A novel cluster-based cooperative MIMO scheme for multi-hop wireless sensor networks2007

Hindawi Publishing Corporation

EURASIP Journal on Wireless Communications and Networking

Volume2006,Article ID72493,Pages1–9

DOI10.1155/WCN/2006/72493

A Novel Cluster-Based Cooperative MIMO Scheme for

Multi-Hop Wireless Sensor Networks

Yong Yuan,1Min Chen,2and Taekyoung Kwon3

1Department of Electronics and Information,Huazhong University of Science and Technology,Wuhan430074,China

2Department of Electrical and Computer Engineering,University of British Columbia,BC,Canada V6T1Z4

3School of Computer Science and Engineering,Seoul National University,Seoul151742,South Korea

Received4November2005;Revised11April2006;Accepted26May2006

A cluster-based cooperative multiple-input-multiple-output(MIMO)scheme is proposed to reduce the adverse impacts caused

by radio irregularity and fading in multi-hop wireless sensor networks.This scheme extends the LEACH protocol to enable the multi-hop transmissions among clusters by incorporating a cooperative MIMO scheme into hop-by-hop transmissions.Through the adaptive selection of cooperative nodes and the coordination between multi-hop routing and cooperative MIMO transmis-sions,the scheme can gain e?ective performance improvement in terms of energy e?ciency and reliability.Based on the energy consumption model developed in this paper,the optimal parameters to minimize the overall energy consumption are found,such as the number of clusters and the number of cooperative nodes.Simulation results exhibit that the proposed scheme can e?ectively save energy and prolong the network lifetime.

Copyright?2006Y ong Yuan et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use,distribution,and reproduction in any medium,provided the original work is properly cited.

1.INTRODUCTION

Due to the limited energy and di?culty to recharge a large number of sensors,energy e?ciency and maximizing net-work lifetime have been the most important design goals for wireless sensor networks(WSNs).However,channel fading, interference,and radio irregularity pose big challenges on the design of energy e?cient communication and routing proto-cols in the multi-hop WSNs.

As the MIMO technology has the potential to dramat-ically increase the channel capacity and reduce transmis-sion energy consumption in fading channels[1],cooperative MIMO schemes have been proposed for WSNs to improve communication performance[2–5].In those schemes,mul-tiple individual single-antenna nodes cooperate on informa-tion transmission and/or reception for energy-e?cient com-munications.Cui et al.[2]analyzed a cooperative MIMO scheme with Alamouti code for single-hop transmissions in WSNs.Li[3]proposed a delay and channel estimation scheme without transmission synchronization for decoding for such cooperative MIMO schemes.Li et al.[4]also pro-posed a STBC-encoded cooperative transmission scheme for WSNs without perfect synchronization.Jayaweera[5]con-sidered the training overhead of such schemes.

However,in the above proposals,the multi-hop rout-ing and distributed operations in WSNs are not taken into consideration,which limits the practical use of the coop-erative MIMO schemes in WSN.In this paper we study the feasibility of a cooperative MIMO scheme in multi-hop WSNs.Radio irregularity of wireless communications and multi-hop routing is considered with the cooperative MIMO scheme.On the other hand,due to its ability of fre-quency reuse and e?ciency in processing highly correlated data,clustering is e?cient in the design of WSNs.There-fore,we incorporate the cooperative MIMO scheme with the LEACH protocol,which is an e?cient clustering protocol due to its energy-e?cient,randomized,adaptive,and self-con?guring cluster formation.As only single-hop communi-cations from cluster heads to the sink are considered in the original LEACH protocol,we modify the LEACH protocol to allow cluster heads to form a multi-hop backbone and in-corporate the cooperative MIMO scheme into each single-hop transmission.Based on the proposed scheme,we investi-gate the energy consumption of each transmission/reception. Then,the overall energy consumption model is developed, and the optimal parameters of the scheme are found such as the number of clusters and the number of cooperative nodes.

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Networking

Sink

Cluster header node

node

Figure1:Multi-hop MIMO-LEACH scheme.

The remainder of the paper is organized as follows.In Section2we describe the design of the proposed cluster-based cooperative MIMO scheme(multi-hop MIMO-LEACH).The overall energy consumption of the proposed scheme is analyzed in Section3.Section4presents simula-tion results and discussions.Section5concludes the paper.

2.THE MULTI-HOP MIMO-LEACH SCHEME

In this section,we will discuss the proposed multi-hop MIMO-LEACH scheme,which is illustrated in Figure1. First,the strategy to?nd appropriate cooperative nodes in the single-hop communications between cluster heads is pro-posed in Section2.1.Based on the strategy,the multi-hop MIMO-LEACH scheme is presented in Section2.2.

2.1.Strategy to choose cooperative nodes

To maximize the performance of single-hop communica-tions between cluster heads,an appropriate strategy should be taken to choose the optimal cooperative nodes.Suppose that the current cluster head will use J cooperative nodes to transmit data to its neighboring cluster head t by the co-operative MIMO scheme.An AWGN channel with squared power path loss is assumed for intracluster communications. For the intercluster communications,we assume the trans-mission from each cooperative node experiences frequency-nonselective and slow Rayleigh fading.Furthermore,the long distance between any two nodes in the network with respect to the wavelength gives rise to independent fading coe?-cients for the cooperative nodes.The rationale behind such channel assumptions is that the inter-cluster transmission distance is much larger than the intra-cluster transmission distance and the transmission environments are more com-plex in the inter-cluster communication.

Denote the distance between node j and its current clus-ter head by d j1.Also,denote the distance and path loss for node j to communicate with t as d jt and k jt,respectively. For each single-hop transmission,the current cluster head will broadcast a data packet to the cooperative nodes.Then, the cooperative nodes will encode and transmit the transmis-sion sequence according to the orthogonal space-time block codes(STBC)to cluster head t toward the sink node.The en-ergy consumption for these two operations in the single-hop transmission will be modeled in the remainder of this sec-tion.Then,a novel strategy will be developed to?nd the op-timal set of cooperative nodes to minimize the overall energy consumption.In developing the strategy,we assume BPSK is adopted as the modulation scheme and the bandwidth is

B Hz.

(1)The energy consumption for the intracluster transmission Denote by E bt(1)the energy consumption for the current cluster head to broadcast one bit to the cooperative nodes.

E bt(1)can be broken down into two main components,the transmit energy consumption E btt(1)and the circuit energy consumption E btc(1).

The BER performance for BPSK is P b=Q(

√

2r).Here r is the signal-to-noise ratio(SNR),which is de?ned as r= P r/(2Bσ2N f)[6]under the assumption of AWGN channel, where P r is the received signal power,σ2is the power density of the AWGN,and N f is the receiver noise?gure.

In the high SNR regime,we can approximate the BER performance as P b=e?r by the Cherno?bound[6].Hence, we obtain P r=?2BN fσ2ln(P b).As the assumption of squared power path loss,E bt(1)can be modelled by

E bt(1)=E btt(1)+E btc(1)

=?2(1+α)N fσ2ln

P b

G1d2max M l+

P ct+JP cr

B

,

(1)

where d max is the maximum distance from the cooperative nodes to the cluster head,αis the e?ciency of the RF power ampli?er,G1is the gain factor at d max=1m,M l is the link margin,N f is the receiver noise?gure,and P ct and P cr are the circuit power consumption of the transmitter and receiver, respectively[2].

Let f1(P b)=?2N fσ2ln(P b)and H(d max)=G1M l d2max. Then,(1)can be rewritten as

E bt(1)=(1+α)f1

P b

H

d max

+

P ct+JP cr

B

.(2)

According to the de?nition,H(d j)can be measured as follows.Let the current cluster head transmit a signal with transmit power P out.Then,the power of the received signal at its cluster member,node j,is P j1=P out/H(d j).Therefore, H(d j)can be measured as

H

d j

=

P out

P j1

.(3)

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From (2),we can ?nd that the energy consumption in the intra-cluster transmission,E bt (1),can be reduced by choos-ing the nearer cooperative nodes.

