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Performance evaluation of spectrum sensing in infrastructure based multihop network is very hard to achieve because of the adverse effects of channel fading. In this paper, performance of a multihop link is studied over Nakagami-m distribution. It provides the exact theoretical methodology for the performance analysis of spectrum sensing by evaluating detection probability. Using a cascaded multihop model, the end-to-end Signal to Noise Ratio (SNR) is given over Nakagami-m distribution. In the analysis, multihop model based on relays are considered over independent and not identically distributed (i.n.i.d) wireless channels. Simulation results show the effect of increase in number of hops on probability of detection for multihop links. Subsequent to the thorough fading severity analysis, it has been accomplished that spectrum hole detection is more crucial at lower SNR values with large number of hops.

Cognitive Radio (CR) may be considered as a truly smart device talented of selecting its own band of action and deciding for certified or uncertified access. The keystone of the practically realizable CR network is the excellent organization of the spectrum sensing. The task of obtaining awareness about the spectrum usage and existence of PU’s in a geographical area is the main concern of spectrum sensing. The parameters viz. SNR in a wireless link, sensing time and threshold level plays an important role for the performance measurement of energy detection based spectrum sensing. Systems in unlicensed frequency bands can achieve great spectrum efficiency but have faced an inference that limits network capacity and scalability. Among all spectrum sensing techniques, energy detection is the most popular method because of its low computational complexities [1-3]. In the literature

[4,5], many research efforts have been expended to analyze the performance of energy detectors. However, an optimal spectrum sensing framework is introduced by [_{i} as a random variable between all nodes. Nakagami-m fading is defined by parametric gamma function for its rapid fading in high frequency long distance propagation. Power of the Nakagami distribution follows the gamma distribution with severity parameter m ranges from 1/2 to ∞. For special case of m = 1, the distribution reduces to Rayleigh distribution [_{d}) as performance metrics.

The remainder of paper is organized as follows: Section 2 briefly describe the system model of cascaded relays based wireless link. In Section 3, performance analysis of energy detection in Additive White Gaussian Noise (AWGN) channel is described and the closed form solution of probability of detection expressions in terms of circular contour integral for multihop cooperative scenario in Nakagami-m fading is derived. The corresponding expressions for equivalent SNR PDF, MGF, CDF and moments for the assumed model are introduced. Section 4 provides performance results and discussion. Finally, in Section 5, we sum up the conclusions of our studies.

Multihop transmission in wireless networks e.g. cellular and ad-hoc networks has been researched in the recent period due to its advantages over traditional networks for a variety of reasons. Relays are used to provide maximum coverage range with high transmission rate. In cellular communications, multihop networks are also very advantageous in terms of deployment and connectivity. Because of the increased spatial reusability of relays, the transmitter power required at the transmitter also reduces. In terms of CRs, the multihop networks can provide better spectrum utilization, if the numbers of relays within a wireless link are suitably chosen. Schematic of a multihop transmission link with primary source (PU-S), Primary receiver (PU-D) and (L-1) intermediate relay nodes is shown in

By applying proper source coding and interleaving techniques, the data is arranged in a block format. Further desired modulation scheme is implemented for multihop transmissions.

When direct path wireless link between (PU-S) and (PU-D) is found to be bad, then the SU senses the wireless link with number of relay node between (PU-S) and (PU-D). In this system model reliable spectrum sensing performance is subjected to channel uncertainty constraint and relay characteristics. Each communication between cognitive relay occurs in orthogonal channels to avoid the inter-channel interference. Frequently used fading distribution to characterize the signal statics is Nakagami-m distribution. We assume that sub-channels between relay nodes and direct path are independent and are Nakagami-m flat fading channels. The noise is added at the relays and this noise is considered as the AWGN signal. In case of severe fading conditions, the transmission from source S to destination D fail if signal at any of the L hops fall below the required threshold SNR γ_{th}. If a symbol transmitted by the source S is successfully decoded by (L-1)^{th} relay i.e. no intermediate relay in the link is down, the SNR at destination D statistically determined by SNR of the last hop. Hence, the statistics of SNR at destination D depends only on Lth hop provided that no other interfering source is present. The cascaded multihop DF relay link can be reduced to an equivalent point-to-point link. By following the approaches proposed in [

