Analysis of mobility impact on interference in cognitive radio networks

Physical Communication 9 (2013) 212–222

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Analysis of mobility impact on interference in cognitive radio networks✩ Ali Rıza Ekti a,∗ , Serhan Yarkan b , Khalid A. Qaraqe c , Erchin Serpedin a , Octavia A. Dobre d a

Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, 77843–3128, United States

b

Department of Electrical and Electronics Engineering, Istanbul Commerce University, Küçükyalı, İstanbul, 34840, Turkiye

c

Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Education City, 23874, Doha, Qatar

d

Faculty of Engineering & Applied Science, Memorial University of Newfoundland, St. John’s, NL, A1B 3X5, Canada

article

info

Article history: Received 2 March 2012 Received in revised form 31 May 2012 Accepted 4 July 2012 Available online 20 July 2012 Keywords: Fading Femtocell Interference Mobility Shadowing

abstract Cognitive radio (CR) technology seems to be a promising candidate for solving the radio frequency (RF) spectrum occupancy problem. CRs strive to utilize the white holes in the RF spectrum in an opportunistic manner. Because interference is an inherent and a very critical design parameter for all sorts of wireless communication systems, many of the recently emerging wireless technologies prefer smaller size coverage with reduced transmit power in order to decrease interference. Prominent examples of short-range communication systems trying to achieve low interference power levels are CR relays in CR networks and femtocells in next generation wireless networks (NGWNs). It is clear that a comprehensive interference model including mobility is essential especially in elaborating the performance of such short-range communication scenarios. Therefore, in this study, a physical layer interference model in a mobile radio communication environment is investigated by taking into account all of the basic propagation mechanisms such as largeand small-scale fading under a generic single primary user (PU) and single secondary user (SU) scenario. Both one-dimensional (1D) and two-dimensional (2D) random walk models are incorporated into the physical layer signal model. The analysis and corresponding numerical results are given along with the relevant discussions. © 2012 Elsevier B.V. All rights reserved.

1. Introduction As wireless communications pervade daily life, many of the ever increasing demands should be met simultaneously. Some of those demands are high performance, improved capacity, better coverage, quality of service (QoS), energy and cost efficiency, and reduced power consumption. Cognitive radio (CR) systems are expected to tackle these demands by applying advanced signal processing techniques [1–5]. Even though there is no formal definition of CR in the literature, a CR is a wireless device which

✩ This paper was presented in part at SPAWC 2012, Cesme, Turkey, June 2012. ∗ Corresponding author. E-mail addresses: [email protected] (A.R. Ekti), [email protected] (S. Yarkan), [email protected] (K.A. Qaraqe), [email protected] (E. Serpedin), [email protected] (O.A. Dobre).

1874-4907/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.phycom.2012.07.004

can be aware of, learn about, and adapt to its surrounding environment [1]. The environment of a CR may include radio frequency (RF) spectrum, user behavior, transmission characteristics and parameters, multi-access interference, and so on [6]. Among all of these, multi-access interference has gained slightly more importance since it degrades the overall wireless communication system performance. This characteristic nature of interference becomes a vital design issue especially in next generation wireless networks (NGWNs) since frequency reuse of one (FRO) is the prominent deployment option. Therefore, it is easy to conclude that modeling and predicting the behavior of future interference levels are two essential tasks for both CRs and NGWNs. Interference behavior is affected mainly by the following four factors: (F.i) environment, (F.ii) network topology/ structure, (F.iii) mobility, and (F.iv) traffic type. Measurement results available in the literature illustrate that different environments affect the wireless signals (therefore,

A.R. Ekti et al. / Physical Communication 9 (2013) 212–222

interference) in different ways [7]. Because each propagation environment presents different characteristics from the fading perspective [6, Table 3], interference in a propagation environment is expected to be different from that in another environment. One of the easiest ways for incorporating the impact of environment into the interference analysis is to use a simple path loss model with different path loss exponents corresponding to different environment types [8]. It is clear that such simplifications cannot provide a detailed description of the interference behavior. Therefore, more comprehensive models encompassing shadowing, small-scale fading, and their higher-order statistical characteristics are required. In contrast to (F.i), (F.ii) refers to a more flexible factor since some properties such as network structure, topology, and deployment can be controlled and/or designed to some extent. From this point of view, the impact of (F.ii) on interference behavior changes with respect to the design options at hand. In traditional cellular systems, one of the most dominant sources of interference is co-channel interference (CCI). CCI is controlled by employing re-use of the frequencies in distant cells (large cluster sizes or reuse factors) at the expense of capacity degradation, such as in Global System for Mobile Communications (GSM) [7,9]. However, especially for NGWNs, FRO seems to be the most prominent deployment option for a general cellular layout, as it improves capacity. Note also that FRO bypasses the frequency planning stage in cellular design, which is a very expensive process. Nonetheless, FRO introduces significant CCI into the system especially for the terminals residing in the vicinity of the cell borders [10]. Similar to centralized and non-centralized schemes, interference behavior in a multi-hop network along with the multiinput multi-output (MIMO) option [11] is not the same as in the single-input single-output (SISO) option. The impact of (F.iii) on interference behavior has more than one aspect. At the microscopic time scale, mobility causes drastic power level fluctuations in the received signal (i.e., interference) [7]. Femtocells constitute one of the best contemporary examples of such scenarios [12–15]. Since the radius of femtocells is noticeably small compared to large-scale networks, the fluctuations in the interference power levels will be drastic due to the mobility of user equipments (UEs). Femtocells are initially designed to operate in licensed spectrum; therefore, a possible RF spectrum scarcity can be expected. Refs. [16,17] proposed an efficient channel reuse approach for femtocells by using the sensing feature of CR. Using this approach, the uplink interference from a macrocell user to a femtocell user can be identified and the proper channel allocation can be established. On the other hand, at macroscopic time scales, the mobility behavior (or pattern) of the transceivers becomes dominant compared to each individual mobility pattern [18–20]. When coupled with multiple interfering sources, mobility behavior gains extra dimensions such as group-cluster behaviors as well as homogeneous–heterogeneous mobility patterns [21]. When the victim nodes are capable of acquiring information about the mobility behavior of interfering sources, they can improve their performances through the use of this knowledge.

