PrototypeFilterDesignBasedonChannelEstimationfor FBMC ...FBMC/OQAM system transmits the real-valued symbols a m,n which are from the constellation mapping of the real components cR

Research ArticlePrototype Filter Design Based on Channel Estimation forFBMCOQAM Systems

Yongjin Liu Xihong Chen and Yu Zhao

Air and Missile Defense College Air Force Engineering University Xirsquoan 710051 China

Correspondence should be addressed to Yongjin Liu liuyongjindw163com

Received 25 September 2019 Revised 31 October 2019 Accepted 8 November 2019 Published 27 November 2019

Academic Editor Francesco Cannizzaro

Copyright copy 2019 Yongjin Liu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A prototype filter design for FBMCOQAM systems is proposed in this study e influence of both the channel estimation andthe stop-band energy is taken into account in this method An efficient preamble structure is proposed to improve the per-formance of channel estimation and save the frequency spectral efficiency e reciprocal of the signal-to-interference plus noiseratio (RSINR) is derived to measure the influence of the prototype filter on channel estimation After that the process of prototypefilter design is formulated as an optimization problem with constraint on the RSINR To accelerate the convergence and obtainglobal optimal solution an improved genetic algorithm is proposed Especially the History Network and pruning operator areadopted in this improved genetic algorithm Simulation results demonstrate the validity and efficiency of the prototype filterdesigned in this study

1 Introduction

As an alternative modulation scheme for the fifth-generation(5G) communication offset quadrature amplitude modu-lation-based orthogonal frequency division multiplexing(FBMCOQAM) outperforms orthogonal frequency di-vision multiplexing (OFDM) with respect to spectral effi-ciency side lobes and sensitivity to asynchronoustransmission [1 2] As one of the important components ofFBMCOQAM systems the prototype filter determines theperformance of the whole system

In recent years many effective methods about prototypefilter design have been brought up All these methods can bedivided into two classes [3] direct approaches and indirectapproaches Indirect approaches mainly include the win-dowing-based method [4 5] and the frequency-samplingmethod [6ndash8] Direct approaches optimize all coefficients ofthe prototype filter Due to many more degrees of designfreedom direct approaches have the potential to achievebetter performance than indirect approaches [9]

However the complexity grows rapidly along with thenumber of coefficients in the direct approach In order toreduce the computational complexity the α-based branch

and bound (αBB) algorithm is used to design the prototypefilter in [10] In [11] the box-based branch and bound (Box-BB) method based on αBB algorithm is proposed to decreasethe number of iterations Spectral factorization is employedto design the final prototype filter in [12] but this method isunstable because the factor is chosen randomly e exis-tence of unknowns in the aforementioned methods makes itdifficult to obtain the global optimum

e genetic algorithm (GA) is a powerful evolutionaryalgorithm which is able to solve this difficulty [13 14] GA isadopted to handle the thinning optimization problem in[15] e Baldwin effect is added into GA via a HistoryNetwork to avoid the inappropriate answer in [16] but thereare so many random operators in this method In [17] therandom assignments of value during initialization andmutation operator are deleted by the pruning operator toreach the answer more quickly In this study an improvedGA based on the History Network and pruning operator isproposed to deal with these random operators

Furthermore accurate channel estimation (CE) at thereceiver is indispensable for reliable signal detection erehave already been some studies on the CE for the FBMCOQAM system

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3730759 11 pageshttpsdoiorg10115520193730759

In [18] two modified CE methods were proposed byplacing guard symbols at different sides of reference pre-amble symbol the existence of guard symbols reducesspectrum efficiency In order to increase spectrum efficiencycoded auxiliary pilot symbols that also carry information aredesigned in [19] However at least two to four pilot symbolsare needed to achieve acceptable performance in practice In[20] an efficient preamble structure is proposed to achieve abetter CE with less preamble consumption Because there isno pilot in the compressed sensing method CE methodsbased on compressed sensing are proposed further to in-crease the spectral efficiency In [21] when prior sparseknowledge is unknown an adaptive regularized com-pressive sampling matching pursuit chooses the support setto achieve better CE accuracy Inspired by the same theory asin [21] two improved CE methods are derived respectivelybased on the interference approximation method (IAM) andpair of real pilots in [22] e key to these methods is thathow to prove that the FBMCOQAM system is sparseHowever whether the system like the scattering channel issparse is debatable

Few studies have taken the CE into account in prototypefilter design In [23] the prototype filter is designed byutilizing the mean square error (MSE) of the CE Howeverzero symbols inserted in the preambles will reduce thesystemrsquos spectral efficiency Inspired by the preamblestructure described in [20] preamble without guard symbolsis considered in these studies to acquire higher transmissionefficiency e proposed structure will improve the accuracyof CE by increasing pseudo pilot power

e influence of noise may be also ignored in prototypefilter design For example in [24] the weight signal-to-in-terference ratio is treated as the utility function to design theprototype filter under highly doubly dispersive channelwithout noise However the noise power is generally equalto or even larger than the interference power in practicalsystems erefore the noise influence should be givensufficient attention in prototype filter design Usually thesignal-to-interference plus noise ratio (SINR) is adopted tomeasure the noise influence A bigger SINR means bettersystem performance erefore it will be difficult to set thethreshold when the SINR is added to the constraints Underthis condition in this study the reciprocal of the signal-to-interference plus noise ratio (RSINR) is adopted to measurethe influence of noise

