Introduction to Orthogonal Frequency Division Multiplexing (OFDM)

Transcription

1 Wireless Iformaio Trasmissio Sysem Lab. Iroducio o Orhogoal Frequecy Divisio Muliplexig OFDM Isiue of Commuicaios Egieerig aioal Su Ya-se Uiversiy

2 OFDM Overview OFDM Sysem Model Orhogoaliy Oulie Muli-carrier Equivale Implemeaio by Usig Iverse Discree Fourier Trasform IDFT Cyclic Prefix CP Summary 2

3 OFDM Overview 2/3 OFDM Orhogoal Frequecy Divisio Muliplexig The basic priciple of OFDM is o spli a high-rae daa sream io a umber of lower rae sreams ha are rasmied simulaeously over a umber of sub-carriers. I elimiaes or alleviaes he problem of muli-pah chael fadig effec, ad low specrum efficiecy. 3

4 OFDM Overview 1/3 Sigle carrier SC vs. muli-carrier MC Sigle carrier : daa are rasmied over oly oe carrier Selecive fadig Muli-carrier : daa are shared amog several carriers ad simulaeously rasmied Fla fadig per subcarrier 4

5 OFDM modulaio OFDM Overview 3/3 Feaures o iercarrier guard bads Overlappig of bads Specral efficiecy Easy implemeaio by IDFTs Very sesiive o sychroizaio Problems of Peak o Average Power Raio PAPR 5

6 OFDM Trasmier a cos 2πf b Ipu f s Daa d Ecoder 1 { a + jb } f f + f, f 1 S/P si cos a 1 b 1 2πf 2πf 1 MUX D Chael si 2πf 1 Figure 1 6

7 OFDM Sysem Model 1/3 A OFDM sysem rasmier is show i Figure 1. The rasmied waveform D ca be expressed as D where f 1 { a cos2πf + b si2πf } 1 f 1 + f ad f Usig a wo-dimesioal digial modulaio forma, he daa symbols d ca be represeed as a + jb a : i-phase compoe b : quadraure compoe 7

8 OFDM Sysem Model 2/3 +1 a T a cos 2πf b Ipu Daa d Ecoder 1 { a + jb } f s S/P si cos a 1 2πf 2πf 1 MUX D Chael +1 1 a T a 1 a 1 b 1 si 2πf 1 8

9 OFDM Sysem Model 3/3 The serial daa elemes spaced by are grouped ad used o modulae carriers. Thus hey are frequecy divisio muliplexed. The sigalig ierval is he icreased o, which makes he sysem less suscepible o chael delay spread impairmes. Small-scale fadig Based o mulipah ime delay spread Fla Fadig 1. BW of sigal < BW of chael 2. Delay spread < Symbol preiod Frequecy Selecive Fadig 1. BW of sigal > BW of chael 2. Delay spread > Symbol preiod 9

10 1 Orhogoaliy 1/3 Cosider a se of rasmied carriers as follows: We ow show ha every wo carriers are orhogoal o each oher. <proof is i ex page> I summary: ,,1, for 2 + e f j π ψ a b q p q p a b d b a q p ad for for * ψ ψ

11 11 Orhogoaliy 2/3 --- Proof a b q p q p j e e q p j e e d e d b a q p j b q p j a q p j b q p j b a q p j b a q p ad for, * π π ψ ψ π π π π π

12 Orhogoaliy 3/3 Very Sesiive o Frequecy Errors 12

13 Mahemaical Expressio of OFDM Sigal 1/2 From above, we kow ha ψ is he orhogoal sigal se. A OFDM sigal based o his orhogoal sigal se ca be wrie as: where f d k, x ψ f a e + k, Re j 2πf T +, k, 1 k jb for T { } d k, ψ kt,1, 2,..., 1 3 T 13

14 Mahemaical Expressio of OFDM Sigal 2/2 T : OFDM symbol duraio d k,, : rasmied daa o he -h carrier of he k-h symbol x Re 1 1 k, ψ kt 2πf kt b si 2πf kt { a cos } k, k, 4 k k C If here is oly oe OFDM symbol i.e. k, i ca be simplified as: x 1 { a cos2πf b si2πf } 5 14

