Saving Energy for Wireless Transmission:An Important Revelation from Shannon Formula

ZHU Jinkang,ZHAO Ming

(1.Key Laboratory of Wireless-Optical Communications,Chinese Academy of Sciences,Hefei 230027,China;2.School of Information Science and Technology,University of Science and Technology of China,Hefei 230027,China)

Abstract:The reduction of power consumption is important for wireless communications and networks.To develop the energy-saving technologies for future wireless transmissions and networks,this paper presents two basic study points:1) The multiple events are merged into a single event;2) the high-order mode is changed to the low-order mode.For this rea?son,we seek that multiple events in wireless transmission links are fused into a single event from Shannon formulas.We also analyze the relationship between the information modula?tion and the error correction,and give a fusion structure of error-corrected modulation.The energy-saving performance of the error-corrected modulation method is further analyzed through comparison with the traditional methods of modulation plus error correction.The re?sults of numerical analysis demonstrate the wireless energy saving methods for wireless sys?tems based on Shannon formulas are the achievable efficient schemes.

Keywords:wireless saving energy;extension of Shannon formula;error-corrected modula?tion;energy-saving performance

1 Introduction

1.1 Motivation

The energy saving,also said as the power efficiency of wireless communications,has always been an important aim pursued for wireless communications and networks.From 3G,4G to 5G,the energy consumption per infor?mation bit has dropped significantly.

However,on the other hand,5G networks are pursuing ex?tremely high peak rates and cloud network uniform manage?ment,which requires high power consumption.The failure to basically seek a solution to this problem will seriously affect the operation of 5G and the future B5G/6G development.Therefore,various energy efficient methods have been re?searched and developed,to optimize and reduce the energy consumption in various links of wireless communications and networks.However,some proposals and methods seem too scattered or specific,and show no systematic support from the basic theory.

In the face of increasing demands for higher transmitting rates and energy,it is necessary to find solutions by seeking enlightenment from expansion and augmentation of the most basic Shannon formula.This is an important issue facing B5G/6G in the future,which is worth studying carefully.

1.2 Related Work

The Green Wireless Conference held in Huangshan,China in 2009 is still vivid in our memory.As an outcome of this con?ference,Ref.[1]summarized the preliminary research on an?tenna design,service transmission,network design and ener?gy-saving function design,reflecting the results and thinking on green wireless communication technology at that time.A re?lated R&D research project supported by National Key Basic Research Program of China (“973”Program) was subsequent?ly launched in 2010 and has made outstanding contributions to promoting the development of energy-saving technology for wireless communications in China.Moreover,among the achievements are several representative important papers such as“Cell Zooming for Cost-Efficient Green Cellular Net?works”[2]and“Traffic-Aware Network Planning and Green Op?eration (TANGO)”[3].The international communities have also been studying green wireless communications actively.

Many research papers focusing on energy efficiency and en?ergy saving have been published around the world.These pub?lications concern three main aspects:fundament research,cel?lular networks and sensor networks.

The basic concepts of energy-efficient communications can be found in Ref.[4],which also summarized some fundamen?tal works and advanced techniques for energy efficiency,in?cluding information-theoretic analysis and multiple transmis?sion technologies.

Based on energy conservation and the Shannon capacity theorem,the capacity-power consumption formula was pro?posed in Ref.[5].The network spectral efficiency and energy efficiency functions of the cellular network were researched,and the relationship between the power consumption and the spectrum efficiency in the cellular networks was also re?vealed[6].A consumption factor theory to analyze and compare energy efficient design choices for wireless communication networks was presented in Ref.[7].These approaches provide new methods for analyzing and comparing the power efficien?cy of communication systems.

For 5G development,the optimization solutions to energy and cost efficiency were investigated for wireless communica?tion systems with a large number of antennas and radio fre?quency (RF) chains[8].The overall power transfer efficiency(PTE) and the energy efficiency (EE) of a wirelessly powered massive multiple input multiple output (MIMO) system were investigated in Ref.[9].Moreover,a novel quadrature spacefrequency index modulation (QSF-IM) scheme was proposed as a promising energy-efficient radio-access technology for 5G wireless systems[10].Using dual antenna constellation,the pro?posed scheme can enhance data rates with no extra cost of en?ergy consumption.

