# Performance Analysis of Ad Hoc Dispersed Spectrum Cognitive Radio Networks over Fading Channels

- KhalidA Qaraqe
^{1}, - Hasari Celebi
^{1}Email author, - Muneer Mohammad
^{2}and - Sabit Ekin
^{2}

**2011**:849105

https://doi.org/10.1155/2011/849105

© Khalid A. Qaraqe et al. 2011

**Received: **1 September 2010

**Accepted: **19 January 2011

**Published: **15 February 2011

## Abstract

Cognitive radio systems can utilize dispersed spectrum, and thus such approach is known as dispersed spectrum cognitive radio systems. In this paper, we first provide the performance analysis of such systems over fading channels. We derive the average symbol error probability of dispersed spectrum cognitive radio systems for two cases, where the channel for each frequency diversity band experiences independent and dependent Nakagami- fading. In addition, the derivation is extended to include the effects of modulation type and order by considering M-ary phase-shift keying ( -PSK) and M-ary quadrature amplitude modulation -QAM) schemes. We then consider the deployment of such cognitive radio systems in an ad hoc fashion. We consider an ad hoc dispersed spectrum cognitive radio network, where the nodes are assumed to be distributed in three dimension (3D). We derive the effective transport capacity considering a cubic grid distribution. Numerical results are presented to verify the theoretical analysis and show the performance of such networks.

## Keywords

## 1. Introduction

Theoretical limits for the time delay estimation problem in dispersed spectrum cognitive radio systems are investigated in [3]. In this study, Cramer-Rao Lower Bounds (CRLBs) for known and unknown carrier frequency offset (CFO) are derived, and the effects of the number of available dispersed bands and modulation schemes on the CRLBs are investigated. In addition, the idea of dispersed spectrum cognitive radio is applied to ultra wide band (UWB) communications systems in [4]. Moreover, the performance comparison of whole and dispersed spectrum utilization methods for cognitive radio systems is studied in the context of time delay estimation in [5]. In [6, 7], a two-step time delay estimation method is proposed for dispersed spectrum cognitive radio systems. In the first step of the proposed method, a maximum likelihood (ML) estimator is used for each band in order to estimate unknown parameters in that band. In the second step, the estimates from the first step are combined using various diversity combining techniques to obtain final time delay estimate. In these prior works, dispersed spectrum cognitive radio systems are investigated for localization and positioning applications. More importantly, it is assumed that all channels in such systems are assumed to be independent from each other. In addition, single path flat fading channels are assumed in the prior works. However, in practice, the channels are not single path flat fading, and they may not be independent each other. Another practical factor that can also affect the performance of dispersed spectrum cognitive radio networks is the topology of nodes. In this context, several studies in the literature have studied the use of location information in order to enhance the performance of cognitive radio networks [8, 9]. It is concluded that use of network topology information could bring significant benefits to cognitive radios and networks to reduce the maximum transmission power and the spectral impact of the topology [10]. In [11], the effect of nonuniform random node distributions on the throughput of medium access control (MAC) protocol is investigated through simulation without providing theoretical analysis. In [12], a 3D configuration-based method that provides smaller number of path and better energy efficiency is proposed. In [13], 2D and 3D structures for underwater sensor networks are proposed, where the main objective was to determine the minimum numbers of sensors and redundant sensor nodes for achieving communication coverage. In [14–16], the authors represent a new communication model, namely, the square configuration (2D), to reduce the internode interference (INI) and study the impact of different types of modulations over additive white gaussian noise (AWGN) and Rayleigh fading channels on the effective transport capacity. Moreover, it is assumed that the nodes are distributed based on square distribution (i.e., 2D). Notice that the effects of node distribution on the performance of dispersed spectrum cognitive radio networks have not been studied in the literature, which is another main focus of this paper.