213

For (F.iv), it is expected that depending on the content of the transmission, different interference scenarios will emerge. Experimental data reveal that voice traffic exhibits quasi-deterministic properties, whereas Internet traffic possesses self-similarity to some extent [20]. Other traffic types such as data, multimedia, and gaming demonstrate distinct stochastic characteristics implying various interference behaviors especially in NGWNs. In the presence of traffic knowledge, interference can be canceled and/or avoided by using interference scheduling techniques [5]. Sufficient statistics should be collected to obtain reliable characterization of the network traffic. Apart from (F.i)–(F.iv), there are some other factors affecting interference as well. Transmission frequency, weather/seasonal conditions and precipitation are just a few factors. For example, the presence of high pressure air can cause unintentional interference to other signals and eventually CCI can occur [22]. However, these factors are outside the scope of this study. In light of the aforementioned discussions, the main contributions of this study can be itemized as follows: (I) a two-dimensional (2D) random walk mobility model is directly incorporated into the interfering fading signal at baseband and (II) all of the main propagation mechanisms (e.g., small- and large-scale fading) along with their higherorder statistical characteristics are taken into account in the signal model. Also, the impact of decorrelation distance on the interference is studied for different propagation environment schemes. The rest of the paper is organized as follows. In Section 2, the system model is presented. This is followed by the interference analysis of the system model, which is described in Section 3. Concluding remarks and further discussions are provided in Section 4. This is followed by Appendices A and B. 2. System model Consider a primary user (PU)–secondary user (SU) simultaneous communication scenario where both transmitter (Tx)–receiver (Rx) pairs are in their close vicinity. In the SU network, assume that there is a secondary user transmitter (SU–Tx) (probably mobile) that is communicating with the secondary user receiver (SU–Rx). Both networks are assumed to operate on the same RF spectrum. Neither frequency division duplexing (FDD) nor time division duplexing (TDD) is allowed. Such scenarios are encountered generally in unlicensed (but regulated) RF bands such as industrial, scientific, and medical (ISM) band. In these scenarios, when a user transmits on one portion of the band of interest, a different user might be receiving on the same portion of the band at the same time, which leads to CCI. A general illustration of the set up considered in this study is depicted in Fig. 1. In Fig. 1, it is assumed also that the separation between primary user receiver (PU–Rx) and SU–Tx is on the order of a couple of meters. With this assumption, femtocell scenarios for NGWNs can also be studied since coverage areas for the femtocells are approximately of this range [12]. SU–Tx exhibits low-speed mobility behavior similar to that of pedestrians. Such mobility behaviors will be modeled as a random walk for the sake of simplicity [23]. The justification for and details of the random walk assumption are

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interference, only I (t ) in (1) will be analyzed. Hence, under the narrowband assumption, the interfering signal leaking onto the PU–Rx can be expressed as I (t ) = m(t )a(t ),

(2)

where m(t ) is the small-scale fading and a(t ) is the largescale fading component [25,26]. Rice and Rayleigh distributions are two of the very frequently used distributions to describe |m(t )| for line-of-sight (LOS) and non-lineof-sight (NLOS) propagation, respectively. As pointed out in [7], a Rayleigh distribution seems to be the natural selection for the fading envelope due to the uniform illumination in all directions in the absence of LOS under different shadowing scenarios. The probability density function (PDF) of the Rayleigh fading, pR (ϖ ), assumes the expression

Fig. 1. Illustration of an interference scenario for CR network.

  ϖ ϖ2 pR (ϖ ) = 2 exp − 2 , σ 2σ

for ϖ ≥ 0.

(3)

In (2), the large-scale fading component, a(t ), can be further decomposed into the following two processes [27]: a(t ) = exp (p(t ) + s(t )) ,

Fig. 2. Illustration of an interference scenario for cellular heterogeneous network.

given in Section 2.1.1. However, at this point it suffices to state that the SU–Tx is assumed to move toward the SU–Rx with probability p, whereas it is assumed to move away from SU–Rx with probability 1 − p. Note that the value of the probability p allows one to assign different levels of randomness in mobility behavior for SU–Tx. Mobility is considered to be with uniform speed in any direction, that is |v|. Therefore, the displacement for the SU–Tx in a unit time period can be represented by ∆x = |v|∆t in Fig. 1. The scenario given in Fig. 1 can be extended to a general (heterogeneous) cellular network topology as shown in Fig. 2 where femtocells are deployed as well. However, neither a horizontal nor a vertical hand-off protocol is assumed in order to focus on the impact of interference. Also, as stated earlier, both the PU and SU networks operate on the same frequency fx ; nevertheless, no information exchange between the networks is permitted and no power control is assumed. The signal received by the PU–Rx can be described by the following equation [24]: r (t ) = I (t ) + ϕ(t ) + n(t ),

(1)

where I (t ) is the interfering signal, ϕ(t ) denotes the desired signal for the PU–Rx, and n(t ) is the ambient noise present at the PU–Rx’s antenna at time instant t. Since the focus of this work is to investigate the impact of