(2)The energy consumption for the intercluster transmission To analyze the energy consumption for inter-cluster trans-missions based on the cooperative scheme,denoted by E bt (2),we re?ne the results in [2].In [2]an equal transmit power allocation scheme is used as the channel state infor-mation (CSI)is not available at the transmitter.If the av-erage attenuation of the channel for each cooperative node pair can be estimated,we can use an equal signal-to-noise (SNR)policy [7]to allocate the transmit power for its e ?ec-tiveness and simplicity.The average energy consumption per bit transmission by BPSK in such a scheme can be approxi-mated by

E bt (2)=(1+α)N 0P 1/J b J j =1(4π)2d k

jt

jt

G t G r λ2

M l N f

+

JP ct +P cr

B

,(4)

where N 0is the single-sided noise power spectral density,P b is the desired BER performance,G t and G r are the transmit-ter and receiver antenna gains,respectively,also,λis the car-rier wavelength [2].The training overhead and transmission rate are not considered in (4),which will be considered in Section 3.

The average attenuation of the channel for node j can be estimated as follows.Assume the channel is symmetric,and t transmits a signal with transmit power P out ,then the power of the received signal at node j ,P jt can be given by

P jt =P out

G t G r λ2

(4π)2d k jt

jt M l N f

=

P out

G

d jt ,k jt

,

(5)

where G (d jt ,k jt )=P out /P jt =((4π)2d k jt

jt /G t G r λ2)M l N f .Therefore,(4)can be reformulated as

E bt (2)=(1+α)N 0P 1/J

b J

j =1

G

d jt ,k jt + JP ct +P cr

=(1+α)f 2(P b )

J j =1

G d jt ,k jt

+

JP ct +P cr

B

.(6)

According to (6),the transmit power of node j to com-municate with cluster head t can be described by

P out jt =G

d jt ,k jt

N 0B

P 1/J b

.(7)

(3)The strategy to choose cooperative nodes

Based on (2)and (6),the overall energy consumption for the single-hop transmission can be written as (8)E bt =E bt (1)+E bt (2)

=(1+α)

f 1

P b H

d max

+f 2

P b

J j =1

G d jt ,k jt

+(J +1) P ct +P cr

B

.

(8)

From (8),the energy consumption for the intraclus-ter transmission E bt (1)and intercluster transmission E bt (2)should be traded o ?to minimize E bt .E bt can be mini-mized by choosing an appropriate set of cooperative nodes,

which can minimize f 1(P b )H (d max )+f 2(P b )

J j =1G (d jt ,k jt ).In order to simplify the distributed strategy design,the cooperative nodes should be chosen as the nodes whose f 1(P b )H (d j 1)+f 2(P b )G (d jt ,k jt )are minimal.In addition,in order to balance the energy consumption,the selection crite-rion is de?ned as

βjt =

E j

f 1

P b H

d j 1

+f 2 P b G d jt ,k jt

,(9)

where E j is the remaining energy in the current round for node j .The rationale behind de?nition of βjt is that the node,which has a good tradeo ?between E bt (1)and E bt (2)and has more remaining energy,should have a larger chance to be selected as cooperative node.Therefore,J nodes with maximum βjt will be chosen as the cooperative nodes to communicate with cluster head t .2.2.

Scheme design

In this section,we will discuss how to enable cluster heads to form a multi-hop backbone by incorporating the cooperative MIMO scheme into the LEACH protocol for each single-hop transmission.As assumed in the LEACH protocol,each node has a unique identi?er (ID).The transmit power of each node can be adjusted,and the nodes are assumed to be al-ways synchronized.Similarly,the operations of the proposed scheme are broken into rounds .Each round consists of three phases:(i)cluster formation phase,during which the clus-ters are organized and cooperative MIMO nodes are selected;(ii)routing phase,during which a routing table in each se-lected node is constructed;and (iii)transmission phase,dur-ing which data are transferred from the nodes to the cluster heads and forwarded to the sink according to the routing ta-ble.

(1)Cluster formation phase

In this phase,each node will elect itself to be a cluster head with a probability p as speci?ed in the original LEACH pro-tocol.After the cluster heads are elected,each cluster head will broadcast an advertisement message (ADV)by transmit power P out using a nonpersistent CSMA MAC protocol.The

4EURASIP Journal on Wireless Communications and Networking

message contains the head’s ID.If a cluster head receives the advertisement message from another head t and the received signal strength(RSS)exceeds a threshold th,it will take clus-ter head t as a neighboring cluster head and record t’s ID.As for the noncluster head,node j,it will record all the RSSs of the received advertisement messages,and choose the cluster head whose RSS is the maximum.Then,it will calculate and save H(d j),G(d jt,k jt),βjt,and P out jt by(3),(5),(7),and(9). Then node j will join the cluster by sending a join-request message(Join-REQ)to the chosen cluster head.This mes-sage contains the information of the node’s ID,the chosen cluster head’s ID,and the corresponding values ofβjt.After a cluster head has received all join-request messages,it will set up a TDMA schedule and transmit this schedule to its mem-bers as in the original LEACH protocol.If the sink receives the advertisement message,it will?nd the cluster head with the maximum RSS,and send the sink-position(Sink-POS) message to the cluster head and mark the cluster head as the target cluster head(TCH).

After the clusters are formed,each cluster head will select corresponding optimal J cooperative nodes for cooperative MIMO communications with each of its neighboring cluster heads.As stated in Section2.1,J nodes with maximumβjt will be chosen to communicate with a neighboring cluster head t.If no such J nodes can be found for t,t will be re-moved from the neighbor list,since too much energy is con-sumed for communicating with t.After selecting the coop-erative nodes,the total energy per bit transmission for com-munications with t,E bt,can be derived by(4).Then,E bt,the ID set of the cooperative nodes for each neighboring cluster head,will be stored.At the end of this phase,the cluster head will broadcast a cooperate-request message(COOPERATE-REQ)to each cooperative node,which contains the ID of the cluster itself,the ID of the neighboring cluster head t, the IDs of the cooperative nodes,and the index of the co-operative nodes in the cooperative nodes set for each cluster head t.Each cooperative node that receives the cooperate-request message(COOPERATE-REQ)will store the ID of t,the index,and the transmit power P out jt and send back a cooperate-ACK message(COOPERATE-ACK)to the cluster head.

(2)Routing table construction

To construct the routing table,the basic ideas of distance-vector-based routing will be used.Each cluster head will maintain a routing table,in which each entry contains desti-nation cluster ID,next hop cluster ID,IDs of cooperative nodes, and mean energy per bit.Initially,only the neighboring clus-ter head will have a record in the routing table.Then each cluster head will simply inform its neighboring cluster heads of its routing table.After receiving route advertisements from neighboring cluster heads,the cluster head will update its routing table according to the route cost and advertise to its neighboring cluster heads the modi?ed routes.After sev-eral rounds of route exchange and update,the routing ta-ble of each cluster head will be converged to the optimal one.Then,TCH will?ood a target announcement message (TARGET-ANNOUNCEMENT)containing its ID to each cluster head to enable the creation of paths to the sink. (3)Data transmission

In this phase,cluster members will transmit?rst their data to the cluster head by multiple frames as in the traditional LEACH protocol.In each frame,each cluster member will transmit its data during its allocated transmission slot spec-i?ed by the TDMA schedule in cluster formation phase,and it will be sleep in other slots to save energy.The duration of a frame and the number of frames are the same for all clusters.Thus the duration of each slot depends on the num-ber of members in the cluster.After a cluster head receives data frames from its cluster members,it will perform data aggregation to remove the redundancy in the data.After ag-gregating received data frames,the cluster head will forward the data packet to the TCH by multiple hops routing.In each single-hop communication,if there exist J-cooperative MIMO nodes,the cluster head will add a packet header to the data packet,which includes the information of source clus-ter ID,next-hop cluster ID,and destination cluster ID.Then the data packet is broadcasted.Once the corresponding co-operative nodes receive the data packet,they will encode the data packet by orthogonal STBC,and transmit the data as an individual antenna with transmission power P out jt in the MIMO antenna array.In the cooperative MIMO scheme,the transmission delay and channel estimation scheme proposed in[3]can be used to solve the problem of imperfect synchro-nization in decoding.

