1 Introduction

LTE femtocells system can provide a satisfied throughput for the users with a good indoor coverage. Thanks to the deployment of the femtocells inside LTE macrocells, the network service can be available anywhere. Moreover, the trapped indoor users can have a satisfied throughput [1,2,3,4,5,6,7]. On the other hand, the femtocells’ deployment inside LTE macrocells leads the available spectrum to be accessed by both macro and femto users. Therefore, a mutual electromagnetic interference (EMI) exists between the macrocells and the femtocells. This EMI value can degrade the performance of the system. This deployment is called a co-channel deployment or a universal frequency reuse one. In Ref. [6, 7], authors simulated a LTE femtocells system and estimated its geometry, signal to interference plus noise ratio, and its throughput applying a co-channel deployment scenario. They developed a recent LTE femtocells system level simulator. Subsequently, Ref. [8] presented a new mathematical model for a co-channel deployment LTE femtocells system. The authors developed a mathematical formula for the users’ geometry and throughput based on a simple capacity concept. Moreover, it presented a novel mathlab simulation framework for a LTE femtocells system which applies the co-channel deployment scenario.

There are a lot of strategies which can reduce the mutual EMI between the femtocells and the macrocells. One of these strategies is the Fractional Frequency Reuse (FFR). This strategy depends on distributing the available spectrum bands on the LTE femtocells system coverage area, in such a way that, the same frequency band can be reused when a minimum separation distance exists. The FFR may be applied in a static or a dynamic manner [9]. Another strategy, used for interference reduction, is the power control. This strategy aims to reduce the radiated power of the femtocells in order to reduce the resulting interference on the neighboring macrocells and femtocells. Authors of Ref. [10] explained, in details, three different power control algorithms. They clarified that the femtocells’ emitted power may be reduced in a random manner, according to a new specified femtocells’ range, or according to a target value of the femto users’ geometry. Subsequently, power control is modified in the self-organizing LTE femtocells network, in such a way that, the femtocells start transmission with a nominal power value then, they can adapt their power in order to optimize and reduce the interference levels affecting the neighboring femtocells [11]. In addition, a novel backoff power control scheme, aiming to discover the fake femto base stations, was handled in Ref. [12]. In general, power control algorithms can reduce the mutual EMI between the femtocells and the macrocells. On the other hand, they can lower the geometry and throughput of the femto users.

Cognitive radio is a dynamic spectrum access technology developed to increase the utilization of a wireless radio spectrum. It allows low priority secondary users to operate at the portion of the radio spectrum, which is not occupied by high priority primary users [13]. The deployment of cognitive radio technology in a LTE femtocellular system is introduced in Ref. [14]. It divided the macro and femto users into primary and secondary users. Moreover, it suggested network architecture for a LTE cognitive overlay femtocells system. This architecture combined the conventional femtocell idea with an infrastructure based overlay cognitive radio network paradigm. In addition, it stated a lot of challenges in the LTE cognitive overlay femtocells system. These challenges may be the future research work in the topic. The concept of deploying cognitive femtocells in LTE networks is extended in Ref. [15]. It handled the two types of interference which are co-tier interference (among the femtocells themselves) and the cross tier interference (between the macrocells and the femtocells). To solve the spectrum scarcity problem, the cognitive femtocells can use both of licensed and unlicended bands from the macrocells and the TV white spaces. Other femtocells’ management strategies, to reduce the mutual EMI between the macrocells and the femtocells, were demonstrated in Ref. [16]. These strategies included; conventional, centralized, and distributed. Furthermore, the paper clarified several cognitive channel assignments for the femtocells. Each femtocell management strategy was simulated and compared with the traditional macrocell system to assure the effect of applying the cognitive radio technology.

