Sumit J Darak
email: sumit.darak@supelec.fr

Sumedh Dhabu
email: sumedh1@e.ntu.edu.sg

Christophe Moy
email: christophe.moy@supelec.fr

Honggang Zhang
email: honggang.zhang@supelec.fr

Jacques Palicot
email: jacques.palicot@supelec.fr

A P Vinod
email: asvinod@ntu.edu.sg

Low Complexity and Efficient Dynamic Spectrum Learning and Tunable Bandwidth Access for Heterogeneous Decentralized Cognitive Radio Networks

Keywords: Decentralized cognitive radio networks, dynamic spectrum learning and access, frequency transformation, multi-armed bandit, variable digital filter

come

Low complexity and efficient dynamic spectrum learning and tunable bandwidth access for heterogeneous decentralized cognitive radio networks

• Low complexity variable bandpass filter (VDF) design • Tunable dynamic spectrum learning and access (DSLA) policy for heterogeneous cognitive radio networks (CRNs) • System level approach to integrate VDF with the DSLA policy to support tunable bandwidth access in heterogeneous CRNs • Higher spectrum efficiency and lower subband switching cost

Introduction

Dynamic spectrum learning and access (DSLA) based Cognitive Radio Networks (CRNs) paradigm has been suggested to improve the overall utilization of the spectrum, which is crowded yet underutilized mainly due to static spectrum allocation policies [START_REF] Mitola | Cognitive Radio: Making Software Radios More Personal[END_REF][START_REF] Ahmad | Wideband Spectrum Sensing Technique Based on Random Sampling on Grid: Achieving Lower Sampling Rates[END_REF]. In CRNs, multiple secondary (unlicensed) users can access the vacant frequency subbands in an opportunistic way with minimal interference to the primary (licensed) users [START_REF] Mitola | Cognitive Radio: Making Software Radios More Personal[END_REF][START_REF] Ahmad | Wideband Spectrum Sensing Technique Based on Random Sampling on Grid: Achieving Lower Sampling Rates[END_REF]. Recently, the term "Green CRN" has been coined, which refers to the CRN which will minimize the CO 2 emissions (by using area and energy efficient radio terminals) along with improving the spectrum utilization [START_REF] Palicot | On The Road Towards Green Radio[END_REF][START_REF] Feng | A Survey of Energy Efficient Wireless Communications[END_REF]. These CRNs can be deployed for smart grid networks, public safety networks, health-care networks, cellular networks [START_REF] Wang | Emerging Cognitive Radio Applications: A Survey[END_REF] etc.

The two representative DSLA approaches are: 1) DSLA for centralized CRNs (centralized DSLA): A central unit is responsible for orthogonalization of M secondary users to optimum subbands (i.e., the subset of M subbands which are identified as superior, in terms of their probability of being vacant, lower interference and noise levels etc., amongst all available subbands in wideband input signal) [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF], 2) DSLA for decentralized CRNs (decentralized DSLA): Each secondary user independently learns, senses and accesses the vacant subbands without any explicit information exchanges or pre-agreements with other users [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF][START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF][START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF]. The centralized DSLA schemes using game theory approach, auction based approach, multi-armed bandit based approach etc. have been proposed and more details can be found in [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF]. Various decentralized DSLA schemes are discussed in detail in Section 2.2. In summary, the decentralized DSLA has the advantages such as ease of implementation, robustness to link or node failures, no communication overhead and lower delay over the centralized DSLA [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF][START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF][START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF]. As a result, decentralized DSLA is preferred choice for public safety networks and proximity-aware social networking services. However, decentralized DSLA suffers from lower utilization of optimum subbands due to the potentially higher number of collisions between the secondary users. This is because of the lack of coordination among secondary users as well as hardware and signal processing constraints at each secondary user terminal.

Existing centralized as well as decentralized DSLA schemes [6-11, 13, 14] assume that the vacant bandwidth requirements of all secondary users are fixed and equal. Such assumption may not be true in the next-generation heterogeneous CRNs such as 5G where multiple wireless communication standards with distinct channel bandwidths co-exist. In order to extend the capabilities of such decentralized CRNs beyond voice to video, data, location and Internet based services demanding distinct bandwidths and to seamlessly integrate multiple services on a single terminal, a tunable decentralized DSLA scheme is desired.

Another constraint for efficient DSLA scheme is the total penalty incurred in terms of delay, energy consumption, hardware reconfiguration and protocol overhead when secondary user switches from one frequency subband to another. Hereinafter, this penalty is referred to as subband switching cost (SSC) [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF]. The total number of subband switchings and hence, SSC should be as small as possible, especially for the resource-constrained battery-operated secondary user terminals. Though the challenges of minimizing the collisions among secondary users are attracting much attention in the research community today, the design of decentralized DSLA scheme with additional constraints of tunable vacant bandwidth and minimum SSC for heterogeneous CRNs has yet to be systematically addressed. The stringent hardware specifications of minimum area and power make the design even more challenging and are the motivations behind the work presented in this paper. The proposed decentralized DSLA scheme consists of:

• A low complexity interpolation and modified frequency transformation based variable digital filter (IMFT-VDF). It provides tunable bandpass responses anywhere over Nyquist band with complete control over the bandwidth as well as the center frequency and offers total gate count savings from 11% to as high as 87% over other VDFs and filter banks.

• Conventional energy detector.

• A proposed tunable decision making policy, ρ t rand , for orthogonalization of secondary users to access optimum subbands. The proposed policy, ρ t rand , differs from existing Multi-Armed Bandit and randomization based policy, ρ rand [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF], in two ways: 1) ρ t rand incorporates an efficient approach to accurately and quickly identify the optimum subset of contiguous, orthogonal vacant subbands of any desired bandwidth from a large number of non-orthogonal subbands that can be filtered using the IMFT-VDF, 2) In ρ t rand , the size of the subband subset is made tunable, as a function of the learned subband vacancy statistics, desired vacant bandwidth and the estimated probability of collisions, compared to a fixed subset size in ρ rand . Both refinements lead to fewer number of collisions and hence, lower SSC.

The simulation results support our claims about the superiority of the proposed decentralized DSLA scheme over the existing decentralized DSLA schemes in terms of spectrum utilization efficiency and SSC. Note that the proposed scheme is not just a straightforward integration of the VDF and existing policy ρ rand . In fact, it is designed by exploiting the exclusive unabridged center frequency control offered by the IMFT-VDF and adapting the ρ rand to efficiently work with the IMFT-VDF.

The work presented in this paper is a significant extension of our preliminary works in [START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF]. For instance, the proposed IMFT-VDF offers complete control over the bandwidth as well as center frequency of bandpass responses compared to the coarse and limited control offered by our VDF in [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF]. Furthermore, the design, architecture as well as gate count complexity are discussed in detail in this paper. Our DSLA scheme in [START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF] was designed for CRNs consisting of single secondary user, which is now extended for more practical and challenging scenarios of multi-standard multi-user CRNs. The rest of the paper is organized as follows. The details of network model and the literature review of different stages of DSLA scheme are presented in Section 2. In Section 3, the design and architecture of the proposed IMFT-VDF are discussed along with suitable design examples and gate count complexity comparisons. The proposed policy, ρ t rand , is presented in Section 4 followed by the simulation results and complexity comparisons in Section 5. Section 6 concludes the paper.

Network Model and Literature Review

Consider the slotted CRN consisting of M secondary users. The desired vacant bandwidth of any k th secondary user is denoted by

B v (k), k ∈ {1, 2, .., M }.
The wideband input signal consists of at most N p active primary users with channel bandwidths, B p (j), j ∈ {1, 2, .., N p }, which are integer multiples of the smallest channel bandwidth, B cmin such that N c = (1/B cmin ), j B p (j) ≤ 1 and M < N c . Note that all the frequency specifications are normalized with respect to half the sampling frequency. Let P vac (i), i ∈ {1, 2, .., N c } be the probability of vacancy for each subband of bandwidth B cmin . It is assumed to evolve as an i.i.d (independent and identically distributed) Bernoulli random process with time, t, stationary and unknown to secondary users [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF][START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF].

Within each secondary user, DSLA scheme can be implemented through filtering stage, detection stage and decision making stage as illustrated in Fig. 1 [START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF]. Depending on the desired vacant bandwidth, B v (k), the decision making stage of corresponding k th secondary user instructs the filtering stage to extract the desired frequency subband of interest via parameters, B v (k) and f center (k) (for center frequency). The extracted subband is then sensed using an imperfect detector i.e., sensing errors may occur. Due to resource and hardware constraints of secondary user terminal, it is assumed that each secondary user can sense only one subband at a time.