(4)

where p(t ) denotes the impact of path loss, and s(t ) represents the shadowing. Notice that the impact of the shadowing process can be well approximated by a log-normal distribution [26]. Before proceeding further, it is important to make a distinction between scenarios with respect to the effects of path loss. Such a distinction is necessary because the impact of path loss is negligible in some scenarios, whereas the impact of path loss cannot be neglected in some other scenarios. Based on this, we can further categorize the communication range between SU and PU with respect to the ratio between ∆x and d as seen in Fig. 1. If ∆dx ∼ = 1 then it is considered a short-range network (SRN); otherwise it is a long-range network (LRN) for d ≫ ∆x. 2.1. Large-scale characterization for SRNs As stated earlier, the impact of path loss cannot be neglected in SRNs; therefore, (4) can be rewritten as



a(t ) = exp γ ln



d + ∆x d0



 + σ g (t ) ,

(5)

where γ denotes the path loss coefficient; g (t ) is a realvalued Gaussian random process with PDF N (0, 1) and σ denotes the standard deviation of the shadowing, which changes with respect to the type of propagation environment; d stands for the transmitter–receiver separation at time instant t; d0 denotes the reference distance in the farfield of the antenna; ∆x represents the total displacement traversed within a unit observation interval ∆t. Empirical results also show that correlation of s(t ) can be approximated by an exponentially decaying function as reported in [28] and is expressed as R(k) = E {S (t )S (t + ∆t )} = σ 2 ρ



|t +k∆t |∆x dρ



,

(6)

where E {·} stands for the statistical expectation, S (t ) = 20 log(s(t )) denotes the observed (sampled) signal, σ 2 is the variance of shadowing, ρ is a model coefficient, ∆x = v ∆t denotes the separation between two points in space assuming the speed of mobile v , and dρ represents the decorrelation distance.

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2.1.1. Mobility model for SRNs The random walk model can be applied to the single PU–Rx and single SU–Tx scenario in Fig. 1 due to the following advantages [29,23]: (I) the random walk model is simple and easy to implement for one-dimensional (1D) and 2D movement, (II) sudden direction changes, which take place in scenarios such as the one described in Section 2, can still be accommodated. For the sake of completeness, first, a 1D random walk will be introduced. Then, the generalized 2D random walk will be outlined. 1D random walk. When mobility is considered in one dimension, since the motions of PU–Rx and SU–Tx are independent of each other, analyzing the motion of only one of the mobile users is sufficient. Note also that one of the most important parameters in the analysis is the relative distance between the PU–Rx and the SU–Tx. Recall that SU–Tx, which is initially at rest, moves toward SU–Rx with the probability p and away from it with the probability 1 − p. Then after N steps, the final position of SU–Tx with respect to its initial position is given by

∆dE = (2pi − 1)N ∆x,

(7)

where i = 1, 2 refers to two different mobiles with different probabilities. Therefore, the distance between SU–Tx and PU–Rx after N steps becomes dN = 2N ∆x (p1 + p2 − 1) + d0 ,

(8)

where N ∈ Z+ . As a special case, if both mobile terminals adopt identical mobility behaviors, (8) degenerates to dN = 2N ∆x (2p − 1) + d0 , where p1 = p2 = p. Generalized 2D random walk. Assume that SU–Tx is able to move in every direction by selecting an angle from [0, 2π) in a uniformly distributed manner. Assume also that mobile SU–Tx traverses ∆xn at the nth step in any direction.1 After N steps, the PDF of the total relative displacement for the SU–Tx being in a circle with a radius of r is given by [32, p. 343] Pr (0 ≤ ∆X ≤ r , n) = r

∞



J1 (rx) 0

 N 

 J0 (x∆xn )

dx, (9)

Pr (0 ≤ ∆X ≤ r , n) = r

∞

J1 (rx) [J0 (x)]n dx.

(10)

0

From the perspective of interference, the most important implication of (9) is the statistical expectation of the total displacement. Since the displacement at every direction is of unit length, the total displacement can be represented as

  N    jθn  ∆X =  e ,  n=1 

where θn is uniformly distributed within [0, 2π ). Based on (11), the desired quantity is E {∆X } for a specific N ≥ 2. It is clear that when N = 2, with the aid of trigonometric identities, the cumulative distribution function (CDF) of ∆X becomes 1

F∆X (x) =

π

 arccos 1 −

x2



2

,

(12)

where, of course, x ∈ [0, 2]. Hence, for the special case of N = 2 (with unit magnitude for each phasor) the expected value of ∆X is calculated to be E {∆X }N =2 = π4 . In order to generalize this idea, the expected value of the magnitude of the addition of two phasors is expressed as E {∆X } =

2     u⃗1  + u⃗2  EII (φ),

(13)

π where EII (·) is the complete elliptic integral of the second kind and assumes the expression

EII (φ) =

π /2





1 − k2 sin2 (θ) dθ ,

(14)

0

with

   u⃗1  u⃗2     , u⃗1  + u⃗2  2

sin2 (φ) = 4 

(15)

where sin2 (φ) = k2 is also referred to parameter m. Thus, if the calculation of the expected value is continued in a recursive manner, one can approximate the expected value of the total displacement as follows:

√ ∆dE = E {∆X } u c0 N , (16) √ where c0 is π /4. However, the performance of the approximation given in (16) matters when N ≪ ∞ since the status of the interference needs to be checked frequently, which implies values of N that are closer to zero [33].

n=1

where Jk (·) represents the kth-order Bessel function of the first kind, ∆xn is the magnitude of the nth step and ∆X is the total relative displacement. When a uniform displacement with unit magnitude is assumed at each step, as in Pearson’s random walk, then (9) degenerates to



215

(11)

1 Note that in the case ∆x = ∆x, ∀n, the generalized version (∆x ̸= n i ∆xj , ∀i̸=j ) degenerates to Pearson’s random walk [30,31].

