成都信工学院：《现代通信原理》（英文版） OFDM Systems N subcarriers

Single carrier system Bandwidth= B Frequency Symbol duration t=1/B Whenτmax>T >Intersymbol interference(sl) (IMHZ Time T=1/B(1μs) Equaliser is needed to reduce si

Time Single Carrier System T=1/B (1s) Frequency B (1MHz) When _max>T →Intersymbol interference (ISI) Bandwidth = B Symbol Duration T=1/B Equaliser is needed to reduce ISI

OFDM Systems(N subcarriers) frequency · Available bandwidth is divided into n subband B Each symbol △f occupies a narrow (kHz) band but longer time lme period T(Ims) For each subcarriers Bandwidth△f=B/N ymbol duration T=NB

Frequency Time T (1ms) B OFDM Systems (N subcarriers) f (1kHz) •Available bandwidth is divided into N subband •Each symbol occupies a narrow band but longer time period For each subcarriers Bandwidth f=B/N Symbol duration T=N/B

已 Source Serial >Encoding>如 Lowpass 个⑧个 Filiter Parallel 2πfN-1 Channel Binary Data Parallel T Decoding to Filter Seria JafAr j2IfN-lt Baseband model of an oFdM communication system For the uniform subcarrier spacing fk=k△k=0,1,…,N-1

e j2 f0t e j 2 fN− 1t e j2 fN− 1t e j2  f0t e j2 fk t e j2 fk t Baseband model of an OFDM communication system f k f k =  k = 0,1,  ,N −1. - - - For the uniform subcarrier spacing

The basic orthogonal function for the kth subcarrier is defined as (,k)=exp(2n/)0≤t<T, Ig(t, k)=O, otherwise The basic functions satisfies the condition of orthogonality g(t,k)·g(t,p)dt=0,k≠p, T (t,k).g(t, p)dt=lg(t, k) dt=T, k=p

( )    = =   ( , ) 0, . ( , ) exp 2 , 0 , g t k otherwise g t k j f k t t T       = = =  =     T T T g t k g t p dt g t k dt T k p g t k g t p dt k p 0 2 0 * 0 * ( , ) ( , ) ( , ) , . ( , ) ( , ) 0, , The basic orthogonal function for the kth subcarrier is defined as The basic functions satisfies the condition of orthogonality

Structure of oFDM systems using FFT X(h) x(n) S(1) binary data encoding Guard Lowpass arrier (BPSK IFFT Insertion filtering modulation nanne y(k) y(n y'(n Binary data decoding lowpass (BPSK.) FFT Guard filtering Carrier deleting demodulation converting Channel OFDM symbol estimation synchronisation

Structure of OFDM systems using FFT Encoding (BPSK…) IFFT Guard insertion Lowpass filtering Carrier modulation Binary data Channel Carrier demodulation Lowpass filtering A/D converting Guard FFT deleting decoding (BPSK…) Binary data OFDM symbol synchronisation Channel estimation X(k) x(n) x’(n) s(t) y’(k) y(n) y’(n) r(t)

10 9 8 7 58 3 6 T ap 210 100-80 60-40 200 100 Frequency (Hz) Scattering function of a medium-range tropospheric scatter channel

Example of OFDM signals data Real part: cos(2Ifit) Imaginary part: sing (2IfiO) OFDM

Example of OFDM signals data Real part: cos(2fk t) Imaginary part: sing(2fk t) 1 -1 -1 1 1 OFDM

Comparison with FDMA 0.6 chI ch2 ch3 s04 802 Frequency guar -0.2 Subcarrier number k

0 1 2 3 4 5 6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Subcarrier Number k Spectra Comparison with FDMA ch1 ch2 ch3 Frequency guard

Assume the information data sequence be X(), k=0, I. N-1, then the transmitted signal s(t) can be expressed as )=∑X(D)p(27) =∑X(plep(12m△)0≤1<7,0≤p≤N and T OFDM Symbol period △f The received signal r(t) is ()=s()*()+() h(t: Channel response; w(O:AWGN signal

Assume the information data sequence be X(k), k=0,1…N-1, then the transmitted signal s(t) can be expressed as ( ) ( ) ( ) ( )exp( 2 ), 0 ; 0 1. exp 2 1 0 1 0 =      − =   − = − = X p j p f t t T p N s t X p j f t N p p N p   f T  = 1 and The received signal r(t) is r(t) = s(t)*h(t)+w(t) h(t):Channel response; w(t):AWGN signal OFDM symbol period

The received on subcarrier ko is y()=7(Oxp(12△f Assume h(=l, w(t=0 (ideal channel), then y() becomes Y()=J 2X(p) 2 rpaf exp(= Afn)

The received on subcarrier k0 is ( ) r(t) ( j k f t)dt T Y k T  = −  0 exp 2 1  Assume h(t)=1, w(t)=0 (ideal channel), then Y(k) becomes ( ) ( ) ( ) ( )   − = =  −  T N p X p j p f t j k f t T Y k 0 1 0 exp 2 exp 2 1  