Digital

T l k ,k (t) = t. Let X l k ,k (t)
denote the number of time slots upto t slots during which transmission over l th k subband by k th user is successful. Let S * t and S t denote the total rewards of the genie-aided policy and the decentralized DSLA policy, respectively. Here, a genie-aided policy means the one where subband vacancy statistics are known a priori and a central unit allocates the secondary users to the M orthogonal optimum subbands. Then, the total regret (i.e., loss due to the selection of occupied subbands and collisions among secondary users), U t , of the CRN up to time t is given by Eq. 1 [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF] and it should be as small as possible.

U t = S * t -S t = M k=1 t-1 x=0 [r * x (k) -r x (k)] (1)
Next, the filtering and decision making stages of decentralized DSLA scheme are discussed in detail.

Filtering Stage

The task of the filtering stage is to extract the frequency subband of any desired bandwidth, B v (k) and center frequency, f center (k). The filtering task is done jointly in the analog front-end and digital front-end of user terminal. Taking into account inter-standard as well as intra-standard channel bandwidth variations and limited flexibility of analog filters compared to its digital counterpart, the analog front-end performs only coarse filtering task and the digital front-end is responsible for the stringent subband filtering task. The digital filtering stage is realized either using digital filter or filter bank such as memory based filters, VDFs, discrete Fourier transform filter bank (DFTFB), fast filter bank etc [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF][START_REF] Yu | Design of Variable Bandedge FIR Filters with Extremely Large Bandedge Variation Range[END_REF][START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF][START_REF] Dhabu | Design of Low Complexity Variable Digital Filter with Large Cutoff Frequency Range based on Second Order Frequency Transformation and Interpolation[END_REF][START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF][START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF][START_REF] Deng | Closed Form Design and Efficient Implementation of Variable Digital Filters with Simultaneously Tunable Magnitude and Fractional Delay[END_REF][START_REF] Yan | A Unified Framework for Designing FIR Filters with Arbitrary Magnitude and Phase Response[END_REF][START_REF] Darak | Efficient Implementation of Reconfigurable Warped Digital Filters with Variable Lowpass, Highpass, Bandpass and Bandstop Responses[END_REF][START_REF] Ambede | A Low Complexity Uniform and Non-Uniform Digital Filter Bank based on an Improved Coefficient Decimation Method for Multi-Standard Communication Channelizers[END_REF][START_REF] Abo-Zahhad | A Novel Algorithm for the Design of Selective FIR Filters with Arbitrary Amplitude and Phase Characteristics[END_REF][START_REF] Koshita | High-performance Variable Band-pass/Band-stop State-space Digital Filters using Gramianpreserving Frequency Transformation[END_REF][START_REF] Pucker | Channelization Techniques for Software Defined Radio[END_REF][START_REF] Lim | Fast Filter Bank (FFB)[END_REF][START_REF] Yli-Kaakinen | Optimization of Flexible Filter Banks Based on Fast-Convolution[END_REF].

Filter Banks

The filter bank [START_REF] Pucker | Channelization Techniques for Software Defined Radio[END_REF][START_REF] Lim | Fast Filter Bank (FFB)[END_REF][START_REF] Yli-Kaakinen | Optimization of Flexible Filter Banks Based on Fast-Convolution[END_REF] and multiple detectors based decentralized DSLA schemes allow quick learning and access to the optimum vacant subband of desired bandwidth, since multiple subbands can be sensed simultaneously. Though such schemes are popularly used in base station receivers, they do not offer an area and power efficient solution for battery-operated resource-constrained secondary user terminals. This is because, the smaller the value of desired subband bandwidth B v (k), higher is the desired resolution (i.e. number of subbands) and hence, higher is the gate count complexity of the filter bank. Furthermore, such terminals can afford only one detector due to the imposed power constraints for longer battery life and hence, only one subband or multiple contiguous subbands in conjunction can be sensed at a time. Since, the other extracted subbands are not used, filter bank is not utilized to its full potential. Also, filter banks are not efficient for wideband input signal with non-contiguous bandwidth. In such cases, VDFs offer an efficient alternative.

Variable Digital Filters

In memory based digital FIR filters, filter coefficients corresponding to all desired bandpass response specifications are stored in memory beforehand. Such filter based scheme is area and power efficient for secondary user terminals due to fewer number of multipliers and adders. However, filter coefficients need to be updated frequently whenever B v (k) and/or f center (k) changes, which incurs huge penalty in terms of sensing time due to multiple memory read/write operations. Higher sensing time means less time for data communication and hence, lower spectrum throughput. Furthermore, memory size increases with the increase in the granularity of B v (k) and f center (k) and is very large. As a result, DSLA using memory based FIR filter is suitable for CRN supporting single communication standard, i.e., B v (k) is fixed and f center (k) is limited to a set of fewer discrete values.

The VDF provides on-the-fly control over B v (k) and f center (k) of the bandpass response through a small number of parameters [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF][START_REF] Yu | Design of Variable Bandedge FIR Filters with Extremely Large Bandedge Variation Range[END_REF][START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF][START_REF] Dhabu | Design of Low Complexity Variable Digital Filter with Large Cutoff Frequency Range based on Second Order Frequency Transformation and Interpolation[END_REF][START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF][START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF][START_REF] Deng | Closed Form Design and Efficient Implementation of Variable Digital Filters with Simultaneously Tunable Magnitude and Fractional Delay[END_REF][START_REF] Yan | A Unified Framework for Designing FIR Filters with Arbitrary Magnitude and Phase Response[END_REF][START_REF] Darak | Efficient Implementation of Reconfigurable Warped Digital Filters with Variable Lowpass, Highpass, Bandpass and Bandstop Responses[END_REF][START_REF] Ambede | A Low Complexity Uniform and Non-Uniform Digital Filter Bank based on an Improved Coefficient Decimation Method for Multi-Standard Communication Channelizers[END_REF][START_REF] Abo-Zahhad | A Novel Algorithm for the Design of Selective FIR Filters with Arbitrary Amplitude and Phase Characteristics[END_REF][START_REF] Koshita | High-performance Variable Band-pass/Band-stop State-space Digital Filters using Gramianpreserving Frequency Transformation[END_REF]. The design of linear phase bandpass VDF for decentralized DSLA is a challenging research problem and highly nontrivial because of the fixed transition bandwidth (TBW) requirement as well as area, power and delay constraints.

A number of bandpass VDF designs are available in the literature [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF][START_REF] Yu | Design of Variable Bandedge FIR Filters with Extremely Large Bandedge Variation Range[END_REF][START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF][START_REF] Dhabu | Design of Low Complexity Variable Digital Filter with Large Cutoff Frequency Range based on Second Order Frequency Transformation and Interpolation[END_REF][START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF][START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF][START_REF] Deng | Closed Form Design and Efficient Implementation of Variable Digital Filters with Simultaneously Tunable Magnitude and Fractional Delay[END_REF][START_REF] Yan | A Unified Framework for Designing FIR Filters with Arbitrary Magnitude and Phase Response[END_REF][START_REF] Darak | Efficient Implementation of Reconfigurable Warped Digital Filters with Variable Lowpass, Highpass, Bandpass and Bandstop Responses[END_REF][START_REF] Ambede | A Low Complexity Uniform and Non-Uniform Digital Filter Bank based on an Improved Coefficient Decimation Method for Multi-Standard Communication Channelizers[END_REF][START_REF] Abo-Zahhad | A Novel Algorithm for the Design of Selective FIR Filters with Arbitrary Amplitude and Phase Characteristics[END_REF][START_REF] Koshita | High-performance Variable Band-pass/Band-stop State-space Digital Filters using Gramianpreserving Frequency Transformation[END_REF]. The allpass transformation (APT) based VDF [START_REF] Darak | Efficient Implementation of Reconfigurable Warped Digital Filters with Variable Lowpass, Highpass, Bandpass and Bandstop Responses[END_REF] offers complete control over the center frequency of bandpass responses but the phase is non-linear which limits its usefulness in DSLA scheme. The frequency response masking based VDFs [START_REF] Yu | Design of Variable Bandedge FIR Filters with Extremely Large Bandedge Variation Range[END_REF] offer complete control over the cut-off frequency of lowpass responses and are suitable for applications requiring sharp TBW. However, their usefulness in the case of bandpass responses with complete control over B v (k) and f center (k) has not been explored and the group delay of the [START_REF] Yu | Design of Variable Bandedge FIR Filters with Extremely Large Bandedge Variation Range[END_REF] is very high. The first and second order frequency transformation based VDFs [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF][START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF][START_REF] Dhabu | Design of Low Complexity Variable Digital Filter with Large Cutoff Frequency Range based on Second Order Frequency Transformation and Interpolation[END_REF] and coefficient decimation method (CDM) based VDFs [START_REF] Ambede | A Low Complexity Uniform and Non-Uniform Digital Filter Bank based on an Improved Coefficient Decimation Method for Multi-Standard Communication Channelizers[END_REF] offer only coarse control over the B v (k) and f center (k). The spectral parameter approximation based VDFs (SPA-VDFs) [START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF][START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF][START_REF] Deng | Closed Form Design and Efficient Implementation of Variable Digital Filters with Simultaneously Tunable Magnitude and Fractional Delay[END_REF] have the advantages of fixed TBW, fewer variable multipliers and lower group delay. However, the gate count complexity of these VDFs is very high and large dynamic range of filter coefficients may impose constraints when fixed-point implementation is desired. Though the SPA-VDF in [START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF] and our SPA-MCDM-VDF [START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF] overcome some of these drawbacks, their gate count complexity is still high.