Simulation results indicate that the proposed preamblestructure achieves outperformance compared to the IAMthe improved GA shows less convergence time and averagegenerations than the proposed GAs in [16 17] the designedprototype filter in this study is better than in [23 24] when itcomes to the performance of the stop-band energy and themagnitude response

e rest of the study is organized as follows In Section 2the FBMCOQAM model is described the preamble isproposed and the CE is obtained In Section 3 the process ofprototype filter design is formulated by minimizing the stop-band energy with constraints on the RSINR and the GAwith the History Network and pruning operator is utilized tosolve the optimization problem e results of the

simulations are discussed in Section 4 and finally a con-clusion is given in Section 5

2 System Model

As one of the contributions of this paper the preamblestructure with larger pseudo pilot power and spectrum ef-ficiency is introduced in this section

At first the FBMCOQAM systemmodel is built Insteadof transmitting complex symbols cmn cRmn + jcImn theFBMCOQAM system transmits the real-valued symbolsamn which are from the constellation mapping of thereal components cRmn and imaginary components cImnAccording to [25] the continuous-time baseband equivalentof the transmitting signal in FBMCOQAM can be written as

s(t) 1113944+infin

nminus infin1113944

Mminus 1

m0amn e

jϕmn ej2πm]0t

g t minus nτ0( 11138571113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981

gmn(t)

(1)

where M denotes the number of subcarriers ]0 1T0

12τ0 denotes the subcarrier spacing T0 denotes the con-stellation interval and τ0 denotes the offset between the realpart and imaginary part of the OQAM symbol g(t) denotesthe prototype filter function ϕmn is an additional phaseterm

ϕmn ϕ0 +π2

(m + n)(mod π) (2)

where ϕ0 can be chosen arbitrarily For simplification in thisstudy ϕ0 0

Considering the multipath channel with an additivewhite Gaussian noise which is denoted as η(t) when thetransmitted FBMCOQAM signal is passed through thebaseband version of the received signal is described as

y(t) 1113946Δ

0h(τ)s(t minus τ)dτ + η(t)

1113944+infin

nminus infin1113944

Mminus 1

m0amn 1113946

Δ

0h(τ)e

minus j2πm]0τejπ(m+n)2

ej2πm]0t

times g t minus τ minus nτ0( 1113857dτ + η(t)

(3)

where h(t) denotes the channel impulse response (CIR) andΔ presents the maximum delay spread of the channel Es-pecially we assume that g(t minus τ minus nτ0) asymp g(t minus nτ0) in thetime interval τ isin [0Δ] for the flat fading channel In thatcase rewrite (3) as

y(t) 1113944+infin

nminus infin1113944

Mminus 1

m0amne

j(π(m+n)2)e

j2πm]0tg t minus nτ0( 1113857

times 1113946Δ

0h(τ)e

minus j2πm]0τdτ + η(t)

1113944+infin

nminus infin1113944

Mminus 1

m0amngmn(t)Hm + η(t)

(4)

where Hm 1113938Δ0 h(t)eminus j2πm]0tdt represents the channel im-

pulse frequency responsee demodulation signal at the (m0 n0) position can be

written as

2 Mathematical Problems in Engineering

ym0 n0langy(t) gm0 n0

rang

1113944+infin

nminus infin1113944

Mminus 1

m0amnHmlanggmn(t) gm0 n0

rang + η1m0 n0

(5)

where langx yrang represents the inner product of x and y andη1m0 n0

langη(t) gm0 n0rang

e orthogonal condition in FBMCOQAM is

R langgmn gm0 n0rang1113966 1113967 δmm0

δnn0 (6)

where R middot denotes real part of the argument and δ(t)

denotes the Diracrsquos delta functionus the FBMCOQAM systemmodel has been built After

that in order to design the new preamble structure the re-lationship between preamble and CE is necessary to be analyzed

21 Relationship between Preamble and CE We definem m0 + p n n0 + q where p q isin Z and then the innerproduct of the pulse shaping filter in equation (5) can berewritten as

langgmn(t) gm0 n0rang 1113946

+infin

minus infingm0+pn0+q(t)g

lowastm0+n0

(t)dt

1113946+infin

minus infine

j(π2) m0+n0p+q( )ej2π m0+p( )vot

middot g t minus n0τ0 minus qτ0( 1113857 times eminus j(π2) m0+n0( )

middot eminus j2πm0v0t

glowast

t minus n0τ0( 1113857dt

j(p+q)

1113946+infin

minus infing t minus n0τ0 minus qτ0( 1113857

middot glowast

t minus n0τ0( 1113857ej2πpv0tdt

(7)

where lowast denotes conjugate operation Since ]oτ0 12t minus n0τ0 minus qτ0 l minus (q2)τ0 and t minus n0τ0 l + (q2)τ0equation (7) can be rewritten as

langgmn(t) gm0 n0rang j

(p+q)1113946

+infin

minus infing l +(minus q2)τ0( 1113857

middot glowast

l minusminus q

2τ01113874 1113875 times e

minus j2πp]0 l+n0τ0+qτ02( )dl

jp+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857

(8)

where Ag(τ ]) denotes the ambiguity function of the filterwhich can be expressed as

Ag(τ ]) 1113946 g t +τ2

1113874 1113875glowast

t minusτ2

1113874 1113875ej2π]tdt (9)