In Ref. [17, 18], authors applied the cognitive radio technique in order to reduce the mutual EMI value between the macrocells and the femtocells. They explained, in details, the system mathematical model. Moreover, they developed the network architecture, which had been stated before in Ref. [7], to support the cognitive capabilities through using a spectrum management subsystem called a spectrum server. It was a data base to include the radio resource map and the sensing results of the cognitive femto users. It was a great chance to completely eliminate the mutual EMI in a LTE femtocells system. Moreover, the users’ throughput is increased as a result of applying the cognitive radio technique. On the other hand, Ref. [18] demonstrated the interference reduction among the femtocells themselves. The outage probability of macro and femto users, over Nakagami-m fading channels, was estimated by applying a novel stochastic model. This model considered the closed access mode and multiple types of cognitive interference management strategies for the femtocells [19]. In general, the overlay cognitive femtocells can operate at 20 dBm which is the maximum allowed transmitted power. The main drawback of this operation is that the femtocells can operate at their maximum power. Moreover, the femto users can only access the free subcarriers, which are not occupied by a macro user. Therefore, the femto users can have a zero throughput when there are no free subcarriers for them, regardless the high values of signal to interference plus noise ratio (SINR). It is a main drawback, in the system, to have a zero value of femto users’ throughput.

In this paper, the femtocells are allowed to operate at the subcarriers even if they are occupied by a macro user. The mutual EMI, which occurs between the femtocells and the macrocells, is allowed to grow on condition that its value cannot affect the operation of the nearest macro users. The femto users can receive the emitted power of each macro user during the uplink transmission. The received power, at each femto user, can give an indication about the macro users’ positions. The received signal strength (RSS), at each femto user, can be sent to the fusion center of the system. At least, three of femto users can actually determine the coordinates of a macro user. When the positions of the macro users are determined, the femtocells can adapt their emitted power in order to reduce the interference on macro users. The reduction of the femtocells’ transmitted power allows them to operate at all subcarriers even if the macro users operate on these subcarriers. The femtocells’ operation is an underlay cognitive one.

The LTE underlay cognitive femtocells system is proposed in this paper. The proposed system is explained in details, mathematically analyzed, and simulated. Moreover, the performance of the proposed system is validated by comparisons with that system which applies the overlay cognitive femtocells. In addition, the determination of macro users’ coordinates is clarified using Cramer rule. Furthermore, the mathematical conditions, which allow the proposed underlay cognitive operation, are derived and approved in the simulation results. Finally, the validation of Cramer rule in macro users’ positioning is verified in the simulation results.

This paper is organized as follows; Sect. 2 describes the proposed system and the proposed network model. Section 3 gives the explanation of the proposed LTE underlay cognitive femtocells algorithm. On the other hand, Sect. 4 handles the path loss models’ calculation. In Sect. 5, the macro users’ localization, using Cramer rule, is explained. Subsequently, the mathematical conditions, which can control the operation of underlay cognitive femtocells, are derived in Sect. 6. The power control techniques, which can be applied as realization tools for the proposed underlay operation, are clarified in Sect. 7. However, Sect. 8 deals with the calculation of macro and femto users’ throughput. Subsequently, Sect. 9 discusses the simulation results. Finally, conclusions are given in Sect. 10.

2 System Architecture

This section explains, in details, the network architecture for a LTE underlay cognitive femtocells system. The system and the detailed network architecture are shown in Fig. 1a, b. The LTE macrocells has a hexagonal pattern. Inside each hexagonal macrocell, the femocells are randomly distributed in order to extend the network coverage inside indoor zones and the other poor service regions. The macrocells can serve the primary macro users. On the other hand, the femtocells can provide service to the secondary femto users only (closed access mode). The macrocells and the femtocells can be connected together through the core network. In fact, this architecture was developed before in Ref. [17, 18]. However, it can fit well in both overlay cognitive and the proposed underlay cognitive operation. There are four parts and they are;

Fig. 1
figure 1

LTE network architecture, which supports the cognitive femtocells [17, 18]

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2.1 Evolved Universal Terrestrial Radio Access Network (E-UTRAN) elements

The Home enhanced Node Base Station (HeNB) is responsible for several functions. These functions include; management of the radio resources, connection establishment between the femto users and the core network, and providing security services through the security gateway. In this network model, the interaction between the femtocells (HeNBs) and the macrocells can be carried out at the core network or at the S1 interface [17, 18].

