Cooperative Spectrum Sensing Using Maximum a Posteriori as a Detection Technique for Dynamic Spectrum Access Networks

Over the past few years, dynamic spectrum access has been gaining an increasing attention as a solution to the spectrum scarcity problem. In this paper, a primary user detection technique based on Maximum A Posteriori estimation is proposed for dynamic spectrum access networks. In the proposed technique, a set of secondary users acting as sensing nodes send their individual decisions about the existence of the primary user to a central fusion center. The fusion center uses the received data to form a codeword then, applies the maximum a posteriori estimation rule to make a final decision regarding the presence of the primary user. The proposed technique takes into consideration the accuracy of the local decisions provided by the secondary users when making a final decision. In this paper, we analyze the performance of the proposed scheme and derive closed-form expressions for the upper bounds of the false alarm and misdetection probabilities. The results show that the proposed technique outperforms other combining techniques in terms of its ability to detect the primary user and, accordingly, minimizes the harmful interference to the licensed network. Moreover, the proposed technique achieves better performance at a lower number of reporting secondary users which compensates for the complexity of the maximum a posteriori estimation.


I. INTRODUCTION
The rapid development in wireless technologies created a large demand on the frequency spectrum, which is not satisfied using the current spectrum management policies [1]. However, studies show that the problem is mainly due to the inefficient utilization of the available spectrum rather than the scarcity of free spectrum bands. For example, in a study on the spectrum utilization in the United States, results showed that around 75% of the reserved spectrum bands are free and not utilized by their incumbent users [2]. Dynamic Spectrum Access (DSA) has been proposed as a solution for the inefficient utilization of the frequency spectrum problem [3]. In DSA systems, unlicensed users (known as Secondary Users (SUs)) are allowed to access and utilize the frequency bands/time slots of the incumbent users (known as the Primary Users (PUs)) for a certain period, and only at a given The associate editor coordinating the review of this manuscript and approving it for publication was Ding Xu . location such that they do not cause service degradation for the PUs [4]. The realization of practical DSA systems is not possible with SUs equipped with conventional transceivers as additional functions have to be performed by the SUs in DSA systems. Cognitive Radio (CR) is the enabling technology behind DSA systems. An SU equipped with CR technologies gains some intelligence as it becomes able to perform additional tasks like spectrum sensing, modulation technique recognition, and adaptive transmission [5]. As the main goal of DSA is to utilize the unused spectrum without causing a harmful interference to the PUs, the ability of SUs to detect the PUs is a crucial function that must be performed by the secondary system. Spectrum Sensing (SS) is the name of the CR process responsible for detecting the presence of the PU in DSA systems, and it is also considered the first step of the cognitive radio cycle [6], [7].
To quantify and evaluate any detection mechanism, false alarm (FA) and misdetection (MD) probabilities must be calculated. The FA probability indicates how frequently the detector wrongly declares that the PU is active utilizing its frequency band. This wrong decision causes the SU to miss the opportunity to utilize the PU's free frequency band. On the other hand, the MD probability indicates how frequently the detector decides that the PU is absent while it is not. This wrong decision may cause transmission collision or harmful interference if the SU transmitted its signal on the occupied band [8].
The presence of the PU is checked by detecting its signal over the targeted frequency band using different methods. These methods are mainly categorized into two approaches. The first one is a blind sensing technique, in which the SUs don't need any pre-known information about the transmitted signal like in the Energy Detection (ED) technique. The ED is widely used for SS function because of its implementation simplicity, and the low design complexity [9], [10]. In the ED technique, the SU needs to estimate only the noise power to set the threshold value and does not require any information about the PU transmission characteristics such as modulation technique or data rate [11], [12].
The other approach is the signal-specific sensing technique, in which the SUs need some knowledge about the characteristics of the PU's signal such as symbol period, modulation type, and carrier frequency, etc. [13]. The most used methods for spectrum sensing in this approach are matched filter detection, and cyclostationary feature detection [14].
The other way to categorize the SS function is based on the level of cooperation between the secondary network members. In this classification, the SS function is carried out either individually or cooperatively. In individual SS, each SU performs the necessary task to detect the PU individually and takes a decision based on its own capability and geographical location without any help from other SUs and without sharing its sensing information with others. On the other hand, SUs may cooperate by sharing their sensing information to achieve a more accurate PU detection result and hence it's named cooperative spectrum sensing. Cooperative spectrum sensing is able to mitigate hidden-node problems [15], and to increase the reliability of detecting PU's activities at low PU signal-to-noise ratios (SNRs) [16].
In cooperative sensing, the final decision about the presence of the PU is taken either by individual SUs in a distributed fashion or by a centralized entity. In distributed sensing, each SU is responsible for taking an individual decision after collecting sensing information from all other neighboring SUs. In centralized sensing, the decision is made at a central entity called the Fusion Center (FC) after collecting individual SU's sensing information. In both categories, the shared sensing information can be in one of three forms: raw-sensing data (samples of the received signal), processed data, or individual decisions. Generally, the sensing results received by the FC or shared with other SUs are combined to perform the cooperative sensing and to obtain the final decision about the PU presence [15].
The most popular combining techniques used in centralized cooperative sensing are Equal Gain Combining (EGC) [17], Maximal Ratio Combining (MRC) [18], and Selection Combining (SC) [19] as will be presented in section II.
In this paper, we propose a centralized cooperative spectrum sensing framework that is based on Maximum A Posteriori (MAP) detection for DSA networks. In this framework, the sensing process occurs in two successive phases. In the first phase, the SUs use the energy detection technique to make local (individual) decisions then, forward the sensing information to the FC. In the second phase, the FC uses the received data to form a codeword then, applies the MAP detection technique to make the final decision about the presence of the PU. The MAP detector determines the most probable hypothesis given the data. Since no other hypothesis is more likely, the decision provided is the optimal one. Moreover, as will be presented in Section IV, the MAP detection takes into consideration the reliability of local sensing information received from the SUs. The reliability of decisions is measured and incorporated into the MAP detection rule using local false alarm and misdetection probabilities (i.e., false alarm and misdetection done by SUs individually). This can be considered as if the FC assigns weights in an optimal manner to individual decisions in order to make the final decision. Therefore, the proposed scheme is expected to outperform other combining techniques used at the FC.
Our contribution in this paper can be summarized as follows: • Proposing a MAP-based centralized cooperative spectrum sensing technique for DSA networks. The proposed scheme is optimal in the sense of minimizing the false alarm and misdetection probabilities at the FC.
• Analyzing the performance of the proposed scheme by deriving a mathematical closed-form expression for the upper bounds of the false alarm and misdetection probabilities.
Additionally, we compare the performance of the proposed detection mechanism with other techniques found in the literature and with the theoretical upper bounds. Organization of paper is as follows: The related work is presented in Section II. The system model is described in Section III. The MAP-based detection technique is described in Section IV. Section V presents the derivation of the upper bounds for the proposed system. Section VI presents the performance evaluation of the proposed technique. Finally, the conclusions are drawn in Section VII.

