Wireless Pers CommunDOI 10.1007/s11277-014-1638-x

Performance Analysis of Time Reversal UWBCommunication with Non-coherent Energy Detector

Dariush Abbasi-Moghadam · Ali Mohebbi ·Zohreh Mohades

© Springer Science+Business Media New York 2014

Abstract Non-coherent receivers, such as energy detectors (ED), are the simplest and themost practical alternatives to coherent receivers for low-rate and low-complexity applicationsin ultra-wideband (UWB) systems. However, these advantages are achieved at the expense ofnon-negligible performance degradation. One solution to improve the performance is to makeuse of time reversal (TR) technique. In this study, the performance of TR technique with non-coherent ED is analyzed in UWB systems. First, we derive an approximate analytical formulafor the error probability of TR-ED which is based on tapped-delay line (TDL) channel model.Next, we theoretically and by simulations analyze the optimum integration interval whichmaximizes the performance of TR-ED. The results show that TR technique, by reducing theintegration interval, considerably improves the performance compared to the conventionalED scheme.

Keywords Ultra-wideband (UWB) · Time reversal · Energy detector · Optimum integrationinterval

1 Introduction

In recent years, ultra-wideband (UWB) communications have received great interest fromboth the research community and industry. UWB has many attractive properties, including low

D. Abbasi-Moghadam (B)Electrical Engineering Department, Shahid Bahonar University of Kerman, Shahab, Kerman, Irane-mail: Abbasimoghadam@uk.ac.ir

A. MohebbiFaculty of Engineering, Islamic Azad University, South Tehran Branch, Tehran, Irane-mail: ali.mohebbi@yahoo.com

Z. MohadesSchool of Electrical Engineering, Iran University of Science and Technology, Tehran, Irane-mail: zohrehmohades@yahoo.com

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power, low sensitivity to fading, easier wall and floor penetration and high performance [1].But in UWB communications, there is the potential to interfere with other systems using thesame frequency bands. Therefore, regulation in some countries requires the implementationof spectrum sensing techniques, which has been widely explored in the context of cognitiveradios, in some bands for the coexistence of licensed primary systems and secondary UWBsystems [2,3]. However, from the receiver point of view, to capture a sufficient amount ofenergy using coherent schemes, such as Rake receivers, a large number of correlators mustbe implemented, which results in a very complex hardware architecture. In this case, channelestimation is a challenging task as it involves intensive signal processing and sampling rates.Such receivers are further burdened with the problem of estimating the amplitude and thedelay of each multipath component. Due to these challenges, there is an impellent need forsimpler receiver structures, capable of exploiting the rich UWB multipath channel diversityat an affordable cost, reasonable power consumption and low complexity.

Because of complexity constraints in practice, non-coherent schemes such as transmittedreference, energy detector (ED) and kurtosis detector (KD) are proposed for UWB systems[4]. In transmitted reference receivers [5], a reference pulse is transmitted before each datapulse and the channel response to the first pulse serves as a noisy template for the detectionof the second pulse. However, there are some issues needed to be addressed in such schemes.For example, to avoid inter-pulse interference and/or perform analog noise averaging overmultiple frames, the required autocorrelation delay needs to be longer than the UWB channellength. Since the maximum delay spread of UWB channels usually ranges from tens tohundreds of nanoseconds, implementing such a long wideband delay line is not practicablefor a realistic UWB system.

Energy and kurtosis detectors are also two proposed non-coherent receivers that do notrequire expensive channel estimation and Rake filters [6–11]. ED is the classical non-coherentreceiver which collects the energy of the received signal over a given interval. Although itssimple architecture is very suitable for low-complexity applications, it suffers from non-trivialperformance degradation due to low signal to noise ratio (SNR) of the received signal inUWB systems [6]. Orthogonal modulation schemes such as on-off keying (OOK) and binarypulse position modulation (BPPM) are typically employed with ED receiver structures. Sincethe noise floor increases linearly in time-bandwidth product (BT), energy detection is lesseffective for UWB signals. But if timing information and channel delay spread are roughlyknown, it is possible to reduce the noise level by choosing an appropriate integration interval.As another solution to mitigate the noise effect, weighted energy detector (WED) schemesare proposed which employ a set of M parallel integrators, each collecting a portion ofthe received signal energy over one of the M non-overlapping time intervals per symbolperiod [7]. The performance improvement of the WED comes with the cost of increasedhardware complexity when performing linear combination over the fractional energies. Forthe implementation of a WED, the clock speed used for energy sampling must be muchfaster than the one used in digital circuits. The difference between the two will be largeras the sub-intervals become shorter, so the weighting operation over the fractional energysamples needs to be carried out in parallel [7,8]. This consequently gives rise to an increasein hardware complexity, especially in the number of multipliers in proportion to the numberof sub-intervals, and so becomes a burden for UWB systems devised for low-power andlow-cost communication systems.