2.2 Evolved Packet Core (EPC) Network Elements

Connection of the different parts of the system is the responsibility of the core network. In addition, it can provide the switching facilities among the different parts of the system [17, 18].

2.3 Closed Subscriber Group (CSG) Provisioning Elements

CSG list servers and CSG admin servers are the femto users’ data base, in such a way that, they can save necessary data about the femto users only [17, 18].

2.4 Packet Data Network (PDN) Elements

The PDN elements can provide connections between the proposed system and Internet or any other IP multimedia system [17, 18].

3 Proposed Underlay Cognitive Femtocells Algorithm

The proposed algorithm for an underlay cognitive operation in a LTE femtocells system passes through a lot of different steps. These steps, which are shown in Fig. 2, can be explained as follow;

Fig. 2
figure 2

The proposed underlay cognitive femtocells’ operation in a LTE femtocells system

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  • During the uplink transmission, each femto user detects the received signal strength (RSS) of the corresponding macro users. The detected RSS values are sent to the fusion center of the system.

  • It is assumed that the fusion center of the system has a priori knowledge about the network such as; femtocells’ locations, femto users’ positions, and much more. The received RSS at the femto users can be converted to corresponding distances by the following relation;

    $$ R = 10^{{\frac{PL(dB) - W - 15.3}{37.6}}} $$
    (1)

    where R is the distance between the transmitting macro user and the detecting femto user, W is the wall loss of the femtocells, PL(dB) is the path loss between the transmitting macro user and the detecting femto user [1]. This path loss value can be calculated by;

    $$ PL(dB) = 21 - RSS(dBm) $$
    (2)

    where 21 dBm is the transmitted power of the macro user (in the uplink) according to the 3GPP standard [21] and RSS(dBm) is the received signal strength at the detecting femto user expressed in dBm. In fact, Eq. 1 represents the path loss map between a trapped femto user and a macro base station but the macro user, in this situation, replaces the macro base station as the transmission is in the uplink.

  • The fusion center can convert each RSS value to a corresponding distance value by using Eqs. 1 and 2. It is obvious that; each femto user can detect the RSS from each macro user. The determined distances, by the femto users, can help in determination of macro users’ coordinates. The fusion center can use Cramer rule to determine the actual coordinates of each macro user. This will be explained in the following sections.

  • As soon as the coordinates of the macro users had been determined, the power control could be applied on the femtocells’ base stations in order to reduce the interference, which affects the nearest macro users. The power of the femtocells is reduced and the mutual EMI level on each macro user, caused by the femtocells’ transmission, is estimated. The mutual EMI is allowed to grow until a certain value. This value will be mathematically determined and approved in the simulation results.

  • The signal to interference plus noise ratio (SINR) and throughput of the users can be obtained.

4 Path Loss Models

Path loss equations are provided by the 3GPP standard releases [1]. These equations can represent the loss in power for the transmitted signals in the downlink. The path loss (PL) value depends on; the distance separating between the base station and the user equipment, the location of the user (indoor or outdoor), and the deployment environment [1, 8, 17]. Assuming an urban environment, the path loss models can be expressed as follow;

  • Macro base station (BS) to user equipment (UE)

    • UE is outside: Both BS and UE are outdoors.

      $$ PL(dB) = 15.3 + 37.6\log_{10} R $$
      (3)

      where R is the distance between the BS and UE expressed in meters.

    • UE is inside an apartment: The UE is indoor but, the BS is outdoor.

      $$ PL(dB) = 15.3 + 37.6\log_{10} R + W $$
      (4)

      where W is the wall attenuation loss.