When g(t) is an even function its instantaneous au-tocorrelation function cg(τ t) g(t + (τ2))glowast(t minus (τ2))

is also an even conjugate Ag(τ ]) is the inverse Fouriertransformation of cg(τ t) and therefore Ag(τ ]) is a real-valued function [26]

We can get equation (10) from equation (5)

ym0 n0 1113944

(pq)isinZj

p+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857

times am0+pn0+qHm0+p + η1m0 n0

am0 n0Hm0

+ 1113944(pq)ne(00)

jp+q+pq+2pn0( )

times Ag minus qτ0 p]0( 1113857am0+pn0+qHm0+p + η1m0 n0

(10)

According to the conventional OFDM CE method thechannel frequency response 1113954Hm0

can be estimated by1113954Hm0 n0

(ym0 n0am0 n0

) when the pilot symbol am0 n0is

transmitted at time-frequency position (m0 n0)We define Cpq j(p+q+pq+2pn0)Ag(minus qτ0 p]0) and Δm0

and Δn0 are respectively the neighborhoods of m0 and n0ΩΔm0 Δn0 (p q) |p|leΔm0 |q|leΔn0 | Hm0+pn0+q asymp Hm0 n0

1113966 1113967

and ΩlowastΔm0 Δn0 ΩΔm0 Δn0 minus (0 0) where ΩΔm0 Δn0 denotes theneighborhood of position (m0 n0) With the increase in |p|

and |q| Cpq becomes very close to zero For example for anisotropic orthogonal transform algorithm (IOTA) functionwhen (p q) notin Ω11 we have [27]

1113936(pq)notinΩ11Cpq

11138681113868111386811138681113868

111386811138681113868111386811138682


11138681113868111386811138681113868

111386811138681113868111386811138682 asymp 002 (11)

From equation (11) we can conclude that one column ofzero symbols is enough to prevent the preamble from beinginterfered by adjacent unknown data symbols However thezero symbols affect the CE performance e structure of theIAM is shown in Figure 1 It can be seen from Figure 1 thatthere are two columns of zero symbols for IAM According to[27] when only one column of zero symbols is inserted in thepreamble its CE performance is worse than the IAM structure

Since am0minus 1n0and am0+1n0

are of inverse signs equation(10) can be rewritten as

ym0 n0 am0 n0

Hm0+ Cminus 10am0minus 1n0

Hm0minus 1 + C10am0+1n0Hm0+1

+ 1113944(pq)ne(00)(pq)ne(plusmn10)qneplusmn1

Cpqam0+pn0+qHm0+p

+ η1m0 n0

(12)

We assume Hm0 Hm0+1 Hm0minus 1 e CE based on the

IAM can be obtained by

1113954Hm0

ym0 n0

bm0 n0

Hm0+ 1113944

(pq)ne(00)(pq)ne(plusmn10)qneplusmn1

Cpqam0+pn0+qHm0+p

bm0 n0

+η1m0 n0

bm0 n0

(13)

Mathematical Problems in Engineering 3

where bm0 n0 am0 n0

+ C10am0+1n0+ Cminus 10am0minus 1n0

bm0 n0can

be treated as the pseudo pilot Equation (13) implies that alarger pseudo pilot power leads to a better CE performancee power of pseudo pilot bm0 n0

for the IAM is shown in thefollowing equation

E1 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0

+ Cminus 10am0minus 1n0+ C10am0+1n0

11138681113868111386811138681113868

111386811138681113868111386811138682

σ2a 1 + 4Ag + 4A2g1113872 1113873

(14)

where σ2a denotes the variance of the transmitted FBMCOQAM symbols amn

Obviously CE based on the IAM has two drawbacks (1)the preamble duration is 3τ0 which leads to the lowspectrum efficiency and (2) the second item in equation (13)cannot be ignored when both the signal-to-noise ratio andconstellation mapping order are high

e fact is that the power of pseudo pilot bm0 n0is mainly

affected by the pilot symbols that surround am0 n0for the

IAM erefore it can be increased by changing thestructure Additionally the zero symbols in the IAM will beredundant if the intersymbol interference is alleviated by thepreamble

In order to overcome two drawbacks of the IAM a newpreamble structure is proposed in this paper It is shown inFigure 2 For sufficient accuracy and concise description onlythree-tap interference of neighborhood symbols is considered

e pseudo pilot power of the proposed structure is

E2 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0

+ 1113944(pq)isinΩlowast33

Cpqam0+pn0+q

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868

2

gtE1

(15)

From (15) we can conclude that the proposed preamblestructure is able to improve CE performance for its largerpseudo pilot power Obviously it also increases the fre-quency spectrum efficiency

22 RSINR In order to take the influence of prototype filteron CE into account while designing the prototype filter theRSINR is chosen to measure these influencese expressionof RSINR based on the CIR is derived in this section

First we rewrite equation (10) as

ym0 n0 am0 n0

Hm0+ a

(i1)m0n0

Hm0+ a

(i2)m0 n0

Hm0+ η1m0 n0

(16)

where

a(i1)m0 n0

1113944(pq)isinΩlowast33 q0

Cpqam0+pn0+q (17)

a(i2)m0 n0

1113944(pq)isinΩlowast33 qne 0

Cpqam0+pn0+q (18)

and then the channel frequency impulse response 1113954Hm0is

estimated by

1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + a

(i2)m0 n0

(19)

