1. Introduction

The alarming growth in wireless devices and technologies coupled with the evolution to multimedia type communications has made the radio spectrum an ever scarce resource. However, surveys made by regulatory bodies in different countries on spectrum utilization have indicated that the actual licensed spectrum islargely underutilized in vast temporal and geographic dimensions [1]-[3]. Hence, innovative technologies that can offer new ways of exploiting the available spectrum are needed [4].Cognitive radio is a novel technology which improves the spectrum utilization by seeking and opportunistically utilizing radio resources in time, frequency and space domains on a real time basis [5] [6].

One of the main tasks of cognitive radio system is spectrum sensing for opportunistic access of the spectrum. Conventionally, spectrum opportunity relates to the three dimensions of the spectrum space: frequency, time, and (geographic) space [4]. In the literature a number of different methods were proposed for identifying the presence of signal transmissions in these dimensions. Some of the most common methods are energy detector based sensing (e.g., [7] [8]), waveform-based sensing (e.g., [9]), cyclostationarity- based sensing (e.g., [10]), and radio identification based sensing (e.g., [11]).

There are other dimensions that need to be explored further for spectrumopportunity. The angle dimension is a typical example [4]. It is to be noted that angle dimension is different than geographical space dimension. In angle dimension, a primary and a secondary user can be in the same geographical area and share the same channel. However, geographical space dimension refers to physical separation of radios in distance such as in different mobile cells. To exploit the angle opportunity, the cognitive transmitter (CT) can be equipped with smart antenna so that it directs its main lobe in the direction of the cognitive receiver (CR) while minimizing or nulling its interference in the direction of the primary receiver (PR).

The authors in [12] proposed opportunistic spectrum access scheme based on angle diversity and smart antenna technology. In their proposed non-intrusive cognitive scheme they showed that, depending on the relative location of the CR and PR, interference from primary transmitter (PT) to CR can be made very small in a large area coined by them as the “decodable zone”. Consequently, the CT and PT can transmit to their respective receivers with high enough SINR when the CRs are deployed in the decodable zone. The present work is an extension of the model developed by [12] to a Rayleigh fading channel.

This paper is organized as follows. The system model is presented in Section II. Section III describes the algorithm for a successful deployment of the CR in the allowable region. In Section IV, we provide some representative numerical results. Finally in Section V concluding remarks and further research topics are hinted.

2. System Model

Assume we have two wireless communication links coexisting in the same geographical region [12]. The primary users (PT and PR) are the licensed users while the secondary or cognitive users (CT and CR) are allowed to opportunistically access the primary user’s spectrum under the conditions to be outlined below. The system setup is shown in Fig. 1. The PT and CT are separated by a distance

Figure 1. Cognitive radio system with primary users

. The PR is randomly distributed within the circle of radius D while the CR is deployed in a small region centered on the CT at . Subscripts and indicate, respectively, primary and cognitive users. and forrepresent, respectively, the distance and azimuth angle from the i’s transmitter to the j’s receiver.

To exploit the angle opportunity, the CT should be equipped with beamforming smart antenna so that it minimizes its interference to the PR while maximizing its radiation in the direction of the CR. Assume that the CT is using a 2-D uniform circular array with M arrayelements and consider, for simplicity, the azimuth angles. The array beamforming gain in the direction of the azimuth angle becomes

(1)

where is the transmit beamforming vector, is the array steering vector, R is the array radius, λ is the carrier wavelength and is the angular location of the array elements with respect to the horizontal axis.

The instantaneous signal-to-interference-plus noise ratio (SINR) at the PR and CR, respectively, are

(2)

(3)

where is the instantaneous received power at receiver j that was transmitted by the ith source and are, respectively, the primary and cognitive transmit beamforming gains.

2.1. Rayleigh Fading Channel

In a Rayleigh fading channel, the received power is exponentially distributed. Let be the average received power (including path loss and shadowing) at receiver j that was transmitted by the ith source. Then

(4)

where is the expectation operator, is the power transmitted by source is the path loss factor and is the large-scale shadowing power with zero mean. The probability density function (pdf) of the instantaneous power is

(5)

Thus, the SINRs (2) and (3) are random variables of the form

(6)

where X and Y are exponentially distributed independent random variables and are real numbers.