3.THE ENERGY CONSUMPTION MODEL OF

THE SCHEME

In this section,we will analyze the energy consumption of the scheme.Based on the result,we will develop an optimization model to?nd the optimal parameters in the scheme,includ-ing the number of clusters k c,and the number of cooperative nodes J.

In analysis,we make the following assumptions.(1) There are N nodes distributed uniformly in an M×M re-gion.(2)An AWGN channel with squared power path loss is assumed for the intracluster communication.(3)A?at Rayleigh fading channel with k th-power path loss is assumed for the intercluster communication.(4)BPSK is used as the modulation scheme and the bandwidth is B Hz.(5)In each frame every node will send a packet with size s to the clus-ter head by probability P.The number of frames in each round is denoted by F n.(6)In maintaining the routing ta-ble in each round,each cluster head will broadcast the rout-ing table,whose size is denoted by R ts for R bt times.(7)The energy consumption for data processing is ignored.

Now,we are ready to model the overall energy consump-tion in each round,denoted by E(k c,J).There are four en-ergy consuming operations in each round.(1)The cluster members transmit data to the cluster head,whose energy consumption is denoted by E s(k c).(2)The cluster heads construct the routing tables,whose energy consumption is

Yong Yuan et al.

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denoted by E r (k c ).(3)The cluster heads transmit aggregated data to the cooperative nodes in each single-hop transmis-sion,whose energy consumption is denoted by E c 0(k c ,J ).(4)The cooperative nodes transmit the data to the next clus-ter head in each single-hop transmission;whose energy con-sumption is denoted by E cs (k c ,J ).3.1.E s (k c )

In order to model E s (k c ),we will ?rst analyze the energy con-sumption for the source nodes to transmit one bit to the clus-ter head,denoted by E bs (k c ).

Under the assumption of BPSK modulation and AWGN channel with squared power path loss,E bs (k c )can be mod-elled in the same manner as E bt (1)in Section 2.1(1),

E bs k c =?2(1+α)N f σ2ln P b G 1E d 2

tc

M l +

P ct +P cr

=?

1πk c (1+α)N f σ2ln P b G 1M 2M l +P ct +P cr

B

,(10)

where d tc is the distance from the node to the cluster head,G 1is the gain factor at d tc =1m.In (10),we use the result in

[8]that E [d 2

tc ]=M 2/2πk c .

On the other hand,when the number of clusters is k c ,the average number of members for each cluster is N/k c .Hence,the total number of bits transmitted to the cluster head for each cluster by each round is S 1(k c )= N/k c F n Ps .Therefore,E s (k c )=k c S 1(k c )E bs (k c ).3.2.E r (k c )

In this section,we will model the energy consumption in constructing the routing table,denoted by E r (k c ).When the number of clusters is k c ,the radius of each cluster can be ap-proximated as radius =M/ πk c [8].Therefore,the distance between each pair of direct neighboring clusters can be ap-proximated as d ctoc =2radius =2M/ πk c .We also assume the number of direct neighbors of each cluster is 4.Under the assumption of ?at Rayleigh fading channel,E r (k c )can be approximated by [2]

E r (k c )=k c R ts R bt

(1+α)N 0P b (4π)2(2M )k

GtGrλ2 πk c k c /2M l N f

+P ct +4P cr

.

(11)

3.3.E c 0(k c ,J )

In this section,we will analyze the energy consumption for the cluster head to transmit aggregated data to the coop-erative nodes,denoted by E c 0(k c ,J ).When the cluster head broadcasts the data,J cooperative nodes will receive it.Sim-ilar to the analysis of E bs (k c ),the energy per bit for this

operation,denoted by E bc 0(k c ,J ),can be described by E bc 0

k c ,J

=?

1πk c (1+α)N f σ2ln P b G 1M 2M l +P ct +JP cr

B

.(12)

We adopt the aggregation model in [9]to describe the ag-gregation operation.The amount of data after aggregation for each round is S 2(k c )=S 1(k c )/( N/k c Pagg ?agg +1),where agg is the aggregation factor.Therefore,E c 0(k c ,J )=k c S 2(k c )E bc 0(k c ,J ).3.4.

E cs (k c ,J )

According to Section 2.1,J cooperative nodes of the current cluster will encode and transmit the transmission sequence according to the orthogonal STBC to the cluster head.In modelling the energy consumption of such operation,we need to consider the impacts of training overhead and trans-mission rate.Suppose that the block size of the STBC code is F symbols and in each block we include pJ training sym-bols,and the block will be transmitted in L symbols du-ration.F/L is called the transmission rate,denoted by R .Then,the actual amount of data to transmit the S 2(k c )bits is S e (k c ,J )=FS 2(k c )/R (F ?pJ ).Therefore,E cs (k c ,J )can be described by E cs

k c ,J

=S e

k c ,J

(1+α)

JN 0P 1/J b (4π)2(2M )k

G t G r λ2 πk c

k/2M l N f +JP ct +P cr

B

.

(13)

Based on the above analysis,the overall energy consump-tion in each round,E (k c ,J )can be described as

E k c ,J =E s k c +E r k c +n k E c 0 k c ,J +n k E cs k c ,J

,

(14)where n k is the average number of hops.In order to simplify the analysis,we assume n k = k c ,which is just the number of clusters along each edge of the sensed region.

Based on (14),we can formulate the optimization model to choose the optimal k c and J as

k ?c ,J ? =argmin E k c ,J

s.t.J ≤10,k c ≤

N

3

,(15)where the ?rst constraint comes from the fact that more co-operative nodes will not improve the transmission energy e ?ciency but cost much circuit energy,and the rationale behind the second constraint is that the size of the cluster should not be too small to make e ?cient aggregation.Since the search space is not large,we can use exhaustive search method to solve (15).

4.SIMULATION RESULTS

In the simulations,400nodes are randomly deployed on a 200×200?eld.The location of the sink is randomly chosen

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EURASIP Journal on Wireless Communications and Networking

Table 1:The system parameters.

α=0.4706M l =40dB G 1=30dB k ∈[3,5]σ2=N

02

=?134dBm/Hz

N f =10dB f c =2.5GHz B =20kHz

P b =10?3P ct =98.2mw P cr =112.6mw F n =2G t G r =5dB i s =2kbits P =0.8R =0.75F =200p =2

R bt =5

R ts =100

in each round.The system parameters are summarized in Table 1.

The meanings of the entries in Table 1are summarized as follows.αis the e ?ciency of the RF power ampli?er,M l is the link margin,G 1is the gain factor at 1m ,k is the path loss,σ2is the power density of the AWGN chan-nel in the intracluster communication,N f is the receiver noise ?gure,f c is the carrier frequency,B is the bandwidth,P b is the desired BER performance,P ct and P cr are the cir-cuit power consumption of the transmitter and receiver,re-spectively,F n is the number of frames per round,G t ,G r are the antenna gains of the transmitter and receiver,s is the packet size,P is the transmit probability of each node,R is the transmission rate,F is the number of symbols in each block,p is the number of required training symbols for each cooperative node,R bt is the times for exchanging the routing table for each round,and R ts is the routing table size.

To simulate the phenomena of radio irregularity,the path loss of the communication between each pair of nodes is dis-tributed randomly from 3to 5.

Each node begins with 400J of energy and an unlimited amount of data to send to the sink.When the nodes use up their limited energy during the course of the simulation,they can no longer transmit or receive data.

During the simulation,we tracked the overall number of packets transferred to the sink,the amount of energy and du-ration required to get the data to the sink,and the percentage of nodes alive.We are interested in the transmission qual-ity and energy saving performance of the proposed scheme.The performance of the proposed multi-hop MIMO-LEACH scheme is compared with the original LEACH and the multi-hop LEACH scheme,in which cooperative MIMO commu-nications is not implemented.The optimal value of k c for the original LEACH is determined by the model in [8].We also develop a similar model to ?nd the optimal k c for the multi-hop LEACH scheme,which will not be discussed here due to the limited space.In the investigated scenario,it is found that the optimal k c for the original LEACH protocol,the multi-hop LEACH scheme,and the proposed scheme are 3,41,and 27,respectively.The optimal J for the proposed scheme is found to be 3.

