Channel Allocation in Cognitive Radios - CiteSeerX

CS270 PROJECT: CHANNEL ALLOCATION FOR COGNITIVE RADIOS, MAY 2005

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Channel Allocation in Cognitive Radios Shridhar Mubaraq Mishra, Member, IEEE,

Abstract— In this project we investigate the problem of channel allocation to a set of users in a centralized Cognitive Radio System. We consider three possible formulations of the problem and explain the relationship of these constraints to physical limiting factors. In the first formulation, we aim to minimize total power used in the system while fulfilling user rate requirements. This problem can be solved as a minimum weight matching problem in a bipartite graph. In the second formulation, we further enforce power constraints per channel. In the last formulation we consider the problem of minimizing impact on each user if the primary user returns. With a few modifications we can map this problem to a Maximum Flow Minimum cost problem, which can be solved by a variation of the Ford-Fulkerson algorithm by augmenting along the minimum cost path. Index Terms— Cognitive Radios, Frequency Allocation

I. I NTRODUCTION T is commonly believed that there is a spectrum scarcity at frequencies that can be economically used for wireless communications. This concern has arisen from the intense competition for use of spectra at frequencies below 3 GHz. As seen in Figure 1, the Federal Communications Commission’s (FCC) frequency allocation chart indicates multiple allocations over all of the frequency bands, which reinforces the scarcity mindset. On the other hand, actual measurements taken at the BWRC (see spectrogram in Figure 2) indicate low utilization especially in the 3-6 MHz bands. This view is supported by recent studies by the FCC’s Spectrum Policy Task Force (SPTF) which reported vast temporal and geographic variations in the usage of allocated spectrum with utilization ranging from 15% to 85% [1]. In order to utilize these ’white spaces’, the FCC has issued a Notice of Proposed Rule Making [2] advancing Cognitive Radio (CR) technology as a candidate to implement negotiated or opportunistic spectrum sharing.

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Fig. 2.

FCC spectrum allocation

with the environment in which it operates’. The key properties of Cognitive radios are: Sensing RF technology that ’listens’ to huge swaths of spectrum. Cognition Ability to identify Primary Users (PU’s) of the spectrum (A Primary user has been allocated the spectrum by the FCC). This identification may be based on power profiles and/or other footprints of primary users. For example, the control channel in GSM system could be used to identify active GSM systems. Furthermore, cognitive radios must be able to determine interference that their communication may cause to passive primary users (for example TV receivers). Adaptability Ability to change parameters to best use unused spectrum. This may include changes to the frequency bands of operation, power levels and modulation parameters. B. System Model

Fig. 1.


A. Cognitive Radios As per the FCC, a Cognitive Radio (CR) is a radio ’capable of changing its operating characteristics based on interaction

Our model of Cognitive Radio systems is very similar to the IEEE 802.11’s infrastructure mode of operation where a group of WLAN users associate with an Access Point (AP). We envision a group of cognitive radios controlled via the AP. The available spectrum is divided into channels. The CR group performs local spectrum sensing on each channel but relies on centralized primary user detection and channel allocation. In order to communicate, the CR group must perform the following tasks: Spectrum Sensing Each CR user senses the spectrum, identifies all primary users and forms a Local Interference Map (LIM). The LIM is a vector which indicates the received interference level as well as the presence of primary users in


each channel. The LIM is sent by the user to the AP. Channel Sounding For each free channel, the AP determines the gain between the AP and each user and thus constructs a channel gain matrix. Channel Allocation The channel gain matrix is used by the AP to allocate channels to users. The above steps are performed repeatedly at fixed time intervals as dictated by PU usage patterns and channel coherence time. In this paper we focus on the third step - Channel Allocation. C. Problem Setup The basic model envisions a set of CR radios/users (N of them) and a set of K channels that have to be allocated to these users. Each channel provides unit bandwidth. We also assume that the channel sounding process provides channel gains (aij is the channel gain for user i, channel k). The users have certain requested rates R1 ...RN . There are P primary users in the system. Each of the K channels belongs to a primary user. These primary users are currently not operating and hence the K channels have be sensed to be free. If a power Pik is used for channel k by user i, the net rate achieveable by user i can be represented as:   K X

