A Low Complexity Suboptimal Energy-Based Detection Method for SISO/MIMO Channels

Wireless Pers CommunDOI 10.1007/s11277-014-1672-8

A Low Complexity Suboptimal Energy-Based DetectionMethod for SISO/MIMO Channels

Mohammad Dehghani Soltani · Hassan Aghaeinia ·Mohammadreza Alimadadi

© Springer Science+Business Media New York 2014

Abstract In this paper, we propose an actually novel and simple method for detection oftransmitted symbols in MIMO channels. This method is based on the energy level of thereceived signals. At the receiver, we assume the knowledge of channel state informationwhich can be estimated by different methods, e.g. by sending pilots. So, we can determineall possible levels of energy. This computation of energy levels is done only once for thequasi-static channels. Energy of the received signals is a criterion by which we can estimatethe transmitted symbols. Detection of transmitted signal is made based on the nearest energylevel and the points which lie on it. In other words, we have restricted our search space to anew smaller space with different levels of energy. Simulation results confirm approximatelythe same performance between the maximum-likelihood detector and the proposed approachespecially in high signal-to-noise ratios with a remarkable reduction in the computationalcomplexity.

Keywords Wireless communications · Energy-based detection · Maximum-likelihooddetector (MLD) · Computational complexity · Quasi-static channel

1 Introduction

Wireless communication systems gain from many assets such as long-range communica-tions, higher throughput and efficiency, more flexibility and mobility. These priorities plus itslower cost of implementation compared to the wired communication systems have persuadedresearchers in the recent decades to concentrate on the extension of wireless communication

M. D. Soltani (B) · H. Aghaeinia · M. AlimadadiElectrical Engineering Department, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Irane-mail: [email protected]

H. Aghaeiniae-mail: [email protected]

M. Alimadadie-mail: [email protected]

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M. D. Soltani et al.

techniques. Moreover, nowadays in wireless communication multiple-input multiple-output(MIMO) systems are widely preferred to single-input single-output (SISO) due to the incre-ment of transmission bit rate and the link reliability without any additional bandwidth andtransmission power. However, high computational complexity is one of the prominent chal-lenges of these systems.

The problem of finding the transmitted symbols in both SISO and MIMO channels bymeans of the maximum-likelihood detector (MLD) is considered as an optimal approach;however, it leads the receiver to suffer from high computational complexity. Optimal MLDnecessitates estimating the transmitted vector by regarding the received vector among the setof all possible vectors. Considering one timeslot at the receiver, this estimated vector shouldminimize the following Euclidean distance:

x = argx

min‖y − Hx‖2, (1)

where x, y,H and x are the estimated vector, the received vector, channel matrix and thetransmitted vector, respectively.

In general, the performance of conventional MIMO detectors in comparison to MLD isundesirable particularly for large size of MIMO systems. Furthermore, the performance ofconventional MIMO systems degrades under the inappropriate condition of channel due tospecial correlation. Unfortunately computational complexity of the MLD increases exponen-tially with the growth of transmitter antennas, say Nt , and constellation size, say M , as M Nt ,because in general MLD necessitates joint detection of an entire block of symbols. Manyattempts have been done in order to reduce the complexity of MLD in both SISO and MIMOchannels [1–5]. In the literatures, some sub-optimum receivers with low to fair complexityhave been proposed [6–8], but still they suffer from their limited performance. Successiveinterference cancellation (SIC) and linear receivers are the most that for successive interfer-ence cancellation (SIC), the overall performance is limited by the quality of the strongestdetected signal. The authors in [9] also have proposed a new method to reduce the complexityof the MLD based on a class of asymptotically good low density parity check codes -expandercodes- over binary symmetric channels (BSCs). Their method has a polynomial complexityorder. However, the proposed method in [9] considers assumptions which are far apart fromreality.