Recently,the energy-saving research on sensor networks has made further progress.A novel inter-cluster routing was proposed in Ref.[11],which simultaneously takes the energy efficiency in both intra-cluster and inter-cluster phases into account.Moreover,a novel concept of energy efficiency wel?fare was introduced[11].The nonlinear fractional programming for the optimal solution to energy efficiency maximization was presented,based on which a particle-swarm optimizationbased solution algorithm was proposed in Ref.[12].An analyt?ical framework for studying the energy efficiency trade-off of cooperation in sensor networks was presented in Ref.[13];this trade-off is shown to depend on several parameters such as the received power,processing power and the power amplifier loss.The analytical and numerical results reveal that for small distance separation between the source and destination,direct transmission is more energy efficient than relaying.

The joint research of spectrum efficiency and energy effi?ciency in wireless communications is also one of the most important topics in the next-generation wireless networking area,which is attracting more and more attention from indus?try,research,and academia[14].In Ref.[15],the energy effi?ciency and spectrum efficiency in underlay device-to-device(D2D) communications enabled cellular networks were in?vestigated.

In summary,since 2009,the research on improving energy efficiency and energy saving has achieved many results.How?ever,compared with the prediction made by Green Touch’s research that the net energy consumption in communications networks would be reduced by up to 98% by 2020 relative to 2010[16]or that the energy efficiency would be increased by a factor of 1 000 compared to the 2010 level[17],it is far from be?ing achieved.Therefore,from the enlightenment of expending Shannon formulas,this paper will study the foundation and new methods of energy saving in wireless transmission and network coverage to meet the needs of future B5G/6G develop?ment.

1.3 Contributions

Energy saving has always been an important aim pursued from 3G,4G to 5G.The energy consumption per information bit has dropped significantly.However,it is still far away from the future B5G/6G development requirement.

To develop the energy-saving technologies for future wire?less transmissions and networks,this paper presents two basic study points:1) The multiple events are merged into a single event,or the opposite;2) The high-order mode is changed to the low-order mode,or the opposite.

Making the joint study of the two points above,we seek that the multiple events are merged into a single event in wireless transmission links,from Shannon channel capacity formula,to obtain a new relationship between the information modulation and the error correction,and give a new method of fusing con?stellation structures of error correction and modulation.Fur?ther,the energy-saving performance of the given fusion struc?ture is analyzed,and compared with traditional method of modulation plus error correction.

The research results indicate the given method of wireless saving energy with the revelation from the Shannon formula has high energy efficiency.

The remainder of this paper is organized as follows.Section 2 is the problem formulation.Section 3 gives a fusion method of error correction and modulation with revelation from the Shannon channel capacity formula.Section 4 analyzes the en?ergy-saving performance of the given method.Finally,in Sec?tion 5,we conclude this paper.

2 Problem Formulation

Facing the future communications,high transmission speed,low energy consumption and short time delay are im?portant requirements that must be met.The historical experi?ence tells us that the solution to major problems must begin from the analysis and demonstration of basic theories.

From the perspective of theoretical analysis of saving ener?gy,the topology structures of two basic study points presented by this paper can be written into two expressions.

The first topology structure is to transform the processing with two or more sub-events into a simple event (or vice ver?sa).It can be expressed as

where thegsub-events of eventAare turned into eventB;the eventBis turned intogsub-events of eventA.

The second expression of topology structure is that the highorder event is transformed into the multiple low-order subevents to improve the energy efficiency (or vice versa),which can be expressed as

where the exponential ordere-1 of eventAe-1is lower than the exponential ordereofAe,and not as complicated asAe.qis the coefficient of the parallel lower-order.

Therefore,this paper discusses the mathematic expressions of energy-saving ability of the two topology structures,includ?ing the performance evaluation of energy saving.

2.1 Evaluation Function of Power Consumption

When wireless communication eventAis considered,such as modulation/demodulation (Mod/Dem),the required power consumptionPa,for achieving transmission capabilitySa,can be expressed as

whereQais other resource consumption items required for achieving expected capabilitySa.This formula represents the energy consumption to realize the transmission capacitySa.In general,the unit of power consumption is mW and the unit of transmission capability is bit.

Given the other resource consumption itemsQa,such as the frequency bandwidth and the time delay,the relationship be?tween the fluctuation in achievable performance and the in?crease or the decrease in power consumption can be derived by the partial differentiation of the power consumption in Eq.(3)as

whereQis a given value of other resource consumption.This formula represents the energy consumption for one-bit in?crease of the transmitted information,which is the incremental relationship between energy consumption and information bits.