Due to the aggregation operation,the number of ef-fective received packets by sink [8]is a good application-independent indication of the transmission quality.The

e ?ective received packets refer to the “real”packets repre-sented by the aggregated packets.I

f no aggregation carries out,the number of e ?ective received packets equals to the number of actual received packets.If the aggregation oper-ation in transmission is information lossless ,the number of e ?ective received packets is just the number of total packets transferred by the source nodes.

Figures 2and 3show the total number of e ?ective pack-ets received at the sink over time and the total number of e ?ective packets received at the sink for a given amount of energy.

Figure 2shows that during its lifetime the LEACH pro-tocol can obtain better latency performance compared to the multi-hop LEACH scheme and the proposed MIMO LEACH scheme.The reason is that the multi-hop oper-ation in the multi-hop LEACH scheme and the multi-hop MIMO-LEACH scheme will increase the latency,and thus result in a less number of data packets sent to the sink for a given period of time.However,the better la-tency performance of the LEACH protocol comes from the more energy consumption compared to the other two schemes.Especially,in the fading channel environment,LEACH protocol will consume much more energy due to its single-hop transmission from the cluster heads to the sink,which will result in less network lifetime and less to-tal number of transmitted packets.Figure 3shows that,with the same amount of energy consumption,the multi-hop MIMO-LEACH scheme can transmit much more data pack-ets compared to the LEACH protocol and the multi-hop LEACH scheme.From these simulation results,we can ?nd that the multi-hop MIMO-LEACH scheme is more suit-able for the application scenario which has large require-ments on network lifetime but little requirements on la-tency.

Figure 4shows the percentage of nodes alive over time.From Figure 4,we can ?nd that the proposed multi-hop MIMO-LEACH scheme can improve the network lifetime greatly.If we de?ne the network lifetime of WSN as the du-ration of more than 70%of network nodes are alive,then we can observe that the network lifetime of WSN with the orig-inal LEACH protocol,the multi-hop LEACH scheme,and the proposed multi-hop MIMO-LEACH scheme is about 0.7×104,8.2×104,and 11×104s.The improvement on network lifetime obtained by the multi-Hop MIMO-LEACH scheme is signi?cant.

However,the percentage of nodes alive over time is not always a good indication to the energy saving performance of a protocol.For example,during the same time,one proto-col transmits less packets than other protocols.Then,though the energy saving performance of the protocol is worse than other protocols,it will still consume less energy.In order to further investigate the energy saving performance,we also simulate the performance in terms of the percentage of nodes alive per amount of e ?ective data packets received at the sink,which is shown in Figure 5.

From Figure 5,we ?nd that the proposed multi-hop MIMO-LEACH scheme needs signi?cantly less energy to transmit the same amount of data packets.Therefore,the

Yong Yuan et al.

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0.5

1

1.5

2

2.5

109

N u m b e r o f e ?e c t i v e d a t a p a c k e t s r e c e i v e d b y s i n k

2

4

6

8

10

12

14104

Time (s)

LEACH

Multi-hop LEACH MIMO LEACH

Figure 2:Total amount of e ?ective packets received at the sink over time.

0.5

1

1.5

2

2.5

109

N u m b e r o f e ?e c t i v e d a t a p a c k e t s r e c e i v e d b y s i n k

024********

16104

Total energy consumption (J )

LEACH protocol Multi-hop LEACH MIMO LEACH

Figure 3:Total amount of e ?ective packets received at the sink per given amount of energy.

improvement on network lifetime obtained by the multi-hop MIMO-LEACH scheme is signi?cant.

On the other hand,the impacts of the parameters,in-cluding the number of cluster heads k c and the number of cooperative nodes J ,are also investigated in the simulation.Figures 6and 7show the percentage of nodes alive over time in di ?erent settings of k c and J .

10

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Figure 4:Percentage of nodes alive over time.

0102030405060708090100P e r c e n t a g e o f a l i v e n o d e s (%)

0.5

1

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Number of e ?ective data packets received by sink LEACH

Multi-hop LEACH MIMO LEACH

Figure 5:Percentage of nodes alive per amount of e ?ective data packets received at the sink.

From the simulation results including those shown in Figures 6and 7,we can ?nd that the energy saving perfor-mance of the proposed scheme is impacted by the param-eters.As for the number of cluster heads,too many cluster heads will reduce the distance for each single hop transmis-sion,which will reduce the transmit energy consumption.More cluster heads will also generate a larger search space

8

EURASIP Journal on Wireless Communications and Networking

0204060

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2

4

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k c =27(opt.)k c =20k c =30

Figure 6:The impact of the number of cluster heads on energy sav-ing performance.

for the routing table construction,which will also reduce the transmit energy consumption further.However,more clus-ter heads will result in more number of hops in transmis-sion to the sink,which will consume more circuit energy for relaying the data packets.Therefore,the number of clus-ter heads should be chosen by trading o ?the transmit en-ergy consumption and circuit energy consumption.As for the number of cooperative nodes,a certain number of co-operative nodes can form the e ?ective independent multi-path transmission so as to energy-e ?ciently combat the fad-ing e ?ects.However,too many cooperative nodes will result in large circuit energy consumption,which will cause large overall energy consumption.Therefore,the number of co-operative nodes should also be chosen to trade o ?the trans-mit energy consumption and the circuit energy consump-tion.5.

CONCLUSION

In this paper,we proposed a cluster based cooperative MIMO scheme to reduce energy consumption and prolong the net-work lifetime.A cooperative MIMO scheme is adopted to mitigate the adverse impacts of fading while clustering is used to facilitate network control and coordination.In the pro-posed scheme,the original LEACH protocol is extended by incorporating the cooperative MIMO communications and multi-hop routing.An adaptive cooperative nodes selection strategy is also designed.Based on the scheme,we investi-gated the energy consumption of each operation.Then,the overall energy consumption model of the scheme is devel-oped,and the optimal parameters of the scheme are found such as the number of clusters and the number of cooperative nodes.Simulation results exhibit that the proposed scheme minimizes energy consumption.

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J =3(opt.)J =5J =2

Figure 7:The impact of the number of cooperative nodes on energy saving performance.

ACKNOWLEDGMENTS

The authors thank the editors and the anonymous reviewers for their valuable suggestions.This work was supported in part by KOSEF Grant no.R01-2004-000-10372-0.REFERENCES

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tradeo?s for data gathering in wireless sensor networks,”in Pro-

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and Communications Societies(INFOCOM’04),vol.1,pp.244–

255,Hong Kong,March2004.

Y ong Yuan received the B.E.and M.E.de-

grees from the Department of Electronics

and Information,Yunnan University,Kun-

ming,China,in1999and2002,respectively.

Since2002,he has been studying at the De-

partment of Electronics and Information,

Huazhong University of Science and Tech-

nology,China,as a Ph.D.candidate.His

current research interests include wireless

sensor network,wireless ad hoc network,

wireless communication,and signal processing.

Min Chen was born on December1980.

He received the BS,MS,and Ph.D degrees

from the Deptartment of Electronic Engi-

neering,South China University of Tech-

nology,in1999,2001,and2004,respec-

tively.He is a postdoctoral fellow in the

Communications Group,Deptartment of

Electrical and Computer Engineering,Uni-

versity of British Columbia.He was a post-

doctoral Researcher in the Multimedia&

Mobile Communications Lab.,School of Computer Science and

Engineering,Seoul National University,in2004and2005.His cur-

rent research interests include wireless sensor network,wireless ad

hoc network,and video transmission over wireless networks.