  • HeNB to UE

    • UE is inside the same apartment with the HeNB.

      $$ PL(dB) = 38.46 + 20\log_{10} R + W $$
      (5)

    • UE is outside the apartment.

      $$ PL(dB) = \hbox{max} ([15.3 + 37.6\log_{10} R],\;[38.46 + 20\log_{10} R]) + W $$
      (6)

    • UE is inside a different apartment.

      $$ PL(dB) = \hbox{max} ([15.3 + 37.6\log_{10} R],\;[38.46 + 20\log_{10} R]) + W $$
      (7)

      To estimate W, the following equations are used;

    • UE is inside the same apartment with the HeNB.

      $$ W = q*L_{iw} + 18.3n^{{\left( {\frac{n + 2}{n + 1} - 0.46} \right)}} + 0.7d_{2D,indoor} $$
      (8)

    • UE is outside the apartment.

      $$ W = q*L_{iw} + L_{ow} + 18.3n^{{\left( {\frac{n + 2}{n + 1} - 0.46} \right)}} + 0.7d_{2D,indoor} $$
      (9)

    • UE is inside a different apartment.

      $$ W = q*L_{iw} + L_{ow1} + L_{ow2} + 18.3n^{{\left( {\frac{n + 2}{n + 1} - 0.46} \right)}} + 0.7d_{2D,indoor} $$
      (10)

      where q is the number of walls separating between the UE and HeNB, R and d 2D, indoor (indoor distance) are in meters, L ow is the penetration loss of the outer wall, L iw is the penetration loss of the inner wall, and n is the number of penetrated floors.

5 Macro Users’ Localization Using Cramer Rule

Cramer rule can be used in the determination of a certain node coordinates [20]. It is used, in this paper, to determine the coordinates of the macro users. The fusion center can determine the macro users’ coordinates with the help of the femto users’ coordinates and the obtained separating distances between the macro users and the femto users. The fusion center, with the help of Cramer rule, can determine a macro user’s coordinates by using the separating distances and the coordinates of the nearest three femto users as shown in Fig. 3. For more clarification, the nearest three femto users, which enclose a macro user, have coordinates; (x 1, y 1), (x 2, y 2), and (x 3, y 3). The separating distances between the three femto users and the macro user, which its coordinates needed to be determined, are; R 1, R 2, and R 3. Then, the coordinates of this macro user, (x u , y u ), can be obtained by the following equations;

$$ \left( {x_{i} - x_{u} } \right)^{2} + \left( {y_{i} - y_{u} } \right)^{2} = R_{i}^{2} \quad {\text{where}}\;i = 1, \, 2, \, 3, \ldots {\text{etc}} $$
(11)
$$ x_{u} = \frac{{\det (A_{1} )}}{\det (A)} $$
(12)
$$ y_{u} = \frac{{\det (A_{2} )}}{\det (A)} $$
(13)
$$ \det (A) = \left| {\begin{array}{*{20}c} {2 \times (x_{3} - x_{1} )} & {2 \times (y_{3} - y_{1} )} \\ {2 \times (x_{3} - x_{2} )} & {2 \times (y_{3} - y_{2} )} \\ \end{array} } \right| $$
(14)
$$ \det (A_{1} ) = \left| {\begin{array}{*{20}c} {(R_{1}^{2} - R_{3}^{2} ) - (x_{1}^{2} - x_{3}^{2} ) - (y_{1}^{2} - y_{3}^{2} )} & {2 \times (y_{3} - y_{1} )} \\ {(R_{2}^{2} - R_{3}^{2} ) - (x_{2}^{2} - x_{3}^{2} ) - (y_{2}^{2} - y_{3}^{2} )} & {2 \times (y_{3} - y_{2} )} \\ \end{array} } \right| $$
(15)
$$ \det (A_{2} ) = \left| {\begin{array}{*{20}c} {2 \times (x_{3} - x_{1} )} & {(R_{1}^{2} - R_{3}^{2} ) - (x_{1}^{2} - x_{3}^{2} ) - (y_{1}^{2} - y_{3}^{2} )} \\ {2 \times (x_{3} - x_{2} )} & {(R_{2}^{2} - R_{3}^{2} ) - (x_{2}^{2} - x_{3}^{2} ) - (y_{2}^{2} - y_{3}^{2} )} \\ \end{array} } \right| $$
(16)
Fig. 3
figure 3