Because am0+pn0+q in equation (18) belongs to a randomsignal its value cannot be determined during transmissionAs a result the term a(i2)

m0 n0is unknown Equation (19) will be

invalid if a(i2)m0 n0

is unknown To solve this problem thepredecision method is adopted e process of estimationbased on the predecision method is shown as follows

m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

ndashj

Figure 2 Proposed structure of preamble

m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

Zero

ndashj

Figure 1 Structure of the IAM


First the initial channel frequency response Hm0can be

obtained from (16)

Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0

Hm0+

a(i2)m0 n0

Hm0+ η1m0 n0

am0 n0+ a

(i1)m0 n0

(20)

Next am0+pn0+q in the term a(i2)m0 n0

can be rebuilt based onzero-forcing equalization Estimate am0+pn0+q based on thepredecision method which is shown as follows

1113954am0+pn0+q Dym0+pn0+q

Hm0

⎡⎣ ⎤⎦ (p q) isin Ωlowast33 qne 0 (21)

Here D[middot] denotes the predecision operator

en a(i2)m0 n0

can be estimated by

1113954a(i2)m0 n0


Cpq1113954am0+pn0+q (22)

Finally since a(i2)m0 n0

has been obtained the CE is obtainedby substituting a(i2)

m0 n0into equation (19) Equation (23)

shows the final channel frequency response expression

1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + 1113954a(i2)

m0 n0

(23)

By inverse Fourier transform the CIR of the FBMCOQAM system is obtained as follows

1113954h(t) 12π

1113946Δ

01113954Hme

minus j2πm]0tdt (24)

We then discretize the CIR 1113954h(t) into form 1113954hl as follows

1113954h(t) 1113944Pminus 1

i0

1113954hiδ τ minus τi( 1113857

1113954hl

1113954hi l lceilτi

Tsrceil

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

where x denotes the smallest integer larger than or equal to x

and Ts (T02M) denotes the sampling periode desired power of the system is closely related to the

transmitted symbol e transmitted symbol at the position(m0 n0) can be estimated as follows

1113954am0 n0 R

ym0 n0

1113954Hm0

⎛⎝ ⎞⎠ (26)

e QAM symbols are reconstructed by combining1113954cm0 n0

1113954am0 2n0+ j1113954am0 2n0+1 (27)

We then define

Hpqm0

1113944

Lhminus 1

l0

1113954hleminus jπ 2m0+p( )lM

Ag minus l minus qM

2 minus p1113876 1113877 (28)

where Lh is the number of channel taps at the sampling rateand Ag[k l] Ag(kTs (lMTs))

e desired power of the system is

Pd m0( 1113857 σ2c RHpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(29)

where σ2c denotes the variance of the complex symbol cAccording to the definition of the variance σ2c is the realnumber

Similarly the interference power and the noise power areexpressed as [28]

PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)

Rej(π2)(p+q+pq)Hpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(30)

Pnoise m0( 1113857 var η1m0 n01113960 1113961

σ2η1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682 (31)

where σ2η denotes the variance of the noise ηus the expression of RSINR is obtained by equations

(29)ndash(31)

RSINR PICI+ISI + Pnoise

Pd m0( 1113857

2σ2a1113936(pq)ne(00) R ej(π2)(p+q+pq)Hpq

m0 1113954Hm0

1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

+ σ2η 1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682

σ2c R Hpqm0 1113954Hm0

1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

(32)

3 Prototype Filter Design

31 ProblemFormulation e prototype filter in the FBMCOQAM system is assumed to be real-valued symmetric andof unit energy According to the previous discussion whenthere is intrinsic interference and noise in the system theprototype filter will influence the CE erefore the RSINRwill be chosen as an important constraint in problem for-mulation At the same time the prototype filter designshould minimize the stop-band energy us the prototypefilter design problem can be formulated as

P1 ming(k)

1113946π

2πMG e

jω1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682dω

st

g[k] g Lg minus 1 minus k1113960 1113961 k 0 1 Lg minus 1

1113944

Lgminus 1

k0g2[k] 1

RSINRleTH

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(33)

where |G(ejω)| |1113936Lgminus 1k0 g(k)eminus jωk| is the magnitude re-

sponse of the prototype filter TH represents a constantthreshold and Lg denotes the length of the prototype filterAccording to [29] g[k]

Ts

1113968g((k minus (Lg minus 12))Ts)

e prototype filter used to be designed without con-sidering the interaction between prototype filter and CE


erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper

ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)

First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]

with a new variable x

x x1 x2 xL1113858 1113859T

g[0] g[1] gLgminus 121113876 11138771113876 1113877

T

(34)

e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors

en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859

T denotes the neworthogonal variable

Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as

P2 min1113944L

j1λjz

2j

stc1(z) minus 1113944

Lg minus 1

k0g2[k] minus 1⎛⎝ ⎞⎠le 0

c2(z) RSINRleTH

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)

All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA

In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]

However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning

operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4

It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation

e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning

e time order of the pruned initialization is shown as

initialization O(NplowastNglowastNc) (36)

where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as

Npruning η ηleNglowastNc

NglowastNc ηgtNglowastNc1113896 (37)

where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by

pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction

ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process

Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer

e time order of the pruned search loop is shown as

search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)

where Ns presents the solution spaceSince the redundant random operators have been

pruned the computational complexity of the whole algo-rithm is certainly reduced

Start Initialization

Fitness

Probability calculation

Increase the ageDRS

Update History NetworkComparison

AboveBelow

Breed individuals

Best individual

Figure 3 Flowchart of the proposed GA in [16]