Taekyoung Kwon is an Assistant Profes-

sor in the School of Computer Science

and Engineering,Seoul National Univer-

sity(SNU),since2004.Before joining

SNU,he was a postdoctoral Research Asso-

ciate at UCLA and at City University New

Y ork(CUNY).He obtained the B.S.,M.S.,

and Ph.D.degrees from the Department

of Computer Engineering,SNU,in1993,

1995,2000,respectively.During his gradu-

ate program,he was a visiting student at IBM T.J.Watson Research

Center and at the University of North Texas.His research interest

lies in sensor networks,wireless networks,IP mobility,and ubiqui-

tous computing.

函数y=f(x)理解与分析周勇关于

关于函数y=f(x)的理解与分析 作者：周勇 （湖南省长沙市第七中学 邮编：410003） 抽象函数y=f(x)是指没有给出函数的具体解析式，只给出了一些体现函数特征的式子的一类函数。由于抽象函数表现形式的抽象性，使得这类问题成为函数内容的难点之一.抽象性较强，灵活性大,解抽象函数重要的一点要抓住函数中的某些性质，通过局部性质或图象的局部特征，利用常规数学思想方法（如化归法、数形结合法等），这样就能突破“抽象”带来的困难，做到胸有成竹.另外还要通过对题目的特征进行观察、分析、类比和联想，寻找具体的函数模型，再由具体函数模型的图象和性质来指导我们解决抽象函数问题的方法。一般以中学阶段所学的基本函数为背景，构思新颖，条件隐蔽，技巧性强。解法灵活，因此它对发展同学们的 抽象思维，培养同学们的创新思想有着重要的作用。 一、关于定义域的理解与分析 例1. 已知函数)(2x f 的定义域是［1，2］，求f （x ）的定义域。 解：)(2x f 的定义域是［1，2］，是指21≤≤x ，所以)(2x f 中的2x 满足412≤≤x 从而函数f （x ）的定义域是［1，4］ 原理：一般地，已知函数))((x f ?的定义域是A ，求f （x ）的定义域问题，相当于已知 ))((x f ?中x 的取值范围为A ，据此求)(x ?的值域问题。已知f(x)的定义域是A ，求()() x f ?的定义域问题，相当于解内函数()x ?的不等式问题。 又如：已知函数f(x)的定义域是[]2,1- ，求函数()? ?? ? ??-x f 3log 21 的定义域。 再如：定义在(]8,3上的函数f(x)的值域为[]2,2-，若它的反函数为f -1 (x)，则y=f -1 (2-3x)的 定义域为 ，值域为 。(]8,3,34,0?? ??? ? 原理：这类问题的一般形式是：已知函数f （x ）的定义域是A ，求函数))((x f ?的定义域。正确理解函数符号及其定义域的含义是求解此类问题的关键。这类问题实质上相当于已知 )(x ?的值域B ，且A B ?，

剖析函数yfx与yftx的奇偶性

剖析函数y=f(x)与y=f[t(x)]的奇偶性 函数的奇偶性是高中数学的重要内容，它与函数的单调性，周期性一起构成研究函数性质的三把钥匙。函数的奇偶性是教学过程中的一个难点，笔者现就教学过程中遇到的问题加以探讨。 教材中奇偶性的定义：一般地，如果对于函数f(x)的定义域内的任意一个x，都有f(-x)=-f(x)则称f(x)为这一定义域内的奇函数。一般地，如果对于函数f(x)的定义域内的任意一个x，都有f(-x)=f(x)则称f(x)为这一定义域内的偶函数。 在函数的奇偶性定义中，若函数y=f（x）的定义域I是关于原点对称（即x?I，则-x?I）且f(-x)=f(x)（或f(-x)=-f(x)）则函数叫偶（或奇）函数。由此可知：函数的定义域关于原点对称是该函数为奇（或偶）函数的必要条件。本文主要谈谈函数y=f（x）与函数y=f(kx+b)(k10)在奇偶性方面的不同表现形式。 例1： y=f(x)是单调递增的奇函数，它的定义域为[-1，1]，求已知函数 1 y. 2 y=f(x) 的定义域为[-1,1] 解：Q 1 x-? \ -1￡231 -1￡x+1￡1 f(23 x-)3-f(x+1) y=f(x)是奇函数 Q 1 \ 2￡2x￡4 -2￡x￡0 x-)3f(-x-1) f(23 y=f(x)是单调递增函数 Q 1 \ -2￡x￡ 23 x-3-x-1

\ -2￡x ￡x ￡-2 或 x 3 1 \x ?{}2- 故2y 的定义域为x ?{}2- 此时2y =0 \ 值域: 2y ?{}0 点评：y=f(x)(x ?R)为奇函数Tf(-x-1)=-f(x+1) (x ?R) 例2： 若函数y=f(2x+1)为奇函数，则f(-2x+1)=___________ 若函数y=f(2x+1)为偶函数，则f(-2x+1)=_____________ 解：由于对应法则后边不是x ，而换成了2x+1,学生无从下手，其实，此题是把自变量x 换成了2x+1后的函数，在运算后的自变量仍然是x ，f(2x+1)是奇函数实质上还是对x 而言的。故y=f(2x+1)是奇函数\f(-2x+1)=-f(2x+1) 同理：若函数y=f(2x+1)为偶函数\ f(-2x+1)=f(2x+1) 点评：若函数y=f(2x+1)为偶函数\ f(-2x+1)=f(2x+1)，即 f(1+2x)=f(1-2x)，这表明函数y=f(a+x) 的图象与y=f(a-x)的图象关于y 轴对称。应注意：y=f(a+x) 的图象与y=f(a-x)的图象关于y 轴对称，这里是两个函数图象间的对称;而函数y=f(a+x)为偶函数，它的图象关于y 轴对称,这里是一个函数图象内部之间的自身的对称。 例3： （2003年安徽三市联考）已知f(2x+1)为偶函数，那么函数f(2x)图像的对称轴方程为_____________ 解法1： Q y=f(2x+1)=f[2(x+12)]的图像向右平移12 个单位得到函数y=f[2(x)]=f(2x)的图像, Q y=f(2x+1)是偶函数\它的图像关于y 轴对称， 即x=0是函数f(2x+1)的图象的对称轴，直线x=0也向右平移12 个单位得直线方程为x=12，故直线x=12 是函数f(2x)的图象的对称轴。 解法2： Q f(2x+1)为偶函数\y=f(-2x+1)=f(2x+1),即f(1-2x)=f(1+2x), Q 定义在R 上的函数y=f(x)满足f(a+x)=f(a-x) (x ?R)，则函数y=f(x)的图象关于直线x=a 对称 \函数y=f(x)的对称轴方程为x=1,故函数y=f(2x)的对称轴方程为x=12 . 点评：y=f(2x+1)为偶函数Tf(-2x+1)=f(2x+1).

怎样求y=Asin(ωx+ψ)的解析式

怎样求y=Asin(ωx+?)的解析式 学习了正弦函数y=Asin(ωx+?)(A>0，ω>0)后，经常会遇到确定其解析式的问题。这里振幅A 常由函数的最值确定，ω则由周期公式T=2π ω 来求得，问题的关键是求初相?。本文介绍确定正弦函数解析式的两种基本方法。 一、待定系数法 分析正弦曲线y=Asin(ωx+?)(A>0，ω>0)满足的几何条件，列出关于A 、ω、?的三个方程，从而解出A 、ω、?，这就是待定系数法。 例1 若函数y=Asin(ωx+?)(A>0，ω>0，00，ω>0)的图象如图1所示，求此函数的解析式。 分析：由图1提供的信息，正弦曲线相邻的最大、最小值之间为周期的 12 。 ∴ 2T =56π－6π=23 π，即T=43π，∴ω=2T π=32 又显然有A=2，下面只须求初相?。 设曲线与x 轴交C ，易知，C(2π,0)将A=2，ω=32，x=2 π ， y=0代入y=Asin(ωx+?)得0=2sin(34 π +?)。 ∴?=k π－34 π ，(k ∈Z)。注意到y=Asin(ωx+?)的图象是由y=sinx 的图象，经过振幅、周 期变换，且向右平移而得，当k=0时，?在区间[－π,π]上有解。∴?=－34 π ，故函数的 解析式是y=2sin(32x －34 π )。 二、平移变换 我们知道，设A>0，ω>0，正弦函数y=Asin(ωx+?)=Asin[ω(x+? ω)] 的图象，可以看成是由函数y=sinx 的图象经过下面变换而得到： y=sinx 的图象 →y=Asinx 图1

初三专题构建y与x的函数关系式——圆.docx

构建y与兀的函数关系式 3 点0为边上的动1.如图，梯形人BCD 中，AD//BC, CD丄BC,已知人B=5, BC=6, COsB = - 5 点，以0为圆心，为半径的OO交边于点P. （1）设03=兀，BP=y,求y与兀的函数关系式，并写出函数定义域； （2）当00与以点D为圆心，DC为半径OD外切时，求的半径； （3）连接OD、AC,交于点E,当ZXCEO为等腰三和形时，求OO的半径.