The basic principle of Cramer rule

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6 Femtocells’ Underlay Operation at Specified Conditions

When the cognitive femtocells are operated in an overlay manner, the macro user can be interfered by the neighboring macrocells [13,14,15,16,17,18,19]. Therefore, the SINR of the received signal of a macro user m, which operates on a subcarrier k, can be expressed as;

$$ SINR_{m,k} = \frac{{P_{M,k} G_{m,M,k}}}{{N_{o} \Delta f + \sum\limits_{{M^{\prime} }}^{{}} {P_{{M^{{\prime}},k}} G_{{m,M^{\prime},k}}}}} $$
(17)

where P M,k and \( P_{{M^{{\prime }} ,k}} \) are the transmitted power of the serving macrocell M and the neighboring macrocell M′ on a subcarrier k, respectively. G m, M, k is the absolute channel gain between the macro user m and the serving macrocell M on a subcarrier k. The absolute channel gain from the neighboring macrocell M′ is denoted by \( G_{{m,M^{{\prime }} ,k}} \). N o is the additive noise power spectral density, and Δf is the subcarriers’ spacing.

By the same way, the SINR of the received signal of a femto user f, which operates on a subcarrier k, can be evaluated by;

$$ SINR_{f,k} = \frac{{P_{F,k} G_{f,F,k} }}{{N_{o} \Delta f + \sum\limits_{{F^{{\prime }} }} {P_{{F^{{\prime }} ,k}} } G_{{f,F^{{\prime }} ,k}} }} $$
(18)

The absolute channel gain can be expressed as a function of the path loss (dB) as;

$$ G = 10^{ - PL/10} $$
(19)

It can be observed that, the EMI occurs among the macrocells themselves and among the femtocells themselves since no mutual EMI exists between the macrocells and the femtocells [17, 18].

In the proposed operation, the underlay cognitive femtocells system, the mutual EMI at the macro user, is allowed to grow on condition that its value guarantee a suitable performance (SINR and throughput) for the macro users. The mathematical condition, which can control the proposed underlay operation, can be given by;

$$ \sum\limits_{F} {P_{F,k} } G_{m,F,k} \le \frac{{N_{O} \Delta f + \sum\nolimits_{{M^{{\prime }} }} {P_{{M^{{\prime }} ,k}} G_{{m,M^{{\prime }} ,k}} } }}{10} $$
(20)

where G m,F,k is the absolute channel gain between the macro user, m, and the interfering femtocell on a subcarrier k. Equation 20 represents the mathematical condition at which the mutual EMI level between the macrocells and the femtocells can grow and guarantee a good performance for the existing macro users. The maximum allowed value for the mutual EMI is;

$$ EMI_{Maximum} = \frac{{N_{O} \Delta f + \sum\nolimits_{{M^{{\prime }} }} {P_{{M^{{\prime }} ,k}} G_{{m,M^{{\prime }} ,k}} } }}{10} $$
(21)

For a single macrocell system, the maximum allowed EMI level can be expressed as;

$$ EMI_{Maximum} = \frac{{N_{O} \Delta f}}{10} $$
(22)

By applying the above condition, the SINR of the proposed underlay cognitive deployment can be approximated to that of the overlay cognitive one. It can be observed that;

$$ SINR_{m,k} \downarrow_{{\Pr oposed_{{}} Underlay}} \approx SINR_{m,k} \downarrow_{{Cognitive_{{}} Case}} $$
(23)

The derived conditions in Eqs. 20, 21, 22, and 23 will be approved and validated in the simulation section.