4 Simulation Result

41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1

Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB

e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE

In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity

42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2

e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them

Table 1 Fundamental parameters of CE

Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols

A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE

Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE


Fitness Crossover

Mutation

Pruning Pruning

DRS

Update History NetworkTerminate

Yes No

Increasethe age

Best individual

Pruning

Pruning


Figure 4 Flowchart of the improved GA in this paper


43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations

e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller

than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11

Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one

In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper

Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study

Without iteration2 iterations3 iterations

4 iterations5 iterationsPerfect bound

ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)

Figure 8 NMSE performance with different number of iterations

A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)

Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM

A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)

Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM

Table 2 Parameters of the improved GA

Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9


is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band

energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter

Table 3 Convergence time of different algorithms

Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s

Method AMethod BMethod C

5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER

Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120

Figure 9 Average generations of different GAs


Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter

5 Conclusion

In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters

ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468

References

[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016

[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016

[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009

[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008

[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures

Table 4 Stop-band energy of different filters

Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB

0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0

Normalized frequency (timesπ radsample)

Mag

nitu

de (d

B)

IOTAFilter designed in [23]

Filter designed in [24]Filter designed in this study

Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study


and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014

[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001

[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003

[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008

[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014

[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995

[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017

[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017

[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019

[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019

[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016

[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998

[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010

[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009

[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016

[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo

SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013

[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016

[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013

[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018

[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017

[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011

[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994

[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007

[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010

[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002

[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995

[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997


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Submit your manuscripts atwwwhindawicom

In [18] two modified CE methods were proposed byplacing guard symbols at different sides of reference pre-amble symbol the existence of guard symbols reducesspectrum efficiency In order to increase spectrum efficiencycoded auxiliary pilot symbols that also carry information aredesigned in [19] However at least two to four pilot symbolsare needed to achieve acceptable performance in practice In[20] an efficient preamble structure is proposed to achieve abetter CE with less preamble consumption Because there isno pilot in the compressed sensing method CE methodsbased on compressed sensing are proposed further to in-crease the spectral efficiency In [21] when prior sparseknowledge is unknown an adaptive regularized com-pressive sampling matching pursuit chooses the support setto achieve better CE accuracy Inspired by the same theory asin [21] two improved CE methods are derived respectivelybased on the interference approximation method (IAM) andpair of real pilots in [22] e key to these methods is thathow to prove that the FBMCOQAM system is sparseHowever whether the system like the scattering channel issparse is debatable

Few studies have taken the CE into account in prototypefilter design In [23] the prototype filter is designed byutilizing the mean square error (MSE) of the CE Howeverzero symbols inserted in the preambles will reduce thesystemrsquos spectral efficiency Inspired by the preamblestructure described in [20] preamble without guard symbolsis considered in these studies to acquire higher transmissionefficiency e proposed structure will improve the accuracyof CE by increasing pseudo pilot power

e influence of noise may be also ignored in prototypefilter design For example in [24] the weight signal-to-in-terference ratio is treated as the utility function to design theprototype filter under highly doubly dispersive channelwithout noise However the noise power is generally equalto or even larger than the interference power in practicalsystems erefore the noise influence should be givensufficient attention in prototype filter design Usually thesignal-to-interference plus noise ratio (SINR) is adopted tomeasure the noise influence A bigger SINR means bettersystem performance erefore it will be difficult to set thethreshold when the SINR is added to the constraints Underthis condition in this study the reciprocal of the signal-to-interference plus noise ratio (RSINR) is adopted to measurethe influence of noise

Simulation results indicate that the proposed preamblestructure achieves outperformance compared to the IAMthe improved GA shows less convergence time and averagegenerations than the proposed GAs in [16 17] the designedprototype filter in this study is better than in [23 24] when itcomes to the performance of the stop-band energy and themagnitude response

e rest of the study is organized as follows In Section 2the FBMCOQAM model is described the preamble isproposed and the CE is obtained In Section 3 the process ofprototype filter design is formulated by minimizing the stop-band energy with constraints on the RSINR and the GAwith the History Network and pruning operator is utilized tosolve the optimization problem e results of the

simulations are discussed in Section 4 and finally a con-clusion is given in Section 5

2 System Model

As one of the contributions of this paper the preamblestructure with larger pseudo pilot power and spectrum ef-ficiency is introduced in this section

At first the FBMCOQAM systemmodel is built Insteadof transmitting complex symbols cmn cRmn + jcImn theFBMCOQAM system transmits the real-valued symbolsamn which are from the constellation mapping of thereal components cRmn and imaginary components cImnAccording to [25] the continuous-time baseband equivalentof the transmitting signal in FBMCOQAM can be written as

s(t) 1113944+infin

nminus infin1113944

Mminus 1

m0amn e

jϕmn ej2πm]0t

g t minus nτ0( 11138571113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981

gmn(t)

(1)

where M denotes the number of subcarriers ]0 1T0

12τ0 denotes the subcarrier spacing T0 denotes the con-stellation interval and τ0 denotes the offset between the realpart and imaginary part of the OQAM symbol g(t) denotesthe prototype filter function ϕmn is an additional phaseterm

