WHAT IS THE PRINCIPLE BEHIND FREQUENCY ALLOCATION?

**Short Answer**

Utilization of radio recurrence groups of the electromagnetic range is directed by governments in many nations, in a Spectrum management process known as frequency distribution or range allotment. Radio proliferation does not stop at national limits. Giving specialized and monetary reasons, governments have looked to orchestrate the assignment of RF groups and their standardization. A number principles bodies take a shot at models for recurrence portion, including:

·         Global Telecommunication Union (ITU)

·         European Conference of Postal and Telecommunications Administrations (CEPT)

·         Between American Telecommunication Commission (CITEL)

So as to enhance harmonization in range usage, the dominant part of administration allotments stipulated in this document were consolidated in national Tables of Frequency Allocations and Utilization which is with-in the obligation of the suitable national organization. The designation may be essential, auxiliary, elite, and shared.

·         essential allotment: is shown by writing in capital letters

·         optional designation: is shown by little letters

·         elite or shared usage: is inside the obligation of organizations.

Be that as it may, military utilization, in groups where there is respectful use, will be as per the ITU Radio Regulations.

Radio spectrum available in our electromagnetic spectrum is a limited natural resource. The demand for radio frequencies is increasing day by day as they are used in long distant communications. To provide effective communication globally and to utilize the available spectrum effectively there is a need for frequency allocation. Any user should have a license to operate on specific frequency band. The utilization of radio spectrum in each country is governed by corresponding government agencies. In conventional technique each user is assigned a license to operate in a certain frequency band of fixed bandwidth. So, most of the time spectrum remains unused and is also difficult to find it. Due to this, allocated bandwidth is not utilized properly. Also most of the networks are subjected to time and regional variations leading to unequal to distribution of traffic along the network. Thus to overcome the spectrum scarcity and utilize band properly new techniques like dynamic spectrum access (DSA) are introduced.  These modernized techniques find their utility in Ad hoc networks, cognitive radios, digital subscriber lines etc.

Objective of Dynamic Spectrum Allocation:

Managing spectrum in a converged radio system and share it among all participating radio networks over space and time to increase overall spectrum efficiency.

Efficiency can be related to technical efficiency, economic efficiency, public policy.

Here technical efficiency refers to minimizing interference, economic efficiency refers to allocating and assigning spectrum to its most economically valuable use and harmonization.

# Introduction:

In Wireless  Ad hoc networks transmitting less power is a crucial objective. Dynamic frequency allocation has a vital role in minimizing transmission power. To achieve this optimization there is a need of centralized processor with full knowledge of the spatial distribution profile of the clusters in the network. In many emerging wireless networks like Ad hoc networks, cognitive radios central frequency allocation authority is not naturally available. In this model , cluster’s activity is described by a stochastic process. Here we assume each cluster has exactly one user and clusters go ON and OFF according to two state Markov model independently. The process of time varying activity of clusters results in a stochastic differential equation for the steady state behaviour of the algorithm.

System Model:

A set of network nodes distributed in space are partitioned into union of clusters. Naturally clusters are already formed in a specified manner. In our network model, clusters are denoted by Ci,  i = 1,2,....N and also each cluster has a cluster head which is responsible for managing some of the network functions. Let dij denote the distance between the cluster heads of Ci and Cj. [5]

Following assumptions are made for this model:

* The ith cluster, Ci, contains ni users.
* For a given time slot, in any cluster at most one user is transmitting and one user is receiving.

For a simplified analysis, we are assumed that at each time slot for any cluster exactly one user is transmitting and one user is receiving. This assumption made a scenario satisfying channel reciprocity between clusters. Because of this reciprocity, the results can be generalized to various other models.

* KP0  is the transmitting power of each user, where K is the normalization constant.
* The size of the clusters is less than the distance between the clusters and bounded below by a distance ẟ.
* This transmission model is a path loss with exponent η. Shadowing and fading are not assumed in this current analysis.
* The spectrum which is accessible is divided into r different bands, denoted by b1, …., br.
* At a given time t, the state of the ith cluster is denoted by si(t) ϵ {1, 2, …., r}, corresponding to the index of the transmission band it is using.
* At the same instant of time the probability of two clusters updating their frequency bands is negligible.
* The processing /transmission rate is greater than the rate of change of the spatial distributions of the clusters in the network .
* The aggregate interference of the network is chosen as our performance metric.