A new grouping-MLD (GMLD) for single carrier-frequency division multiple access (SC-FDMA) systems is proposed in [10]. In order to reduce the MLD complexity of the receiver,the GLMD performs local ML detection by means of grouping the received symbol blockbased on an orthogonal projection. Consequently, the GMLD has a lower complexity thanthe MLD. However, its performance approaches to the performance of the MLD in cost ofhigher computational complexity.

In this paper, we have proposed a new detection technique with a low computationalcomplexity in comparison with the MLD and in cost of a very negligible performance lossin medium and high SNRs. We consider the energy of received vector as a criterion to detecttransmitted symbols. Using this feature, we first estimate the energy level of the transmittedsignal according to the energy of received signal. Many points will be omitted in this step tobe checked for next step. Then, we limit our search to those vectors which lie in this smallersearch space, i.e. we ignore the vectors with desultory energy levels from our search space.It is notable that finding the energy level of the transmitted signal has a very low complexitysince no multiplication should be calculated and it just has a few limited additions. Therefore,the major complexity of this method returns to search the vectors which lie in the same energy

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Low Complexity Suboptimal Energy-Based Detection Method

level. We will also show that our proposed method does not depend on any coding and it canbe treated as a general method since it considers the energy of received signals.

The rest of this paper is organized as follows. Section 2 describes a preliminary of levels ofenergy and complexity criteria which are used through this paper in abundance. System modelof the proposed model is brought in Sect. 3. Furthermore, our proposed method for SISOand MIMO channels along with analytical results are given in Sect. 4. Finally, simulationresults, future works and conclusions are presented in Sects. 5, 6 and 7, respectively.

Notation: In the rest of the paper, vectors and matrices are denoted using lower-case andupper-case boldface letters, respectively. Moreover, notations ε(s) and C

m×n indicate theenergy of signal s and m × n dimensional complex space, respectively.

2 Preliminaries

Through this paper, we use some terms that are necessary to be explained here:

2.1 Level of Energy

Constellation points which have equal energy lie in the same level, each constellation has one(just 4-QAM constellation has one level of energy) or more levels of energy. For instance,consider a 16-QAM constellation which is depicted in Fig. 1. As this figure shows, there arethree levels of energy, ε1 = 1

2 d2min, ε2 = 5

2 d2min and ε3 = 9

2 d2min , i.e. ε{si }i=1:4 = 1

2 d2min ,

ε{si }i=5:12 = 52 d2

min and ε{si }i=13:16 = 92 d2

min and dmin represents the minimum distancebetween the points of the constellation. One can easily compute the total number of energylevels for any 2n-QAM constellation by utilizing equation 2n−3 + 1. Table 1 represents thetotal number of energy levels for some M-QAM constellations.

Fig. 1 16-QAM constellationwith three energy levels ofd2

1 = 12 d2

min, d22 = 5

2 d2min and

d23 = 9

2 d2min, i.e. E{si }i=1:4 =

d21, ε{si }i=5:12 = d2

2 and

E{si }i=13:16 = d23

Table 1 Number of energy levels for 2n -QAM constellation

M 4 8 16 32 64 128 2n

Number of energy levels 1 2 3 5 9 17 2n−3 + 1

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Fig. 2 General block diagram of MIMO system model with Nt transmit and Nr antennas

2.2 Complexity Criterion

Here, we consider a criterion to measure the complexity of our method and to do a comparisonwith other approaches. The criterion which we use through this paper is the total numberof required additions and multiplications to find the ML solution of our proposed method.Clearly, the same operations must be repeated for every point of the search space, and the totalnumber of operations is a linear function of Ns , where Ns is the search space size. Therefore,in some parts, to make a simple comparison between our proposed decoding method and thatof the MLD, we consider the number of constellation points that are selected to compute theEuclidean distance with the received signal as the complexity criterion.

We also presume the channel to be quasi-static, such that the computational complexityof any preprocessing steps like computation of energy levels is negligible.