7 Power Control Techniques

The main idea of power control algorithms is the reduction of the femtocells’ emitted power [10, 17, 18]. This power can be reduced by a lot of techniques which can be explained as follow;

  • The first technique is based on assigning a fixed level of power to every femtocell. This level is less than 20 dBm which is the maximum allowed level of power for any femtocell.

    $$ P_{f} \prec 20dBm $$
    (24)

    where P f is the transmitted power of the femtocell after employing power control.

  • In the second technique, the transmitted power of the femtocell is varied in order to cover a certain range. The emitted power of the femtocell can be calculated as follow;

    $$ P_{f} = \hbox{min} ([P_{m} + G_{\theta } - PL_{m} (d) + PL_{f} (r)],[P_{\hbox{max} } ]) $$
    (25)

    where r is the femtocell radius, P m is the macro base station transmission power, G θ is the antenna gain, PL m (d) is the macrocell path loss at distance d, and PL f (r) is the path loss at distance r.

  • On the contrary, the third technique depends on varying the emitted power of the femtocell in order to guarantee a target SINR for a user specified range. Suppose that, SINR t and SINR c are the target and the current SINR respectively. The algorithm can be implemented as;

    $$ P(k + 1) = \frac{{SINR_{t} }}{{SINR_{c} }}P(k) $$
    (26)

    where P(k) is the power level of the femtocell at kth iteration.

8 Calculation of Users’ Throughput

The throughput of a macro user, m, which is operated on a subcarrier, k, can be evaluated by Eq. 27 [17, 18] as follows;

$$ C_{m,k} = \Delta f \cdot \log_{2} (1 + \alpha \cdot SINR_{m,k} ) $$
(27)

On the other hand, the throughput of a femto user, f, which is operated on a subcarrier, k, can be expressed as [17, 18];

$$ C_{f,k} = \Delta f \cdot \log_{2} (1 + \alpha \cdot SINR_{f,k} ) $$
(28)

where α is a constant factor and it depends on the target bit error rate (BER) as; α = −1.5/ ln (5·BER).

9 Simulation Results

In this section, the LTE femtocells system is simulated applying the previous overlay cognitive femtocells’ concept, and the proposed underlay cognitive femtocells’ operation. Moreover, the derived mathematical conditions, which allow the proposed underlay cognitive femtocells’ operation, are approved and validated. In addition, the performance of Cramer rule in macro users’ localization in the proposed system is estimated and validated. The proposed system is simulated using the parameters which are included in Table 1. There are 19 macrocells, each of them can serve eight macro primary users. There are four femtocells inside each macrocell. Each femtocell serves two cognitive femto users. The LTE macrocell is plotted in a hexagonal pattern with a radius value of 250 m. On the other side, the femtocell has a radius value of 20 m. The macrocell and the femtocell base station can emit power till 46 and 20 dBm, respectively. The transmitted data is mapped with 16-QAM modulation over 10 MHz bandwidth. The LTE overlay cognitive femtocells system is simulated and its performance is clarified. The CDF is used in order to describe the system performance, geometry and throughput. It is the Cumulative Distribution Function for a specified distribution. It can be given for a real valued random variable, X, by;

