Problem Formulation in Communication Systems

Transcription

1 Problem Formulaion in Communicaion Sysems Sooyong Choi School of Elecrical and Elecronic Engineering Yonsei Universiy

2 Inroducion Problem formulaion in communicaion sysems Simple daa ransmission sysem : Digial communicaions Code division muliple access (CDMA) sysems Orhogonal frequency division muliplexing (OFDM) sysems Muliple-inpu muliple-oupu (MIMO) sysems Problem Formulaion in Communicaion Sysems by Sooyong Choi 2

3 Problem formulaion Digial communicaion sysems ransmier (XM or x) Par Receiver (RCV or Rx) Par : Reverse sep of x par Modem = modulae + demodulae Problem Formulaion in Communicaion Sysems by Sooyong Choi 3

4 Problem formulaion Simple daa ransmission h( ) c( ) g ( ) Modulaed informaion symbol a Channel h w : addiive whie Gaussian noise y Equalizer g ~ a Esimaed informaion _ signal a -D Problem Formulaion in Communicaion Sysems by Sooyong Choi 4

5 Problem formulaion Simple daa ransmission Modulaed informaion symbol x Channel h n : addiive whie Gaussian noise r Equalizer g ~ x Esimaed informaion _ signal x -D r h x w i i i M r h x n i i i M : Number of channel aps, r h x hx n h h h x x xmn, M r Hx n, r r r r N, H h n n n nn, h h h M h h h hm h h h h h Problem Formulaion in Communicaion Sysems by Sooyong Choi 5 M

6 Problem formulaion Code division muliple access (CDMA) sysems Binary message b() Binary PN signal s() Signals in ime and Frequency Domain Modulaor (BPSK) Carrier Asin(2f c +) DS/SS-BPSK signal s()=ab()s()sin(2f c +) C Daa Spreading Code(ransmier) Modulaed daa Problem Formulaion in Communicaion Sysems by Sooyong Choi 6

7 Problem formulaion Code division muliple access (CDMA) sysems Received signal r() =Ab(-)s(- ) sin(2f c + ) PN signal synch. s(- ) Local PN generaor r() = Ab(-)s(- ) sin(2f c + ) ^ b( ) Ab ( -τ ) s( - τ )sin(2 f c Symbol iming recovery Carrier recovery ') s( - τ sin(2f c + ) Mached Filer )sin(2 f c ') d ˆ ( )d b ( ) 2 Ab ( -τ ) s( - τ ) (.5){ - cos(4 f c 2 ')} d 2.5 Ab ( -τ ) s( - τ ) d Ab ( -τ ) Problem Formulaion in Communicaion Sysems by Sooyong Choi 7

8 User User 2 User 3 User 4 User User 2 User 3 User 4 Problem formulaion Code division muliple access (CDMA) sysems User Daa User Daa Spreading Spread Code (Modulaed) daa Spreading 4 h 3 rd 2 nd s 4 h 3 rd 2 nd s Deecion Sum Despread daa h 3 rd 2 nd s Sum & Normalizaion Mached filering s 2 nd 3 rd 4 h Channel & AWGN Problem Formulaion in Communicaion Sysems by Sooyong Choi 8

9 Problem formulaion Code division muliple access (CDMA) sysems r r K A ( ) b s n Consider he wo User CDMA Sysem 2 A ( ) b s n A ( ) b s A ( ) b s n( ) y A b, A b z y A b A b z , y=rab+z Correlaor : A( ) b s s d r() A ( ) b s s d s n( ) d A() b A () b z() 2, 2 2 y y y=hx+z 2 2, Correlaor # Correlaor #2 Correlaor #K, A 2 A 2 b b Linear Operaor 2 L z z 2 b b 2 b K Problem Formulaion in Communicaion Sysems by Sooyong Choi 9

10 Problem formulaion Orhogonal frequency division muliplexing (OFDM) sysems { X } { Y } Frequency domain ime domain N N j2 m m xm ( ) X exp XW N NIDF( ), m,,..., N N X N N j2 m m X x( m)exp x( m) WN DF( ),,,..., N, N m N N m N X Equivalen model WN 2 exp j N Problem Formulaion in Communicaion Sysems by Sooyong Choi

11 Problem formulaion Orhogonal frequency division muliplexing (OFDM) sysems x ang. # Sub sig. # 5 Sum(real) 5 IFF(real) x ang. #2 Sub sig. #2 5 Sum(image) 5 x -6 IFF(image) x ang. #3 -.5 Sub sig. # Comparison(real) x ang. #4 -.5 Sub sig. # Comparison(image) Problem Formulaion in Communicaion Sysems by Sooyong Choi

12 Inersymbol Inerference ransmied signal: Received Signals: Line-of-sigh: Refleced: he symbols add up on he channel Disorion! Delays Problem Formulaion in Communicaion Sysems by Sooyong Choi 2

13 Concep of Parallel ransmission Channel impulse response ime Frequency Channel ransfer funcion Channel (serial) Frequency Signal is broadband 2 Channels Frequency 8 Channels Frequency Channels are narrowband Problem Formulaion in Communicaion Sysems by Sooyong Choi 3