3 MIMO System Model

A high level multiple-input multiple-output system model is presented in Fig. 2 which isconsists of Nt and Nr antennas at the transmitter and receiver, respectively. Data stream ismodulated by means of a M-array constellation and then it divided to Nt sub-stream. Beforethe modulation, channel coding may be utilized, however we have not considered channelcoding in this paper.

The principal goal of spatial multiplexing systems is diverse stream transmission of dataover different transmit antenna on the same carrier frequency and at the equal symbol rate.Transmitted symbols can be shown by a vector of Nt dimensions. Suppose that the data streamon the pth transmit antenna as a function of time t is shown by sp(t). When a transmissionoccurs, the transmitted signal from the pth transmit antenna meets diverse paths to reachthe qth receive antenna. In other words, it reaches the qth receive antenna through a straightpath and other indirect paths. Let the bandwidth BW of the system to be selected suitablythat the delay between the first and last received signal is lower than 1/BW. In this case, thesystem is a narrowband one. For this kind of systems, all multipath components betweenthe pth transmit antenna and the qth receive antenna could be summed and represented by aterm hqp(t). Since signals are transmitted on just one carrier frequency through all transmitantennas, the qth receive antenna receives the multipath signals of the pth transmit antennaplus the other Nt − 1 transmit antennas which can be shown by the following equation :

yq(t) =Nt∑

p=1

hqp(t)sp(t) + ηq . (2)

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To show all Nr received signal in one equation, matrix notation can be utilized as below

y(t) = H(t)s(t) + η(t), (3)

where s(t), y(t) and H(t) matrices are equal to

s(t) =

⎡

⎢⎢⎢⎣

s1(t)s2(t)...

sNt (t)

⎤

⎥⎥⎥⎦ , y(t) =

⎡

⎢⎢⎢⎣

y1(t)y2(t)...

yNr (t)

⎤

⎥⎥⎥⎦ and H(t) =

⎡

⎢⎢⎢⎣

h1,1(t) h1,2(t) · · · h1,Nt (t)h2,1(t) h2,2(t) · · · h2,Nt (t)...

......

...

hNr ,1(t) hNr ,2(t) · · · hNr ,Nt (t)

⎤

⎥⎥⎥⎦ , (4)

where hi, j is the channel gain (a complex value) between j th transmit antenna and i th

receive antenna. Moreover, η = [η1,η2, . . . ,ηNr]T ∈ C

Nr ×1 is the complex zero meanwhite Gaussian noise vector with i.i.d entries ηi ∼ CN (0, N0) for iε{1, 2, . . . , Nr }.

To have a fair comparison between MIMO and SISO systems, the transmitting powerfrom each transmit antenna at a MIMO system should be equal to 1/Nt transmitting powerof a SISO system. Therefore, the total transmission power at the transmitter are equal to thetransmission power at a SISO system without considering the transmit antennas.

4 Our Proposed Method

4.1 Our Proposed Method for a SISO System

We can model a system with single receiver and single transmitter antenna as below

y(ti ) = h(ti )x(ti ) + η(ti ). (5)

At the receiver, by the knowledge of h(ti), we first compute number of energy levels and theircorresponding values. For this purpose, these levels can be computed easily by multiplicationof the constellation energy levels by the fading coefficient, i.e. εj = |h(ti)|2 εconstellation

j , j =1, . . . , 2n−3 + 1 for a 2n-QAM constellation. For instance, a 16-QAM constellation hasthree energy levels: 1

2 d2min,

52 d2

min and 92 d2

min. The receiver should construct the followingenergy levels: 1

2 |h(ti)|2 d2min,

52 |h(ti)|2 d2

min, and 92 |h(ti)|2 d2

min. In the next step, the receivercomputes the energy of the received signal y(ti), compares this value with all possible energylevels and chooses the nearest one. As mentioned, this step has a negligible complexity.For example, a 16-QAM constellation has three energy levels and therefore receiver needsto compare the energy of received signal y(ti) just two times to determine its energy level.After finding the energy level, the receiver checks all points and selects the point that has theminimum Euclidean distance with y(ti), i.e.

mins∈D⊂�

‖y(ti) − h(ti)s‖2, (6)

where s ∈ D are all symbols that lie in the same energy level and � stands for a 2n-QAMconstellation. In this step, receiver must check at least 4 and at most 8 points for a 16-QAM constellation instead of exactly 16 comparisons. Thus, we have low multiplicationsand additions in the detection process.