The Algorithm:

By using the above assumptions, the interference experienced by Ci caused by all other clusters is given by

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{\displaystyle I\_{c\_{i}}(N,\{d\_{ij}\},t)=\sum\_{j\neq i}\frac{P\_{o}}{{d}^{n}\_{ij}} \delta(s\_{i}(t),s\_{j}(t)) }

Where ẟ(x,y) is the Kronecker delta function, defined as

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For a given time t = tn, one of the clusters, say Ci, updates its transmission frequency band. The update process is done in an asynchronous manner for all the clusters. We assume that the updates are taking place at times t1, t2, …. . The discrete-time version interference model is given by

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{\displaystyle I\_{c\_{i}}(N,\{d\_{ij}\},l)=\sum\_{j\neq i}\frac{P\_{o}}{{d}^{n}\_{ij}} \delta(s\_{i}(l),s\_{j}(l)) }

Where l denotes to the time t = tl, when updates are taking place. Let Ck(N, {dij}, l) denote the set of clusters transmitting in band bk. Also, Ici,k(N, {dij}, l) denote the interference experienced by Ci caused by all the clusters in Ck(N, {dij}, l) if Ci  was using band bk for k = 0, ….. , r-1. Ici,k(N, {dij}, l) can be written

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{\displaystyle I\_{c\_{i} k}(N,\{d\_{ij}\},l)=\sum\_{c\_{j}\epsilon C\_{k}(N,\{d\_{ij}\},l),j\neq i}\frac{P\_{o}}{{d}^{n}\_{ij}} }

At a given time l, the aggregate interference is given by

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{\displaystyle I(N,\{d\_{ij}\},l)=\sum\_i I\_{c\_{i}}(N,\{d\_{ij}\},l)=\sum\_i \sum\_{j\neq i}\frac{P\_{o}}{{d}^{n}\_{ij}} \delta(s\_{i}(l),s\_{j}(l)) }

For notational convenience, time dependence functions I(N, {dij}, l), Ici(N, {dij}, l) and Ici,k(N, {dij}, l) are replaced by I(N, {dij}), Ici(N, {dij}) and Ici,k(N, {dij}) respectively. The worst-case aggregate interference scenario, optimal scenario and output of the algorithm are denoted by Iw(N, {dij}), I0(N, {dij}) and Ia(N, {dij}) respectively.

Main Algorithm:

“Each cluster scans all the frequency bands b1, …., br  asynchronously over time . A cluster chooses the frequency band in which it experiences the minimum aggregate interference from other clusters. The cluster updates its state according to the following rule [1]

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{\displaystyle s\_{i} (l+1)=argmin\_{k} I\_{c\_{i}}(N,\{d\_{ij}\},t) }

Where si(l + 1) is the new state of cluster head ci updated at time l.

Main Results:

* Convergence

       For any given reciprocal channel model, the Main Algorithm converges to a local minimum in polynomial time in N.

* Performance Bounds

          The aggregate interference of all the clusters Ia(N, {dij}) corresponding to the state of the algorithm following convergence is less than the worst case aggregate interference scenario.

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{\displaystyle I\_{a}(N,\{d\_{ij}\})\leq \frac{1}{r} I\_{w}(N,\{d\_{ij}\}) }

If all the clusters are collinear, then it is referred to as a linear array in which di, i+k = kd, for 1≤ i, i+k≤ N.

* Optimal Strategy

         I0(N, {dij}) denotes the aggregate interference of optimal strategy for a given linear array of clusters located in [0, (N-1)d]. Then,

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{\displaystyle \lim\_{N \rightarrow \infty} \frac{1}{N} I\_{o}(N,\{d\_{ij}\})\geq\frac{1}{r^{n}} 2\zeta (\eta)\frac{P\_{o}}{d^{n}} }

where ζ(η) is the Riemann zeta function.

* If r = 2 and η ≥ 2 then the optimal strategy is the alternating assignment for any N.[2]

Time Varying Setup:

For given N clusters, to represent the state of each cluster an activity indicator state ai(l), is considered. ai(l) = 1 corresponds to active state of the cluster at time l and ai(l) = 0 corresponds to inactive state of cluster at time l. The evolution of probabilities is given by

The above equation corresponds to a two state symmetric Morkov model. Here we can observe that the probability of cluster in future state depends only on present state but not on the past. We assume P1ci(0) = 1 for all ci, i = 1, ….. , N.

In this algorithm, upper bound does not hold in presence of time varying statistics. The update process of clusters is modeled by Poisson process of rate 1/ΔT.

The number of users going to active/sleep in each time slot is much less than the total number of users.[6]

Simulation Results:

In the above result we can observe that, in both the cases the algorithm still performs within the upper bound almost all the time. The normalized aggregate interference is being reduced gradually over time. We can also see that the normalized aggregate interference is less for higher 𝛼.

Dynamics of the Algorithm, r = 2

Let I(t) denote a continuous time approximation to the aggregate interference of the network at time t. Here, two accessible frequency bands, b0 and b1 are considered. For convenience, we associate ϵi = -1 and ϵi = 1 to clusters in band b0 and b1 respectively.

If Ci is active at time t, it will experience an interference

In the above equation Ii is the worst case interference experienced by cluster Ci .

An appropriate band bj is assigned for cluster Ci . If the cluster Ci  is not assigned in its appropriate band at time t, then the aggregate interference will be increased by

If M(t) users in a cluster Ci are not in appropriate frequency bands, the aggregate interference is given by [4]

Where Ia is the target performance of the algorithm.