2.1.2. Interference analysis for SRNs In order to see how interference evolves in time, it is reasonable to check its status at equally spaced time instants such as {I (t + k∆t )}k with k ∈ Z+ . It is very useful to investigate first the envelope of the interference, |I (t )|. Since m(t ) is a complex process, the envelope of the interference is expressed as

|I (t )| = |m(t )| exp (γ ln (d) + σ g (t )) ,

(17)

where d0 is assumed to be 1 m for the sake of having an easier analysis. Assume that one can obtain the next interference status as I (t + ∆t ) and express the envelope as in (17). Then

|I (t + ∆t )| = |m(t + ∆t )| exp (γ ln (d ± ∆dE ) + σ g (t + ∆t ))

(18)

is obtained, where ±∆dE refers to the increase/decrease of the initial (PU–Rx)–(SU–Tx) separation. In order to quantify the change in interference power level in time, the

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A.R. Ekti et al. / Physical Communication 9 (2013) 212–222

ratio between the interference envelopes can be calculated as

∆I ( t ) =

|I (t + ∆t )| . |I (t )|

(19)

If the logarithm operator is applied to (19), it follows that L|I | (t ) = ln (∆I (t ))

= L∆|m| (t ) + γ (ln (d ± ∆dE ) − ln(d)) + σ (g (t + ∆t ) − g (t )) ,

(20)

where

|m(t + ∆t )| L∆|m| (t ) = ln |m(t )| 



TII

 





E L|I | (t ) = p L∆|m+ | (t ) − L∆|m− | (t )



  d + ∆dE  + γ p ln  d − ∆dE  

TI

   + L∆|m− | (t )

  d − ∆dE  + ln   d  





TIV

 

   + σ p g + (t + ∆t ) − g − (t + ∆t )     + σ g − ( t + ∆t ) − g ( t ) ,   

(21)

TIII

with L∆|m± | (t ) = ln

|m(t )|

,

where ()+ and ()− denote the cases in which the (PU–Rx)– (SU–Tx) separation increases or decreases, respectively, after N steps. In (21), assuming that the small-scale fading component is governed by a complex Gaussian process, one can conclude that TI refers to the difference of two independent and identically distributed log-Rayleigh processes, whereas TII represents the difference of two TI , and TIII corresponds to the difference of two real-valued normal processes with PDF N (0, 1). Also, it is reasonable to ∆d think that 0 ≤ d E < 1 with a known path loss exponent

γ . Based on this reasoning, by letting u =   1+u TIV = p ln + ln (1 − u) 1−u

∆dE d

where p(t ) is assumed to be unity as explained above. 2.2.1. Interference analysis for LRNs Consider a simple SU–Tx and PU–Rx CR communication scenario as illustrated in Fig. 1. In such a scenario, assume that the interference spilled over the victim by the source is represented by the following random process: I (t ) = m(t )a(t ).

(23)

After an observation interval of ∆t, the interference is given by (24) +

TIII



(22)

I (t + ∆t ) = m(t + ∆t )a(t + ∆t ).



  m± (t + ∆t )

The actual relationship between the displacement and the relative distance to neighboring PU–Rx includes also the position of the interference source. For instance, in Fig. 1, the actual relationship between the relative distance and the displacement should include both (d ± ∆x) and ∆x. Since d ≫ ∆x is assumed, the path loss term mentioned in (4) can be ignored. In this regard, for the sake of completeness, the large-scale fading component in (4) can be expressed as follows: a(t ) = exp(σ g (t )),

is satisfied. After some algebraic manipulations, the expected value of L|I | (t ) in (20) can be expressed as



2.2. Large-scale characterization for LRNs

,

is obtained. From the practical point of view, the path loss exponent can be assumed to satisfy γ ≥ 2, including the free space conditions. Since p is known to be within [0, 1], TIV can be approximated by a simple linear equation with respect to p for values of u closer to zero. With a known p, then TIV represents only a constant whose value changes with u. In fact, for practical purposes, N should be assumed to be close to zero since diverging N implies greater uncertainty in the next interference status to be observed. Therefore, it is reasonable and safe to assume that u takes values that are generally close to zero.

In general, after a sequence of k observations, k ∈ Z , (24) can be defined as I (t + k∆t ) = m(t + k∆t )a(t + k∆t ). It is assumed that the decorrelation distance dρ is not exceeded within the observation duration, i.e., kv ∆t ≤ dρ . Note that there are two distinct characteristics embedded in (23). First, the shadowing process is governed by an exponential function which implies a logarithmic conversion in terms of its PDF. Second, both the small-scale and large-scale fading processes are of multiplicative form. Due to these two characteristics, a logarithmic transform can be very useful in terms of yielding easier analysis. Therefore, it is convenient to define the logarithmic transformation through the operator R as follows:

R (f (·)) = ln (f (·)) .

(25)

For the sake of brevity, let any function to which R is applied be represented with f (·), where f (·) denotes any positive real-valued function. In order to satisfy this requirement for operator R along with the fact that I (·) is a complex-valued process in general, in the remainder of the paper the envelope notation (|·|) will be used. Consider the change of interference between two observations be defined in terms of their envelopes as follows:

|∆I (k)| =

|I (t + k∆t )| . |I (t + (k − 1) ∆t )|

(26)

Then, the following relation can be written as a direct consequence of (26):

|∆I (K )| =

k |I (t + k∆t )|  |∆I (i)| , = |I (t )| i =1

(27)

where |∆I (i)| is the rate of change of interference observed by the PU–Rx. (27) directly relates the initial and final interference conditions after a sequence of k observations.