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Fig. 3 Error region of symbol S1for the first detection algorithm

4.2 Improvement of the Proposed Algorithm via Searching in Two Energy Levels

To improve the performance of our algorithm, we propose an algorithm which searchestwo layers instead of one layer. We will see that this algorithm has approximately the sameperformance as the MLD.

Consider error region for the symbol Si as the area in which our proposed algorithm leadsto error in detection of this symbol. The shaded area in Fig. 3 depicts the error region forsymbol S1. This figure depicts that if the transmitted symbol is S1 and the received signal yfalls in the shaded area, error occurs. Since in this region the Euclidean distance between thereceived signal y and the symbol S1 is less than Euclidean distance between other symbols,MLD detects S1 as the transmitted symbol. But our algorithm first detects the closest energylayer to the energy of the received signal y (in this case it selects the second layer), thenthe decision is made based on the Euclidean distance between symbols of this layer andthe received signal y. After selecting the second level, our proposed energy-based algorithmdetects either S2 or S3 according to the Euclidean distance between them and the receivedsignal y, and therefore the error occurs.

To omit this error, we have improved our algorithm in expense of a little complexitycomputation. If we want to have nearly the same performance as the MLD, our algorithmshould check points of two energy levels with the nearest value to energy of the receivedsignal y. Suppose the received signal y has an energy between 1

2 d2min and 5

2 d2min , i.e. 1

2 d2min ≤

ε(y) ≤ 52 d2

min . Then, according to this improved algorithm, Euclidean distance betweenreceived signal y and the points which lie in both layers 1 and 2 should be checked. Thepoint with minimum Euclidean distance is the solution of this improved algorithm and alsothe solution of the MLD. Using this approach, we indeed improve the performance of systemto be nearly matched with the performance of the MLD, in expense of a bit complexityincrement.

Now let us compute the total complexity of our proposed method in terms of total mul-tiplications and additions. Although complexity of finding the energy level is negligible, ifwe consider it the complexity of the proposed method can be found as below:

CPM = Nadd × Cadd + Nadd × Cmul + CEL

≈ Nadd × Cadd + Nadd × Cmul . (7)

where CPM denotes the complexity of proposed method. Besides, Cadd , Cmul and CEL rep-resent the complexity of computing one addition, the complexity of computing one multi-plication and the complexity of computing the energy level of received signal, respectively.Moreover, Nadd and Nmul stand for total required additions and multiplications of checkinga constellation point using (6), respectively.

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The last equation is due to the fact that CEL � Nadd × Cadd + Nmul × Cmul . This isthe maximum amount of computation for M-QAM. For instance, consider 64-QAM con-stellation. Our algorithm should carry out at most 20 comparisons, (12 comparisons for thesixth energy level and 8 comparisons for the fifth or the seventh energy level), and at least12 comparisons, (4 comparisons for the first energy level and 8 comparisons for the secondenergy level, there are five other similar cases). Therefore, maximum complexity of findingthe transmitted signal is equal to:

20 × Cadd + 20 × Cmul + CEL,

which can be approximated as

≈ 20 × Cadd + 20 × Cmul .

Comparing to the complexity of MLD which is equal to:

64 × Cadd + 64 × Cmul . (8)

It can be perceived that we reduced 44 multiplications and additions, i.e. (44Nmul + 44Nadd)

which is a considerable value.

