calculate f(x/H1) and f(x/H0). Specht has proposed
a probabilistic neural network based Baysian classifier for Radar detection
The project was conducted in SIMPA laboratory at the University of Science and Technology of Oran during the last semester of 2009.
Performance of the RKNN for radar target detection in background noise (unwanted
clutter) (Vassileva, 2006) has been evaluated in terms
of probability of detection versus Signal to Noise Ratio (SNR) in different
environments (noises). The classifier is also compared to MLP, PNN and RBF neural
networks for the same conditions. The RBF neural network is used with and EM
(Expectation Maximization) training algorithm for better performance. This network
is widely used for classification and approximation problems with different
alternatives (Lu and Ye, 2007; Alippi
et al., 2001) and it can be used even with high-dimensional problems
(Joo et al., 2002). For all networks we use a
window size of 15 points. The training samples are generated from -10 to 20
dB SNR (signal to noise ratio) with 100 samples for each 1 dB. The probability
of false alarm is set to 10-5; we can find this value in many radar
references (McDonough and Whalen, 1995). For all these
results we used Matlab for the training, testing and simulation of results.
This latter presents the advantage of incorporated functions for signal and
neural network processing. Hence, it is easier to test our technique in comparison
with other techniques without wasting a long time in programming every single
In log-normal noise
||For σ = 0.1: The RKNN detector outperforms the
other detectors for values of SNR>8 dB and reaches a probability of detection
of 1 (certainty) when the SNR is greater than 16 dB (Fig.
3). However, this is not the case for SNR<8 dB where the RKNN have
poor performance in front of the other detectors
||Performance curves in log-normal noise σ = 0.1
||Performance curves in log-normal noise σ = 0.5
The other techniques behave the same way for all values of SNR but do not reach
a probability of detection of 0.9 except for the PNN when SNR>12 bB. RBFNN
and MLPNN are quite similar in this case.
||For σ = 0.5: RKNN and PNN behave the same way
as for the preceding case (σ = 0.1). PNN gives better performance for
SNR<15 dB. The two networks reach 0.9 for SNR>10 dB (Fig.
Performance of the MLP decreases significantly in this case. The network has
poor results in this type of noise. This may be caused by a wrong choice of
the networks initial parameters. This problem is one of the most important
disadvantages of MLP neural networks. There is actually no rule when it comes
to choosing the parameters except repeating tests until satisfaction.
Weibull noise: In Weibull noise, it is obvious that RKNN outperforms other neural network based detectors (Fig. 6) when the SNR is greater than 5 dB. But when the SNR is less than 5 dB, the response is not really in favour of the RKNN since the probability of detection is not very reliable (i.e., the system can not decide whether there is a target or not, unless we accept a high probability of false alarm). In practice this has a great impact on the detecting system, since we speak no more about a human-independent (automatic) system, but about a human-assisted detection.
From Fig. 5, we can observe that the best results were generated
by the RBF neural network for almost all values of SNR.
||Performance curves in Weibull noise α = 0.75
||Performance curves in Weibull noise α = 1
The RKNN has better performance for values of SNR greater than 15 dB.
In this case of noise, we can see that results presented by the RBF neural network with an EM training algorithm are good and close to those of a RKNN in some cases.
Both RKNN and RBF networks presented here give better performance for low values
of SNR and outperform most traditional Radar target detectors in presence of
background unwanted clutter (Vassileva, 2006; Sangston
and Gerlach, 1994). The problem here is that the clutter signals are often
as strong as or stronger than the signals returned from the desired target and
makes it difficult to separate the two wave forms.
In many applications it is common for the SNR to be low. In these situations,
accurate and robust estimation of features from the spectrogram or its derivatives
is very difficult and leads to poor performance (Gurbuz
et al., 2007).
When we talk about Radar target detection in non Gaussian noise, the methodology
of resolution is most of time not evident. In this document we presented the
RKNNs applied to Radar target detection. Simulation results presented here are
very promising and show that neural networks can be applied successfully where
other techniques fail or find serious difficulties due to the complexity of
the problem (Vassileva, 2006). For almost all cases
presented in this document, the RKNN has shown a stable behaviour when the other
classifiers behaved according to the nature of the noise. Hence, the RKNN can
be applied to radar detection regardless to the nature of the noise. This point
should be addressed in more depth for further studies. The use of a boosting
algorithm with probabilistic neural networks can increase and smooth performance.
Results given by the RBF are also good and deserve more attention especially
with the EM training algorithm. Many neural networks can be used with a modified
boosting algorithm to increase performance and overcome some of their weaknesses
(Bolat and Yildirim, 2003).
The authors would like to thank all members of the SIMPA laboratory at the University of Sciences and Technology for their support and great patience.