Decision Making Stage

Various decision making policies have been proposed for the decentralized DSLA in single user as well as multi-user CRNs [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF][START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF][START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF]. These policies have shown to outperform the conventional random selection based policies under self-play i.e. when implemented at all secondary users [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF][START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF][START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF]. They mainly differ in the type of online learning algorithms they employ for decision making such as ε-greedy algorithm, Multi-Armed Bandit based upper confidence bound (UCB) algorithms such as U CB 1 , U CB H , U CB t , U CB v etc. [START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF] or combination of these algorithms [START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF] as well as different strategies for orthogonalization of secondary users [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF]. Analytically, the UCB based algorithms [START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF] are mathematically proved to be optimal with logarithmic regret bound, which is the best we can expect when secondary users do not have any information about P vac (i) ∀i. Hence, the discussion in this paper is limited to the UCB algorithm.

CRN with Single Secondary User (M =1)

The direct application of UCB algorithm for single user CRN (M =k=1) is discussed in [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF][START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF][START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF][START_REF] Jouini | Upper Confidence Bound Algorithm for Opportunistic Spectrum Access with Sensing Errors[END_REF]. In brief, secondary user sequentially but randomly senses all N r (1) subbands once at the beginning. Then, for t > N r (1), the subband, I t (1), with the maximum value of g-statistic, B l1,1 (t), is chosen [START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF].

The g-statistic, B l1,1 (t), for channel l 1 and time t is the sum of the learned mean availability of the subband, Xl1,1 (t) (exploitation factor) and UCB-bias, A l1,1 (t) (exploration factor) [START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF]. Mathematically [START_REF] Auer | Finite-time Analysis of the Multiarmed Bandit Problem[END_REF],

I t (1) = arg max l1 B l1,1 (t) (2)
where

B l1,1 (t) = X l1,1 (t) T l1,1 (t) Xl 1 ,1 (t) + 2 • ln(t) T l1,1 (t) A l 1 ,1 (t) ∀l 1 (3)
In [START_REF] Darak | Dynamic Spectrum Access with Tunable Bandwidth for Multi-standard Cognitive Radio Receivers[END_REF], we proposed DSLA scheme using UCB to support the tunable vacant bandwidth requirements for CRNs with single secondary user.

CRN with Multiple Secondary Users (M >1)

For practical scenario where CRN consists of M (> 1) secondary users, notable research on the integration of learning algorithms such as UCB algorithm with different access strategies for orthogonalization of secondary users to optimum subbands has been pursued in [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF].

In particular, the ρ rand policy in [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF], achieves orthogonalization by randomly and independently assigning rank to each secondary user. For example, k th secondary user with the rank, p k ∈ {1, 2, ..M }, selects the subband having p th k maximum value of g-statistic, B l k ,k (t). When collision occurs, the corresponding secondary users randomly and independently change their rank. Another access policy in [START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF] follows time division fare share approach where the rank of each secondary user is not fixed but is updated in round robin fashion to allow equal access to optimum subbands among all secondary users. Analytically, both the policies are simple, order-optimal and their regret, U t , is logarithmic with time t. However, the SSC of the policy in [START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF] increases linearly with t since each secondary user switches the subband in each time slot compared to the policy in [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF] where subband switching occurs only when secondary user experiences collision. Thus, policy in [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF] is preferred over the policy in [START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF].

The mixed strategy based policy in [START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF] employs soft-max action selection method that uses the Boltzmann distribution. The performance of this method is similar to [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF] but its SSC is high and convergence has not been proved analytically. The policy in [START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF] is designed by combining access strategies in [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF] with two learning algorithms to take into account different P vac distributions. However, the SSC of [START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF] is high for the P vac distributions where it performs better than [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF]. The proposed policy is discussed later in Section 4.

Proposed IMFT-VDF

The IMFT-VDF is the first contribution of this paper and constitutes the filtering stage of the proposed decentralized DSLA scheme shown in Fig. 1.

IMFT-VDF Design

Consider a linear-phase prototype lowpass filter, H(Z), of order 2N and symmetric filter coefficients h n , n ∈ {0, 1, .., 2N }, i.e., h n =h 2N -n ∀n. The transfer function of the prototype filter, H(Z), in Taylor form is given as [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF],

H(Z) = N n=0 a n Z -N Z + Z -1 2 (4)
where the coefficients a n are related to the impulse response coefficients h n through the Chebyshev polynomials [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF]. The second order frequency transformation based VDF, H 2 (z), is then given by [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF],

H 2 (z) = N n=0 a n Z -N D(z) (5)
where Z → z represents the frequency transformation [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF]. The term D(z) denotes the second order frequency transformation and is given by [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF],

D(z) = Z + Z -1 2 = 2 v=0 A v z + z -1 2 v ( 6
)
where A v , v ∈ {0, 1, 2}, are the controlling parameters of D(z) and they decide the cut-off frequency and the TBW of the H 2 (z) [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF]. Substituting Eq. 6 into Eq. 5, we get

H 2 (Z) = N n=0 a n z -2(N -n) 2 v=0 A v z v-2 1 + z -2 2 v n (7)
Let ω c and Ω c be the cut-off frequencies of the H(Z) and H 2 (z), respectively. Similarly, T BW p and T BW 2f t are the TBW of the H(Z) and H 2 (z), respectively. Then, Ω c and T BW 2f t are given by [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF],

Ω c = cos -1 -A 1 ± A 2 1 -4A 2 (A 0 -cosω c ) 2A 2 (8)
T BW 2f t = T BW p • A 1 sinΩ c + A 2 sin(2Ω c ) sinω c (9)
if the following constraints are met [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF]:

A 0 + A 1 + A 2 = 1 (10a) 0 ≤ A 1 ≤ 1 (10b) A 2 1 -4A 2 (1 -A 1 -A 2 -cosω c ) ≥ 0 (10c)
In [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF], A 1 = 1 and hence, according to Eq. 10(a), A 2 = -A 0 , which eventually leads to reduction in the number of variable multipliers from 3 to 1. However, the detailed analysis of Eq. 8 and Eq. 10 indicate that, by restricting A 1 = 1, the cut-off frequency range of H 2 (z) is limited to approximately 25% of the Nyquist band. This in turn narrows the bandwidth and the center frequency range of the bandpass responses. Our modified second order frequency transformation based VDF in [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF] allows A 1 to take any reciprocal of power-oftwo value instead of unity. The resultant VDF is then combined with the CDM to obtain variable lowpass, highpass, bandpass and bandstop responses from a fixed-coefficient lowpass prototype filter. However, it can not provide complete control over the center frequency and the bandwidth of the bandpass response over the entire Nyquist band. Furthermore, the TBW deteriorates significantly at higher cut-off frequencies. To overcome these drawbacks, especially for the DSLA application, a new IMFT-VDF based on the modified frequency transformation and interpolation technique is proposed. The main differences between IMFT-VDF and our VDF in [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF] are :

1. The parameter A 1 is tunable as against to fixed A 1 in [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF] but restricted to the reciprocal of power-of-two values and zero. 2. The architecture consists of two branches of D(z) (discussed in detail later in Section 3.3) compared to single branch of D(z) in [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF].