ϕmn ϕ0 +π2

(m + n)(mod π) (2)

where ϕ0 can be chosen arbitrarily For simplification in thisstudy ϕ0 0

Considering the multipath channel with an additivewhite Gaussian noise which is denoted as η(t) when thetransmitted FBMCOQAM signal is passed through thebaseband version of the received signal is described as

y(t) 1113946Δ

0h(τ)s(t minus τ)dτ + η(t)

1113944+infin

nminus infin1113944

Mminus 1

m0amn 1113946

Δ

0h(τ)e

minus j2πm]0τejπ(m+n)2

ej2πm]0t

times g t minus τ minus nτ0( 1113857dτ + η(t)

(3)

where h(t) denotes the channel impulse response (CIR) andΔ presents the maximum delay spread of the channel Es-pecially we assume that g(t minus τ minus nτ0) asymp g(t minus nτ0) in thetime interval τ isin [0Δ] for the flat fading channel In thatcase rewrite (3) as

y(t) 1113944+infin

nminus infin1113944

Mminus 1

m0amne

j(π(m+n)2)e

j2πm]0tg t minus nτ0( 1113857

times 1113946Δ

0h(τ)e

minus j2πm]0τdτ + η(t)

1113944+infin

nminus infin1113944

Mminus 1

m0amngmn(t)Hm + η(t)

(4)

where Hm 1113938Δ0 h(t)eminus j2πm]0tdt represents the channel im-

pulse frequency responsee demodulation signal at the (m0 n0) position can be

written as



rang

1113944+infin

nminus infin1113944

Mminus 1


rang + η1m0 n0

(5)





δnn0 (6)






+infin


lowastm0+n0

(t)dt

1113946+infin

minus infine




glowast


j(p+q)

1113946+infin


middot glowast


(7)



(p+q)1113946

+infin


middot glowast

l minusminus q

2τ01113874 1113875 times e



(8)


Ag(τ ]) 1113946 g t +τ2

1113874 1113875glowast

t minusτ2

1113874 1113875ej2π]tdt (9)




ym0 n0 1113944

(pq)isinZj



am0 n0Hm0

+ 1113944(pq)ne(00)

jp+q+pq+2pn0( )


(10)



(ym0 n0am0 n0




1113966 1113967




11138681113868111386811138681113868

111386811138681113868111386811138682


11138681113868111386811138681113868

111386811138681113868111386811138682 asymp 002 (11)




ym0 n0 am0 n0




Cpqam0+pn0+qHm0+p

+ η1m0 n0

(12)



1113954Hm0

ym0 n0

bm0 n0

Hm0+ 1113944


Cpqam0+pn0+qHm0+p

bm0 n0

+η1m0 n0

bm0 n0

(13)


where bm0 n0 am0 n0


bm0 n0can



E1 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0


11138681113868111386811138681113868

111386811138681113868111386811138682

σ2a 1 + 4Ag + 4A2g1113872 1113873

(14)








E2 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0


Cpqam0+pn0+q

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868

2

gtE1

(15)




ym0 n0 am0 n0

Hm0+ a

(i1)m0n0

Hm0+ a

(i2)m0 n0

Hm0+ η1m0 n0

(16)

where

a(i1)m0 n0


Cpqam0+pn0+q (17)

a(i2)m0 n0


Cpqam0+pn0+q (18)


estimated by

1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + a

(i2)m0 n0

(19)





m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

ndashj


m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

Zero

ndashj




obtained from (16)

Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0

Hm0+

a(i2)m0 n0

Hm0+ η1m0 n0

am0 n0+ a

(i1)m0 n0

(20)




Hm0



en a(i2)m0 n0

can be estimated by

1113954a(i2)m0 n0


Cpq1113954am0+pn0+q (22)





1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + 1113954a(i2)

m0 n0

(23)


1113954h(t) 12π

1113946Δ

01113954Hme



1113954h(t) 1113944Pminus 1

i0

1113954hiδ τ minus τi( 1113857

1113954hl


Tsrceil

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)




1113954am0 n0 R

ym0 n0

1113954Hm0

⎛⎝ ⎞⎠ (26)


1113954am0 2n0+ j1113954am0 2n0+1 (27)

We then define

Hpqm0

1113944

Lhminus 1

l0


Ag minus l minus qM

2 minus p1113876 1113877 (28)



Pd m0( 1113857 σ2c RHpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(29)



PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)

Rej(π2)(p+q+pq)Hpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(30)

Pnoise m0( 1113857 var η1m0 n01113960 1113961

σ2η1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682 (31)


(29)ndash(31)


Pd m0( 1113857


m0 1113954Hm0

1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

+ σ2η 1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682


1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

(32)



P1 ming(k)

1113946π

2πMG e

jω1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682dω

st


1113944

Lgminus 1

k0g2[k] 1

RSINRleTH

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(33)



Ts








x x1 x2 xL1113858 1113859T

g[0] g[1] gLgminus 121113876 11138771113876 1113877

T

(34)





P2 min1113944L

j1λjz

2j


Lg minus 1


c2(z) RSINRleTH

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)






















Fitness


Increase the ageDRS


AboveBelow

Breed individuals

Best individual



4 Simulation Result









A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE



Fitness Crossover

Mutation

Pruning Pruning

DRS


Yes No

Increasethe age

Best individual

Pruning

Pruning












ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018









rang

1113944+infin

nminus infin1113944

Mminus 1


rang + η1m0 n0

(5)





δnn0 (6)