2如图，在半径为5的OO中，点A、B在QO ±, ZAOB=90。，点C是弧AB上的一个动点，AC与0B的延长线和交于点D,设AC=x, BD=y. (2011.静安区二模) (1)求y关于x的函数解析式，并写出它的定义域； (2)如果00】与O0相交于点4、C, RQO｛与OO的圆心距为2,当BD = -OB时，求的 3 半径； (3)是否存在点C,使得△ DCBs\DOC2如果存在，请证明；如果不存在，请简要说明理由.

?3?

3 在梯形ABCD中，AD//BC, AB丄AD, 4B=4, AD=5, CD=5. E为底边BC上一点，以点E为圆心，BE为半径画OE交线段DE于点F. （1）如图，当点F在线段DE .L时，设BE=x, DF=y,试建立y关于x的函数关系式，并写出口变量x 的取值范围； （2）当以CD为直径的与OE相切时，求x的值； （3）连接AF、BF,当是以AF为腰的等腰三角形吋，求x的值.（201b徐汇区二模）

4.如图1,已知OO的半径长为1, PQ是00的直径，点M是PQ延长线上一点，以点M为圆心作圆, 与交于4、3两点，连接用并延长，交OM于另外一点C. (1)若恰好是?0的直径，设0M=x, AC=y,试在图2中画出符合要求的大致图形，并求y关于x的断数解析式； (2)连接04、MA. MC,若04丄M4,且△0M4与Z\PMC相似，求0M的长度和0M的半径长； (3)是否存在OM,使得人3、4C恰好是一个正五边形的两条边？若存在，试求0M的长度和的 半径长；若不存在，试说明理由.(2011.嘉定区二模) 图 2

自变量x和因变量y有如下关系

自变量x和因变量y有如下关系： y=kx+b （k为任意不为零实数，b为任意实数） 则此时称y是x的一次函数。 特别的，当b=0时，y是x的正比例函数。 即：y=kx （k为任意不为零实数） 定义域：自变量的取值范围，自变量的取值应使函数有意义；若和实际相反，。 一次函数的性质 1.y的变化值和对应的x的变化值成正比例，比值为k 即：y=kx+b（k≠0) （k为任意不为零的实数b取任何实数） 2.当x=0时，b为函数在y轴上的截距。 3.k为一次函数y=kx+b的斜率,k=tg角1(角1为一次函数图象和x轴正方向夹角) 形。取。象。交。减 一次函数的图像及性质 1．作法和图形：通过如下3个步骤 （1）列表[一般取两个点,根据两点确定一条直线]； （2）描点； （3）连线，可以作出一次函数的图像——一条直线。因此，作一次函数的图像只需知道2点，并连成直线即可。（通常找函数图像和x轴和y轴的交点）2．性质：（1）在一次函数上的任意一点P（x，y），都满足等式：y=kx+b(k≠0)。（2）一次函数和y轴交点的坐标总是（0，b)，和x轴总是交于（-b/k，0）正比例函数的图像总是过原点。 3．函数不是数，它是指某一变量过程中两个变量之间的关系。 4．k，b和函数图像所在象限： y=kx时 当k＞0时，直线必通过一、三象限，y随x的增大而增大； 当k＜0时，直线必通过二、四象限，y随x的增大而减小。 y=kx+b时： 当k>0,b>0, 这时此函数的图象经过一，二，三象限。 当k>0,b<0, 这时此函数的图象经过一，三，四象限。 当k<0,b<0, 这时此函数的图象经过二，三，四象限。 当k<0,b>0, 这时此函数的图象经过一，二，四象限。 当b＞0时，直线必通过一、二象限； 当b＜0时，直线必通过三、四象限。 特别地，当b=0时，直线通过原点O（0，0）表示的是正比例函数的图像。 这时，当k＞0时，直线只通过一、三象限；当k＜0时，直线只通过二、四象限。 4、特殊位置关系 当平面直角坐标系中两直线平行时，其函数分析式中K值（即一次项系数）相等 当平面直角坐标系中两直线垂直时，其函数分析式中K值互为负倒数（即两个K 值的乘积为-1）

y=a(x-h)平方+k图像性质和求解析式备课讲稿

y=a(x-h)平方+k图像性质和求解析式

()k h x a y +-=2 图像性质和求解析式 平移规律： 1、将二次函数2x y =的图像向右平移1个单位长度，再向上平移3个单位长度，所得的图像解析式为（ ） A.()312+-=x y B.()312++=x y C.()312--=x y D.()312 -+=x y 2、把抛物线22 1x y -=向_____平移_____个单位，再向_____平移____个单位，就得到抛物线()112 12-+-=x y 。 3、关于二次函数()2 14+-=x y 的说法正确的有（ ） ①顶点坐标为（1，3）；②对称轴为x=1-；③1-

1、二次函数()432 12+-= x y 的图像的开口方向、对称轴、顶点坐标分别是（ ） A.向上，直线x=3，（3，4） B.向上，直线x=3-，（3-，4） C.向上，直线x=3，（3，4-） D.向下，直线x=3，（3，4） 2、一般地，抛物线()k h x a y +-=2 的图像的特点是（ ） A.a ＞0，开口向上；对称轴是直线x=h ；顶点坐标是（h ，k ） B.a ＜0，开口向下；对称轴是直线x=h ；顶点坐标是（h ，k ） C..a ＞0，开口向上；a ＜0，开口向下；对称轴是直线x=h ；顶点坐标是（h ，k ） D.a ＞0，开口向上；a ＜0，开口向下；对称轴是直线x=ah ；顶点坐标是（ah ，k ） 3、抛物线()623 12-+=x y 的开口向_____，顶点坐标为_____，对称轴是______，当2-

y=a(x-h)平方+k图像性质和求解析式

()k h x a y +-=2 图像性质和求解析式 平移规律： 1、将二次函数2x y =的图像向右平移1个单位长度，再向上平移3个单位长度，所得的图像解析式为（ ） A.()312+-=x y B.()312++=x y C.()312--=x y D.()312 -+=x y 2、把抛物线221x y - =向_____平移_____个单位，再向_____平移____个单位，就得到抛物线()112 12-+- =x y 。 3、关于二次函数()214+-=x y 的说法正确的有（ ） ①顶点坐标为（1，3）；②对称轴为x=1-；③1-