II. RELATED WORK
In this section, we discuss recent related work in cooperative spectrum sensing for DSA systems. In cooperative sensing, combining individual decisions from SUs can be performed in two different ways according to the available bandwidth for the control channel [4], [15]:  [20]. The main benefit of soft combining is its higher accuracy result compared to hard combining which consumes less bandwidth in reporting the SUs detection results. In MRC combining technique, the signals from each SU are co-phased and weighted before being combined. The applied weights have to be adjusted according to the estimated channels between the SUs and the FC [17]. The EGC technique is similar to MRC except for the weighting circuits [18]. A performance comparison between the MRC and EGC as combining techniques for cooperative spectrum sensing in cognitive radio networks is provided in [21]. In the SC approach, the output performance of the combining process is the same as the highest SNR among all received signals. Using the soft combining technique for making a decision using the one-bit individual decision from each SU is also used. A quantized equal gain combining (QEGC) technique, presented in [22], gives a good balance between the two benefits. Moreover, in [23], authors show how the MRC technique can be used to get a better performance than the majority voting technique when used at the FC. Furthermore, Hybrid techniques are usually used to achieve a trade-off between the benefits of the good performance of the soft combining and the low communication cost of the hard combining. An example of such a hybrid system is presented in [24] where a clustered distributed detection system using a fuzzy logic system and a fuzzy c-means clustering algorithm is developed.
The authors of [25] proposed a cooperative blind Bayesian-based detection framework for spectrum sensing in CR networks to overcome the noise variance uncertainty problem which severely degrades the performance of the ED. In [25], M SUs calculate the power of observed signals and forward it to the FC. Then, the FC utilizes the proposed algorithm to blindly make the final decision about the existence of the primary user. The proposed algorithm is designed mainly for low SNR scenario (e.g., −22dB) and, accordingly, it requires a large window size for power calculations (at least 1000 time instants). Although the proposed framework provides a good performance in terms of false alarm and misdetection probabilities, the authors assume error-free links between the SUs and the FC.
MAP rule was used in SUs to enhance the ability of detection by solving the uncertain noise energy problem that appears when the energy detection technique is used. In this technique, the spectrum sensing is performed in two steps.
First, the posterior probability of each state is estimated using the structure of the Bayesian network, then the posterior probability can be used as the prior probability for the MAP approach [26].
The MAP estimator has been used at the FC by Zhou et.al. [27]. Progressive MAP algorithm was used at the FC as a binary energy detection technique to recover the transmitted decisions stream received from the SUs, which can be considered as passing a two-state Markov chain. However, the final decision is made using hard combining techniques (majority voting/OR techniques).
As a combining technique, authors of [28] developed a MAP-based technique in the relay-based network to improve the recovery of the originally transmitted signal. The detector constructs a codeword from both the relayed and the directly transmitted signals. It uses MAP to estimate the originally transmitted data. In our proposed system, the MAP algorithm is used alone at the FC and independent of any other decision techniques. This is different from the other mentioned systems that use MAP at the SUs or the FC to enhance the detection of the received bits and use other combining techniques to obtain the final decision.