3. The VDF in [START_REF] Darak | Design of Variable Linear Phase FIR Filters Based on Second Order Frequency Transformations and Coefficient Decimation[END_REF] is combined with the CDM to obtain variable bandpass responses from variable lowpass responses. In IMFT-VDF, the interpolation and conventional lowpass to highpass conversion by reversing the sign of every alternate coefficient are used to increase the tunable cut-off frequency range of lowpass responses.

IMFT-VDF Design Procedure

Consider the design of the IMFT-VDF which provides complete control over the cut-off frequency of lowpass response from Ω c1 to Ω c2 with the desired TBW of T BW d , desired passband and stopband ripples of δ pd and δ sd , respectively. The desired bandpass response with lower and upper cut-off frequencies Ω bpc1

and Ω bpc2 , respectively, is obtained by subtracting a lowpass response with cutoff frequency of Ω bpc1 from another lowpass response with cut-off frequency of Ω bpc2 using the same prototype filter and two branches of D(z). Without loss of generality, we have Ω bpc1 ≥ Ω c1 , Ω bpc2 ≤ Ω c2 , Ω c1 ≈ 0 and Ω c2 ≈ 1. The design steps of IMFT-VDF are:

1. The Ωc1 is chosen such that Ωc1 = min{Ω c1 , (1-Ω c2 )} and Ωc2 = (1-Ωc1 )

to maintain the symmetry with respect to center of the Nyquist band. 2. The lowpass cut-off frequency range from Ωc1 to Ωc2 is divided into two equal parts : a) Ωc1 to 0.5, b) 0.5 to Ωc2 . 3. For the desired frequency specifications of T BW d , δ pd and δ sd , the cut-off frequency of the prototype filter, ω c , the parameters A 1 and A 2 are chosen using the Eq. 8-10 such that lowpass responses with desired frequency specifications and cut-off frequency range from Ωc1 to 0.5 are obtained. This is a simple optimization problem to obtain Ωc1 to be as close to Ωc1 as possible with the constraints of meeting the desired frequency specifications. 4. When Ωc1 ≤ Ωc1 , interpolation is not required. Otherwise, lowpass responses with cut-off frequency Ω c where Ωc1 ≤ Ω c ≤ Ωc1 or Ωc2 (= 1 -Ωc1 ) ≤ Ω c ≤ Ωc2 are obtained using conventional interpolation technique. Corresponding interpolation factor is p ≈ ( Ωc1 / Ωc1 ) such that p ∈ {2, 4, 8...} and masking filter, H m (z), is required to mask the undesired subbands. 5. Finally, the lowpass response with 0.5 ≤ Ω c ≤ Ωc2 is obtained from the corresponding lowpass response with cut-off frequency of (1-Ω c ) by changing the sign of every alternate filter coefficients and then taking the complementary response. Note that Ωc1 ≤ (1 -Ω c ) ≤ 0.5.

The IMFT-VDF is not just a straightforward integration of the frequency transformation and interpolation. In fact, the IMFT-VDF is carefully designed by exploiting the architectural advantages such as lower deterioration in TBW and lower multiplier complexity of the proposed second order modified frequency transformation as well as exclusive multiband response capability of the conventional interpolation technique, without compromising on the total gate counts. The architecture of the IMFT-VDF is discussed in the next section.

IMFT-VDF Architecture

The architecture of the IMFT-VDF is shown in Fig. 2 where the modified second order frequency transformation is realized through D(z) shown in Fig. 3. The IMFT-VDF architecture consists of prototype filter of order 2N with coefficients, a 0 , a 1 ,..,a N , obtained from the original filter coefficients, h 0 , h 1 ,..h 2N , as discussed in Section 3.2. These coefficients are fixed and can be hardwired. In the IMFT-VDF architecture, the Taylor structure of filter implementation is redesigned by shifting each D(z) block adjacent to the corresponding delay block. This is done to obtain multiple lowpass responses, each with distinct cut-off frequency, using single fixed-coefficient prototype filter and multiple branches of D(z). The proposed structure is referred to as transposed Taylor structure.

Even though the number of delay blocks are higher in the transposed Taylor structure compared to conventional Taylor structure [START_REF] Roy | Frequency Transformations for Linear-phase Variable-Cutoff Digital Filters[END_REF], the total group delay of both structures is equal for a given N . Moreover, in the FPGA implementation of the VDF, delays can be implemented without significant hardware cost. The D(z) is given by,

D(z) = A 0 z -2 + A 1 1 + z -2 2 + A 2 1 + z -2 2 2 (11)
From Eq. 10(a), we have A 0 =1-A 1 -A 2 . Substituting this relationship into Eq. ( 11) and simplifying, we obtain

D(z) = z -2 + A 1 1 + z -2 2 z -1 -z -2 + A 2 1 + z -2 2 2 -z -2 (12)
Using Eq. 12 and interpolation factor p, D(z) can be efficiently implemented as shown in Fig. The output logic unit is depicted in Fig. 4. It consists of N th m (N m << N ) order fixed-coefficient masking filter, H m (z), to mask the undesired subbands in multiband response when p > 1 as discussed in Section 3.2. The cut-off frequency, TBW, passband and stopband ripples of H m (z) are (1/p), 2[(1/p) -Ωc1 -(0.5 • T BW d )], δ pd and δ sd , respectively. The masking filter outputs are denoted by l (response obtained using lowpass masking filter with cut-off frequency of (1/p)), h (response obtained using highpass masking filter with cut-off frequency of [1 -(1/p)]), h c (response obtained using lowpass masking filter with cut-off frequency of [1 -(1/p)]), and I (appropriately delayed version of input signal). In Fig. 4, y 1c is a complementary response of y 1 obtained by subtracting y 1 from an appropriately delayed version of the input signal. The multiplexer control signal, sel y, is a 6 bit signal which controls two 2:1 multiplexers and two 4:1 multiplexers as shown in Fig. 4. The IMFT-VDF architecture provides variable lowpass, highpass, bandpass and bandstop responses using fixed-coefficient lowpass filter. The control signals to obtain variable bandpass responses are given in Table 1 where "X" denotes "don't care"condition. Similarly, variable lowpass responses can be obtained using only a single branch of D(z) with output y 2 . The corresponding values of sel y are {101100,101111,101111,101110} for cut-off frequency lying in the range Ωc1 to Ωc1 , Ωc1 to 0.5, 0.5 to Ωc2 and Ωc2 to Ωc2 , respectively. The bandstop and highpass responses can be obtained as the complementary responses of the corresponding bandpass and lowpass responses. and T BW d = 0.01π with δ sd = -50 dB and δ pd = 0.1 dB, the gate count requirements of memory based FIR filter, MCDM-VDF [START_REF] Ambede | A Low Complexity Uniform and Non-Uniform Digital Filter Bank based on an Improved Coefficient Decimation Method for Multi-Standard Communication Channelizers[END_REF], non-linear phase APT-CDM-VDF [START_REF] Darak | Efficient Implementation of Reconfigurable Warped Digital Filters with Variable Lowpass, Highpass, Bandpass and Bandstop Responses[END_REF], SPA-VDF [START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF], SPA-MCDM-VDF [START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF] and IMFT-VDF are given in Table 2. The term "NLP"in Table 2 stands for non-linear phase. For the first design example, the order of an FIR filter is 90 and coefficients 395 corresponding to different bandpass responses are stored in memory beforehand.

The MCDM-VDF consists of prototype filter of order 2000 and decimation factor, D, varies from 1 to 25. Our APT-CDM-VDF consists of prototype filter of order 160 and variable responses are obtained by integrating the first order APT with the reduced second order APT. The SPA-VDF consists of 2 subsets of 6 sub-filters, each of order 90. Our SPA-MCDM-VDF consists of 6 sub-filters each of order 90 and a masking filter of order 14.