+infin


lowastm0+n0

(t)dt

1113946+infin

minus infine




glowast


j(p+q)

1113946+infin


middot glowast


(7)



(p+q)1113946

+infin


middot glowast

l minusminus q

2τ01113874 1113875 times e



(8)


Ag(τ ]) 1113946 g t +τ2

1113874 1113875glowast

t minusτ2

1113874 1113875ej2π]tdt (9)




ym0 n0 1113944

(pq)isinZj



am0 n0Hm0

+ 1113944(pq)ne(00)

jp+q+pq+2pn0( )


(10)



(ym0 n0am0 n0




1113966 1113967




11138681113868111386811138681113868

111386811138681113868111386811138682


11138681113868111386811138681113868

111386811138681113868111386811138682 asymp 002 (11)




ym0 n0 am0 n0




Cpqam0+pn0+qHm0+p

+ η1m0 n0

(12)



1113954Hm0

ym0 n0

bm0 n0

Hm0+ 1113944


Cpqam0+pn0+qHm0+p

bm0 n0

+η1m0 n0

bm0 n0

(13)


where bm0 n0 am0 n0


bm0 n0can



E1 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0


11138681113868111386811138681113868

111386811138681113868111386811138682

σ2a 1 + 4Ag + 4A2g1113872 1113873

(14)








E2 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0


Cpqam0+pn0+q

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868

2

gtE1

(15)




ym0 n0 am0 n0

Hm0+ a

(i1)m0n0

Hm0+ a

(i2)m0 n0

Hm0+ η1m0 n0

(16)

where

a(i1)m0 n0


Cpqam0+pn0+q (17)

a(i2)m0 n0


Cpqam0+pn0+q (18)


estimated by

1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + a

(i2)m0 n0

(19)





m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

ndashj


m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

Zero

ndashj




obtained from (16)

Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0

Hm0+

a(i2)m0 n0

Hm0+ η1m0 n0

am0 n0+ a

(i1)m0 n0

(20)




Hm0



en a(i2)m0 n0

can be estimated by

1113954a(i2)m0 n0


Cpq1113954am0+pn0+q (22)





1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + 1113954a(i2)

m0 n0

(23)


1113954h(t) 12π

1113946Δ

01113954Hme



1113954h(t) 1113944Pminus 1

i0

1113954hiδ τ minus τi( 1113857

1113954hl


Tsrceil

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)




1113954am0 n0 R

ym0 n0

1113954Hm0

⎛⎝ ⎞⎠ (26)


1113954am0 2n0+ j1113954am0 2n0+1 (27)

We then define

Hpqm0

1113944

Lhminus 1

l0


Ag minus l minus qM

2 minus p1113876 1113877 (28)



Pd m0( 1113857 σ2c RHpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(29)



PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)

Rej(π2)(p+q+pq)Hpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(30)

Pnoise m0( 1113857 var η1m0 n01113960 1113961

σ2η1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682 (31)


(29)ndash(31)


Pd m0( 1113857


m0 1113954Hm0

1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

+ σ2η 1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682


1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

(32)



P1 ming(k)

1113946π

2πMG e

jω1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682dω

st


1113944

Lgminus 1

k0g2[k] 1

RSINRleTH

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(33)



Ts








x x1 x2 xL1113858 1113859T

g[0] g[1] gLgminus 121113876 11138771113876 1113877

T

(34)





P2 min1113944L

j1λjz

2j


Lg minus 1


c2(z) RSINRleTH

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)






















Fitness


Increase the ageDRS


AboveBelow

Breed individuals

Best individual



4 Simulation Result









A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE



Fitness Crossover

Mutation

Pruning Pruning

DRS


Yes No

Increasethe age

Best individual

Pruning

Pruning












ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018








where bm0 n0 am0 n0


bm0 n0can



E1 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0


11138681113868111386811138681113868

111386811138681113868111386811138682

σ2a 1 + 4Ag + 4A2g1113872 1113873

(14)








E2 bm0 n0

11138681113868111386811138681113868

111386811138681113868111386811138682

1113882 1113883 a2m0 n0


Cpqam0+pn0+q

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868

2

gtE1

(15)




ym0 n0 am0 n0

Hm0+ a

(i1)m0n0

Hm0+ a

(i2)m0 n0

Hm0+ η1m0 n0

(16)

where

a(i1)m0 n0


Cpqam0+pn0+q (17)

a(i2)m0 n0


Cpqam0+pn0+q (18)


estimated by

1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + a

(i2)m0 n0

(19)





m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

ndashj


m0 minus 3

m0 minus 2

m0 minus 1

m0

m0 + 1

m0 + 2

m0 + 3

+1

ndash1

j

Data

Zero

ndashj




obtained from (16)

Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0

Hm0+

a(i2)m0 n0

Hm0+ η1m0 n0

am0 n0+ a

(i1)m0 n0

(20)




Hm0



en a(i2)m0 n0

can be estimated by

1113954a(i2)m0 n0


Cpq1113954am0+pn0+q (22)





1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + 1113954a(i2)

m0 n0

(23)


1113954h(t) 12π

1113946Δ

01113954Hme



1113954h(t) 1113944Pminus 1

i0

1113954hiδ τ minus τi( 1113857

1113954hl


Tsrceil

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)




1113954am0 n0 R

ym0 n0

1113954Hm0

⎛⎝ ⎞⎠ (26)


1113954am0 2n0+ j1113954am0 2n0+1 (27)