III. SYSTEM MODEL
The system and network models are shown in Fig. 1, and the symbols used in this paper are listed in Table 1. In the secondary network, we have the FC and M SUs acting as sensing nodes which cooperate to detect the PU status. The detection of the PU signal is performed in two phases. In the first one, each SU makes a local decision about the existence of the PU signal individually and forwards the sensing results to the FC. The SU detects the existence of the PU using ED technique where the energy of the received signal is calculated and compared to a pre-defined threshold TH . The SU transmits +1 symbol if the PU is detected (i.e., the energy of the received signal is greater than or equal to TH ) and −1 if not.
In the second phase, the FC constructs a codeword from the data received from all SUs and applies the MAP-based detector to make a final decision about the presence of the PU. In this paper, we assume that the PU and all the SUs transmit their data using Binary Phase Shift Keying (BPSK) modulation and that the channels between network nodes experience Rayleigh flat fading with Additive White Gaussian Noise (AWGN). The signal received from the PU at the n-th SU is given by and the signal received from the n-th SU at the FC is given by where: signal corresponding to the n-th SU decision bit C SU n ∈ {−1, +1} where E SU n is the energy of the n-th SU bit. • h SU n and h FC n are Rayleigh flat fading channel gains of the links between the PU and the n-th SU and between the n-th SU and the FC, respectively, where E[h 2 • w SU n and w FCn are AWGN with zero mean and variances N 0SU n /2, and N 0FC n /2 at the SU receiver and at the FC receiver, respectively. The signals received at the FC from M SUs can be written in a matrix form as follows

and the channel matrix H is given by
The received vector Y is used by the MAP-based detector to make the final decision about the PU status. This decision depends on the quality of the channels between the PU and the SUs, the detection thresholds used by the SUs, and the quality of the channels between the SUs and the FC.

IV. MAP-BASED DETECTION TECHNIQUE
Unlike the majority voting and MRC techniques, the MAPbased detection technique takes into consideration the quality of the detection links between the PU and the SUs. The quality of these links is measured using false alarm and misdetection probabilities. These probabilities are passed to the MAP detection algorithm together with the received vector Y to make the final decision about the existence of the PU. The MAP detection technique is optimal in the sense that it minimizes the false alarm and misdetection probabilities at the FC [29]. To describe the proposed algorithm, we consider a secondary network composed of three SUs and one FC. However, the same procedure can be generalized to support an arbitrary number of SUs. Let CW = C SU 1 C SU 2 C SU 3 T be the codeword of the individual detection decisions sent by the SUs to the FC. If the channels between the PU and the SUs have the same quality (i.e equal received SNRs), and all of the SUs use the same detection threshold TH , this codeword will take one of two values [+1 Since, in general, these channels don't necessarily have the same quality, this codeword will take one of eight possible values. The probability of sending a codeword depends on the status of the PU (active or idle) as illustrated in Table 2 where P D n , P MD n , and P FA n are the local probabilities of detection, misdetection, and false alarm, respectively, at the n th SU. The values of these local probabilities are calculated according to the following equations [30]- [32] where σ 2 w n = N 0SU n /2 is the noise variance at n-th SU, γ SU n = E s /N 0SU n is the received SNR at n-th SU, and (., .) is the incomplete Gamma function. For mathematical traceability, we assign two different indices to the same codeword in order to represent the difference in its probability (i.e., CW i+8 = CW i but P(CW i+8 ) = P(CW i )).
The MAP detection technique estimates the transmitted codeword CW as follow: where the maximization process occurs over the sixteen possible codewords provided in Table 2, and X i is the transmitted vector corresponding to the codeword CW i , e.g., Substituting from (9) into (8) yield After estimating the codeword CW , the FC makes a decision about the PU status as follows  9 PU status ← PU is Active 10 else 11 PU status ← PU is Idle 12 end 13 return PU status