The IMFT-VDF offers total gate count savings of {30%, 87%, 18%, 60%, 37%} and {22%, 78%, 11%, 55%, 27%} over FIR filter, MCDM-VDF [START_REF] Ambede | A Low Complexity Uniform and Non-Uniform Digital Filter Bank based on an Improved Coefficient Decimation Method for Multi-Standard Communication Channelizers[END_REF], APT-CDM-VDF [START_REF] Darak | Efficient Implementation of Reconfigurable Warped Digital Filters with Variable Lowpass, Highpass, Bandpass and Bandstop Responses[END_REF], SPA-VDF [START_REF] Löwenborg | Minimax Design of Adjustable Bandwidth Linear-Phase FIR Filters[END_REF] and SPA-MCDM-VDF [START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF] for the first and second design examples, respectively. The group delay of the IMFT-VDF is slightly higher than other VDFs. However, it must be taken into account that the total latency of the physical layer in each secondary user terminal is small compared to the latency of the CRNs [START_REF] Xu | Decision-Theoretic Distributed Channel Selection for Opportunistic Spectrum Access: Strategies, Challenges and Solutions[END_REF][START_REF] Bkassiny | A Survey on Machine-Learning Techniques in Cognitive Radios[END_REF]. This means that little penalty in group delay is acceptable for more than 27% savings in total gate count. For delay sensitive applications, our SPA-MCDM-VDF [START_REF] Darak | Linear Phase VDF Design with Unabridged Bandwidth Control over the Nyquist Band[END_REF] offers good alternative at the cost of higher gate count.

Proposed Decision Making Policy, ρ t rand

The second contribution of this paper is a tunable decision making policy, ρ t rand , designed to take full advantage of exclusive bandwidth and center frequency control offered by IMFT-VDF. The objective is to strike a balance between the number of collisions among secondary users and the total time spent on the non-optimum subbands, in order to achieve lower regret, S t as well as lower SSC for different distributions of P vac (i) ∀i and distinct B v (k) ∀k requirements in heterogeneous decentralized CRNs. It is assumed that all secondary users employ the same policy but do not exchange any information with other users.

The advantages and disadvantages of different policies are discussed in Section 2.2. Accordingly, the proposed policy, ρ t rand , is designed via three refinements to existing policy, ρ rand [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF], which are: 1) Low complexity tunable filtering stage (discussed in Section 4.1), 2) An efficient approach to identify the optimum subset of contiguous orthogonal subbands (discussed in Section 4.2), 3) Tunable subset size (discussed in Section 4.3).

Low Complexity Tunable Filtering Stage

In the literature, the input signal scenarios with N c ≤ 16 are considered.

As per the assumption of stationary vacant subband statistics, the subband bandwidth must be narrow which means that the total bandwidth of the input signal, when N c ≤ 16, is not sufficiently wide. On the other hand, the practical scenarios usually involve wideband input signal and the requirement of tunable

B v (k) indicates large N c (≫ 16).
When the filtering stage of each secondary user is implemented using filter bank, the desired resolution (i.e., number of subbands) of the filter bank is equal to (1/B v (k)). For example, if N c =16 and B v (k) = B cmin (or B v (k) = 2• B cmin , respectively), then 16 subband (or 8 subband, respectively) filter bank is required. To support tunable bandwidth requirement, an efficient approach would be to implement N c subband filter bank and then, (N c /2), (N c /4) or (N c /8) subband filter bank responses can be obtained by combining adjacent subbands. However, higher the value of N c , higher the resolution and hence the gate count, power consumption as well as group delay of the filter bank are. Hence, filter bank is not an area and power efficient choice when N c is large (≫ 16). We propose a low complexity solution by replacing the filter bank with the IMFT-VDF. The gate count, power consumption and group delay of the IMFT-VDF are independent of N c and B v (k).

Subsets of Orthogonal Subbands

In the existing policies [8-11, 13, 14], it is assumed that, B v (k) = B cmin , ∀k, and hence, N r (k) = N c ∀k, i.e., all the subbands are orthogonal. This means that the performance of existing schemes [8-11, 13, 14] The IMFT-VDF, due to its finer granularity of center frequency of bandpass responses, offers large number of subsets of orthogonal contiguous subbands. Higher the number of subsets, higher is the time spent in the non-optimum subbands and hence, higher is the regret, U t . Thus, the number of subbands choices are limited to two in the proposed policy where the first subset starts with the subband numbered as 1 while the second subset starts with the subband numbered as (⌈B v (k)/2 • B cmin ⌉+1), except the case B v (k) = B cmin where only one subset is available. The idea behind such subset division is to keep less than 50% overlapping between the corresponding subbands of two subsets so that the subsets are sufficiently apart in terms of their combined vacancy statistics which in turn leads to fewer errors in the selection of optimal subset by secondary users.

In the proposed ρ t rand policy, all M secondary users independently learn the vacancy statistics of these two subsets during initial phase spanning up to t s (k) time slots. This means that the secondary user spends half the number of time slots, i.e., (t s (k)/2) slots, in the first subset and remaining half in other subset. Then, at t = t s (k)+1, secondary user independently choses the optimum subset based on the past sensing events. The number of time slots, t s (k), needs to be chosen carefully since very small or very large value of t s (k) would lead to the higher U t as well as higher SSC as discussed before. Based on empirical observations, we have,

t s (k) = 2λ • L t (k), ∀k (13)
where L t (k) is the tunable subset size for k th user (discussed in Section 4.3) and λ is a constant between 5 to 10.

Tunable Subset Size, L t

In the existing policies for multi-user CRNs [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF], the size of the subset, comprising of optimum subbands, is fixed, i.e., L t (k) = M ∀k. This means that the distinct and tunable vacant bandwidth requirements of each secondary user as well as the different types of distributions of P vac are not taken into consideration while choosing L t (k). Also, the number of collisions and hence, SSC are bound to be large in the beginning as well as for higher M since it is unlikely that secondary users will always agree on the order of the optimum subbands [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF]. The solution to these issues can be provided by making the subset size, L t (k) ∀k, tunable as discussed below.

The minimum and maximum subset sizes, L tmin (k) and L tmax (k), respectively, for k th secondary user are given by,

L tmin (k) = M u=1 N c • B v (u) N c • B v (k) (14)
L tmax (k) = N c • B cmin B v (k) -e (15)
where e=1 if the first subset is chosen, i.e. sub(k)=1, otherwise e=0. It can be observed that L tmin (k) = L tmax (k) = M when B v (k) = B cmin ∀k, same as that in [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF][START_REF] Liu | Distributed Learning in Multi-Armed Bandit with Multiple Players[END_REF][START_REF] Wang | Exploration vs Exploitation for Distributed Channel Access in Cognitive Radio Networks: A Multi-user Case Study[END_REF][START_REF] Zandi | Decentralized Spectrum Learning and Access Adaptive to Channel Availability Distribution in Primary Network[END_REF].

If L t (k) ≈ L tmin (k), then the number of collisions among secondary users, U t and SSC are all high. On the other hand, very large value of L t (k) ≈ L tmax (k), offers lesser number of collisions and lower SSC, but leads to higher regret U t due to the selection of the sub-optimum subbands i.e. the subbands with lower P vac . By taking into account the fact that the U t and SSC due to the collisions are much worse than the same due to the selection of the sub-optimum subbands in the DSLA scenario compared to conventional machine learning scenarios, a new method to make the value of L t (k) ∀k tunable is proposed as follows:

1. For each integer value from L tmin (k) to L tmax (k), the total predicted reward, St (k, q k ) ∀k and q k ∈ {1, 2, .., (L tmax (k) -L tmin (k) + 1)}, at any given time t, are calculated as,

St (k, q k ) = Ltmin(k)+q k -1 u=1 Pvac (k, j k,u ) • L tmin (k) L tmin (k) + q k -1 (16)
where

j k,u = sub(k) + (u -1) • B v (k) B cmin (17)
Here Pvac (k, j k,u ) ∀u denote the learned subband availability statistics arranged in the descending order, i.e., Pvac (k, 1)

≥ Pvac (k, 2) ≥ • • • ≥ Pvac (k, L tmax (k)).
2. For each integer value from L tmin (k) to L tmax (k), the total predicted number of collisions, Ct (k, q k ) ∀k and ∀q k , at any given time t, are calculated as

Ct (k, q k ) = 1 - (L tmin (k) + q k -1)! (q k -1)!(L tmin (k) + q k -1) Ltmin(k) (18)
3. Finally, the optimum value of L t (k) is given by

L t (k) = L tmin (k) + s -1 (19)
where s = arg max

q k St (k, q k ) • t -Ct (k, q k ) • t (20)
Each secondary user independently calculates the value of L t (k). To minimize the computational complexity, the value of L t (k) is updated only when corresponding secondary user experiences collision with other secondary users. The proposed learning policy, ρ t rand , is described in detail in Algorithm 2.