It can be proved that

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{\displaystyle \frac{\text{d}{\xi [I(t)]}}{\text{d}t}=-\frac{\rho}{\tau}(\xi[I(t)]-\zeta[I\_{a}]}

Where 𝜺 denotes the ensemble average over different update patterns, 𝜏 = NΔT and ⍴ is a geometrical constant showing the effective number of  interacting neighbors to a cluster including itself. Above equation shows an approximate differential equation which describes aggregate interference decreasing exponentially with rate ⍴ /𝜏 near the equilibrium point. [3][8]

Simulation Results:

The above graph represents dynamics of the algorithm for a uniform linear array of 100 clusters.

Here a theoretical estimate for ρ in this case is 3, since every cluster has two nearest neighbors.

Discussion of the Results:

The convergence of main algorithm in polynomial time in N, regardless of the configuration of nodes in the network is guaranteed. An upper bound on the performance of the algorithm is independent of the topology of the network and is also a direct consequence of the structure of the main algorithm. A class of linear arrays are chosen for the analysis to obtain further performance bounds for the main algorithm. The Riemann zeta function in the statement is nearly a consequence of pathloss model for the transmission and at each time slot, only one transmitter and one receiver are active in each cluster. In dynamics of the algorithm, when r = 2 states that optimal strategy is the alternating frequency allocation for finite N.

The stochastic analysis for time varying case shows that we can design the update rate to guarantee a finite steady state variance.

For r > 2, generalization of the result is not straight forward because the combinatorial possibilities of the assignments grow exponentially without r. Any model with channel reciprocity suffices for the convergence of the algorithm. [7]

Simulation Results:

Fig(a) Normalized aggregate shannon capacity and normalized aggregate interference for arrays of 100 clusters vs time for an uniform linear array.

Fig(b) Normalized aggregate shannon capacity and normalized aggregate interference for arrays of 100 clusters vs time for a rectangular lattice.

Fig(c) Normalized aggregate shannon capacity and normalized aggregate interference for arrays of 100 clusters vs time for a hexagonal lattice.

The above simulation results show the performance of algorithm on different arrays of 100 clusters in one and two dimensions. The normalized aggregate interference for a uniform linear array with r = 2, rectangular lattice with r = 4 and hexagonal lattice with r = 4 are predicted. The computation of the optimal frequency band for the rectangular and hexagonal lattices is difficult. So, we have compared the performance of the algorithm to that of 1:4 frequency reuse pattern as a near optimal candidate. In the above all cases more than 90% of the capacity of the near optimal centralized frequency assignment is obtained.

Fig(4) Normalized aggregate shannon capacity and normalized aggregate interference performance for 2 dimensional vs N with r = 4. (a) rectangular lattice (b) hexagonal lattice

The above figure represents the hexagonal and rectangular arrays of clusters, performance of the main algorithm, worst case and 1:4 frequency reuse pattern with d = 1, r = 4, P0 = 1 and η = 2.

The above figure(5) shows dynamics of the algorithm for a linear array of 100 clusters arranged uniformly for 𝞪 = 1 which is a no time varying statistics. The empirical curve is averaged over 500 various update patterns.

Fig(6) Normalized steady state variance vs switching rate of linear array of 100 clusters distributed uniformly.

From the above figure it is observed that the theoretical variance with ρ = 3 is same as that of empirical variance.

All the above simulation results taken from  B. Babadi, and V. Tarokh, “A Distributed Dynamic Frequency Allocation Algorithm”,

Conclusion:

* A framework is introduced to analyse the performance of distributed dynamic frequency allocation algorithm and time varying statistics.
* In the presence of time variation in the activity of clusters near the equilibrium point a stochastic differential equation is derived.
* A trade-off inequality stabilizes the performance of algorithm.
* The possibilities of both open loop and closed loop stochastic controls are opened by stochastic modelling framework, research process is going on all these problems which helps for the further purposes.

References:

 [1]  Cendrillon, Raphael, et al. "Autonomous spectrum balancing for digital subscriber lines." Signal Processing, IEEE Transactions on 55.8 (2007): 4241-4257.

 [2] Jorgenson, Dale W. "Rational distributed lag functions." Econometrica: Journal of the Econometric Society (1966): 135-149.

  [3] Itô, Kiyosi. "109. Stochastic Integral." Proceedings of the Imperial Academy20.8 (1944): 519-524.

  [4] Peng, Chunyi, Haitao Zheng, and Ben Y. Zhao. "Utilization and fairness in spectrum assignment for opportunistic spectrum access." Mobile Networks and Applications 11.4 (2006): 555-576.

   [5] Ramanathan, Subramanian. "A unified framework and algorithm for channel assignment in wireless networks." Wireless Networks 5.2 (1999): 81-94.

   [6] Srivastava, Vivek, et al. "Using game theory to analyze wireless ad hoc networks." IEEE Communications Surveys and Tutorials 7.1-4 (2005): 46-56.

   [7] Yu, Wei, George Ginis, and John M. Cioffi. "Distributed multiuser power control for digital subscriber lines." Selected Areas in Communications, IEEE Journal on 20.5 (2002): 1105-1115.

   [8] W. Feller, Probability Theory, Vol. I, John Wiley, New York, 1957 and Vol. II, 1966