Kurtosis detector is also a non-coherent receiver which, instead of measuring the energy,estimates the fourth order statistics, i.e., kurtosis value, of the received signal. Despite hav-ing a better performance than ED, kurtosis detector is shown to be very sensitive to theunderestimation of the integration interval [9–11].

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Time reversal (TR) technique, which is basically a pre-filtering technique, is anotherattractive approach which has shown promising results for improving the performance inUWB systems, especially in multiple antenna applications [12–15]. The most remarkableeffect of this pre-filtering is that the received signal is focused both in time and space at theintended receiver. Through temporal focusing, TR system is capable of effectively mitigatinginter-symbol interference (ISI), and by spatial focusing, multi-user interference can be sig-nificantly reduced. By making use of TR technique and employing non-coherent detectionat the receiver, the cost versus data-rate issue can be alleviated to a greater extent.

In this paper, the performance of non-coherent ED with TR filter is analyzed. In the firstpart, an analytical expression for the probability of error of TR-ED for a tapped-delay line(TDL) channel model is proposed. TDL is a suitable channel model for dense multipathenvironments such as UWB indoor channels. Since the parameters of this channel model indifferent propagation scenarios are known, the designer would have a closed form equationfor performance evaluation. The analytical bit error rate (BER) performance of TR-ED iscompared with the BER obtaining from the simulation results of the standard IEEE UWBchannel models. Because the performance of ED is highly dependent on the integrationinterval, we evaluate the optimum integration interval for TR-ED both theoretically and bysimulations in the next part.

The remainder of the paper is organized as follows. The principle of TR technique withcoherent detection is addressed in Sect. 2. In Sect. 3, the performance of TR-ED and optimumintegration interval are theoretically analyzed. Performance evaluation and simulation resultsare presented in Sect. 4. Finally, we conclude the paper in Sect. 5.

2 Coherent Time Reversal UWB Communication

Time reversal technique uses the UWB channel impulse response (CIR) as a pre-filter atthe transmitter. Therefore, there is a channel estimation stage in this method in which thereceiver sends an impulse to the transmitter and the receiver-transmitter CIR, i.e., h(t), isestimated at the transmitter. This estimated CIR is then time reversed and complex conjugated,i.e., h∗(−t), and is utilized as a pre-filter at the transmitter for the data transmission stage.Therefore, the received signal can be expressed as

r(t) = x(t) ⊗ h∗(−t) ⊗ h(t) + n(t) = x(t) ⊗ heq(t) + n(t), (1)

where ⊗ denotes convolution operation, n(t) is additive white Gaussian noise (AWGN) withzero mean and two-sided power spectral density N0/2, and heq(t) is the equivalent CIRfrom the transmitter viewpoint. Furthermore, x(t) is the transmitted signal which in BPPMmodulation can be defined as

x(t) = √Eb

∞∑

i=−∞p(t − iTb − biδ), (2)

where p(t) is the UWB pulse with unit energy, Eb is energy per bit, Tb is the symbol periodand δ is the BPPM shift. Moreover, the binary symbols bi ∈ {0, 1} determine the position ofp(t) either at time iTb or iTb + δ.

The determination of BER is one of the most fundamental tasks in performance evaluationand optimization of any communication systems. Assuming there is no ISI, the performance

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of a TR-UWB communication system with BPPM signaling is described as [12]

Pe = Q(√

SN R)

= Q

(√Eb

N0

)

, (3)

where Q(x) = ∫∞x

(1/

√2π

)e−y2/2dy is the Q-function.