2.2. Distribution of SINR

As shown above, the SINR is expressed as a ratio of two random variables. The distribution of the ratio of two random variables has been extensively investigated by many authors especially when X and Y are independent and identically distributed (i.i.d.) random variables [13, 14].

To determine the pdf of (6), define an auxiliary random variable

(7)

The joint distribution of Z and W becomes

where is the Jacobian of the transformation. We assume and to be i.i.d. exponential random variables with mean and .

Hence, integrating over W, the marginal pdf of Z becomes

(8)

The mean of Z becomes

(9)

where is the incomplete gamma function.

The probability of becomes

(10)

3. Problem Statement

Consider a scenario where all users except the CT use omnidirectional antennas. For communication of cognitive radio systems to be established, the CT equipped with beamforming smart antenna has to guarantee that its average interference to the PR is below a threshold I_o while at the same time keeping high enough SINR_c at the CR.

The cognitive radio has to be aware of the approximate location of the PR. Assume that the distance to the PR, , is known exactly while its angle information is uncertain by . Let be the worst interference gain in the direction of PR, then

(11)

The optimal CT beamforming can be optimized as:

(12)

where , -the first constraint, guarantees that the worst interference to the PR has to be kept below I_o, is a constraint on the sidelobe leakage of the CT antenna beam with half-power beamwidth ,is the maximum transmission power of CT, , where is the average SNR requirement of the CR.

From the optimization problem (12), is determined. Note that when the gain becomes

After determining the optimal for the CT, the CT’s transmission power becomes

(13)

The first term in (13) comes from the interference constraint while the second term is

For successful communication of the cognitive system, the SINR_c at the CR has to be greater than a minimum value T with a high probabilityp.

P_T in (10) for the CR with , and referring to (3) and (6), we have , and so that

(14)

To meet the power constraint (13),

(15)

(16)

where and . Therefore,

(17)

The CT, subject to (12) and (17), can establish communication with the CR at high probability p when

(18)

For a fixed CT location, the area in space where (18) is satisfied defines the region where the CRs can be deployed to have permissible communication.

If only large-scalepath loss (non-fading channel) is considered, communication is established when with

(19)

4. Numerical Results

Assume location of the PR is fixed so that d_pp, d_cp, and θ_cp are fixed. The CR is to be randomly deployed with d_cc and θ_cc uniformly distributed, i.e., and Further assumed_min=10m, d_max=D, the path loss factor to be and

The allowable region where the CR can be deployed is simulated using the Monte-Carlo simulation for different combinations of Tandp.

For and the allowable region, both for non-fading and Rayleigh fading channels, is shown in Fig. 2. The second simulation, shown in Fig. 3, is obtained as in above except that Fig. 4 shows the effect of varying p.

Figure 2. Comparison of non-fading and Rayleigh channels for

Figure 3. Comparison of non-fading and Rayleigh channels for

Figure 4. Effect of varying p in Rayleigh fading channel

Define the angle spectral efficiency as

(20)

Thus represents the improvement on angle spectral usage. Spectral efficiency of 80.89% and 71.80% is achieved in simulations shown in Fig. 2(a) and 2(b), respectively. We observe that the angle spectral efficiency decreases in a Rayleigh fading channel as compared to a non-fading channel with the same as expected. Similarly 64.27% and 10.38% is achieved as shown in Fig. 3(a) and 3(b), respectively, while 34.22% and 2.0% is achieved in Fig. 4(a) and 4(b), respectively.

Fig. 5 shows variation of as a function of for non-fading channel and Rayleigh fading for different p. From the figure, we infer that the efficiency jumps from zero to approximately 70% in almost 5 dB interval and then saturates.

Figure 5. Angle spectral efficiency, as a function of

5. Conclusions

In this paper we have shown that the angle spectrum in cognitive radio system can be greatly exploited using smart antenna and thus leading to multiplexing of multiple users in the same channel at the same time in the same geographical area. The resulting efficiency in spectrum utilization greatly relieves the already congested radio spectrum. The region where the CRs are to be deployed for a given PT, PR and CT locations is determined for Rayleigh fading channel and compared with the non-fading channel. The angle spectral efficiency for Rayleigh channel is lower as compared to a non-fading channel with the same power levels and radio terminal locations. We have also shown that increasing the CT power beyond a certain level has negligible effect on the spectral efficiency.

Future work might include other types of fading channels. Fast fading and frequency selective Rayleigh channels can also be considered.