4.3 Our Proposed Method for MIMO System

Let us consider the MIMO system presented in Fig. 2 in which the channel is assumed tobe quasi-static and Rayleigh. Supposed that space time block code (STBC) with an arbitrarycoding is utilized, thus S = [s1, s2, . . . , st ] ∈ C

Nt ×t is the STBC matrix, which is transmittedby Nt antennas over t time slots and sl = [s1, s2, . . . , sNt ]T for lε{1, 2, . . . , t} are transmittedvectors at the l-th time slot through Nt antennas. Since the channel is supposed to be quasi-static, H should be constant during t time slots transmission.

4.3.1 Energy-Based Detection Algorithm for MIMO Systems

At the receiver, we determine the energy of all possible transmitted vectors, si for i ∈{1, 2, . . . , t} just one time by using H due to the assumption of quasi-static channel. Afterttransmissions, we should compute new energy vectors, again. So we compute energy ofri ,∀i ∈ {1, 2, . . . , Nr } for each receiver antenna as follows:

ε = [ε(r1) ε(r2) · · · ε(rNr )]T ,

where

r1 = [h11 h12 h13 · · · h1Nt

] [s1 s2 s3 · · · sNt

]T = h1sT ,

r2 = [h21 h22 h23 · · · h2Nt

] [s1 s2 s3 · · · sNt

]T = h2sT ,

...

rNr = [hNr 1hNr 2hNr 3 · · · hNr Nt

] [s1s2s3 . . . sNt

]T = hNr sT , (9)

where hi , ∀i ∈ {1, 2, . . . , Nr } are the channel matrix coefficients between the receiverantenna i and Nt transmitter antennas. As mentioned before, these energy vectors are com-puted once and the receiver just computes energy of received signal y = [y1y2 . . . yNr

]T. Letus represent this energy vector as

ε(y) = [ε(y1) ε(y2) . . . ε(yNr )

]T.

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Since s j ,∀ j ∈ {1, 2, . . . , Nt } is chosen from the constellation set, it could have k possiblevalue and k ∈ {1, 2, . . . , M}. Therefore, transmitted vector sl = [s1, s2, . . . , sNt ]T can haveM Nt different values. Consequently, there are M Nt possible values for any of ε(ri ),∀i ∈{1, 2, . . . , Nr }.

To find the transmitted vector si for i ∈ {1, 2, . . . , t}, our proposed algorithm is presentedas below:

1) Compare ε(y1) with all M Nt possible energy ε(r1) and select N1 vectors which have theclosest energy to that of ε(y1) among M Nt vectors.

2) Compare ε(y2) with all N1 possible energy ε(r2) and select N2 vectors which have theclosest energy to that of ε(y2) among N1 vectors.

3) Continue this process till NNr -th step in which ε(yNr ) is compared with all NNr −1 possibleenergy ε(rNr ) and select NNr closest vectors among NNr −1 vectors.

4) Computes Euclidean distance between received signal y and these NNr vectors.5) Select the vector with the minimum Euclidean distance as the transmitted vector.

Since these NNr steps have no multiplication, computational complexity of this method ismostly related to step 4. We should mention that selection of large Ni, i ∈ {1, 2, . . . , Nr }increases the computational complexity and it is recommended to choose it properly. Wehave chosen these parameters adaptively according to noise variance in our simulation. It issuggested to choose great value for Ni at initial steps and then decrease it in next steps.

To reduce the complexity of our proposed algorithm in high SNRs, we decrease the finalNNr according to SNR, i.e. in higher SNRs due to the weak effect of noise we can omit lowprobable candidates (vectors) to be searched by the ML criterion.