A.R. Ekti et al. / Physical Communication 9 (2013) 212–222

217

Note that future interference levels can be expressed in terms of previous ones using (27) as follows:

As in Case 1, if all of the steps (33) through (34) are performed, one ends up having the following:

|I (t + k∆t )| = |I (t )| |∆I (K )| .

|∆I (Z )| = (ln (|m(t + z ∆t )|) − ln (|m(t )|))

(28)

The sequence of changes in (28) implies that future interference levels are a function (random process) of the differences between observations for a given initial observation value I (t ). When the operator R is applied to (28), the multiplicative form is converted into an additive form yielding:

|I (t + k∆t )| = ln (|I (t )|) + ln (|∆I (K )|) .

  |I (t )| = ln (|I (t )|) = ln m(t )eσ g (t )  (30)

When induction for unit observation interval ∆t is applied to some k ∈ Z+ ,

|I (t + k∆t )| = ln (|m(t + k∆t )|) + σ g (t + k∆t ),

(31)

is obtained. Here, for an arbitrary k value, there are two possibilities in regard to the comparison dρ Q k∆x:

D :=

 

Din ,



Dout ,

 if k ≤

dρ



∆x otherwise,

(32)

where D denotes the status of the mobile whether it is still within the correlation region (Din ) or not (Dout ), and ⌊·⌋ is the floor operator. In what follows, these two possibilities will be investigated. Case 1: Inside the Correlation Region. Here, the interference conditions will be investigated for D = Din . Let (29) be expressed as

|∆I (K )| = ln (|I (t + k∆t )|) − ln (|I (t )|) .

 1. Given the initial interference condition I (t ) Proposition or |I (t )| , the variance of the next interference level I (t +   k∆t ) or |I (t + k∆t )| will always be lower inside the correlation distance and Din asymptotically converges to Dout as k → ∞. Proof. See Appendix A for the proof.



It is noteworthy that in both Case 1 and Case 2, the interfering signal I (·) is always shadowed at both the initial and final sampling instants, at t, t + k∆t, and t + z ∆t, respectively. However, it is also possible that after a duration of zL ∆t from the initial observation instant t, the interfering signal might experience LOS rather than experiencing a different shadowing in another region. Formally, this is expressed in terms of (24) as follows: I (t + zL ∆t ) = m(t + zL ∆t ),

(36)

since g (t + zL ∆t ) = 0 in the special case of LOS. Thus, this special case yields the following result. Proposition 2.   Given the initial interference conditions I (t ) or |I (t )| , for correlation values a ≤ 0.5, it is impossible to achieve a lower variance for the next interference level inside the correlation region if LOS is established outside the correlation region. Proof. See Appendix B for the proof.



Both Propositions 1 and 2 imply that decorrelation distance of the propagation environment and the correlation coefficient are key to understanding the future behavior of interference. 3. Numerical results

(33)

If both ln (|I (t + k∆t )|) and ln (|I (t )|) are replaced with the expressions given in (30) and (31), respectively, and the terms are regrouped, the following is obtained:

|∆I (K )| = (ln (|m(t + k∆t )|) − ln (|m(t )|)) + σ (g (t + k∆t ) − g (t )) .

be statistically independent. In order to contemplate the difference between the cases stated in Case 1 and Case 2, the following is required.

(29)

In order to have an insight into interference behavior, first, three critical parameters must be recalled: the decorrelation distance of shadowing, dρ ; the speed of the mobile, v ; and the observation interval, ∆t. These parameters are crucial, since they are related to each other in terms of both space–time translations as in ∆x = v ∆t and comparisons as in dρ Q k∆x for any k. It is appropriate to begin with the linearized envelope of I (t ), that is |I (t )|:

= ln (|m(t )|) + σ g (t ).

+ σ (g (t + z ∆t ) − g (t )) , (35)   dρ where z ≤ ∆x and ⌈·⌉ is the ceil operator. In contrast to (34), in (35) note that g (t + z ∆t ) and g (t ) are assumed to

(34)

Note that, in (34), g (t + k∆t ) and g (t ) are correlated because of (6) and D = Din . The consequence of this situation will be introduced later. However, for the sake of having a complete understanding of this case, the following needs to be given first. Case 2: Outside the Correlation Region. Here, the interference conditions will be investigated for D = Dout .

3.1. Numerical results for SRNs Before proceeding further, it is worth first characterizing the impact of path loss for short distances along with proportionally large displacements. In Fig. 3, the simulated values of TIV in (21) for the path loss with respect to the ∆d displacement ratio u = d E for different probability values are plotted, where d = 5 m and γ = 2 represent the free space path loss. It is seen that when 0 ≤ u ≤ 0.5, TIV exhibits almost a linear behavior. This linear behavior is illustrated in Fig. 3 with a least-squares fit applied to the case where p = 0.1. Another critical and very interesting observation is that for the maximum uncertainty case (i.e., p = 0.5), the expected value of the path loss (by neglecting both small-scale fading and shadowing) with

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Fig. 3. The evolution of path loss with respect to u =

Fig. 4. The difference of interference levels with respect to the probability p for different u values.

these settings has a tendency to increase with an increase in u, which implies N → ∞ for a 2D random walk. However, for a 1D random walk, such an observation cannot be made. In light of the previous results, the behavior of (20) can be investigated. The results for different u values with respect to the probability p are given in Fig. 4, where d is selected to be again 5 m with γ = 2. As can be seen from Fig. 4, as u approaches unity, the interference level difference between the two observations becomes more drastic. This is expected, since the impact of path loss for such short separations but for relatively larger displacements leads to drastic power changes in the signal mean where the presence of a strong correlation is assumed in the shadowing. Finally, we evaluate the performance of the approximation for the expected value of the total displacement in the 2D random walk model. The results for the proposed approximation are given in Fig. 5. However, one should keep in mind that even though many of the approximations focus on the divergent values of N, peculiar to the interference scenario considered in this work, the values of N that are closer to zero are very important. This emanates from the fact that divergent N implies less frequent interference updates yielding greater uncertainties in future interference levels.