V. SYSTEM PERFORMANCE
In this section, in order to analyze the performance of the proposed technique, we derive closed-form expressions for the upper bounds on the final false alarm and misdetection probabilities (i.e., P FA , and P MD at the FC). The derived expressions are used to determine the theoretical limits of the system performance and for the sake of comparison with other techniques.
For M SUs, the probabilities of false alarm and misdetection are given by and respectively, where P (CW k → CW i ) is the pairwise error probability of confusing CW k with CW i when CW k is transmitted and when these are the only two hypotheses. P(CW k ) can be calculated by substituting from (5), (6), and (7) into codewords probabilities given in Table 2. In order to calculate the upper bounds on P FA and P MD (i.e., the right hand sides of (11) and (12) ), we need to derive an analytical expression for P (CW k → CW i ) for all possible values of i, k. In the mathematical analysis, we continue with the assumption of having three SUs. However, following the same outlines, the derivation can be generalized to support an arbitrary number of SUs.
When the transmitted codeword is CW k , the received vector would be Y k = HX k + W . According to (10), the probability P (CW k → CW i ) is given by (13), as shown at the bottom of the next page, where W , H (X k − X i ) denotes the dot product of the two vectors W and H (X k − X i ). When be a zero mean Gaussian random variable with variance where ϑ ki = P (CW k ) /P (CW i ). In order to find P (CW k → CW i ), (14) has to be averaged over the distribution of the random variable H (X k − X i ) . Since gains of all channels have the same distribution, then, the distribution of the H (X k − X i ) depends on the hamming distance between CW k and CW i , i.e., the number of positions at which CW k and CW i are different. In what follows, the hamming distance between CW k and CW i will be denoted by δ ki where δ ki ∈ {0, 1, 2, 3}. Hence, the distribution of the H (X k − X i ) would be the same for any pair of codewords with the same δ ki . Therefore, the probability P (CW k → CW i ) depends on: 1) The hamming distance δ ki 2) The ratio between codewords probabilities ϑ ki . Let ρ δ ki (ϑ ki ) = P (CW k → CW i ). Then, we need to derive ρ δ ki (ϑ ki ) for δ ki ∈ {0, 1, 2, 3}.
• Derivation of ρ 0 (ϑ ki ) Since ρ 0 (ϑ ki ) = P (CW k → CW i ) when CW k = CW i (but P(CW k ) = P(CW i )), we can not use (14) to continue our derivation and we have to go one step backwards. From (13), Since P(CW k ) and P(CW i ) are deterministic variables, then • To derive ρ 1 (ϑ ki ), we consider the case when k = 3 and i = 7 that can be generalized to all values of k and i such that δ ki = 1. From (14), Averaging (17) over distribution of h 2 SU 1 , yields where = √ γ FC /(1 + γ FC ), and the derivation of (18) is provided in Appendix VII.
Averaging (19) over the distribution of Z 2 which is given in (20) yields to (23), as shown at the bottom of the 8th page; the derivation is given in the Appendix VII.
and averaging (21) over (22) yields (24), as shown at the bottom of the next page; the derivation is given in Appendix VII.