$$ CDF = P(X \le A) $$
(29)
Table 1 The simulation parameters
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It can be defined as the probability of a random value, X, takes on a value less than or equal to A [22]. As a result, the CDF is the probability of a user’s geometry and throughput becomes below a certain value. Figure 4 shows the SINR of the macro users applying the cochannel deployment and the overlay cognitive deployment. On the other hand, Fig. 5 displays the SINR of the femto users applying the cochannel deployment and the overlay cognitive deployment. It can be observed that, applying the overlay cognitive radio concept in a LTE femtocells system can increase the SINR for both of macro and femto users in comparison with that system which applies the cochannel deployment. Moreover, the LTE overlay cognitive femtocells system can provide a great macro users’ throughput, as shown in Fig. 6, especially when the macro users are allowed to operate at all available subcarriers. The macro users are primary users so they have the priority to operate at any subcarrier anywhere at any time. The femto users are secondary users so they can operate at the subcarriers which are vacated from the macro users. Really, the macro users’ throughput is very good as displayed in Fig. 6. On the other hand, the femto users’ throughput becomes zero as there are no free subcarriers for them as clarified in Table 2, which presents the average of the femto users’ throughput when the macro users operate at all subcarriers. It is a great disadvantage in the LTE overlay cognitive femtocells system to have a zero femto users’ throughput value, however the high values of the SINR.

Fig. 4
figure 4

The SINR of the macro users applying the cochannel deployment and the overlay cognitive one

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Fig. 5
figure 5

The SINR of the femto users applying the cochannel deployment and the overlay cognitive one

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Fig. 6
figure 6

The throughput of the macro users applying the overlay cognitive deployment when they operate at all available subcarriers

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Table 2 The average of macro and femto users’ SINR and throughput
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The performance of the proposed LTE underlay cognitive femtocells system is simulated. The power control is applied and the femto users are allowed to operate at the subcarriers which are occupied by the macro users. The mutual EMI between the macrocells and the femtocells exists. This EMI level can grow and build up on condition that it cannot exceed the maximum value which is mathematically determined in Eqs. 21 and 22. The maximum value of the femtocells’ emitted power is examined and tested. It can be observed that, the maximum value of the femtocells’ emitted power becomes −6.4 dBm. In other words, the femtocells can operate at the subcarriers, which are occupied by the macrocells, on condition that their transmitted power cannot exceed the value of −6.4 dBm. In fact, the permission of the mutual EMI to exist and grow between the macrocells and the femtocells can decrease the SINR of the macro users. The challenge is to make the EMI effect on the macro users very small, in such a way that, it can be neglected. The proposed underlay cognitive operation is simulated. It can be observed that, the proposed underlay operation can solve the problem of a zero value of femto users’ throughput which may occur when the overlay cognitive femtocells are employed as shown in Fig. 7. The reason for the existence of femto users’ throughput is that the femto users are allowed to operate at all subcarriers even if they were occupied by macro users. On the contrary, the previous overlay cognitive deployment allows the femto users to operate at the free subcarriers only. The femto users’ throughput, in the proposed cognitive underlay operation, exists and has a suitable value. On the other hand, the EMI level on the femto users increases and the SINR of the femto users decreases as a result of applying the proposed underlay cognitive operation as displayed in Figs. 8, and 9. For more clarification, applying the proposed underlay cognitive operation can increase the EMI level on the femto users. Therefore, it can decrease the SINR of the femto users. On the other hand, the increased level of the EMI and the loss of the SINR can be paid as a price for the existence and the suitable value of the femto users’ throughput.

Fig. 7
figure 7

The throughput of the femto users in the proposed underlay cognitive deployment

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Fig. 8
figure 8

The EMI effect on the femto users applying the overlay cognitive and the proposed underlay cognitive deployment

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Fig. 9
figure 9

The SINR of the femto users applying the overlay cognitive and the proposed underlay cognitive deployment

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The performance of the proposed LTE underlay cognitive femtocells system is validated by comparisons with that system which applies the overlay cognitive deployment. From Figs. 10, 11, it can be concluded that, the proposed LTE underlay cognitive femtocells’ system can provide a better macro users’ throughput than the corresponding one which applies the overlay cognitive femtocells when the macro users are allowed to operate at half of the subcarriers of the system. Moreover, the proposed LTE underlay cognitive femtocells system can provide a satisfied value of femto users’ throughput and it can overcome the problem of a zero femto users’ throughput value which could exist when deploying the overlay cognitive femtocells.