Several channel models have been proposed for UWB systems, e.g., tapped-delay line(TDL) and the modified Saleh-Valenzuela channels [16,17]. In this paper, a carrier-less TDLchannel model, which is a simple but very practical model, is adopted for UWB system. Itis based on the physical understanding of the received signal which is the sum of multiplereplicas of the transmitted signal due to reflection, refraction and scattering. This model hasbeen used in mobile radio applications and is widely applied to the UWB indoor channels.According to this model, the CIR of the UWB channel can be written as [18]

h(t) =L−1∑

i=0

hiδ(t − i�) =L−1∑

i=0

piαiδ(t − i�), (4)

where δ(·) is the Dirac delta function, L is the total number of paths, � is the multipathresolution which is assumed to be the same as the time domain pulse width, hi = piαi , pi ∈{−1,+1} is the equiprobable phase of the i th path and αi is the corresponding amplitudewhich is modeled as an independent Rayleigh random variable with probability density

function fαi (x) = xσ 2

ie−x2/2σ 2

i . The mean value of αi is E {αi } =√

π2 σ 2

i =√

πγ i

2 and the

second moment is E{α2i } = 2σ 2

i = γ i in which γ = e−�/� . Besides, � is the mean RMSdelay spread and the average power of α0 is 1. TDL channel model with equidistant tapsis suitable for the dense multipath environment, such as an indoor channel. Since the timedelay between different incoming paths could be very small, it is straightforward to treat allpropagation paths within one time-resolution as one effective channel tap [16,17].

Considering (1) and (4), the equivalent CIR (heq(t)) can be written as [15]

heq(t) =2L−2∑

i=0

heqi δ(t − i�). (5)

Based on the definition of convolution, heqi can be expressed as

heqi =

⎧⎨

⎩

∑ij=0 h j pL−1+ j−i αL−1+ j−i 0 ≤ i ≤ L − 1

∑L−1j=i−L+1 h j pL−1+ j−i αL−1+ j−i L ≤ i ≤ 2L − 2

. (6)

By substituting h j = p jα j into (6), and by letting k = L − 1 + j − i in the second term, wehave

heqi =

{∑ij=0 p j α j pL−1+ j−i αL−1+ j−i 0 ≤ i ≤ L − 1

∑2L−2+ik=0 pk+i−L+1αk+i−L+1 pk αk L ≤ i ≤ 2L − 2

. (7)

Consequently, the power of the received signal after using TR technique can be written as[15]

STR = P0 E(heq

L−1

)2 = P01 − e− L�

�

1 − e− ��

(1 + e− L�

�

1 + e− ��

+ 1 − e− L��

1 − e− ��

)

, (8)

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Performance Analysis of Time Reversal UWB Communication

where P0 is the transmitted power before applying TR technique. Using γ = e−�/� andafter some manipulations, we can rewrite (8) as

STR = P0(1 − γ L

) (1 − γ L+1

)

(1 − γ )(1 − γ 2

) . (9)

Therefore, the SNR of the received signal can be expressed as

SNR = STR

(N0/2)= 2P0

(1 − γ L

) (1 − γ L+1

)

N0 (1 − γ )(1 − γ 2

) . (10)

Now, considering (3) and (10), the probability of error for coherent TR receiver in TDLchannel model can be expressed as

Pe,TR = Q

(√2P0

(1 − γ L

) (1 − γ L+1

)

N0 (1 − γ )(1 − γ 2

)

)

(11)

3 Time Reversal Energy Detector

Energy detector is one of the most popular non-coherent schemes for UWB signal reception.It performs squaring operation, integration over a given time window TI , and thresholddecision as shown in Fig. 1. The signal at the output of integrate and dump device is

zED(ts) =ts+TI∫

ts

r2(t)dt, (12)

where ts and TI are the sampling time and integration interval, respectively.To detect BPPM signals, the receiver samples the signal z(t) at two instances, ts = iTb

and ts = iTb + δ. If we denote the corresponding energy estimates at these two instances byzED,0(i) = z(iTb) and zE D,1(i) = z(iTb + δ), then we have

zED,0(i) =iTb+TI∫

iTb

r2(t)dt, (13)

zED,1(i) =iTb+δ+TI∫

iTb+δ

r2(t)dt, (14)

where the sub-indices “ED, 0” and “ED, 1” denote energy estimates for the bit positions “0”and “1”, respectively. The decision rule in ED is stated as

b̂i ={

0, zED,0(i) ≥ zED,1(i)1, zED,0(i) < zED,1(i)