VI. PERFORMANCE EVALUATION
In this section, the performance of the proposed MAP-based cooperative spectrum sensing technique is evaluated. We started by studying the effect of the detection threshold value at the SUs on the performance of the proposed technique compared to the traditional techniques. Then, we characterize the performance of the system in terms of its probabilities (P FA and P MD ) over the feasible Range of Operation (RO) [35]. Then we study the effect of different links quality in terms of different SNR values on the system performance. Finally, the effect of the number of reporting SUs is investigated. The network model used in the evaluation process is depicted in Fig. 1. The default simulation parameters are listed in Table 3 unless otherwise clearly mentioned. All experiments were carried out using MATLAB. The upper bounds for P FA and P MD are shown in Fig. 2 and Fig. 3, respectively. The figures show how  much the proposed technique performance is very close to its mathematical upper bounds.

A. THE PROPOSED SYSTEM DETECTION CAPABILITY
In this experiment, we study the PU detection ability of the proposed technique. Fig. 4 shows the value of the FC detection probability P D against the detection threshold TH . As can be inferred from the figure, the MAP-based technique outperforms other techniques, especially at higher threshold values. Fig. 5 shows the Receiver Operating Characteristic (ROC) of the proposed technique compared to the other combining techniques and its theoretical upper bounds. As can be noticed, the performance of the proposed technique is better than the other combining techniques especially at lower P FA where there are enhancements  in the P D of about 10%. This means that the proposed system is able to achieve a higher probability of detection at low false alarm probability values compared to the other techniques. To complete the performance analysis, we compare the proposed MAP-Based technique with the progressive MAP proposed in [27]. As can be inferred from Fig. 6, the proposed technique outperforms both the progressive MAP-OR and the progressive MAP-majority techniques especially at low (acceptable) values of P FA . The relationship between P MD and P FA can illustrate the overall system performance, which characterizes the ability of the system to make a trade-off between the high ability to detect the presence of the PU and the possibility of triggering a false alarm. The RO, as defined by [35], is a certain region where both P MD and P FA are suitable to be used with general cognitive radio systems, this region extends from 10 −3 to 10 −1 for both probabilities. A comparison between the proposed MAP and the other combining techniques, at different values of SNR at the SUs side, is shown in Fig. 7. The comparison shows that the proposed technique outperforms the other techniques, even that for low SNRs where the other combining techniques failed to achieve acceptable results as shown in Fig. 7.b.
VOLUME 8, 2020 The same result can be observed if we compared the system performance in terms of P D against average SUs' SNR values at a given value of P FA (e.g., P FA = 10 −2 ). As shown in Fig. 8, the proposed technique has a much better performance compared to other combining techniques even at lower SNR values. For example, the MAP-based system achieved P D = 90% at SNR value = 9dB. However, in order to achieve the same value of P D with other techniques, the SNR has to reach 13 dB. Also, the P D value of other combining techniques doesn't exceed 96% when the SNR equals 20 dB; While the proposed MAP-based technique can achieve it at 12 dB SNR.

B. THE EFFECT OF LINKS QUALITY
In this experiment, we investigate the effect of the detection link (PU→ SU) and the reporting link (SU → FC) quality on the system performance. The link quality is expressed by the value of the SNR at the receiving node. Fig. 9 shows the effect of the detection link and the reporting link on the value of P D .  Two curves are plotted, the first one (dashed line) shows the effect of the detection link quality represented by the received SNR at the SU when the SNR at the FC (representing the SU → FC link status) is kept constant at 10dB. In the same way, the second curve (solid line) shows the effect of the reporting link between the SU and the FC at constant SU's SNR. By comparing the two curves, we can conclude that the detection link has a much higher effect on the overall detection process compared to the reporting one.
The full study of the effect of the two links and their combined effect on P D can be obtained from Fig. 10 which shows the performance of the system at different links status. It generalizes the previous finding that the detection link has a higher effect on the system performance compared to the reporting link.

C. THE EFFECT OF NUMBER OF REPORTING SUs
In this experiment, we investigate the effect of the status and the number of reporting SUs on the detection performance of the system. In the first experiment, we examine the effect  of adding an additional SU to the cooperative set of SUs on the overall performance. We assume a set of two SUs with an average SNR value, for which the system achieves a certain probability of detection P D 2 . Fig. 11 shows the enchantment in the system performance, in terms of the value of P D , due to the contribution of the third SU at different average SNR values for the original two SUs. As can be inferred from the figure, there is a threshold value for the third SU's SNR value after which the contribution of third SU becomes noticeably effective in the detection process and enhances the overall system performance. Also, this threshold value has an inversely proportional relation with the average SNR of the original SUs. From this result, it can be inferred that adding a new SU to the cooperative set may not be always useful taking into consideration the additional communication overhead and the increase in detection algorithm computational complexity as will be discussed later.
In the second experiment, we evaluate the effect of the number of reporting SUs on the achieved value of P D by comparing the performance of the proposed MAP-based technique with that of the other combining techniques for the same number of reporting SUs. As shown in Fig. 12 the MAP-based technique outperforms the other techniques at the same number of SUs especially at low values of average SNR at the SUs.