Simulation Results and Complexity Analysis

In this section, we presents the simulation results and compare the performances of the proposed decentralized DSLA scheme (IMFT-VDF and ρ t-rand ) with the existing decentralized schemes (DFTFB and ρ rand [START_REF] Anandkumar | Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret[END_REF]) in terms of total vacant spectrum utilization (average reward), SSC and gate count complexity. The wideband input signal consists of 128 (=N c ) subbands i.e., B cmin = case 2, respectively.

Next, we consider the scenario where the desired vacant bandwidth is not same for all users i.e., B v (k) ≥ B cmin ∀k. In Fig. 7c and Fig. 7d, the plots of average reward, S t , in percentage vs. time slot t, are shown for case 1 and case 2, respectively. Here, M = 25 with the desired bandwidth of first 4 users is (1/256), next 2 users is (2/256), next 8 users is (3/256), next 3 users is (5/256) and (6/256) for the last user. It can be observed that the total reward (vacant spectrum utilization) of the proposed scheme is higher than other schemes. In Fig. 7f, the plot of average reward at t=20000 is plotted for different values of M . It can be observed that the proposed scheme outperforms other schemes for all M . Furthermore, the proposed scheme offers 8% and 17% lower average SSC than 128-DFTFB based scheme for case 1 and 2, respectively. The simulation results presented here are more realistic compared to those in the existing literature, since N c is very large and the desired vacant bandwidths, B v (k) ∀k, of secondary users are not constrained to B cmin . In addition, the scenarios of different distributions of P vac (i) ∀i are considered.

The total gate counts of 128-DFTFB and 64-DFTFB for the desired specifications of T BW d =0.06π, δ sd = -50 dB and δ pd = 0.1 dB, which corresponds to the prototype filter of order 90, are 1925070 and 873230, respectively. Then, according to Table 2, the IMFT-VDF, with total gate counts of 465385, offers 76% and 47% saving over 128-DFTFB and 64-DFTFB, respectively. However, the group delay of IMFT-VDF is 104 which is higher than the group delay of 45 in case of DFTFBs.

Conclusion and Future Works

In this paper, a low complexity and efficient dynamic spectrum learning and access (DSLA) scheme for next-generation heterogeneous decentralized cognitive radio networks is proposed. The proposed decentralized DSLA scheme consists of new low complexity modified second order frequency transformation and interpolation technique based variable digital filter and tunable decision making policy, ρ t rand . The simulation results and gate count complexity analysis verifies the superiority of the proposed scheme over existing schemes. The future research direction includes the efficient implementation of the proposed scheme on the USRP-GNU-FPGA platform and analyzing the performance using real radio signals.

  z )= Modified second order frequency transformation, *p= Interpolation factor.

Figure 2 :

 2 Figure 2: Architecture of the IMFT-VDF using transposed Taylor structure.

3 .

 3 The IMFT-VDF comprises of two parallel branches of D(z), controlled via signals S xp , C xn , A 1x and A 2x where x ∈ {1, 2}. Each branch provides either lowpass response (when p=1) or multiband response (when p >1) depending on the value of corresponding S xp . A 1x and A 2x control the values of parameters A 1 and A 2 , respectively. C xn controls adder-subtractor block where C xn =1 corresponds to the addition and C xn =-1 corresponds to the subtraction. In general, C xn = (-1) n when the desired cut-off frequency lies in the range 0.5 to Ωc2 . Otherwise, C xn = 1 ∀n. The outputs of these two branches, y 1 and y 2 , are given to the output logic unit.

Figure 3 :

 3 Figure 3: Modified second order frequency transformation, D(z).

Figure 4 :

 4 Figure 4: Output logic unit.

3. 4 .Figure 5 :

 45 Figure 5: Variable lowpass responses using IMFT-VDF with Ω c1 = 0.05π, Ω c2 = 0.95π, T BW d ≤ 0.06π, δ sd = -50 dB and δ pd = 0.1 dB.

Figure 6 :

 6 Figure 6: Input signal of bandwidth B with Pvac distribution.

Figure 7 :

 7 Figure 7: (a) Case 1: Reward St in % vs. time t for M =25 and Bv(k) =0.012 ∀k, (b) Case 2: Reward St in % vs. time t for M =25 and Bv(k) =0.012 ∀k, (c) Case 1: Reward St in % vs. time t for M =20 with Bv(k) ={(1/256),(1/256),(1/256),(1/256),(2/256),(2/256),(3/256),(3/256),(3/256),(3/256),(3/256), (3/256),(3/256),(3/256),(5/256),(5/256), (5/256),(6/256)}, (d) Case 2: Reward St in % vs. time t for M =20 and same Bv(k) as in (d), (e) Reward St in % vs. number of secondary users, M for t = 20000 and Bv(k) = 0.012 ∀k where C1 and C2 denote case 1 and case 2, respectively, (f) Reward St in % vs. number of secondary users, M for t = 20000 and different Bv(k).

Filtering Stage Detection Stage Decision Making Stage Wideband digitized input signal B B f center

  Let T l k ,k (t) denote the number of time slots where the subband l k ∈ {1,2, ..,N r (k)} is sensed by k th secondary user in t slots. Then, l k

	the number of subbands choices for the corresponding k th secondary user are
	N r (k) = [N c + 1 -⌈(B v (k)/B cmin )⌉] non-orthogonal subbands. When multiple
	secondary users transmit on the non-orthogonal subbands, collision occurs and
	no user is able to transmit successfully. When no collision occurs and the sub-
	band is vacant, it is assumed that the secondary user transmits successfully and
	gets a reward, r t (k)=⌈(B v (k)/B cmin )⌉ ∀k.
	Figure 1: Decentralized DSLA scheme in k th secondary user terminal.
	When B v (k) = B cmin , then the corresponding k th secondary user can choose
	any one of N r (k) = N c orthogonal subbands. However, when B v (k) > B cmin ,

Table 1 :

 1 Control Singals for Bandpass Response

	Ω bpc1	Ω bpc2	S 1p S 2p	C 1n	C 2n	sel y
		Ωc1 to Ωc1	p	p	1 ∀n	1 ∀n	X11100
	Ωc1 to Ωc1	Ωc1 to 0.5 0.5 to Ωc2	p p	1 1	1 ∀n 1 ∀n	1 ∀n (-1) n 001000 001000
		Ωc2 to Ωc2	p	p	1 ∀n	1 ∀n	X10101
		Ωc1 to 0.5	1	1	1 ∀n	1 ∀n	X11111
	Ωc1 to 0.5	0.5 to Ωc2	1	1	1 ∀n	(-1) n X11111
		Ωc2 to Ωc2	1	p	1 ∀n	1 ∀n	100010
	0.5 to Ωc2	0.5 to Ωc2 Ωc2 to Ωc2	1 1	1 p	(-1) n (-1) n X11111 (-1) n 1 ∀n 100010
	Ωc2 to Ωc2 Ωc2 to Ωc2	p	p	1 ∀n	1 ∀n	X11101

Table 2 :

 2 Gate Count Complexity Comparison

				TBW d = 0 .06 π			TBW d = 0 .01 π
				No. of		GroupTotal GroupTotal
	VDFs	Multi-Adders Mux'sMemoryDelay Gate	Delay Gate
							Count		Count
		pliers			Words
	FIR-	46	90	0	23000	45	668850 250	3651600
	Filter
	MCDM-	1003 8003	8002 0	1000 3685545 3500 12885545
	VDF [24]
	APT-	241	640	480	810	NLP 566650 NLP 3178850
	CDM-
	VDF [23]
	SPA-	562	1093	830	0	45	1174175 250	6346325
	VDF [19]
	SPA-	294	1109	564	0	52	739665 256	3898715
	MCDM-
	VDF [20]
	IMFT-	155	903	406	0	104	465385 622	2835085
	VDF

  under tunable vacant bandwidth requirements (B v (k) > B cmin ) and wideband input signal scenarios (N c ≫ 16) has not been analyzed yet.Consider an input signal bandwidth B as shown in Fig.6. It is divided into 16 (= N c ) uniform subbands, each with bandwidth B cmin = B/16 andP vac (i), i ∈ {1, 2, ..16}. When B v (k) > B cmin ,then the corresponding subbands of bandwidth B v (k) are non-orthogonal. For example, if B v (k)=2•B cmin ∀k, then there are 15 non-orthogonal subbands of bandwidth B v (k). Since each subband overlaps in frequency with at most [2 • (⌈B v (k)/B cmin ⌉ -1)] number of other subbands of bandwidth B v (k), clustering of non-orthogonal subbands into different subsets is necessary to reduce the number of collisions among secondary users. Furthermore, quick and accurate selection of the optimum subset will lead to smaller total regret, U t , and the SSC. For example, let us consider the two possible subsets of orthogonal subbands when B v (k)=2•B cmin :

	{(1-2),(3-4),...,(15-16)} or {(2-3), (4-5),...,(14-15)}. Using the relation P (AB) =
	P (A) + P (B) -P (A) • P (B), corresponding combined probability of vacancies
	are obtained as {0.39, 0.11, 0.19, 0.52, 0.62, 0.36, 0.87, 0.28} and {0.14, 0.03,
	0.21, 0.8, 0.06, 0.59, 0.84}, respectively. It can be easily observed that the
	second subset has higher number of optimum subbands than the first for this
	design example considered here. Thus, timely selection of the second subset will
	definitely lead to the lower regret, U t .