where, H represents joint probability that outage occur with first (L-1) hops and represent the SNR of L^{th} hop. Assuming channels of L hops as independent and not identically distributed (i.n.i.d) and relays in

where, is the Cumulative Distribution Function (CDF) of γ_{l}. The primary signal is assumed to be s(t) with multiple cascaded relays employ different relaying mechanisms to transmit the received signal.

In the energy detection in AWGN channel, the received signal x(t) takes the form x(t) = h*s(t) + n(t). Where h*s(t) is the channel impulse response and n(t) is the AWGN. The received signal is first pre-filtered by an ideal band-pass filter with transfer function to limit the average noise power and normalize the noise variance. The squared and integrated output is used to finally construct a measure of the energy of the received waveform. The detected energy is further compared with a predefined threshold which depends on the noise floor, and if detected energy is below than the predefined threshold then it is assumed that licensed spectrum is free otherwise if the detected energy is above than the predefined threshold value then it is assumed that licensed spectrum is occupied by the PUs hence determining the presence or absence of the licensed user’s spectrum. The performance of energy detection can be efficiently characterized in terms of detection probability and false alarm probability. In this model, Nakagami-m flat fading environment is considered. The distribution function of Nakagami-m fading is given by

where Γ(.) is the Gamma function, Ω = y^{2} is the average power where y is Nakagami distribution envelope and m is fading parameter. Since Nakagami distribution encompasses scattered, reflected and direct components of the original transmitted signal, the output of the energy detector acts as the test statistic to test the two hypothesis H_{0} and H_{1} at particular instant of time (t) and can be represented as

The hypothesis H_{0} describes that only noise is the present without signal within a channel and hypothesis H_{1} describes that the signal and noise both are present within the channel. The probability density function (PDF) of the received signal for hypothesis H_{0} and H_{1} is given by

where Г(.) is the gamma function, I_{v}(.) is the v^{th}^{ } order modified Bessel function of the first kind, and u = TW is the time bandwidth product. is the signal to noise ratio at the cognitive coordinator. The detection probability (P_{d}) and probability of false alarm (P_{f}) are denoted as

A circular contour (Ω) integral representation given by [

where W is the circular contour of radius r that encloses the origin. The singularities of the integrand are Z = 0 and Z = 1 therefore radius of the contour ranges from 0 to 1. The received signal’s SNR changes randomly in a faded channel; because of this the detection probability is also random in nature. So, the average detection probability Pd(Avg) is given by

where is the moment generating function (MGF) of SNR and E(.) is the expectation.

In cascaded multihop system, all relays are serially connected with each other. The output of first relay is the input of the second and so on. The end to end signal SNR is dependent upon individual hop SNR. The relays within a wireless link can work on the condition, which is based upon the fact that the multihop branch is dominated by the weakest SNR within the hops of relayed link. This estimation has been referred by various authors working for the recital analysis of multihop systems [27,28]. According to this approximation, the end-to-end SNR of the relayed branch is given by

This SNRγ_{l}, is averaged over Nakagamim PDF and can be expressed as

Direct link can be established when there is LOS distance between (PU-Tx) and (PU-Rx). This direct link incorporation would have influence on the detection performance. The P_{d} can be improved with the association of direct link. The total SNR will now dependent upon both the links and is the summation of MGF of both links. The destination combines the received signals from the source to destination (direct link) and source to relays and then to the destination link (relay link). The combination of direct and relayed wireless links will result in improved system performance. So, for further analysis both the links has been considered.