  aik Pik   log2 1 + N  P   k=1 ajk Pjk + No

(1)

j=i+1

We would like this rate to be greater than Ri . The above equation asssume the presence of a Successive Interference Cancelletion (SIC) decoder at the receiver and hence allows multiple users to use the channel. For this project we make the following assumptions: • To avoid the need of a SIC decoder we wish to ensure that a channel is allocated to a single user. • We wish to remain in the low SNR regime where we can approximate the log function (log2(1 + x) ≈ xlog2(e)). The first two assumptions reduces the rate constraint to, K X

aik Pik log2 (e) ≥ Ri , ∀i

(2)

k=1

Fo rthe remainder of the paper we shall combine the log2 (e) factor as a part of the requested rate. • The channel exhibits slow fading and hence the allocation process takes place within the channel coherence time. • There exists an underlay control channel that is used by group members to communicate with the access point. • We assume that the system has full knowledge of the channel gains. The rest of this paper is organized as follows: In Section II we describe related work in this area. In Section III we present the ’Minimize total power of CR system’ formulation

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and describe a method to solve this problem using minimum weight matching. In Section IV we present a different variant of the problem - the ’Minimize Total Power with power constraints per channel’ formulation. This is a integer variant of the Quadratic Problem, Quadartic Constraints (QPQC) problem. Next, in Section V we describe the ’Minimize impact of returning primary users’ problem and describe a method to solve this using Maximum Flow. Finally Conslusions and future work are offered in Section VI. II. L ITERATURE S URVEY There has been a great deal of work in the area of channel allocation. The problem constraints dictate the type of solution achieved. In the first type of problem formulation there are no individual requested rates only a limit on the sum capacity. The optimal (minimum power) solution for the sum capacity allocates the channel to the user that has the largest gain for that channel. This problem has been studied in [3] and its polymatroid structure has also been proven. This type of system cannot guarantee fairness or service quality. The rates assigned to the user depends on the channel gain matrix. In some sense, these rates are the ’social optimal’ for the given channel matrix. That is, these are the rates that minimize power for a given total capacity. If the capacity (rates) is distributed among the users in a different way, then the whole system has to pay a penalty and spend additional power to attain the same sum capacity. In real systems, fairness and services guarantees are very important. Thus, we considered problems where each user has a requested rate. When each user has a requested rate that must be met, the optimal solution no longer yields a single user per channel. Multiple users share a channel and a SIC decoder has to be used. This problem has also been studied in [3]. There has been a great deal of work in the area of channel allocation for cellular systems. The current cellular standard (CDMA, IS-95 also [4]) uses a power-controlled interference system to allocate the same rate to every user. The channel allocation is optimized for voice users and does not focus on minimizing power or maximizing throughput. For an OFDM system, [5] introduces a method to maximize the average rate with partial knowledge of the channel. In an OFDM system, users randomly hop from one channel to the next, using Hadamard hopping patterns. However, random hopping is not compatible with a system that must avoid interfering with PU’s. [6] introduces a power adaptive scheme for OFDM users to maximize the data rate for the system. Most channel allocation schemes allocate channels once according to the stationary channel statistics (termed Fixed Channel Allocation -FCA [7] [8]). FCA schemes are optimal under very high load conditions. However, in a system with PU’s and bursty loads, FCA schemes are not optimal. [9] introduces an auction method that allows users to fairly compete for channel. The work done in [10] is the closest to our model for channel allocation. The ’one-user-per-channel’ problem is formulated in [10] and an approximation algorithm is developed. This is extended in [11] to minimize both channel use and power simultaneously.


III. M INIMIZING TOTAL P OWER OF THE CR SYSTEM Since Cognitive Radios are secondary users, the FCC is looking at ways of limiting their impact. One of the ways to achieve this, is to limit the total power of a CR system. This line of reasoning leads to the problem of minimizing the sum power of the system while meeting user rates.