4.3.2 Analytical Results

It is worth mentioning that since the energy of signal will be affected by the energy of noisein low SNRs, performance of our proposed method degrades, i.e. in this case noise can affectthe energy of transmitted symbol. Due to this reason a gap between the performance of MLDand the performance of our proposed method in low SNRs is expected. However, in highSNRs, since the energy of transmitted signal is remarkably larger than the energy of noise,then noise has not suffice power to affect the energy of transmitted signal and thereforeperformance of our proposed method does not degrade and it is similar to the performanceof MLD with an acceptable approximation in high SNRs. This result is also compatible withintuitive sense. Thus, the BER of MLD is the lower bound of our proposed method’s BER,that is,

PProposeds ≥ 1 −

(1 −

(1 − 2(

√M − 1)√

M

)Q

(√3γs

M − 1

))2

, (10)

where PProposeds represents the symbol error rate of our proposed method and M is the size of

the constellation. Besides, in this non-equality Q(.) stands for the Q-function and γs = EsN0

is

the average SNR per symbol in which Es denotes the average energy per symbol. The proofof the right side of equation (10) is given in [11].

Since in high SNRs the BER of our proposed method and the BER of MLD are nearlysimilar with an acceptable approximation, the following equation is held

PProposeds

∼= 1 −(

1 −(

1 − 2(√

M − 1)√M

)Q

(√3γs

M − 1

))2

for SNR � 1. (11)

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Fig. 4 A comparison between our proposed energy-based method and the MLD for 16-QAM constellationin a Rayleigh fading SISO channel

The nearest neighbor approximation to probability of symbol error depends on whether theconstellation point is an inner or outer point. Inner points have four nearest neighbors, whileouter points have either two or three nearest neighbors; in both cases the distance betweennearest neighbors is 2dmin [11]. If we take a conservative approach and set the number ofnearest neighbors to be four, we obtain the nearest neighbor approximation

PProposeds

∼= 4Q

(√3γs

M − 1

)for SNR � 1. (12)

5 Simulation Results

In this section, we compare the BER of our proposed method with MLD in a Rayleigh andquasi-static channel for both SISO and MIMO systems. Figure 4 depicts the performanceof our proposed method utilizing one level of energy and two levels of energy and theperformance of MLD. As this figure shows using two levels of energy leads to lower BER,however its computational complexity is larger. A gap between the performance of ourproposed method and the performance of MLD can be observed in low SNRs. However, asthe SNR increases this gap disappears, especially in high SNRs BER curve of the proposedmethod matches the BER curve of MLD. As mentioned previously, in low SNRs, noise hasa considerable energy in comparison to the energy of transmitted signal therefore it couldaffect the detection performance. However, in high SNRs, since the energy of transmittedsignal is much larger than the energy of noise, therefore it could not affect the performanceof our proposed method and it is approximately the same as the performance of MLD in highSNRs.

One may be worry about our energy-based method while the energy level of the receivedsignal is more different than the energy level of transmitted signal, our proposed method

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Table 2 Maximum number of constellation points to be compared with received signal for detection oftransmitted symbol

Scheme Maximum number of constellation points to be checked

16-QAM 32-QAM 64-QAM 128-QAM

Energy-based detector with one level ofenergy

8 8 12 16

Energy-based detector with two levels ofenergy

12 16 20 24

MLD 16 32 64 128

degrades highly. But as a significant note, we should mention that when the energy level ofthe received signal is more different than the energy level of transmitted signal, with a greatprobability ML detector also falls into error.

When we use more levels of energy for detection, the performance of our proposedmethod approaches to the performance of ML detector; however the computational com-plexity increases.

Table 2 represents the maximum number of constellation points to be compared withreceived signal to detect the transmitted symbol for our proposed energy-based method usingone level or two levels of energy and the MLD. According to Table 2 and Fig. 4 one caneasily see that our proposed method has an acceptable performance.

For a MIMO system with two transmitter antennas and two receiver antennas (Nt = Nr =2), we have presented a comparison between our proposed algorithm in the previous sectionand MLD. We have used Alamouti’s [12], Golden [13] and Sezginer’s [14] codes in oursimulations with the following code matrix codes

SAlamouti =[

s1 s2

−s∗2 s∗

1

], (13)

SGolden = 1√5

[α(s1 + θ s2) α(s3 + θ s4)γ(s3 + θ s4) α(s1 + θ s2)

], (14)

where θ = 1+√5

2 , θ = 1−√5

2 , α = 1 + j(1 − θ), α = 1 + j(1 − θ) and γ = ejπ6 .