In this section, the average detection probability in multihop wireless link over Nakagami-m fading channel is investigated. In the investigations, both independent and identically distributed (i.i.d) and independent and nonidentically distributed (i.n.i.d) channels for L-hops are assumed. Also, it is assumed that the link is operating in L-hops and relays are located at equal distances between (PU-S) and (PU-R), and the normalized threshold SNR is also scaled by L. Channel state information (CSI) is assumed to be available at each receiving node and all cascaded nodes within a wireless link are coordinated, i.e., transmission within whole chain occurs without any delay. Based on the system model, the simulation results are given in this section. The analysis covers the receiver operating curves (ROC). It is shown that the spectrum sensing employed in the cognitive radio network depends upon channel distributions. The results shown in this section are for average detection probability of the multihop wireless link. Each relay node act as a pseudo random bit sequence generator (PRBS). The PRBS is modulated by digital modulation techniques for the simulation. The modulated information is transmitted in the channel as symbols. Each symbol is assigned specific energy E as per the transmitted power requirements. The transmitted symbol is corrupted by (AWGN) and fading effects in the channel. The AWGN is additional to the transmitted symbol to set required (SNR) over the wireless communication link. The symbol with SNR

is multiplied by random Nakagami-m fading coefficients generated from the Nakagami-m distribution. Further, energy detection can be used to analyze the spectrum availability. The symbol in the channel is now faded according to the Nakagami-m fading. Nakagami-m fading with arbitrary fading parameters and arbitrary average SNR levels has been used to carry simulation through MATLAB. The AWGN can also be added to the symbol as per required SNR with the help of dedicated function. _{d} verses P_{f} for four hop cascaded multihop system with different values of average SNR. As the value of average SNR within wireless link are increased, the P_{d} starts increasing. The interesting fact is the value of targeted P_{d} 0.9 is reported at average SNR of 16 dB with Nakagami-m fading severity parameter m = 2 and P_{f} = 0.2.

We also have analyzed the spectrum sensing with different values of fading severity parameter m. The P_{d} has direct relation with the severity parameter m. To explore the other side, the receiver operating curves can also be dependent upon fading severity parameter m. The practical value for m lies between 0 - 5 in typical wireless communication. We considered cascaded multihop channel consisting of L hops with desired probability of detection (0.9) and probability of false alarm (0.1 - 0.2).

_{d} also increasing within a three hop wireless link. The value of targeted P_{d} = 0.9 is reported at severity parameter m = 4 for average SNR of 12 dB at the P_{f =} 0.2. Further, we analyze the spectrum sensing with number of hops in cascaded multihop network.

_{d} is dependent upon the number of cascaded relays in the wireless link.

and (PU-D), then we get the lower bound value of 𝑃_{d}. If one relay (L = 1) is placed between (PU-S) and (PU-D), there is an abrupt change in P_{d} for the fixed value of 𝑃_{f} = 0.2. That value of P_{d} act as an upper bound value. The upper bound value clearly represents that the detection performance improves a lot with the incorporation of single relay between transmitter and receiver. But, it doesn’t go on increasing with increase in number of relays. As the relays within a wireless link starts increasing, the value of P_{d} lies in between the upper and lower bound P_{d} values.

_{d}) varies with average SNR per hop for the fixed value of m. It has been concluded that if the average SNR within the hop start increasing the P_{d} also increasing for a fixed value of P_{f}.

The closed form expressions for end to end SNR have been given for cascaded multihop communication link over Nakagami-m fading channels. The method presented in this contribution is useful for the P_{d} analysis. The performance evaluation is done by keeping three main objectives in mind i.e. number of relays, type of relays and fading severity. It is to be mentioned that the several investigations have been performed to determine the effect of increasing the number of hops on the performance of multihop network. Further, results clearly include evaluation of P_{d} under different channel conditions with variable m. This work carried to evaluate the

performance analysis of energy detection system in Nakagami-m fading channel for a cascaded multihop wireless link and has yield an optimum value of fading parameter m for the desired detection probability. Further the detection performance is improved by deploying the diversity (multi-branch) scheme. On the basis of investigation, it is to be noted that the increasing number of relays/hops deteriorates the performance of the system for the given average SNR and fading parameter m. Inspection also reveals that if the channel conditions are good, the greater number of hops can be used to achieve target P_{d}.