∆dE d

for different probability values p.

Fig. 5. The performance comparison for the  approximation proposed in (16) and asymptotic approximation with 2n .

3.2. Numerical results for LRNs When solely Din is of interest, Proposition 1 mentions an asymptotic behavior regarding the variance of the next interference level. For this purpose, a set of simulations is performed and each scenario is run 10,000 times in order to obtain reliable statistics. As stated in Section 2, a process m(t ) whose envelope is Rayleigh distributed is generated with the classical Jakes spectrum for v = 3 m/s to emulate a mobile pedestrian at 1.8 GHz. For the process s(t ), the standard deviation of shadowing is chosen to be σ = 7.5 dB as reported in [28]. The log-normal shadowing process is generated through the use of the Gauss–Markov assumption. Fig. 6 plots the variances with respect to observation index k. As can be seen from the Fig. 6, there is an increase tendency in the variance with respect to k. This stems from (A.7) as stated in Proposition 1. The relationship between Din and Dout needs to be checked as well in terms of the variance to be observed for the change of interference conditions in the next sampling instant. For this purpose, the same setup given above is used. However, instead of employing a correlated shadowing, uncorrelated shadowing is imposed on the process while keeping the small-scale fading statistics the

A.R. Ekti et al. / Physical Communication 9 (2013) 212–222

Fig. 6. Asymptotic behavior of interference for Din . As k increases, the variance of the difference between the previous observation and the next observation increases as well.

Fig. 7. The PDFs for both the interference distributions inside and outside the correlation region. Note that the distribution is more spread for Dout compared to that for Din , as stated in Proposition 1.

same. For comparison purposes, both PDFs are plotted in Fig. 7. In the simulations, all of the indices k are evaluated for the case corresponding to Din . However, due to space limitations, an arbitrary value is chosen among them for display purposes. Peculiar to Fig. 7, the PDF corresponding to Din refers to the PDF calculated for k = 7. As can be seen from Fig. 7, the difference process |∆I | corresponding to the case for Dout exhibits more spread compared to that for Din in accordance with Proposition 1. 4. Final remarks and further discussions In the first part of this study, it is shown that relatively larger displacements in short (PU–Rx)–(SU–Tx) separations lead to drastic power level fluctuations in the observed interference power levels. It is observed that the impact of path loss is one of the major factors changing the future interference conditions for low-speed mobility scenarios especially within short communication ranges.

219

In the second part, it is shown that large-scale fading plays a crucial role for the evolution of interference in time for long-range CR networks. The results show that the decorrelation distance of large-scale fading (i.e., shadowing) plays a crucial role in predicting future interference conditions in LRNs. Given the initial interference measurement, it is shown that the future interference levels are highly dependent on maintaining the decorrelation distance throughout the observation interval. In a mobile environment, if the observation interval is kept too short, then it is not expected to have a drastic fluctuation in the new interference level. In contrast, if the observation interval is kept too long, then correlation loses its significance; furthermore, the path loss might dominate and the analysis will be different. Another interesting relationship between interference and observation interval is the correlation coefficient. It is shown that for larger correlation coefficients, the observation interval can be prolonged since a drastic fluctuation in the next observation instant is not expected. This implies that NGWNs should take the correlation coefficient into account for minimizing the amount of interference measurements and reporting, which leads to reduced signaling and processing power consumption. Even though this study focuses on the analysis of a single mobile SU scenario, it can be extended to many interesting and diverse problems some of which include investigating the impact of group mobility behavior of the SU–Tx and/or PU–Rx and cluster mobility behavior in which the terminals adopt heterogeneous mobility patterns. It is worth emphasizing also that this study assumes a perfect desired signal recovery in the presence of both ambient noise and interference. However, remnants of the desired signal extraction process and the impact of noise must both be taken into account especially from the practical point of view. Therefore, another interesting research topic related to this study is the analysis of evolution of interference from the perspective of a victim under such practical scenarios including imperfect signal recovery. Acknowledgments This publication was made possible by NPRP grant 09–341–2–128 and 4–1293–2–513 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. Appendix A. Proof of Proposition 1 Note that in both (34) and (35), the first terms to the right of the equal signs exhibit the same statistics. At the sampling instants, small-scale fading processes are assumed to be independent of each other, since the distance traversed, kv ∆t, is considered to be on the order of several wavelengths. In order to capture the impact of the correlation for (34), the relationship between g (t + k∆t ) and g (t ) must be defined. Since g (t ) is a normal process with N (0, 1),

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A.R. Ekti et al. / Physical Communication 9 (2013) 212–222

the Gauss–Markov assumption can be used for defining the discrete relationship as follows: x[i + 1] = ax[i] +



1 − a2 n[i],

(A.1)

where x[i] is the process observed, a is the correlation coefficient, and n[i] is assumed to be additive white Gaussian noise (AWGN) with N (0, 1). Here, in order for the desired correlation properties to be captured, the following is necessary: a = bv ∆t /dρ .

(A.2)

It is worth mentioning that the selection of b depends on the environment and there are several values for several different environments such as bsuburban = 0.3 for dρ = 100 m, whereas burban = 0.82 for dρ = 10 m [28]. Consider (34) along with (A.1) for k = 1. Then, the following holds:

|∆I | = (ln (|m(t + ∆t )|) − ln (|m(t )|))    + σ (a − 1) g (t ) + 1 − a2 n(t ) .