SSez =[

as1 + bs3 −cs∗2 − ds∗

4as2 + bs4 cs∗

1 + ds∗3

], (15)

where a = c = 1√2, b =

(1−√

7)+j(1+√

7)

4√

2and d = −jb. We have also considered a spatial

multiplexing case in our simulations.The results are depicted in Fig. 5 for the 16-QAM constellation (s1, s2, s3 and s4 in the code

matrix are chosen from the 16-QAM constellation). As this figure shows our energy-basedalgorithm has an acceptable performance especially in high SNRs since its performanceapproaches to the performance of MLD. As mentioned in previous section, if we considerNNr as the complexity of our proposed method, Fig. 6 represents the variation of complexityin terms of SNR for the curves of Fig. 5. This complexity is arbitrary and one can increaseit to get a better performance. Note that if we choose NNr = 256 for all SNRs, our proposedalgorithm shows exactly the same performance as MLD.

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Fig. 5 A comparison between our proposed energy-based algorithm and MLD for 16-QAM constellation ina Rayleigh fading MIMO channel with Nt = Nr = 2 and utilizing Alamouti’s, Golden STBC and spatialmultiplexing (SM)

Fig. 6 The complexity of our proposed method for 16-QAM, 2 × 2 MIMO channel

6 Future Works

As mentioned in Sect. 3, we have reduced NNr according to the SNR but fully arbitrary,therefore one can proposed and improved the selection of this parameter properly to reducethe complexity of the system, more than this.

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7 Summary and Concluding Remarks

This paper proposed a new method to detect transmitted symbols based on the energy ofreceived signals. We have presented this algorithm for both SISO and MIMO systems. Fora SISO system, first we proposed a one level energy-based detection method. Then, weincreased the involved levels of energy which shows a really near optimal performance asthe MLD in high SNRs. To benefit from lower computational complexity, we just used twolevels of energy which is indeed sufficient in high SNRs. For MIMO system, we presentedan algorithm which computes energy of the received vector in each receiver antenna and itselect NNr vectors through five aforementioned steps as the final candidates for transmittedvectors. Decision will be made according to Euclidean distance between the received signalsand these NNr vectors. Simulation results show that our algorithm in both SISO and MIMOcases had a good performance which its performance approaches to the performance of MLDin high SNRs.

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Mohammad Dehghani Soltani received his B.Sc. degree and gainedas the first rank in Electrical Engineering from Shahid Bahonar Univer-sity, Kerman, Iran, in 2010 and his M.Sc. degree from the Departmentof Electrical Engineering, Amirkabir University of Technology, Tehran,in 2012. His current research interests include wireless MIMO systems,MIMO-OFDM systems channel modeling and wireless Ad-Hoc.

Hassan Aghaeinia received his B.Sc. degree in Electrical Engineer-ing from Amirkabir University of Technology (Tehran Polytechnic),Tehran, Iran, in 1987 and his M.Sc. degree in Electrical Eng. fromAmirkabir University of Technology (Tehran Polytechnic), Tehran, Iranin 1989. He has also received M.Sc and his Ph.D. degree in Electri-cal Eng. from Valenciennes University (UVHC), Valenciennes, France,respectively in 1992 and 1996. He is currently the associate professorof Amirkabir University. His current research interests include wirelessMIMO systems, MIMO-OFDM systems channel modeling and imageprocessing.

Mohammadreza Alimadadi received his B.Sc. degree in ElectricalEngineering from Shahid Beheshti University, Tehran, Iran, in 2009and his M.Sc. degree from the Department of Electrical Engineer-ing, Amirkabir University of Technology, Tehran, in 2012. His currentresearch interests include wireless MIMO systems, channel modeling,wireless Ad-Hoc and sensor networks.

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