. (15)

r2(t) r(t) Bandpass

Filter ( ) 2⋅ IT

dt0

Energy Estimation & Decision

ED (t)

Fig. 1 Energy detector receiver

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Due to non-linear behavior of ED, the probability density function (PDF) of noise at thedetector output is obviously not Gaussian. It can be shown that r2(t) in (12) has a central ornon-central chi-square distribution depending on whether only noise or signal-plus-noise ispresent at the receiver input. However, zED(t) can be approximated as a Gaussian distributionusing central limit theorem, if time-bandwidth product is large enough, i.e., TI B 1.Therefore, using Gaussian approximation, the error probability of ED with BPPM is givenby Dubouloz et al. [6]

Pe,ED ≈ Q

⎛

⎜⎝

√√√√

(Ecap (T0, TI )/N0

)2

2BTI + 2Ecap (T0, TI )/N0

⎞

⎟⎠ , (16)

where B is the bandwidth of receiver front end, T0 and TI are the starting point and the lengthof integration interval, respectively, and Ecap(T0, TI ) is the amount of energy captured withinthe integration window.

To be able to fully collect the energy of the received pulses and maximize the performance,the integration interval has to be as long as the channel maximum delay spread. But thereceived signal energy is small in the tail of the channel response, and the receiver gathersmore noise energy than signal energy in longer integration intervals. On the other hand, if theintegration time is too short, not enough signal energy can be collected and the performancedecreases. In order to maximize the performance, we need to find the optimum values of T0 andTI which minimizes Pe,ED or equivalently maximizes the average output SNR. Considering(16), this can be formulated as [19]

(T0−Opt , TI−Opt

) = arg Min(Pe,ED

)

T0,TI

= arg MaxT0,TI

((Ecap(T0,TI )/N0)

2

2BTI +2Ecap(T0,TI )/N0

).

0 < TI ≤ Tb/20 < T0 < TI

(17)

The above equation is a nonlinear optimization problem with two variables, T0 and TI ,which can be solved by using the Lagrange method. However, due to high number of channelparameters, it is extremely difficult to solve (17) analytically. To simplify the computationtask, we assume that T0 has a fixed value and then we perform the maximization over theother variable, i.e., TI . Since T0 is defined as the arrival time of the first significant tap, andit is known that the correlation has the maximum value at zero in TR-UWB, we can assumeT0 = 0. And the optimum TI for ED is

TI−Opt = arg MaxTI

(E2

cap(0,TI )

2N 20 BTI +2N0 Ecap(0,TI )

)= arg Max

TI

(E2

cap(0,TI )

N0 BTI +Ecap(0,TI )

).

0 < TI ≤ Tb/2(18)

Considering TI = K� and TDL channel model and using γ = e−�/� , the amount ofcaptured energy can be expressed as

Ecap (0, TI ) = Ecap (0, K�) � P0

K∑

l=0

γ l

= P0

K∑

l=0

e− l�� = P0

1 − e− (K+1)��

1 − e− ��

= P01 − γ (K+1)

1 − γ. (19)

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Performance Analysis of Time Reversal UWB Communication

By substituting (19) in (16), the approximate error probability of ED in TR system withBPPM can be formulated as

Pe,TR−ED = Q

⎛

⎜⎜⎝

√√√√√√

(P0N0

)2 (1 − e− (K+1)�

�

)2

2BK�(

1 − e− ��

)2 + 2 P0N0

(1 − e− �

�

) (1 − e− (K+1)�

�

)

⎞

⎟⎟⎠ . (20)

In the same way by replacing (19) in (18), TI−Opt for the ED after some manipulations canbe described as