In this paper, performance analysis of dispersed spectrum cognitive radio systems is carried out under practical considerations, which are modulation and coding, spectral resources, and node topology effects. In the first part of this paper, the performance analysis of dispersed spectrum cognitive radio systems is conducted in the context of communications applications, and average symbol error probability is used as the performance metric. Average symbol error probability is derived under two conditions, that is, the scenarios when each channel experiences independent and dependent Nakagami- fading. The derivation for both cases is extended to include the effects of modulation type and order, namely, M-ary phase-shift keying ( -PSK) and M-ary quadrature amplitude modulation ( -QAM). The effects of convolutional coding on the average symbol error probability is also investigated through computer simulations. In the second part of the paper, the expression for the effective transport capacity of ad hoc dispersed spectrum cognitive radio networks is derived, and the effects of 3D node distribution on the effective transport capacity of ad hoc dispersed spectrum cognitive radio networks are studied through computer simulations [17].

The paper is organized as follows. In Section 2, the system, spectrum, and channel models are presented. The average symbol error probability is derived considering different fading conditions and modulation schemes in Section 3. In Section 4, the analysis of the effective transport capacity for the 3D node distribution is provided. In Section 5, numerical results are presented. Finally, the conclusions are drawn in Section 6.

## 2. System, Spectrum, and Channel Models

Since there is not any complete statistical or empirical spectrum utilization model reported in the literature, we consider the following spectrum utilization model. Theoretically, there are four random variables that can be used to model the spectrum utilization. These are the number of available band ( ), carrier frequency ( ), corresponding bandwidth ( ), and power spectral density (PSD) or transmit power ( ) [18]. In the current study, is assumed to be deterministic. We also assume that PSD is constant and it is the same for all available bands, which results in a fixed SNR value. Additionally, since we consider baseband signal during analysis, the effect of such as path loss are not incorporated into the analysis. Ergo, the only random variable is the bandwidth of the available bands which is assumed to be uniformly distributed [18] with the limits of and , where and are the minimum and maximum available absolute bandwidths, respectively. In addition, we assume perfect synchronization in order to evaluate the performance of dispersed spectrum cognitive radio systems. The analysis of the system is given as follows.

where denotes the real part of the argument, is the carrier frequency, and represents the equivalent low-pass waveform of the transmitted signal.

where , , and are the gain, delay, and phase of the th path at th band, respectively. Slow and nonselective Nakagami- fading for each frequency diversity channel are assumed.

Notice from (5) that dispersed spectrum utilization method can provide full SNR adaptation by selecting required number of bands adaptively in the dispersed spectrum. This enables cognitive radio systems to support goal driven and autonomous operations.

where and . We assumed single-cell and single user case in this study. However, the analysis can be extended to multiple cells and multiuser cases, which is considered as a future work. At this point, we have obtained the total SNR, and in order to provide the performance analysis the average symbol error probability for two different cases, independent and dependent channels, are derived in the following section.

## 3. Average Symbol Error Probability

In this section, we derive the average symbol error probability expressions of dispersed spectrum cognitive radio systems for both independent and dependent fading channel cases considering -PSK and -QAM modulation schemes. We selected these two modulation schemes arbitrarily. However, the analysis can be extended to other modulation types easily.

### 3.1. Independent Channels Case

where is the fading parameter and , in which is a function of modulation order . Therefore, for -QAM and -PSK modulation schemes, is and , respectively.

#### 3.1.1. M-QAM

#### 3.1.2. M-PSK

### 3.2. Dependent Channels Case

To show the effects of dependent case in our system, we just need to use the covariance matrix that shows how the
bands are dependent. To the best of our knowledge, unfortunately there is not empirical model or study on the dependency of dispersed spectrum cognitive radio or frequency diversity of channels, and determining such covariance matrix requires an extensive measurement campaign. However, there are studies on the dependency of space diversity channels [20, 21]. Therefore, we use two arbitrary correlation matrices for the sake of conducting the analysis here. These two arbitrary correlation matrices are linear and triangular, and they are referred to as *Configuration A* and *Configuration B*, respectively, in the current study.

where , and are eigenvalues of covariance matrix for .

where is the eigenvalue of covariance matrix for the th band.

#### 3.2.1. M-QAM

#### 3.2.2. M-PSK

## 4. Effective Transport Capacity

- (i)
Each node transmits a fixed power of , and the multihop routes between a source and destination is established by a sequence of minimum length links. Moreover, no node can share more than one route.

- (ii)
If a node needs to communicate with another node, a multihop route is first reserved and only then the packets can be transmitted without looking at the status of the channel which is based on a MAC protocol for INI: reserve and go (RESGO) [14]. Packet generation, with each packet having a fixed length of bits, is given by a Poisson process with parameter (packets/second).