Fig. 10
figure 10

The macro users’ throughput applying the overlay cognitive and the proposed underlay cognitive deployment

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Fig. 11
figure 11

The femto users’ throughput applying the overlay cognitive and the proposed underlay cognitive deployment

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Equations 21 and 22 represent the derived mathematical conditions, which allow an underlay operation of the cognitive femtocells in a LTE system. These conditions are verified in Table 3. This table studies the average of EMI value, which affects the macro users, in the proposed system with and without the mutual EMI level due to the proposed underlay operation. In other words, this table demonstrates the level and the effect of the resulted mutual EMI, which appears because of the proposed underlay operation, on the macro users’ SINR and throughput. It can be noticed that, applying an underlay operation for the cognitive femtocells system can increase the EMI level from −62.4018 to −62.2592 dBm. On the other side, this EMI value cannot decrease the macro users’ SINR and throughput. The value of 0.1425 dB loss in the macro users’ SINR, because of the underlay operation, can be neglected. Moreover, the value of 0.2005 Mbps loss in the macro users’ throughput, because of the underlay operation, can be neglected. It can be concluded that, the macro users’ SINR and throughput cannot be affect by allowing the mutual EMI to grow because of the proposed underlay cognitive femtocells’ operation in a LTE system. Therefore, the derived mathematical conditions are valid and effective.

Table 3 The average of macro users’ EMI, SINR, and throughput of the proposed LTE underlay cognitive femtocells system with and without the mutual interference value which appears due to the proposed underlay operation
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The validation of Cramer rule in the determination of macro users’ coordinates is tested. Table 4 presents the real coordinates of the macro users, especially which are inside the central macrocell, and the estimated ones using Cramer rule. It can be concluded that, the estimation of the coordinates, at the fusion center of the system, using Cramer rule can provide a good accuracy since the difference between the real coordinates and the estimated ones is very small and cannot be observed. For more clarification, Tables 5, 6, and 7 study the validation of Cramer rule in the estimation of coordinates in a LTE femtocells system at cochannel deployment, overlay cognitive deployment, and underlay cognitive deployment, respectively. From these tables, it can be noticed that, Cramer rule can fit well in the estimation of coordinates in a LTE femtocells system.

Table 4 The real coordinates of the macro users, which are inside the central macrocell, and the estimated ones using Cramer rule
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Table 5 The average of macro users’ EMI, SINR, and throughput in a cochannel deployment LTE femtocells system using the real coordinates of the macro users and the estimated ones using Cramer rule
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Table 6 The average of macro users’ EMI, SINR, and throughput in a LTE overlay cognitive femtocells system using the real coordinates of the macro users and the estimated ones using Cramer rule
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Table 7 The average of macro users’ EMI, SINR, and throughput in the proposed LTE underlay cognitive femtocells system using the real coordinates of the macro users and the estimated ones using Cramer rule
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10 Conclusions

The proposed LTE underlay cognitive femtocells system is explained. The system is mathematically analyzed and simulated. Moreover, the mathematical conditions, which can allow a good underlay operation for the cognitive femtocells, are derived and approved by simulation. Furthermore, the power control algorithms, which are the realization tools of an underlay cognitive operation, are explained in details. In addition, Cramer rule, which is used in the macro users’ positioning, is tested and validated. The proposed LTE underlay cognitive femtocells system can overcome the limitation of the overlay one when there are no available subcarriers to be accessed by the femto users. In the proposed system, the femto users’ throughput has an acceptable value as the femto users can operate at all available subcarriers. Moreover, the macro and the femto users’ throughput are higher than the corresponding system which is operated by overlay cognitive femtocells. It can be observed that, the mutual EMI, which affects the macro users because of the femtocells’ underlay operation, cannot decrease the macro users’ SINR and throughput. In addition, Cramer rule can best fit in the coordinates’ determination in a LTE femtocells system at different deployments.