(A.3)

If (A.3) is generalized to any arbitrary k value for k > 1 through the recursive relationship defined in (A.1), the following is obtained:

|∆I (K )| = (ln (|m(t + k∆t )|) − ln (|m(t )|)) +σ    ×

a − 1 g (t ) +



k





1−

a2

 k−1 

 a

k−1−i

n(t + i∆t )

.

i=0

(A.4) In order to proceed further, consider the mean and the variance of |∆I (K )|. The mean of |∆I (K )| is calculated as E |∆I (K )| = E M (K ) + σ ak − 1 E {g (t )}







+ σ 1 − a2



 k−1 







 ak−1−i E {n(t + i∆t )} .

2 2 σ|2∆I (Z )| = σM (Z ) + 2σ .

(A.9)

Finally, recall that for the duration z ∆t, M (·)s are governed by the same process along with 0 < a < 1; thus

σ|2∆I (K )| < σ|2∆I (Z )| .

(A.10)

Furthermore, for k → ∞, σ|2∆I (K )| → σ|2∆I (Z )| , which shows the asymptotic behavior and completes the proof. Appendix B. Proof of Proposition 2 In order to prove the Proposition 2, it is sufficient  to 

examine the statistics of |I (t + zL ∆t )|, where zL ≤

dρ

∆x

as in Section 2.2.1 except for the fact that there is LOS between the interference source and the victim. Later on, the results will be compared with those of the case Din for the statistics of |I (t + k∆t )|, namely with (A.6) and (A.7). Under the assumption that considers LOS for D = Dout , the interference is given by I (t + zL ∆t ) = m(t + zL ∆t ).

(B.1)

Since there will be a LOS component at the instant t + zL ∆t, the small-scale fading component can be expressed as follows:

M (K )



Similarly, if the variance of |∆I (Z )| is calculated, then the following holds:

(A.5)

i=0

m(t + zL ∆t ) = gC (t + zL ∆t ) + qejθ ,

(B.2)

where gC (·) = gℜ (·) + jgℑ (·)

(B.3)

is a complex-valued Gaussian process N 0, 2σ ; both gℜ (·) and  gℑ (·) are real-valued Gaussian processes N 0, σC2 ; q and θ are the amplitude and phase of the LOS component, respectively. In order to proceed further to relate (B.1) with the difference operator, the envelope of (B.1) is necessary:



2 C



Since both g (t ) and n(t ) are zero-mean processes, (A.5) degenerates to

|I (t + zL ∆t )| = |m(t + zL ∆t )| .

E |∆I (K )| = E M (K ) .

Now the linearized difference operator can be applied as follows:









(A.6)

Note  that ifM(K ) is replaced with the difference in (A.4), E |∆I (K )| = 0, since both m (t + k∆t ) and m (t ) are assumed to be governed by the same process during the interval k∆t. Bearing in mind that processes M (K ), g (t ), and n(t ) are mutually independent of each other, after some mathematical manipulations the variance is given by

σ|∆I (K )| = σM(K ) + 2σ 2

2

2



1−a

k



.

(A.7)

The analysis of (35) is easier compared to that of (34), since g (t + z ∆t ) and g (t ) are assumed to be statistically independent, as stated in Section 2.2.1. In light of these pieces of information, the expected value of |∆I (Z )| is given by

    E |∆I (Z )| = E M (Z ) ,

(A.8)

since g (t ) is a normal process N (0, 1), as stated earlier.

|∆I (ZL )| = ln (|I (t + zL ∆t )|) − ln (|I (t )|) .

(B.4)

(B.5)

Thus, the expected value of (B.5) is given by E |∆I (ZL )| = E {ln (|I (t + zL ∆t )|)}





− E {ln (|I (t )|)} .

(B.6)

Following the same reasoning and steps defined in Section 2.2.1, the variance of |∆I (ZL )| is expressed as follows:

σ|2∆I (ZL )| = σL2 (Z ) + σ 2 ,

(B.7)

where L(Z ) is defined as

L(Z ) = ln (|m(t + zL ∆t )|) − ln (|m(t )|) .

(B.8)

Note that L(ZL ) and M (Z ) are governed by different processes. This stems from the fact that M (Z ) contains two instants of a single process, whereas L(ZL ) contains

A.R. Ekti et al. / Physical Communication 9 (2013) 212–222

two different processes which are instants of, again, two different processes. Hence, in order to make a better comparison, the variance of |∆I (K )| needs to be decomposed as follows in accordance with (A.4): 2 2 2 σM (K ) = σ|m(t +k∆t )| + σ|m(t )| .

(B.9)

The variance of overall process then becomes

  σ|2∆I (K )| = σ|2m(t +k∆t )| + σ|2m(t )| + 2 1 − ak σ 2 .

(B.10)

In light of the decomposition defined in (B.9), now, (B.8) can be rewritten as follows:

L(Z ) = σ|2m(t +zL ∆t )| + σ|2m(t )| ;

(B.11)

therefore, (B.7) becomes

σ|2∆I (ZL )| = σ|2m(t +zL ∆t )| + σ|2m(t )| + σ 2 .

(B.12)

Since both the mean and the variance of |∆I (ZL )| are in hand, the comparison can be established. First, consider the mean values of both |∆I (K )| and |∆I (ZL )|. As stated in Section 2.2.1, E |∆I (K )| = 0. However, E |∆I (K )| ̸= 0









because of (B.5). Furthermore, E |∆I (K )| > 0, since it is known that the LOS component introduces a positive bias into the mean value of the envelope of the received signal. As for mean values, there is also a difference in variances as well. First, consider (B.12). Note that in (B.2), as q → ∞ while 2σC2 remains the same (i.e., as the Ricean factor increases), σ|2m(t +z ∆t )| → 0, since the LOS compo-





L

nent dominates. This shows that if LOS is established at the instant t + zL ∆t, σ|2m(t +z ∆t )| < σ|2m(t +k∆t )| . Under these cirL





cumstances, note that the term 2 1 − ak in (B.10) gives an insight into the inequalities between variances σ|2∆I (K )| and σ|2∆I (Z )| . Because it has already been shown that the inL

equality σ|2m(t +z

L ∆t )|

< σ|2m(t +k∆t )| is always valid under LOS

scenarios, it is sufficient to solve the following inequality to complete the proof of Proposition 2: 1 ≤ 2 1 − ak





and

a≤e

ln 0.5 k

.