- (iii)
The INI experienced by the nodes in the network is mainly dependent on the node distribution and the MAC protocol.

- (iv)
The condition , where is transmission data rate of the nodes, needs to be satisfied for network communications.

### 4.1. Average Number of Hops

where is the node volume density.

where is the diameter of sphere and represents the integer value closest to the argument.

Assuming that the destination node is in the center, we try to calculate all the interference powers transmitting from all nodes by clustering the nodes into groups in order to find out the general formula for .

- (i)
- (ii)
The interference power at the destination node received from one of eight nodes, at a distance , is .

- (iii)
The interference power at the destination node received from one of twelve nodes, at a distance , is .

- (iv)
The interference power at the destination node received from one of twenty nodes, at a distance , where , and , is .

- (v)
The interference power at the destination node received from one of twenty nodes, at a distance , is .

- (vi)
The interference power at the destination node received from one of twenty nodes, at a distance , where , , is .

## 5. Numerical Results

^{3}, ,

*μW*, and . In order for the numerical results to be comparable to the results in [14], we choose the value of for Nakagami- fading channels, which represents Rayleigh fading channels. The effects of 3D node distribution on the effective transport capacity of ad hoc dispersed spectrum cognitive radio networks are investigated through computer simulations considering dispersed channels between two nodes, and the results are shown in Figure 7. In ad hoc model the dependency of channels is assumed to be the same as dependent channels case in Section 3.2. This figure represents the relationship between the bit rate and the effective transport capacity considering 3D node distribution. It is shown that at low and high values, the effective transport capacity is low. However, at intermediate values, the effective transport capacity is saturated. This is due to the fact that the average sustainable number of hops is defined as the minimum between the maximum number of sustainable hops and the average number of hops per route. Full connectivity will not be sustained until reaching the average number of hops. Having reached the average number of hops, full connectivity will be sustained until the number of hops is greater than the threshold value as defined by an acceptable BER, since a low SNR value is produced by low and high values. It can be seen that the correlation between fading channels degrades the performance of the system and it can also be noted that Configuration A case performs better than Configuration B case.

It is known that the deployment of an ad hoc network is generally considered as two dimensions (2D). Nonetheless, because of reducing dimensionality, the deployment of the nodes in a 3D scenario are sparser than in a 2D scenario, which leads to decrease of the internodes interference, thus increasing the effective transport capacity of the system. This can be observed by comparing Figures 7 and 8.

In addition, the 3D topology of dispersed spectrum cognitive radio ad hoc network can be considered in some real applications such as sensor network in underwater, in which the nodes may be distributed in 3D [13]. The 3D topology is more suitable to detect and observe the phenomena in the three dimensional space that cannot be observed with 2D topology [25].

## 6. Conclusion

In this paper, the performance analysis of dispersed spectrum cognitive radio systems is conducted considering the effects of fading, number of dispersed bands, modulation, and coding. Average symbol error probability is derived when each band undergoes independent and dependent Nakagami- fading channels. Furthermore, the average symbol error probability for both cases is extended to take the modulation effects into account. In addition, the effects of coding on symbol error probability performance are studied through computer simulations. We also study the effects of the 3D node distribution along with INI on the effective transport capacity of ad hoc dispersed spectrum cognitive radio networks. The effective transport capacity expressions are derived over fading channels considering -QAM modulation scheme. Numerical results are presented to study the effects of fading, number of dispersed bands, modulation, and coding on the performance of dispersed spectrum cognitive radio systems. The results show that the effects of fading, number of dispersed bands, modulation, and coding on the average symbol error probability of dispersed spectrum cognitive radio systems is significant. According to the results, the effective transport capacity is saturated for intermediate bit rate values. Additionally, it is concluded that the correlation between fading channels highly affects the effective transport capacity. Note that this work can be extended to the case where the number of available bands change randomly at every spectrum sensing cycle, which is considered as a future work.

## Appendix

## Declarations

### Acknowledgment

This paper was supported by Qatar National Research Fund (QNRF) under Grant NPRP 08-152-2-043.

## Authors’ Affiliations

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