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in a bipartite graph where a group of N users is matched to a set of K channels. If K 6= N , this problem is a rectangular assignment problem which has be solved by a variation of the Munkres algorithm [12]. C. Complexity Analysis In [12] it is shown that minimum weighted matching for a rectangular matrix of size m × n is O(a2 b), where a = min(m, n) and b = max(m, n). If K > N the algorithm can solve the channel assignment problem in O(N 2 K) steps.

A. Problem Formulation •

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N Users K Channels aik - Channel Gain for user i, channel k

D. Example

Minimize, N X K X

1ik Pik

(3)

Figure shows a toy example with 2 users and 4 channels. The requested rates are 1 bits/s each.

i=1 k=1

Subject to, 1ik ∈ {0, 1}

0≤

N X

(4)

1ik ≤ 1,

∀k

(5)

i=i K X

1ik aik Pik ≥ Ri ,

∀i

(6)

k=1

B. Minimun Weight Matching Solution First we wish to make the following observation: under the assumption that channel capacity is a linear function of power, we notice that we would like to allocate a single channel (the best channel) to each user. Lemma 1: The optimal (minimum power) solution is one in which each user is allocated a single channel Proof: By condradiction: Assume that the optimal solution is obtained by allocating two channels to user i (say channel j and channel k) which are of lower value that another channel l i.e. aij < aik ≤ ail (Note that l could be the same as k, which is a specific instance of this formulation). Then it must be true that Pij + Pik < Pil since we selected channels j and k. Since all choices meet the rates constraints we have: Pij aij + Pik aik = Pil ail . Dividing throughout by ail , we get: Pil

=

Pil

P × D. If P >= r, then we would like to make cuts along (xclj +h , t), (1 ≤ h ≤ cj and lj is the index of the first channel in Sj ) for all sets Sj . Hence the capacity of the cut will be K. Since P × D ≤ N × D ≤ K, we have shown that the size of the minimum cut for fixed size sets is N × K.

The proof for variable size sets is still in progress.

Fig. 5.

Theorem 5: If N ×D < K the minimum cost flow of value N × D is the optimal solution to the original problem Proof: From Lemma 4 we know that finding a flow of value N × D corresponds to finding a solution in the original problem. Since the flow found is also minimum cost this corresponds to the optimal solution in the original problem


Proof: First we show that, if MaxFlow finds a maximum flow of capacity N × D, then it is satisfies (18), (19) and (21). Since each edge between nodes s and vi , 1 ≤ i ≤ N is of capacity D this corresponds to allocating D channels to each user (since a channel is allocated to one user only - see Lemma 2). From Lemma 3 we know that these D channels must belong to different sets. Hence the flow satisfies equations (18), (19) and (21). Next we show that, if N × D < K then MaxFlow must find a maximum flow of capacity N × D. The value of the maximum flow is given by the value of the minimum cut. First, assume that the minimum cut does not include any edges (s, vi ), 1 ≤ i ≤ N . For the set of edges between each node and set j i.e. (vi , ui,j ), 1 ≤ i ≤ N , we have two choices. If cj ≤ N then we would prefer to include edges (xclj +h , t), (1 ≤ h ≤ cj and lj is the index of the first channel in Sj ) in the cut. If cj > N then we would prefer to include edges (vi , uij ). This set of edges contribute a capacity of N . In our case, we have assumed that the number of elements in each set is less than or equal to N . Hence the size of the min J P cut is cj = K. j=1

Next we consider the case where we include N − P of the edges (s, vi ) and exclude P of the these edges. For the P edges not included we must make subsequent cuts. We are left with the same set of choices as before. If cj ≤ P then we would prefer to include edges (xclj +h , t), (1 ≤ h ≤ cj and lj is the index of the first channel in Sj ) in the cut. If cj > P then we would prefer to include edges (vi , uij ). Each such set of edges contributes a capacity of P . Lets assume that there are S sets with cardinality less than P . Then, the total capacity of the cut is: X (N − P ) × D + cj + (J − S) × P (21) cj