(B.13)

+

Recalling that k ∈ Z − {0}, it is concluded that in order for (B.13) to hold, the greatest value that the correlation coefficient can reach is a = 0.5. In other words, unless 0.5 ≤ a, it is impossible to achieve σ|2∆I (K )| < σ|2∆I (Z )| , and L

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A.R. Ekti et al. / Physical Communication 9 (2013) 212–222 Ali Riza Ekti is currently a Ph.D. student in the Department of Electrical Engineering and Computer Science at Texas A&M University. He received his B.S. degree in electrical and electronics engineering from Mersin University, Mersin, Turkey (September 2002–June 2006), studied at Universidad Politechnica de Valencia, Valencia, Spain, on an Erasmus Student Exchange Program in 2005, and received his M.Sc. degree from the University of South Florida, Tampa, Florida in Electrical Engineering (August 2008–December

2009). His current research interests include statistical signal processing, wireless propagation channel modeling, minimax problems, optimization, cognitive radio and interference management in next generation wireless networks. Serhan Yarkan received his B.S. and M.Sc. degrees in computer science from Istanbul University, Istanbul, Turkey, in 2001 and 2003, respectively, and his Ph.D. degree from the University of South Florida, Tampa, in 2009. From 2010 to November 2011, he was a Postdoctoral Research Associate with the Department of Computer and Electrical Engineering, Texas A&M University, College Station. Since December 2011, he has been an Assistant Professor with the Department of Electrical and Electronics Engineering, Istanbul Commerce University. His research interests include statistical signal processing, cognitive radio, wireless propagation channel modeling, cross-layer adaptation and optimization, and interference management in next generation wireless networks. Khalid A. Qaraqe (M’97–S’00) was born in Bethlehem. He received his B.S. degree (with honors) in electronics engineering from the University of Technology, Baghdad, Iraq, in 1986, his M.S. degree in electronics engineering from the University of Jordan, Amman, Jordan, in 1989, and his Ph.D. degree in electronics engineering from Texas A&M University, College Station, in 1997. From 1989 to 2004, he held a variety of positions with many companies, and he has over 12 years of experience in the telecommunication industry. He has worked for Qualcomm, Enad Design Systems, Cadence Design Systems/Tality Corporation, STC, SBC, and Ericsson. He has worked on numerous Global System for Mobile Communications, code division-multiple access (CDMA), and wideband CDMA projects and has experience in product development, design, deployments, testing, and integration. Since July 2004, he has been with the Department of Electrical Engineering, Texas A&M University at Qatar, Doha, Qatar, where he is currently a Professor. His research interests include communication theory and its application to design and performance, analysis of cellular systems, and indoor communication systems. Particular interests are in the development of thirdgeneration Universal Mobile Telecommunications Systems, cognitive

radio systems, broadband wireless communications, and diversity techniques.

Erchin Serpedin (SM’04) received his specialization degree in signal processing and transmission of information from Ecole Superieure D’Electricite (SUPELEC), Paris, France, in 1992, his M.Sc. degree from the Georgia Institute of Technology, Atlanta, in 1992, and his Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, in January 1999. He is currently a Professor in the Department of Electrical and Computer Engineering, Texas A&M University, College Station. He is the author of two research monographs, 75 journal papers, and 120 conference papers. His research interests include statistical signal processing, information theory, bioinformatics, and genomics. He currently serves as the Associate Editor for the IEEE Transactions on Information Theory, IEEE Transactions on Communications, Signal Processing (Elsevier), and European Association for Signal Processing Journal on Bioinformatics and Systems Biology.

Octavia A. Dobre received her Dipl. Ing and Ph.D. degrees in Electrical Engineering from the Polytechnic University of Bucharest (formerly the Polytechnic Institute of Bucharest), Romania, in 1991 and 2000, respectively. During her Ph.D. studies, she was with the Department of Remote Control and Electronics in Transports. Between 1998 and 2001 she was an Assistant Professor in the same department. In 2001 she joined the Wireless Information Systems Engineering Laboratory at Stevens Institute of Technology in Hoboken, NJ, as a Fulbright fellow. Between 2002 and 2005, she was with the Department of Electrical and Computer Engineering at New Jersey Institute of Technology (NJIT) in Newark, NJ, as a Research Associate. Since 2005 she has been with the Faculty of Engineering and Applied Science at Memorial University, Canada, where she is currently an Associate Professor. Her research interests include cognitive radio systems, spectrum sensing techniques, blind signal recognition and parameter estimation techniques, transceiver optimization algorithms, dynamic spectrum access, cooperative wireless communications, network coding, resource allocation, underwater communications, and optical OFDM. Dr. Dobre is an Editor for IEEE Communications Letters and IEEE Communications Surveys and Tutorials, and has served as a Guest Editor for the IEEE Journal of Selected Topics on Signal Processing, and Lead Guest Editor of the ELSEVIER PHYCOM ‘‘Cognitive Radio: The Road for its Second Decade’’ special issue. She has also been the technical program co-chair for the Signal Processing and Multimedia Symposium of the IEEE Canadian Conference on Electrical and Computer Engineering in 2009 and for the Signal Processing for Communications Symposium of the International Conference on Computing, Networking, and Communications in 2012.