D. SYSTEM COMPLEXITY
In this section, we calculate the system complexity (O) as a function of the number of reporting SUs for the proposed system as well as the traditional combining techniques. Also, we compare the performance of the proposed techniques (value of P D ) with other traditional techniques. To explain how to get the order of complexity, we need first to calculate the complexity of basic mathematical operations; If we have two numbers of n 1 and n 2 bits, so the complexity order of their addition operation is O (max(n 1 , n 2 )), their multiplication operation is O (n 1 n 2 ), and for taking the log operation of nbit number is O n 2 ln (n) [36].
For the proposed algorithm, let l h , l s , and l p be the number of bits representing the channel coefficient h, transmitted signal X from the SU, and the P(CW i ) respectively, and M is the number of SUs. The computational complexity of the proposed MAP technique can be calculated by analyzing the terms of (10) as follows: Since X i is the BPSK modulated signal corresponding to the i th SU decision bit and only be either {+1, −1}, the complexity order of this term is O 2M (l s l h ) 2 .
• log (P (CW i )): The complexity order of this term will be O 2 M l 2 p ln(l p ) . As the two parts are summed, the complexity order of (10) can be shown as, Using the same method for the traditional combining techniques, as defined in [17] and [18], the computational complexity order can be calculated as follows: While the previous set of computational complexities show that the MAP-based technique has a higher complexity, it is more fair to compare the complexity for the same performance level. In our case, we should compare the complexity of different systems that achieve the same value of P D . Assuming that l h = 4, l s = 6, the average SNR value of SUs = 10dB, and the SNR value at the FC = 20dB will be used during the comparison. Fig. 13 shows a comparison between the achieved value of P D for MAP, EGC, and MRC techniques at nearly the same complexity values. 1 The MAP technique achieves a higher value for P D for the same complexity. 2 Moreover, for the same complexity value, the MAP-based technique utilizes a smaller set of reporting SUs which means less communication overhead. From the previous analysis, we can notice that the MAP-based technique provides a fair trade-off between the complexity and the performance.

VII. CONCLUSION
In this paper, we presented a Maximum A Posteriori (MAP) based cooperative spectrum sensing technique for dynamic spectrum access networks. The proposed technique was implemented at the central fusion center where the individual sensing decisions, regarding the presence of the primary user, are transmitted from the secondary users. The fusion center applies the MAP technique using the collected results to obtain a final estimation regarding the presence of the primary user and disseminated it to the secondary users. Besides the algorithm design, the mathematical upper bounds in terms of the probability of detection and probability of false alarm have been derived for the sake of comparison. The performance of the proposed technique has been compared to that of other techniques found in literature. The results show that our proposed technique outperforms the others in terms of the ability to detect the presence of the primary user represented by enhancing the probability of detection (10% improvement) for the same false alarm probability. Also, the proposed system achieves a better performance in terms of the required number reporting secondary users compared to other techniques for the same probability of detection which compensates for the higher complexity of the system. Also, MAP is considered as a green communication technique by having less power consumption by reducing the number of reporting SUs.

APPENDIX B DERIVATION OF ρ 2 ϑ ki
Consider the case when k = 4 and i = 6 (which have two different bits belong to SU 1 ,and SU 3 ), and averaging (19) over the distribution of Z 2 (20) yields γ FC dz (37) where C = log(ϑ ki )

4
. Integrating (37) using integration by parts where: dV where Using Mathematica to solve this integration yields Substituting back for r,d,and C then using (40) in (38), and after some mathematical manipulations, (23) is obtained.

APPENDIX C DERIVATION OF ρ 3 ϑ ki
Consider the case when k = 2 and i = 7 (which have three different bits belong to SU 1 , SU 2 , and SU 3 ). Averaging (21) over the distribution of Z 3 (22) yields γ FC dz (41) where C = log(ϑ ki )

4
. Integrating (41) using integration by parts where: dV Using Mathematica to solve this integration and substituting back in (43), and after some mathematical manipulations, (24) is obtained.