Therefore,we define the energy-saving evaluation function of eventAas

whereηais the amount of information that can be obtained per added unit of power,and it must be greater than zero.As long asηa＞1,the performance improvement will be greater than the increased energy consumption,and it is possible for im?proving the energy-saving effect.The largerηa,the greater the energy efficiency,or vice versa.

Obviously,Eqs.(3),(4)and(5)are also suitable for eventB.

As in Eq.(1),wireless eventAconsists ofgsub-events and the energy consumption isPa=Pa1+...+Pa g.Then the in?cremental relationship between energy consumption and infor?mation bits is

and the energy-saving evaluation function of eventAis rewrit?ten as

Therefore,in wireless communications,how to seek an achievable technical method to obtain high energy-saving effi?ciency is an important problem.

2.2 Energy Saving of Combining Multiple Events

Now,we consider to develop a new event(eventB),which is synthesized by thegsub-events of eventA.Also we will com?plete the design for selecting eventBor original eventAde?pending on the consumed energyPb.Based on the principle of minimum energy expenditure,Eq.(8) can be used to choose the best design of a new event according to the prin?ciple of consuming less energy.

In fact,the design above is not that simple.For example,are the changes of energy consumption of eventAand eventBfor the change of transmission ability the same? When the transmission capabilitySof eventAand that of eventBare the same,the answer to this problem is

and then we must choose eventB,and vice versa.

Therefore,in-depth research is needed to find a better method and effective design for achieving the given wireless event,which facilitates minimizing the energy expenditure.

2.3 Energy Saving of High-Order Event

There is a wireless event withe-order,denoted asAe,of which the power consumption isPae.For the order reduction processing,we transform eventAeinto eventAe-1,reducing the event frome-order to (e-1)-order.Generally,the energy consumption of eventAe-1will be less than that of eventAe,and its performance will also be less than the performance of eventAe.

Thence,we need to confirm how many eventsAe-1have the same performance with the single eventAe,and carefully study if their power consumption is less than that of the sin?gle eventAe.The comparison of the achievable performance and the energy consumption between the high-order event andqlow-order events,when the transmission capabilitySof eventAand that of eventBare the same,can be ex?pressed as

and then,we must choose eventB,and vice versa.

Eq.(10) represents that the higher the energy efficiency,the lower the energy consumption required for performance improvement and the better the design.

Thence,the energy-saving issue of wireless communica?tions and networks is divided into two research topics:

1) When an event having multiple sub-events compares with another single event,which one has smaller energy con?sumption?

2) Comparing a high-order event and multiple low-order sub-events,which one has smaller energy consumption?

Here,we have presented the mathematical expressions of two types of energy-saving problems.The next sections will show the revelation from the Shannon formula and accord?ingly provide energy-saving solutions to the problems.

3 Revelation from Shannon Channel Ca?pacity Formula

As is well known,the channel capacity formula of Shan?non theory is a very important theoretical foundation of wire?less communications.It is also very important for our re?search on energy saving for wireless transmission links,wire?less area coverage,wireless networking,etc.

Here,we discuss energy-saving issues of the wireless transmission link that includes two parts:the error correc?tion coding/decoding (codec) and the modulation/demodula?tion (Mod/Dem),as shown in Fig.1.This link is a stable transmission flow for a given channel.In this regard,some researchers have made considerable efforts,trying to com?bine the error correction and the modulation into one event.However,they have not yet obtained good usable results.From the perspective of saving energy,it is worth deep studying.

Therefore,we suggest seeking the inspiration and meth?ods by extending the Shannon formula,study the fusion of the error correction codec and the modulation/demodulation,and analyze the relationship between the information rate and power consumption in the fading channel.

3.1 Fusion of Error Correction and Modulation

If the input signal isx(t) and the output signal isy(t)through the Gaussian fading channel,the characteristic of the Gaussian channel ish,and the channel noise isN0,the relationship between input and output is

According to the Shannon formula of channel capacity[18],the wireless transmission capacityC(x,y)can be written as

▲Figure 1.Current wireless transmission link

whereC(x,y)is the entropy of the output signalywhen the in?put isx,i.e.,the channel capacity;H(N0)is the lost entropy due to channel noise;P(x) is the statistics function deter?mined by the transmitted signal sourcex(t).Generally,MaxH(y)is the entropy of the input signalx,i.e.MaxH(y)=H(x).H(x) is an integral from negative infinity to positive infinity,which is unavailable in practical applications.

For this reason,we can define the confidence of the cumu?lative probability distribution as a reference variable,which is denoted asω,and get the accurate entropy under the giv?en confidences.