15 Muli-carrier Equivale Implemeaio by usig IDFT 1/6 Accordig o he srucure of he rasmier, rasmiig a OFDM sigal requires oscillaors, a exremely high cos for hardware implemeaio. Soluio: A equivale mehod for rasmiig OFDM sigals is usig iverse discree Fourier rasform IDFT. 15

16 Muli-carrier Equivale Implemeaio by usig IDFT 2/6 I geeral, each carrier ca be expressed as: S c A c e j 2π f +φ c c 6 We assume ha here are carriers i he OFDM sigal. The he oal complex sigal S s ca be represeed by: where ad of - A f, φ, h carrier, 1 Ss f + f f 1 A are ampliude, respecively. 16 e j 2πf + φ phase, 7 carrier frequecy

17 Muli-carrier Equivale Implemeaio by usig IDFT 3/6 The we sample he sigal a a samplig frequecy 1, ad A ad φ becomes: S s k φ A 1 1 φ A A e The he sampled sigal ca be expressed as: j π f + f k + φ π Ss k j 2πf k + φ j2 k f A e e 11 17

18 Muli-carrier Equivale Implemeaio by usig IDFT 4/6 The iverse discree Fourier rasform IDFT is defied as he followig: f k 1 1 F f Comparig eq.11 ad eq.12, he codiio mus be saisfied i order o make eq.11 a iverse Fourier rasform relaioship: f 1 e j 2πk

19 Muli-carrier Equivale Implemeaio by usig IDFT 5/6 If eq.13 is saisfied, j 2π fk +φ A e is he frequecy domai sigal S s k is he ime domai sigal f is he sub-chael spacig is he symbol duraio i each sub-chael This oucome is he same as he resul obaied i he sysem of Figure 1. Therefore IDFT ca be used o geerae a OFDM rasmissio sigal. 19

20 Muli-carrier Equivale Implemeaio by usig IDFT 6/6 a cos 2πf b Ipu f s d 1 { a + jb } f f 1 + f, f si cos a 1 b 1 d d 1 d 2 si 2πf 2πf 1 2πf 1 D Chael Ipu f s d 1 { a + jb } f f 1 + f, f f d 1 D Chael 2

21 Cyclic Prefix I mulipah chael, delayed replicas of previous OFDM sigal lead o ISI bewee successive OFDM sigals. Soluio : Iser a guard ierval bewee successive OFDM sigals. 21

22 Cyclic Prefix Guard ierval leads o iercarrier ierferece ICI i OFDM demodulaio I DFT ierval, differece bewee wo subcarriers does o maiai ieger umber of cycles loss of orhogoaliy. Delayed versio of subcarrier 2 causes ICI i he process of demodulaig subcarrier 1. 22

23 Cyclic Prefix Cyclic prefix CP : A copy of he las par of OFDM sigal is aached o he fro of iself. Ipu daa symbols d[ ] d[ ] d[ 1] d[ 2] d[ 1] D D[] D[] 1 D[] 2 [ g ] [ ] D 1 D ~ [ k] 23

24 Cyclic Prefix All delayed replicas of subcarriers always have a ieger umber of cycles wihi DFT ierval o ICI 24

25 OFDM Receiver r~ r Oupu daa Symbols 25

26 Cyclic Prefix Oe of he mos impora reasos o do OFDM is he efficie way i deals wih mulipah delay spread. To elimiae ier-symbol ierferece ISI, a guard ime or cyclic prefix is iroduced for each OFDM symbol. The legh of he guard ime or cyclic prefix is chose o be larger ha he delay spread. 26

27 Badwidh Efficiecy I a classical parallel sysem, he chael is divided io o-overlappig sub-chaels o avoid ier-carrier ierferece ICI. The diagram for badwidh efficiecy of OFDM sysem is show below: 27

28 Summary The advaage of he DFT-based OFDM sysem : The use of IDFT/DFT ca reduce he compuaio complexiy. The orhogoaliy bewee he adjace sub-carriers will make he use of rasmissio badwidh more efficie. The guard ierval or cyclic prefix is used o resis he iersymbol ierferece ISI. The mai advaage of he OFDM rasmissio echique is is high performace eve i frequecy selecive chaels. The drawbacks of he OFDM sysem : I is highly vulerable o sychroizaio errors. Peak o Average Power Raio PAPR problems. 28