TI−Opt = arg MaxK

⎛

⎜⎝

(P0N0

)2

2B

(1−e− (K+1)�

�

)2

K�

(1−e− �

�

)2

+ P0B N0

(1−e− �

�

)(1−e− (K+1)�

�

)

⎞

⎟⎠ ,

0 < K ≤ �N/2 (21)

where N is the number of channel bins in one symbol period, i.e., N = Tb/�. This is aninteger optimization problem which can be solved through an exhaustive search for K =1, 2, . . . , �N/2 , or through more computationally efficient techniques, which is too complex.Modeling K as a continuous variable, and by taking the derivative of (21) with respect to Kand setting it equal to zero, we have

d

d K

⎛

⎜⎝

(1 − e− (K+1)�

�

)2

K�(

1 − e− ��

)2 + P0N0 B

(1 − e− �

�

) (1 − e− (K+1)�

�

)

⎞

⎟⎠ = 0, (22)

where dd K (·) denotes derivative operation. By defining

γ̄ = P0

N0 B, (23)

we have

d

d K

⎛

⎜⎝

(1 − e−(K+1) �

�

)2

K�(

1 − e− ��

)+ γ̄

(1 − e−(K+1) �

�

)

⎞

⎟⎠ = 0

⇒(

1 − e−(K+1) ��

)×

(2�

�e−(K+1) �

�

[K�

(1 − e− �

�

)+ γ̄

(1 − e−(K+1) �

�

)]

−(

1 − e−(K+1) ��

) [�

(1 − e− �

�

)+ γ̄

�

�e−(K+1) �

�

])= 0. (24)

Solving the first term (1 − e−(K+1)�/�) = 0 gives K = −1 which is unacceptable since Kis a positive integer. Solving the second term in (24) yields the following solution

K =(

1 − e−(K+1) ��

) (1 − e− �

� − γ̄�

e−(K+1) ��

)

2 ��

e−(K+1) ��

(1 − e− �

�

) . (25)

As it is seen, optimum integration interval can be derived by solving (25). Unfortunately, aclosed-form solution for (25) does not exist. However, we can observe that TI−Opt can beexpressed as a function of channel parameters and SNR.

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4 Simulation Results

This section is dedicated to verify and evaluate the performance of TR-ED. We compareour proposed theoretical performance of TDL-based TR-ED with the performance of EDand TR-ED obtained by simulations over UWB channel models CM1 and CM2 proposed byIEEE 802.15.4a [20]. CM1 corresponds to an indoor environment with line of sight (LOS)component while CM2 proposed for an indoor environment without the LOS propagation.The mean values of RMS delay spread for CM1 and CM2 are 16.66 ns and 19.38 ns, respec-tively. The parameters and assumptions for simulating the performance of ED and TR-ED inCM1 and CM2 are as follows: The BPPM modulation scheme is used for both the receivers.The second order derivative of Gaussian pulse is used as the transmitted pulse which ismathematically defined as

p(t) =(

t2 − σ 2p

)e− t2

2σ2p

√2πσ 4

p

(26)

where σp determines the pulse width (Tp), which in our simulation Tp = 1.5 ns. The energyof the channel impulse response is normalized to unity and the symbol period is Tb = 400 ns.Compared to the maximum delay spread of both CM1 and CM2 channels, Tb is long enoughto avoid ISI. Besides, the number of symbols used for simulations is 106 and we assume thatthe synchronization is perfect.

In Fig. 2, we have shown the theoretical performance of TR-ED proposed in (20) andcompared it with the simulated performance of TR-ED in CM1 and CM2 channels. Theparameters of the adopted TDL channel model have the following values [21]: � = 0.5 nsand � = 7.5 ns for CM1 and � = 10 ns for CM2 channel. Moreover, the receiver bandwidthis B = 4GHz. As it is seen, there is a satisfying agreement between theoretical and simulationresults for practical SNRs. It is worth mentioning that the observed difference between thetheoretical and simulated curves is mainly due to the type of channel model we have adoptedin deriving the error probability of TR-ED. Although TDL channel model is a simplified and

Fig. 2 Performance comparison of TR-ED obtained theoretically and by simulations for TI = 200 ns in CM1and CM2

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Performance Analysis of Time Reversal UWB Communication

Fig. 3 Performance comparison of ED and TR-ED for different integration intervals at Eb/N0 = 18 dB

Fig. 4 Performance comparison of ED and TR-ED for their optimum integration intervals

practical model, there are some differences between this channel model and CM1 and CM2proposed by IEEE 802.15.4a standard. Furthermore, we have used some approximations inderiving (20). Considering the above, the applied TDL channel model and the approximationsjustify the difference between the theoretical and simulation results.