14 SCM and MCM SCM (Single Carrier Modulaion) MCM (Muli Carrier Modulaion) Problem Formulaion in Communicaion Sysems by Sooyong Choi 4

15 Principles of Muli-Carrier Modulaion Channel impulse response Daa on single carrier f f Single carrier - Shor pulse duraion - ISI over several symbols - A equalizer wih a large number of aps subch. f Mulicarrier wih 4 subchannels subch. 2 subch. 3 subch. 4 f f 2 f 3 f f f 2 f 3 f Muli-carrier - N parallel longer duraion pulses - Shorer ISI relaive o he symbol duraion - Guard band (a) ime domain (b) Frequency domain Problem Formulaion in Communicaion Sysems by Sooyong Choi 5

16 OFDM Principle Problem Formulaion in Communicaion Sysems by Sooyong Choi 6

17 OFDM Principle Orhogonaliy Principle j2 f j j2 f j : e e d d j2 f 2 j f j j: e e d e d e j2 f f j2 f f j2 f f j2 f f j2 f f j2 f f j2 f f j j j j e j j e j2 f f j2 f f j j 2 j 2 j 2 j e e e e e j f f j f f j f f j j e j2 f f j2 f f j j2 f f j2 n Problem Formulaion in Communicaion Sysems by Sooyong Choi 7 e j f f j n

18 Efficien OFDM Modem OFDM Modem : IFF/FF pair Modem of OFDM symbol OFDM modem n N n j n n N n n s s e X X x ) ( 2,,, ) ( ) ( s s s n n j s n s n d e x d x X ) ( 2 *,, ) ( ) ( ) ( 2 e j 2 ( ) e N j () x X X N X () n 2 e j 2 ( ) e N j ˆX ˆX ˆ N X Problem Formulaion in Communicaion Sysems by Sooyong Choi 8

19 Efficien OFDM Modem ransmied nh OFDM symbol x n () Sampling rae: s /N x( ) N n X,, ( ) n n N n X n, e 2 j ( ns ) s, N N j2 m m xm ( ) X exp XW N NIDF( ), m,,..., N N X N N j2 m m N N m N N m N X X x( m)exp x( m) W DF( ),,,..., N 2 WN exp j N Problem Formulaion in Communicaion Sysems by Sooyong Choi 9

20 Orhogonaliy ime domain Freq. domain Problem Formulaion in Communicaion Sysems by Sooyong Choi 2

21 Vecor Channel Model Consider a frequency selecive wireless lin and bandwidh B Channel: L impulse response g[l] (l=,,,l-) Daa symbols: s[] ( =,, 2,, N-) o be ransmied, uni average energy (N: power of 2) s=[s[] s[] s[n-]] Problem Formulaion in Communicaion Sysems by Sooyong Choi 2

22 Vecor Channel Model ransmier firs performs an IFF operaion H s D s where D is an N N marix whose mnh ( m, n, 2,... N) elemens is given by j2 ( m)( n) N [ D] mn, e For example, N 8, N j j j j j j j e e e e e e e j j j j j j j e e e e e e e j j j j j j j e e e e e e e D j j2 j3 j4 j5 j6 j7 8 e e e e e e e j j j j j j j e e e e e e e j j j j j j j e e e e e e e j j j j j j j e e e e e e e Problem Formulaion in Communicaion Sysems by Sooyong Choi 22

23 Vecor Channel Model A new sequence, s is now consruced by appending a cyclic prefix (CP) of lengh L- o he vecor s and ransmied OFDM symbol s and symbol duraion OFDM s sn [ L] sn [ ] OFDM N L s ', s s [] B sn [ ] Problem Formulaion in Communicaion Sysems by Sooyong Choi 23

24 Vecor Channel Model Received vecor y is lengh N+2L-2 ha comprises he OFDM symbol convolved wih he channel of lengh L y E ' s Gs n n is he addiive ZMCSCG noise vecor wih covariance marix NoIN G is an N( N L) oepliz marix derived from he channel impulse response s(7) For example, L=3 and N=8, s(8) g(2) g() g() s() g(2) g() g() s(2) g(2) g() g() s(3) g(2) g() g() s ' G s(4) g(2) g() g() s(5) g(2) g() g() s(6) g(2) g() g() s(7) g(2) g() g() s(8) Problem Formulaion in Communicaion Sysems by Sooyong Choi 24

25 Vecor Channel Model Firs L- samples of s are idenical o he las L- samples We can reduce he complexiy in compuing by changing G g[] g[2] g[] g[] g[] g[2] g[2] g[] g[] g[2] g[] g[] G c g[2] g[] g[] g[2] g[] g[] g[2] g[] g[] g[2] g[] g[] y can be simplified as y E G s n s c H Now, G is circulan and eigendecomposiion may be expressed as G = D ΩD c s() s(2) s(3) s(4) s ' s(5) s(6) s(7) s(8) c Problem Formulaion in Communicaion Sysems by Sooyong Choi 25