按右图所示的流程,输入一个数据x,根据y与x的关系式就输出一个数据

1.按右图所示的流程，输入一个数据x，根据y 与x 的关系式就输出一个数据y，这样可以将一组数据变换成另一组新的数据，要使任意一组都在20～100（含20 和100）之间的数据，变换成一组新数据后能满足下列两个要求：（Ⅰ）新数据都在60～100（含60 和100）之间；（Ⅱ）新数据之间的大小关系与原数据之间的大小关系一致，即原数据大的对应的新数据也较大。（1）若y 与x 的关系是y＝x ＋p(100－x)，请说明：当p＝ 1 2 时，这种变换满足上述两个要求；（2）若按关系式y=a(x－h) 2 ＋k (a>0)将数据进行变换，请写出一个满足上述要求的这种关系式。（不要求对关系式符合题意作说明，但要写出关系式得出的主要过程） 2. 如图，抛物线经过ABC △的三个顶点，已知BC x ‖轴，点A 在x 轴上，点C 在y 轴上，且．（1）求抛物线的对称轴；（2）写出A B C ，，三点的坐标并求抛物线的解析式；（3）探究：若点P 是抛物线对称轴上且在x 轴下方的动点，是否存在PAB △是等腰三角形．若存在，求出所有符合条件的点P 坐标；不存在，请说明理由．开始y 与x 的关系式结束输入x 输出y 龙文学校个性化辅导资料二次函数综合题田老师第 2 页共7 页 3.如图，抛物线 交x 轴于A、B 两点，交y 轴于点C，点P 是它的顶点，点A 的横坐标是，点B 的横坐标是1．(1)求m 、

n 的值；(2）求直线PC 的解析式；(3）请探究以点A 为圆心、直径为 5 的圆与直线PC 的位置关系，并说明理由．(参考数：， 3 1.73 ，如图，对称轴为直线x＝ 2 7 的抛物线经过点A（6，0）和B （0，4）．（1）求抛物线解析式及顶点坐标；（2）设点E （x，y）是抛物线上一动点，且位于第四象限，四边形OEAF 是以OA 为对角线的平行四边形，求四边形OEAF 的面积S 与x 之间的函数关系式，并写出自变量x 的取值范围；（3）①当四边形OEAF 的面积为24 时，请判断OEAF 是否为菱形？②是否存在点E，使四边形OEAF 为正方形？若存在，求出点E 的坐标；若不存在，请说明理由．O E F x= 7 2 B(0,4) A(6,0) x y 龙文学校个性化辅导资料二次函数综合题田老师第 3 页共7 页图1 F E P D y x B A C O 图2 O C A B x y D P E F 5. 已知：如图，在平面直角坐标系中，四边形ABCO 是菱形，且∠AOC=60°，点B 的坐标是(0, 8 3) ，点P 从点C 开始以每秒1 个单位长度的速度在线段CB 上向点 B 移动，设秒后，直线PQ 交OB 于点D. （1）求∠AOB 的度数及线段OA 的长；（2）求经过A，B，C 三点的抛物线的解析式；（3）当 4 3, 3 3 时，求t 的值及此时直线PQ 的解析式；（4）当a 为何值时，以O，P，Q，D 为顶点的三角形与 相似？当a 为何值时，以O，P，Q，D 为顶点的三角形与OAB

下列曲线反映了变量y与变量x之间的关系

下列曲线反映了变量y与变量x之间的关系，其中y是x的函数的是（） A．B．C．D． 如图，某市一天的温度变化的图象，通过观察图象可知，下列说法错误的是（） A．这天15时的温度最高B．这天3时的温度最低 C．这天最高温度和最低温度的差是10℃D．这天21时的温度约31℃ 一段导线在0℃时的电阻为3Ω，温度每增加1℃，电阻增加0.009Ω，那么电阻R（Ω）与温度t（℃）的函数关系式为（） A．R＝0.009t B．R＝3＋0.009t C．R＝3.009t D．R＝3t＋0.009 函数中自变量x的取值范围是（） A．x≤2B．x＝3 C．x＜2且x≠3D．x≤2且x≠3（2009·河北）如图所示的计算程序中，y与x之间的函数关系所对应的图象应为（）A．B．C．D．

如图所示，直线y＝kx＋b经过点A，B，则k的值为（） A．3 B．C．D． 已知一次函数和的图象都经过点A（－2，0）且分别交y轴于B、C两点，那么△ABC的面积是（） A．2 B．3 C．4 D．6 如图，一次函数y＝kx＋b的图象经过A、B两点，则kx＋b＞0的解集是（） A．x＞0 B．x＞2 C．x＞－3 D．－3＜x＜2 一次函数y＝kx＋2经过点（1，1），那么这个一次函数中（） A．y随x的增大而增大B．y随x的增大而减小 C．图象经过原点D．图象不过第二象限 （2010年连云港）某公司准备与汽车租赁公司签订租车合同，以每月用车路程x km计算，甲汽车租赁公司每月收取的租赁费为y1元，乙汽车租赁公司每月收取的租赁费为y2元，若y1、y2与x之间的函数关系如图所示，其中x＝0对应的函数值为月固定租赁费，则下列判断错误的是（） A．当月用车路程为2000 km时，两家汽车租赁公司租赁费用相同 B．当月用车路程为2300 km时，租赁乙汽车租凭公车比较合算 C．除去月固定租赁费，甲租赁公司每公里收取的费用比乙租赁公司多 D．甲租赁公司平均每公里收到的费用比乙租赁公司少 把直线y＝2x＋1向下平移2个单位，相当于把它向右平移了（） A．1个单位B．2个单位C．3个单位D．4个单位

例1、下面的表分别给出了变量x与y之间的对应关系,判断y是

例1. 解：（1）y 是x 的函数； （2）y 是x 的函数； （3）y 不是x 的函数，因为对于变量x=1，变量y 有1与-1两个值与它对应； （4）y 是x 的函数 说明：对于x 的每一个值，y 都有唯一的值与它对应.第四个是常数函数它符合函数的定义. 例 2、判断下列关系是不是函数关系？ （1）长方形的宽一定时，其长与面积； （2）等腰三角形的底边长与面积； （3）某人的年龄与身高； （4）关系式| y |=x 中的y 与x. 分析：判断一个关系是不是函数关系，第一要看是不是一个变化过程；第二要看在这个变化过程中，是不 是有两个变量；第三要看自变量每取一个确定值，函数是不是都有唯一确定的值与它对应. 解：（1）长方形的宽一定时，其长所取的每一个确定的值，面积都有唯一确定的值与它对应，所以长与 面积是函数关系. （2）因为三角形的面积受底和高两个因素的影响，当等腰三角形的底取一个定值时，它的面积又受 高的影响，不能有唯一确定的值和底相对应，所以底边长与面积不是函数关系. （3）人的任意一个确定的年龄，都有唯一确定的身高与之相对应，所以某人的年龄与身高是函数关 系. （4）x 每取一个正值，y 都有两个值与它对应，所以| y | = x 不是函数关系. 说明：年龄与身高的变化不按某种规律，但某人每一个确定的年龄，必有唯一确定的身高和 它相对应， 因此函数关系是一定的，所以不要以为存在一定比例关系或一定规律，能用解析式表示的才是函数关系. 例 3、汽车由北京驶往相距850千米的沈阳，它的平均速度为80千米/小时，求汽车距沈阳的路程S （千 米）与行驶时间t （小时）的函数关系式，写出自变量的取值范围. 分析：北京距沈阳850千米，汽车距沈阳的路程等于全程减去已行驶的路程，已行驶的路程等于速度乘以 时间. 解：85080S t =- 0S t ≥??≥? 得850800 t t -?? ≥?

高三数学(文)函数y=f(x)对称性与周期性关系人教版知识精讲

高三数学(文)函数y=f(x)对称性与周期性关系人教版 【本讲教育信息】 教学内容： 函数y f (x)对称性与周期性关系 【典型例题】 1.定义在R上的函数f (x)，若总有f (a x) f (a x)成立，贝U函数f(x)的图象是关于直线x a成轴对称图形。反之，若函数 f (x)的图象关于直线x a成轴对称图形，则 必有f (a x) f (a x) 推论，对于定义在R上的函数，若有f (a x) f (b x)，则f (x)图象关于直线 x旦b 成轴对称图形，反之亦真。 2 证明：若对x R，总有f (a x) f (a x)，设点(x°, f(x。))，在y f (x)的图象上，点(x o,f (x o))关于 x a 的对称点(2a x°, f(x°))，由f(x°) f[a (a x。)] f[a (a X o)] f (2a x°)，则点(2a X o,f(x。))在函数y f (x)的图象上，由x°的任意性知f (x)的图象关于直线x a对称，反之证明略。 ◎ x t 2 a b a b 推论，由f (a x) f (b x) f (- - t) f (— - t)显然 2 2 [例1]已知f(x) x2 bx c，满足f( 1 x) f( 1 x)且f(0) 3，当x 0 时， 比较f (b x)与f (c x)的大小。 解：由f( 1 x) f( 1 x)知f(x)关于x 1对称，故b 2，又由f(0) 3知 c 3，贝U f (x)在(，1]递减，在[1,)上递增。 当x 0 时，3x 2x 1 ??? f (3x) f(2x)即f(b x) f (c x) 当x 0 时，0 3x 2x 1 ???f(3x) f(2x)，即f(b x) f (c x)