We assumeωis the achievable confidence of the signalx(t),the reliable channel capacityCωunder the given confi?dence is the difference of the entropy of input signalH(x)and the entropy of noisewhich contains the noise entropy and the out-of-confidence discarding entropy.There?fore,the achievable transmission capacityCωunder the con?fidenceωis expressed as

If＞0,Cω＜Hxandω＜1.It was only whenω=100% andN0=0 that we may achieve the lossless capac?ity,Cx,ω=Hx.

When the input signalx(t) hasnsymbols,(x1,...,xn) and the Gaussian channel is a normal distributed channel,the mean probability of errors appearing at the Gaussian chan?nel is,whereis the number of combina?tions ofiinn.Then the entropy ofN0,which causesierror symbols in the output,is expressed as

In this way,the reliable transmission capacityCωbased on the confidenceωcan be expressed as

wherekis the maximum number of the error symbols that can be corrected at the same time.

If there is only one error or error-free innoutput symbols,then+1 symbol combination states in the output will be on?ly received,and the reliable transmission capacityCωcan be simplified to

The channel capacityCωis the amount of receivable infor?mation (denoted asm) transmitted bynsymbols.For exam?ple,if the signalxhas three symbols (n=3),x1,x2andx3will be treated as a block,including one information symbol(m=1),and input into the Gaussian fading channel.The pos?sible states received are one error-free state,and three states with a single error.The total is four combination out?put states(Fig.2).

Here we express the probability of the right symbol and the wrong symbol aspiandq1,respectively.If the appearing probabilities of the four output states are all the same,i.e.,pi=q1=1/4 fori=1,2,3,the confidence of this block isω=3/4=75% and the reliable channel capacity of correct?ing one error isCω=3-log2(1+3)=1 bit.

Therefore,based on Eqs.(15) and (16),we can give an er?ror-corrected modulation method.For example,in the 1/3 code block shown in Fig.2,x2is the information bit while the others are check symbols.In this way,one symbol error can be corrected and the transmission efficiency is 1/3.Sim?ilarly,we can build error-corrected modulation of 2/5 code,3/7 code,4/9 code,5/11 code,6/13 code,7/15 code,etc.

3.2 Constellation Diagram of Error-Corrected Modulation

Based on the above modulation method,we can combine the error-correction function with modulation structure.

The error-corrected modulation method is a constellation modulation structure with the error correction capability.

To construct the constellation diagram of the error-correct?ed modulation,the processing steps are divided into three parts:1) planning the constellation point with the information bit plus check bit as a code;2) choosing a location of the con?stellation point suitable for transmitting information bits;3)di?viding the constellation area of the correctable error,where the erroneous information bit can be directly detected by the receiver.

▲Figure 2.Modulation block with one error correction

Fig.3 shows the (3,1) modulation code with 3 symbols as an example,where one information bit and two check symbols are included and the single error can be corrected.Therefore,the information symbol of the (3,1)code is 0 or 1 and the added er?ror check symbols can be 00,11,or 01,10.The modulation coding has one of two structures with no error:(0,0,0)(1,1,1)or(0,0,1) (1,1,0).This modulation code with the constellation points can correct one error.

4 Analysis of Energy-Saving Efficiency

4.1 Energy Consumption of Two Modulation Methods

Based on the above processing,this section analyzes the pow?er consumption of two structures of the error correction plus modulation and the error-corrected modulation for wireless transmission links,to find which method saves more energy.

First,let us consider the traditional transmission link,in which the error correction codec is eventA1and the Mod/Dem is eventA2,to analyze the energy-saving efficiency.

The power consumption of eventA1can be expressed as

In general,the coding process is addition operation depend?ing on the coding lengthn,and the power consumption of the coding process can be expressed as the function ofiT-order coding lengthn.For simplicity,the power consumption of the decoding process can be expressed as the function ofiR-order coding lengthn.Then,the power consumption of the coding and decoding processing of the(n,m) code can be respective?ly simplified to

Therefore,the total power consumption of the coding and decoding processing of the (n,m) code for the corrected errork=1,i.e.,the total of eventA1,is

where the subscriptTmeans the transmitting process,Rmeans the receiving process,andP0is the power consumption of a single addition operation(i=1) of one symbol.Moreover,i=1 means the addition operation,andi=2 andi=3 are re?spectively the multiplication operation and the convolution or iteration operation.