Figure 3 compares the performances of ED and TR-ED in terms of different integrationintervals (TI ) in CM1 and CM2 channels. As it is seen, for each receiver scheme, there isan optimum integration interval that minimizes the BER. It is clearly observed that applyingTR technique, remarkably improves the performance of ED for short integration intervals.This is due to the temporal focusing property of TR technique in which a large amount of

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Fig. 5 Performance comparison of TR-ED for different integration intervals and SNRs in CM1

Fig. 6 Performance comparison of TR-ED for different integration intervals and SNRs in CM2

received signal power is concentrated within a few taps and therefore the optimum value ofintegration time for TR-ED can be very short.

Figure 4 illustrates the performance of ED and TR-ED for their optimum values of inte-gration intervals in CM1 and CM2. As it is expected, the performance of ED in TR schemeis better than the conventional scheme. TR-ED outperforms about 1.5 dB over ED in CM1,and more than 1 dB in CM2 at BER of 10−3. Moreover, it is observed that the performance ofthe receivers in CM1 is superior than their performance in CM2 and that is due to the strongLOS component in CM1 channel model which leads to more concentration of energy in theoptimum integration interval.

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Performance Analysis of Time Reversal UWB Communication

Figures 5 and 6 are dedicated to analyze the effect of SNR on the optimum integrationinterval of TR-ED. These figures show that changing the values of SNRs has a very littleeffect on the optimum integration interval in both CM1 and CM2 channels. Therefore, itis concluded that applying TR technique can remarkably reduce the sensitivity of optimumintegration interval to SNR.

5 Conclusion

In this paper, based on TDL channel model, we derived a new approximate analytical expres-sion for the error probability of TR-ED. It was shown that TR-ED outperforms conventionalED due to its temporal focusing property. Besides, we theoretically analyzed the optimumintegration interval for TR-ED and demonstrated that it can be expressed in terms of channelparameters and SNR. Moreover, the simulation results showed that the optimum integrationinterval has a very little sensitivity to SNR in TR scheme.

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Dariush Abbasi-Moghadam was born in Kerman, Iran on July 21,1976. He received the B.S. degree in electrical engineering from ShahidBahonar University, Kerman, Iran, in 1998 and the M.S. and Ph.D.degree in Iran University of Science and Technology, Tehran, Iran, in2001 and 2011, respectively, both in electrical engineering. He was pri-mary with the Advanced Electronic Research Center—Iran from 2001–2003 and worked on the design and analysis of satellite communica-tion systems. In September 2004, he joined Iranian Telecommunica-tions Company, Tehran, as a Research Engineer. He is currently withDepartment of Electrical Engineering at Shahid Bahonar University,Kerman, Iran. His research interests are in the area of wireless commu-nications, Ultra Wideband communication systems, satellite communi-cation systems, Power linecommunications, and Radar signal process-ing.

Ali Mohebbi received the B.S. degree in electrical engineering fromIslamic Azad University of Karaj, Karaj, Iran, in 2006, and the M.S.degree in electrical engineering from Islamic Azad University, SouthTehran Branch, Tehran, Iran, in 2011. His current research inter-ests include ultra-wideband systems and signal processing techniquesfor wireless communications, with emphasis on synchronization algo-rithms in OFDM systems.

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Performance Analysis of Time Reversal UWB Communication

Zohreh Mohades was born in Kerman, Iran on July, 1985. shereceived the B.S. degree in electrical engineering from Shahid BahonarUniversity, Kerman, Iran, in 2008 and the M.S. degree in Iran Uni-versity of Science and Technology, Tehran, Iran, in 2011, in electricalengineering. Her research interests are in the area of wireless commu-nications, communication theory, and image.

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