求y关于x的函数关系式

如图9，在Rt △ABC 中，∠ACB ＝90°.半径为1的圆A 与边AB 相交于点D ，与边AC 相 交于点E ，连结DE 并延长，与线段BC 的延长线交于点P . （1）当∠B ＝30°时，连结AP ，若△AEP 与△BDP 相似，求CE 的长； （2）若CE=2，BD=BC ，求∠BPD 的正切值； （3）若1 tan 3 BPD ∠=，设CE=x ，△ABC 的周长为y ，求y 关于x 的函数关系式. 图9 图10(备用) 图11(备 用) H H A B C D E P 图3 图2 P E D C B A 图1 P E D C B A 【答案】解：（1）如图1，∵∠ACB=90°, ∠B=30°,∴∠BAC=60°,∵AD=AE=1∴ΔADE 为等边三角形，∴∠ADE=∠AED=60°,∴∠BDE=∠AEP=120°,∠CEP=60°,∴∠EPC=30°=∠B, ∴ΔBDP 为等腰三角形，∵ΔAPE 与ΔBPD 相似，∴ΔAPE 为等腰三角形，AE=EP=1, ∴CE=1 2 EP=12 ． （2）设BC=BD=x ，∠ACB=90°,∴222(1)3x x +=+，∴x =4 ，BC=BD=4， 过D 作DH ⊥BC 交BC 于H,如图2，∴DH ∥AC,∴BD DH BA AC = ,∴453DH =,∴12 5 DH =, 同理可得4 5CH = ,∵DH ∥AC,∴CE PC DH PH =,212455 CP CP =+,∴CP=4, ∵∠ECP=90°,∴ tan BPD ∠=1 2 ． （3）如图3，当1tan 3BPD ∠=时，设CE=x ，∴CP=3x ,由（2）BD DH BA AC = ，CE PC DH PH = ∴设BD=m ，∴(1)1m x DH m += +，3(1)1m x PH m +=+，331 m x CH m -=+，33BC m x =-， ∴222(1)(33)(1)m m x x +=-++，∴1251 (),4 x m x m +==舍，∴y =m ＋1＋x ＋1＋3m －3x=3x ＋3．

y=a(x-h)平方图像性质和求解析式

()2 h x a y -= 平移规律： 1、将抛物线23x y =向右平移2个单位，所得抛物线是（ ） A.()223+=x y B.()223-=x y C.232-=x y D.232 +=x y 2、抛物线()2n x m y +=向左平移2个单位后，得到的函数关系式是()244--=x y ，那么m=______，n=_____。 3、下了二次函数的图像不能通过函数2 3x y =的图像平移得到的 是（ ） A.232+=x y B.()213-=x y C.()2132+-=x y D.22x y = 4、将抛物线22 -=x y 向左平移3个单位，所得抛物线的函数表达式为_____。 开口方向、对称轴、单调性、顶点坐标 1、抛物线()222-=x y 的顶点坐标是_____，在（ ） A.第一象限 B.第二象限 C.x 轴上 D.y 轴上 2、二次函数()243 2-= x y 的开口方向、对称轴、顶点坐标分别是（ ） A.向上，直线x=4，（4，0） B.向上，直线x=4-，（4-，0） C.向上，直线x=4，（0，4） D.向下，直线x=4-，（0，4-） 3、抛物线()233 2+=x y 的开口_______，对称轴是________，顶点坐标为_____；当x ＞3-时，y________；当x=3-时，y 有_____值，是_____。 4、关于抛物线①221x y =；②1212+-=x y ；③()2221-=x y ，下列结论正确的是（ ） A.顶点相同 B 对称轴相同 C.形状相同 D.都有最高点 5、已知二次函数()n x y +--=2 2的图像上有三个点A （1-，1y ），B （2，2y ），C （4，3y ），则1y 、2y 、3y 的大小关系为（ ） 与一次函数图像位置关系 1、在平面直角坐标系中，函数1--=x y 与2)1(2 3--=x y 的图像大致是（ ） A. B. C. D.

设关于x的一次函数与,则称函数

阅读题（函数类） 1、（07绍兴中考）2 2、设关于x 的一次函数11b x a y +=与22b x a y +=，则称函数 )()(2211b x a n b x a m y +++=（其中1=+n m ）为此两个函数的生成函数． （1）当x=1时，求函数1+=x y 与x y 2=的生成函数的值； （2）若函数11b x a y +=与22b x a y +=的图象的交点为P ，判断点 P 是否在此两个函数的生成函 数的图象上，并说明理由． 2、（08绍兴中考）22、定义[]p q ，为一次函数y px q =+的特征数． （1）若特征数是[]22k -，的一次函数为正比例函数，求k 的值； （2）设点A B ，分别为抛物线()(2)y x m x =+-与x y ，轴的交点，其中0m >，且OAB △的面积为4，O 为原点，求图象过A B ，两点的一次函数的特征数． 3、 （09绍兴六校联考）23、定义{a ，b ，c}为函数y=ax2+bx+c 的“特征数”．如：函数y=x2-2x+3的“特征数”是{1，-2，3}，函数y=2x+3的“特征数”是{0，2，3}，函数y=-x 的“特征数”是{0，-1，0} （1）将“特征数”是{0，3 3 ，0}的函数图象向上平移2个单位，得到一个新函数，这个新函数的解析式是 。 （2）在（1）中，平移前后的两个函数分别与y 轴交于O 、A 两点，与直线x= —3分别交于C 、B 两点，判断以A 、B 、C 、O 四点为顶点的四边形形状，请说明理由； （3）若（2）中的四边形（不包括边界）始终覆盖着“特征数”是{1，－2b,b 2 + 2 1} 的函数图象的一部分，求满足条件的实数b 的取值范围？

如何理解y=f(x) 的一些问题

如何理解“()x f y =”的一些问题 王德明 函数概念在初中是这样叙述的： 设在一个变化过程中有两个变量x 与y ，如果对于x 每一个值，y 都有唯一的值与它对应，那么就说y 是x 的函数，x 叫做自变量。这是学生认识函数概念的第二个阶段（算术基础之上），即作为“变化过程”的函数．在高中，函数的概念则是建立在对应基础上的，即作为“对应关系”的函数： 设A ，B 是非空数集，如果按某个确定的对应关系f ，使对于集合A 中的任意一个数x ，在集合B 中都有唯一确定的数()x f 和它对应，那么就称：f A B → 为从集合A 到集合B 的一个函数，记作 ()x f y = ，A x ∈． 其中，x 叫做自变量，x 的取值范围A 叫做函数的定义域；与x 的值相对应的y 的值叫做函数值，函数值的 集合(){}A x x f ∈叫做函数的值域． 在此基础上，将函数定义中的两个集合从非空数集扩展到任意的集合，引出了映射的概念，从而函数成了一种特殊的映射，也顺利拓展了函数的表达方式，对函数的理解产生了质的飞跃．通过对函数()x f y =这一抽象关系式的认识，十分有益于抽象思维能力的提升．当然这一跃，难免使高一学生对()x f y =的理解产生一些疑虑和偏差．为此应该想明白： 1 ()x f y =中的符号f 表示什么 ()x f y =中的f 表示的是确定这个函数的 映射的对应法则．也就是表示y 与x 之间的 函数关系（显然不是f 与x 的积）．由于函数关系可以用解析法、列表法、图象法等表示，所以不能把()x f y =单纯地理解是由解析式给出的函数；其次，当函数关系是一个解析式时，“f ”指的就是运算法则． 如：()11 +=x x f ，“f ”是指对（）内的对象x 进行倒数运算并加1．当然函数的运算法则未