Second,eventA2is then-order quadrature amplitude modu?lation(QAM)modulation.By the Shannon theory,the transmit?ted signal symbol rate in unit bandwidth and unit time is

▲Figure 3.Error-corrected modulation constellation of(3,1)coding

wherenis the number of transmitted signal symbols in a block.The receiving demodulation of eventA2is similar to code demodulation,with only multiplication and comparison;the power consumption can be expressed as

Therefore,the power consumption of Mod/Dem withnsym?bols is

whereN0is the power of channel noise;fR(nr) is the power consumption of the receiver,which is proportional to the code lengthn.

Therefore,the total power consumption of coding/decoding plus Mod/Dem can be expressed as

WheniT=iR=r=2,all power operations inP0item of Eq.(22) are multiplication operation.The power consumption of the operation of one symbol is denoted asP0,and is equal to the unit noise powerN0.IfP0=N0,Eq.(22) can be simpli?fied as

Fig.4 shows the power consumptions of eventAunder dif?ferent sending/receiving parameters withk=1 andN0=P0=1 μW.Obviously,the power consumption increases exponentially asnincreases.With the increase of the sending/receiving parameters,the power consumption also increases significantly.

Fig.4 demonstrates that for every additional bit of informa?tion in error correction coding,frommtom+1,the code length must be increased by two symbols at least,fromnton+2,that ism=(n-1)/2.Therefore,(3,1) code,(5,2) code,(7,3) code,(9,4) code,(11,5) code,(13,6) code,(15,7) code,etc.are all such coding.

Based on Eq.(5) in Section 2,the evaluation function of power consumption of eventAcan be expressed as

IfαT=βT=1andαR=βR=3,the evaluation function of power consumption of eventAcan be expressed as

Based on different lengths of symbol blocks and information bits (n,m),the evaluation function of the power consumption for error correction and modulation separation in the tradition?al transmission link is shown in Fig.5.

Now,let us consider the fusion structure of error-corrected modulation of eventB.Only Mod/Dem processing is taken forcombination states shown in Eq.(15),wherej=0,1,...,k(kis the number of error correction symbols of a block).Then,the power consumption of eventBcan be simplify as

wherei=2 andP0=N0.

The power consumptions of eventBunder different sending/receiving parameters are shown in Fig.6,where the parame?ters areαT,βT,αR,andβR,k=1,andN0=P0=1 μW.Obvi?ously,the power consumption increases exponentially withm.Along with the increase of the sending/receiving parameters,the power consumption also increases.However,compared with eventAshown in Fig.4,the power consumption is signifi?cantly reduced.

Similar to Eq.(25),the evaluation function of the power con?sumption of eventBcan be expressed as

▲Figure 4.Power consumption of event A under different parameters

▲Figure 5.Energy-saving evaluation function of the traditional link

▲Figure 6.Power consumption of event A under different parameters

WhenαT=βT=1 andαR=βR=3,the evaluation func?tion of power consumption of eventBcan be simplified as

Based on different length of symbol blocks and information bits (n,m),the evaluation function of power consumption of the given modulation method for eventBis shown in Fig.7.

4.2 Energy-Saving Comparison of Two Modulation Methods

According to the above analysis,the modulation link of (n,m)code structure is divided into two modes,eventAand eventB.The amount of information transmitted in a block with a coding lengthnism,that is,the transmission rate ism/n.Then,the power consumption of eventAisPa≈N0((αT+αR+βR)n2+βT2n) and that of eventBisPb≈N0(βRn2+βT(1+n)2m).

Therefore,the improved energy-saving degree from eventAto eventBat the same information ratem/nis defined as

WhenαT=βT=1 andαR=βR=3,the improved energysaving degree of conversion of eventAinto eventBis

The improved degree of energy saving is shown in Fig.8,which demonstrates that the longer the code length,the higher the improved degree in energy saving.If the length of coding is 15,the improved energy-saving degree reaches up to 35%in theory.

5 Conclusions

With the widespread deployment and application of 5G net?works,the requirements for wireless energy saving are getting higher and higher.This paper introduces two basic study points for wireless energy saving and gives the error-corrected modulation method and its fusing constellation structure based on extending the Shannon formulas.

This paper also analyzes and compares the energy-saving performance of two wireless transmission chains,the traditional and the proposed.The numerical analysis shows that the pro?posed error-corrected modulation method improves the energysaving effect of the traditional method by 35%in theory.

▲Figure 7.Energy-saving evaluation function of the proposed modula?tion mode

▲Figure 8.Error-corrected modulation constellation of(3,1)coding