We then define

Hpqm0

1113944

Lhminus 1

l0


Ag minus l minus qM

2 minus p1113876 1113877 (28)



Pd m0( 1113857 σ2c RHpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(29)



PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)

Rej(π2)(p+q+pq)Hpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(30)

Pnoise m0( 1113857 var η1m0 n01113960 1113961

σ2η1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682 (31)


(29)ndash(31)


Pd m0( 1113857


m0 1113954Hm0

1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

+ σ2η 1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682


1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

(32)



P1 ming(k)

1113946π

2πMG e

jω1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682dω

st


1113944

Lgminus 1

k0g2[k] 1

RSINRleTH

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(33)



Ts








x x1 x2 xL1113858 1113859T

g[0] g[1] gLgminus 121113876 11138771113876 1113877

T

(34)





P2 min1113944L

j1λjz

2j


Lg minus 1


c2(z) RSINRleTH

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)






















Fitness


Increase the ageDRS


AboveBelow

Breed individuals

Best individual



4 Simulation Result









A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE



Fitness Crossover

Mutation

Pruning Pruning

DRS


Yes No

Increasethe age

Best individual

Pruning

Pruning












ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018









obtained from (16)

Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0

Hm0+

a(i2)m0 n0

Hm0+ η1m0 n0

am0 n0+ a

(i1)m0 n0

(20)




Hm0



en a(i2)m0 n0

can be estimated by

1113954a(i2)m0 n0


Cpq1113954am0+pn0+q (22)





1113954Hm0

ym0 n0

am0 n0+ a

(i1)m0 n0 + 1113954a(i2)

m0 n0

(23)


1113954h(t) 12π

1113946Δ

01113954Hme



1113954h(t) 1113944Pminus 1

i0

1113954hiδ τ minus τi( 1113857

1113954hl


Tsrceil

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)




1113954am0 n0 R

ym0 n0

1113954Hm0

⎛⎝ ⎞⎠ (26)


1113954am0 2n0+ j1113954am0 2n0+1 (27)

We then define

Hpqm0

1113944

Lhminus 1

l0


Ag minus l minus qM

2 minus p1113876 1113877 (28)



Pd m0( 1113857 σ2c RHpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(29)



PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)

Rej(π2)(p+q+pq)Hpq

m0

1113954Hm0

⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(30)

Pnoise m0( 1113857 var η1m0 n01113960 1113961

σ2η1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682 (31)


(29)ndash(31)


Pd m0( 1113857


m0 1113954Hm0

1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

+ σ2η 1113954Hm0

11138681113868111386811138681113868

111386811138681113868111386811138682


1113966 111396711138681113868111386811138681113868

111386811138681113868111386811138682

(32)



P1 ming(k)

1113946π

2πMG e

jω1113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682dω

st


1113944

Lgminus 1

k0g2[k] 1

RSINRleTH

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(33)



Ts








x x1 x2 xL1113858 1113859T

g[0] g[1] gLgminus 121113876 11138771113876 1113877

T

(34)





P2 min1113944L

j1λjz

2j


Lg minus 1


c2(z) RSINRleTH

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)






















Fitness


Increase the ageDRS


AboveBelow

Breed individuals

Best individual



4 Simulation Result









A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE



Fitness Crossover

Mutation

Pruning Pruning

DRS


Yes No

Increasethe age

Best individual

Pruning

Pruning












ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018












x x1 x2 xL1113858 1113859T

g[0] g[1] gLgminus 121113876 11138771113876 1113877

T

(34)





P2 min1113944L

j1λjz

2j


Lg minus 1


c2(z) RSINRleTH

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)






















Fitness


Increase the ageDRS


AboveBelow

Breed individuals

Best individual



4 Simulation Result









A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE



Fitness Crossover

Mutation

Pruning Pruning

DRS


Yes No

Increasethe age

Best individual

Pruning

Pruning












ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018








4 Simulation Result









A16QAMB16QAM

A4QAMB4QAM

5 10 15 20 25 30 350SNR (dB)

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

NM

SE



Fitness Crossover

Mutation

Pruning Pruning

DRS


Yes No

Increasethe age

Best individual

Pruning

Pruning












ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018
















ndash20

ndash18

ndash16

ndash14

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

NM

SE

2 4 6 8 10 12 14 16 18 200SNR (dB)


A256B256

A512B512

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 250SNR (dB)


A256B256

B512A512

ndash15

ndash10

ndash5

NM

SE

2 4 6 8 100SNR (dB)










5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018













5 10 15 20 25 300SNR (dB)

10ndash3

10ndash2

10ndash1

100

BER


10ndash3

10ndash2

10ndash1

100

BER

5 10 15 20 25 300SNR (dB)



The s

tand

ard

GA

The a

ccel

erat

edG

A in

[16]

The p

rune

dG

A in

[17]

The i

mpr

oved

GA

in th

is stu

dy

0

20

40

60

80

100

120




5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018









5 Conclusion



Data Availability




Acknowledgments


References








0 0005 001 0015 002 0025 003 0035 004

ndash80

ndash60

ndash40

ndash20

0


Mag

nitu

de (d

B)









































Journal of












Journal of







Volume 2018






Volume 2018












































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

PrototypeFilterDesignBasedonChannelEstimationfor FBMC ...FBMC/OQAM system transmits the real-valued symbols a m,n which are from the constellation mapping of the real components cR