Acknowledgment

This work has received a French state support granted to the CominLabs excellence laboratory and managed by the National Research Agency in the "Investing for the Future"program under reference No. ANR-10-LABX-07-01. The authors would also like to thank Region Bretagne, France, for its support of this work.

Consider

. This scenario is similar to the one considered in the existing literature in the sense that the vacant bandwidth requirement of all users is identical. In Fig. 7a and Fig. 7b, the plots of average reward, S t , in percentage vs. time t, are shown for two different types of input distributions, case 1 (C1) and case 2 (C2), respectively. Here, M = 25, B v (k) = 0.012 ∀k and S t % = [100 • (S * t -S t )/S * t ] where S * t is the average reward of the genie-aided policy (i.e., centralized DSLA). It can be observed that the total reward (vacant spectrum utilization) of the proposed scheme is higher than other schemes. In Fig. 7e, the plot of average reward at t=20000 is plotted for different values of M . It can be observed that the proposed scheme outperforms other schemes for all M . Furthermore, the proposed scheme offers 15% and 26% lower average SSC than 128-DFTFB based scheme for case 1 and He was working as Assistant System Engineer in Tata Consultancy Services (TCS), Pune, India from September 2007 to December 2008. Currently, he is pursuing postdoctoral research at the CominLabs Excellence Center, Université Europèenne de Bretagne (UEB) and Supélec, Rennes, France. His current research interests include design of area and power efficient variable digital filters and reconfigurable filter banks as well as integration of such architectures with the efficient decision making and learning algorithms for various wireless communication applications.

Sumedh


Grégoire Lefebvre

Christophe Laurent

Julien Ros

Christophe Garcia

Supervised Image Classification by SOM Activity Map Comparison

This article presents a method aiming at quantifying the visual similarity between two images. This kind of problem is recurrent in many applications such as object recognition, image classification, etc. In this paper, we propose to use self-organizing feature maps (SOM) to measure image similarity. To reach this goal, we feed local signatures associated to salient patches into the neural network. At the end of the learning step, each neural unit is tuned to a particular local signature prototype. During the recognition step, each image presented to the network generates a neural map that can be represented by an activity histogram. Image similarity is then computed by a quadratic distance between histograms. This scheme offers very promising results for image classification with a percentage of 84.47% of correct classification rates.

Introduction

In many computer vision applications such as multimedia data mining, pattern recognition, etc., evaluating the inter-image similarity is fundamental. Whereas human beings are able to compare immediately two images, by automatically extracting discriminative image features, this project in computer vision stays unsolved.

Measuring the similarity between two images is a very challenging problem. In fact, the image similarity measurement is tightly linked with the image content representation. Three approaches appear in the literature. First, the image description can be global with one or more representations describing the whole content, in a compact structure. Classical color histograms [START_REF] Manjunath | Color and texture descriptors[END_REF] are an example of such an approach. In this case, image similarity is generally computed by classical Minkowski metrics. The second issue is based on a preliminary segmentation step. A low level descriptor is affected to each region. Image similarity corresponds thus to an attributed graph matching problem [START_REF] Berretti | Efficient matching and indexing of graph models in content-based retrieval[END_REF]. The third way to represent image content is to extract several interest points (IP) and to consider related patches, called regions of interest (ROI) [START_REF] Mikolajczyk | Scale and affine invariant interest point detectors[END_REF]. These salient points are considered as perceptually important and their neighborhood can be described by local descriptors. In this way, the whole image content is represented by a set of local regions. Due to the lack of order between detected salient areas, the similarity is determined by registration-based methods.

Based on some psycho-visual experiments [START_REF] Hoffman | The role of visual attention in saccadic eye movements[END_REF], our approach focuses on the last technique. Indeed, human vision system executes saccadic eye movements between salient locations to capture image content. Likewise, Tversky studies [START_REF] Tversky | Features of similarity[END_REF] showed that when we compare two images, we detect common and distinct concepts between these regions. Our method tries to reproduce this extraction and distinction concept with a codebook learning strategy based on SOM algorithm [START_REF] Kohonen | Self-Organizing Maps[END_REF] and an activity histogram distance. We firstly search salient locations in the images to be compared. Local visual features are then extracted from salient regions and projected onto a set of SOM-based learned visual prototypes, resulting in a visual activity map. This activity map is represented by a neural activity histogram coding the frequency of prototype appearance. By construction, the inter-bins similarity is embedded in this vector. Finally, we measure the image similarity by a quadratic distance [START_REF] Hafner | Efficient color histogram indexing for quadratic from distance functions[END_REF] between neural activity histograms. This distance allows to quantify common and distinct concepts between the compared images.

This method has been experimented for two kinds of applications: a supervised image classification problem where the system detects 75.60% of correct classification rules on a database containing 4200 photos for 5 categories; an adult content filtering method where the correct classification rate reaches 84.47%.

This paper is organized as follows: In Section 2, we first present our image classification scheme based on SOM learning from ROI descriptions. Then, Section 3 contributes to some experimental results. And finally, conclusions are discussed in Section 4.

2 Supervised Image Classification Scheme

System Architecture

As outlined in [START_REF] Duda | Pattern Classification[END_REF], a classification scheme is generally composed of three main steps: pre-processing, feature extraction and classification. In this paper, we mainly focus our attention on the two first items, the last being performed by a k-nn algorithm. In our approach, the pre-processing step consists of detecting some salient points in the image to be compared, reducing thus the zones of interest to a limited number of regions considered as been perceptually relevant. From each detected salient point, a salient patch is extracted (Figure 1) and a local feature vector is calculated. Each local feature vector is then fed into a SOM network resulting a neural activity map composed of all winning cells. Finally, to complete the feature extraction step, the obtained activity map is converted into a neural activity histogram measuring the frequency of SOM prototype appearance. Assessing the similarity between two images is then reduced to compute the quadratic distance [START_REF] Hafner | Efficient color histogram indexing for quadratic from distance functions[END_REF] between the associated neural activity histograms. Using this distance is offered by the topology preservation property of SOM [START_REF] Kohonen | Self-Organizing Maps[END_REF]. Indeed, as neighboring cells in the SOM network are tuned to similar visual features, an inter-bin similarity matrix can be easily constructed, collecting the similarity between SOM cells.

The different computational steps used in this method are detailed in the next sections.

Regions of Interest Detector

According to the active vision mechanisms, the goal of salient point detectors is to find perceptually relevant image locations. Many detectors have been proposed in the literature [START_REF] Harris | A combined corner and edge detector[END_REF] [2] [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF]. The salient locations selected by human visual system contain generally high contrast, lines and edges [START_REF] Hoffman | The role of visual attention in saccadic eye movements[END_REF]. Following this observation, we focus our interest on the Harris detector [START_REF] Harris | A combined corner and edge detector[END_REF], the contrast detector [START_REF] Bres | Detection of interest points for image indexation[END_REF] and a wavelet salient point detector [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF].

The Harris's detector [START_REF] Harris | A combined corner and edge detector[END_REF] aims at locating salient zones on corner by searching for the maxima of a function based on the local autocorrelation matrix of the signal. The second descriptor [START_REF] Bres | Detection of interest points for image indexation[END_REF] proposes to locate salient points in high contrasted area. For this purpose, a multi-resolution contrast pyramid is built and can be viewed as a saliency map. The third salient point detector [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] uses a wavelet analysis to find pixels on sharp region boundaries.

Self Organized Map Learning

The Kohonen model [START_REF] Kohonen | Self-Organizing Maps[END_REF] is based on the construction of a neuron layer in which neural units are arranged in a lattice L. Usually, the lattice is two dimensional (rectangular or hexagonal). The neural layer is innervated by d input fibers, called axons , which carry the input signals and excite or inhibit the cells via synaptic connections. As underlined in the previous section, the Kohonen network aims at preserving the topology of the input space and at tuning each cell to a particular set of stimuli. To reach these goals, the excitation of neurons has to be restricted to a spatially localized region in L and the location of this region has to be determined by those neurons that respond most intensively to a given stimulus. Moreover, L acts as a topographic feature map if the location of the most strongly excited neurons is correlated in a regular and continuous fashion with a restricted number of signal features of interest [START_REF] Ritter | Neural Computation and self-Organizing Maps : an introduction[END_REF]. Neighboring locations in L correspond thus to stimuli with similar features. To satisfy these properties, a neighboring function between cells must be added in the network model. For this purpose, each cell i ∈ L is connected to a set N L (i) of neighboring cells, defining thus a topological ordering. The goal of the Kohonen learning algorithm is then to adapt the shape of L to the distribution of the input vectors. As shown in Figure 2, the 2D lattice shape changes during the learning process to capture the input information and the topology existing in the input space. Those two properties can be considered as a competitive learning and a topological ordering. Let us now describe the SOM algorithm by assuming a SOM lattice structure composed of N × N neural units. Let M be the input space and X = x(t) be a set of observable samples with x(t) ∈ M ⊂ d , t ∈ {1, 2, . . .} being the time index. Supposing M = m i (t) is a set of reference vectors with m i (t) ∈ d , i ∈ {1, 2, . . . , N × N }. For a linear initialization, we compute the two eigenvectors of the autocorrelation matrix of x that have the largest eigenvalues. The rectangular lattice is then defined along a 2D linear subspace spanned by this two eigenvectors.

If x(t) can be compared simultaneously to all m i (t) by using a distance measure d(x(t), m i (t)) in the input space, then the best matching unit (BMU) m c (t) is defined by : m c (t) = arg min i d(x(t), m i (t)), ∀i = 1, 2, . . . , N × N.

(1) A kernel-based rule is used to reflect the topological ordering observed in the human visual cortex. The updating scheme aims at performing a stronger weight adaptation at the BMU location than in its neighborhood. This kernelbased rule is defined by :

m i (t + 1) = mi(t) + α(t)h ci (t)[x(t) -m i (t)], (2)
where α(t) designates the learning rate i.e. a monotonically decreasing sequence of scalar values with 0 < α(t) < 1. h ci (t) represents the neighborhood function that governs the strength of weight adaptation as well as the number of reference vectors to be updated. Classically, a Gaussian function is used, leading to :

h ci = exp - ||r c -r i || 2 2δ(t) 2 . ( 3
)
Here, the Euclidian norm is chosen and r i is the 2D location for the i th neuron in the network. δ(t) specifies the width of the neighborhood during time t.

Similarity Evaluation

When the SOM learning is over, the last step consists in quantifying the visual similarity between two images I 1 and I 2 . The extracted patches from I 1 and I 2 are presented to the learned SOM. Each patch actives a particular cell (i.e., the BMU) and increments an activity histogram corresponding to answer responses for each neuron : ∀i ∈ {1, 2, . . . , N × N },

H 1 (i) = card{p ∈ I 1 , ||p-m i || < ||p-m j ||, ∀j = i} (4)
To obtain a probability distribution, the histogram is then normalized. The same strategy is applied to image I 2 to build the H 2 histogram. To quantify the image similarity we compare these two histograms. This is made possible by the preserving topology property of SOM : two input vectors closed in the observation space are now close in the SOM lattice and activate two cells in a small neighborhood. We use here an efficient quadratic histogram distance from Hafner and al. studies [START_REF] Hafner | Efficient color histogram indexing for quadratic from distance functions[END_REF]. In particular, the quadratic distance D Q uses an inter-bins similarity matrix A. This quadratic distance is defined by :

D Q (H 1 , H 2 ) = (H 1 -H 2 ) t A(H 1 -H 2 ), A = |a ij | ∈ N ×N , (5)
where a ij represents the similarity between the i th element of H 1 and the j th element of H 2 . We have to keep in mind that indices i and j are the activated SOM neurons from I 1 and I 2 stimuli. The weight normalization denotes a value closed to 1 for similarity and closed to 0 for dissimilarity.

a ij = 1 - ||r i -rj|| max i,j ||r i -r j || . (6)
Finally, for determining the test image category, we compute the k-nearest learning images based on the previous criterion ( 5) and a k-nn classification is performed.

Experimental Results

For all the experiments, we configure our SOM network with the following rules to reach good learning results in terms of accurate input data representation [START_REF] Kohonen | Self-Organizing Maps[END_REF] :

• the learning steps are 500 times the number of cells ;

• the available samples are applied cyclically ;

• α(t) = T T +99t forms a monotonically decreasing sequence, with the number T of learning steps ;

• the width of neighborhood function δ(t) decreases linearly from

√ 2N 2 2
to 0.5 .

The first experiment is to test how our system behaves with large databases (1200 learning images, 3000 test images). These images are divided into 5 categories: buildings, flowers, motorbikes, mountains and planes. We decide to evaluate the IP detector influence on the learning step. The results are exposed in Figure 3. This experimentation shows that the best configuration for clustering these 5 categories is a wavelet based detector [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] with 7x7 patches associated to a 20x20 SOM. The patch dimension and the SOM lattice size are chosen by experimental results. Indeed, the classifier offers a correct detection rate averaged across categories of 75.60%. Some correctly classified examples are shown in Figure 4.

The confusion matrix of the image classification system with the best configuration is shown in Table 1. The best classified category is motorbikes: 83.17% are correctly detected; The worst case happened to the flower category. This can be explained by the large variety of color and shape in this cluster. In a context of web content filtering, we try to detect harmful content to censure pornographic images. The database is downloaded from Internet and is composed of 1110 adult images and 1200 benign images. The second category, known to be the rest of the world, is mainly constituted of landscapes, portraits and life scenes. 733 images of each category is preserved for the learning database. This study illustrates, with the salient point detector previously determined, interesting results with 7x7 patches. Indeed, 78.59% of adult images are correctly classified 79.31% of benign images are recognized (Table 2). By adding the mean color descriptor (MCD) [START_REF] Manjunath | Color and texture descriptors[END_REF], we increase performances comparatively to a simple RGB patch. We can note that our approach gives better results than a classical MCD strategy.

Conclusion

In this paper, we proposed an original classification system using directly patches information. Based on the two main properties of SOM -which are dimension reduction and topology preservation -this architecture features image categories with activity histograms. In order to quantify the visual similarity between two images, we only need to compare their individual histogram. This solution implemented for image classification gives us very promising results. However, a growing and pruning strategy or a hierarchical SOM could be useful for learning large databases. Further improvements may be applied by coupling SOM algorithm and vector quantization methods as LVQ. Another possible issue is to learn activity maps built by SOM with a supervised neural network to get more robustness.

Figure 1 .

 1 Figure 1. System architecture.

Figure 2 .

 2 Figure 2. (a)Linear initialization. (b)Final topological evolution. (c)Patches projection on SOM lattice.

Figure 3 .

 3 Figure 3. SOM with different IP detectors.

Figure 4 .

 4 Figure 4. Correct and misclassified images.

Table 1 . Confusion matrix for 5 clusters

 1

	A=building	72.00	6.17	9.67	5.17	7.00
	B=flower	2.17	67.83 23.67	4.67	1.67
	C=motorbike	2.67	12.50 83.17	0.67	1.00
	D=mountain	4.17	6.33	5.67	78.17	5.67
	E=plane	8.50	4.33	4.17	6.17	76.83

Classified as → A (%) B (%) C (%) D (%) E (%)

Table 2 . Adult and benign detection rates

 2

	Approach	Descriptor	Adult Benign	Mean
	Patch3x3	Haar + RGB + SOM 20x20	72.94	75.38	74.16
	Patch7x7	Haar + RGB + SOM 20x20	78.59	79.31	78.35
	Patch7x7	Haar + MCD	77.98	75.68	76.83
	Patch7x7	Haar + MCD + SOM 20x20	88.86	80.09	84.47