26 Vecor Channel Model Gc H D ΩD Ω diag[ [], [],..., [ N-]], wih L l j2 l N [ ]= g[ l] e,,, 2,..., N [ ] is he sampled frequency response of he channel, where represens he one index. Receiver performs a FF y Dy E s DGcs Dn H E DG D s Dn E s s y [ ] E[ s ] [ ] n [ ],,, 2,..., N s c Ωsn Problem Formulaion in Communicaion Sysems by Sooyong Choi 26

27 Problem formulaion OFDM Received samples in he ime domain yn ( ) xn ( ) hn ( ) wn ( ) Frequency domain signal model Y( m) FF( y( n)) X() X() XN ( ) X( m) H( m) W( m) H() H() HN ( ) W() W() WN ( ) Y() Y() YN ( ) Y Y() Y() YN ( ) H Vecor form Y XH W X() X() X XN ( ) H() H() HN ( ) W W() W() WN ( ) Problem Formulaion in Communicaion Sysems by Sooyong Choi 27

28 Problem formulaion MIMO MIMO : Spaial Muliplexing Muliple anennas a boh ends of a radio lin is required. Increase in daa rae by ransmiing simulaneously independen daa sreams hrough differen anennas. No channel nowledge a ransmier is required. Channel capaciy is maximized if H becomes orhogonal. H H is a marix channel Problem Formulaion in Communicaion Sysems by Sooyong Choi 28

29 Problem formulaion MIMO Parallel decomposiion of he MIMO channel Obained by a ransformaion : ransmi precoding & receiver shaping ransmi precoding Linear ransformaion on he channel inpu vecor x~ as x V~ x Receiver shaping Muliplying he channel oupu y wih U H ransmi precoding and receiver shaping ransform he MIMO channel ino R H parallel single-inpu singleoupu (SISO) channels wih inpu x~ and oupu ~ y ransmi Precoding and Receiver Shaping. Problem Formulaion in Communicaion Sysems by Sooyong Choi 29

30 Problem formulaion MIMO Parallel decomposiion of he channel H y U ( Hxn) H H U ( UΣV xn) H H U ( UΣV Vx n) H H H U UΣV Vx U n Σx n H n ~ U n (2) Parallel Decomposiion of he MIMO Channel : Diagonal marix of singular values of H wih σ i on he ih diagonal and zeros everywhere else Muliplicaion by a uniary marix does no change he disribuion of he noise n and n ~ H U n are idenically disribued. he ransmi precoding and receiver shaping ransform he MIMO channel ino R H parallel independen channels ih channel : Inpu =, Oupu =, Noise = n ~ x~i y~i i, Channel gain = σ i Problem Formulaion in Communicaion Sysems by Sooyong Choi 3

31 Problem formulaion MIMO Spaial Muliplexing If scaering is rich enough (i.e., H has a high ran), several daa pipes are creaed wihin he same bandwidh. SVD H H=UΛV w V V H m w m U U H Muliplexing gain a no exra bandwidh or power, bu only from spaial domain expansion. Problem Formulaion in Communicaion Sysems by Sooyong Choi 3

32 Problem formulaion MIMO Spaial Muliplexing h 2 h 2 h H h h h 2 h 2 22 h 22 C MIMO = log 2 de[i +(SNR/2) HH * ]= Inerpreaion: ransmier Where he i are he eigenvalues o HH * SNR SNR log 2 log min(n r, n ) parallel channels : ransmi simulaneously independen daa sreams he he min(n r, n ) channel channels spaial muliplexing Receiver Problem Formulaion in Communicaion Sysems by Sooyong Choi 32

33 Problem formulaion MIMO MIMO Channel Capaciy Specular Channel: small angular spread or LOS Only one pipe is used C beamforming = log 2 (+SNR ) [bi/(hz s)] Scaering Channel: large angular spread or NLOS C m MIMO 2 i SNR log i n Capaciy increases linearly wih min(n r,n ). Change he muliplexing order depending on he channel sae Problem Formulaion in Communicaion Sysems by Sooyong Choi 33

34 Problem formulaion MIMO Degree of Freedom (Muliple Anennas) Signals arrived in muliple direcions provide muliple degrees of freedom for communicaion. Differen signals from he same direcion canno provide muliple degrees of freedom. Problem Formulaion in Communicaion Sysems by Sooyong Choi 34

35 Problem formulaion MIMO Degree of Freedom (MIMO) Signals arrived in muliple direcions provide muliple degrees of freedom for communicaion. he same effec can be obained via rich scaering even when anennas are close ogeher. m by n channel wih rich scaering : min(m, n) degrees of freedom y y Hxn y M r h h M x n hmr h MrM x M n Mr H Problem Formulaion in Communicaion Sysems by Sooyong Choi 35

36 Problem formulaion Simplified communicaion sysem Linear equaion Modulaed informaion symbol x Channel H n: addiive whie Gaussian noise r Receiver w x~ Deecion Esimaed informaion signal x _ r r Hx n r r rn n h h hm h h hm SISO mulipah 2, A H : H channel,2 A 2 h h hm h h h h OFDM M H :, MIMO fla channel H : h channel N 2 hmr h MrM h N, n n nn, ; CDMA Problem Formulaion in Communicaion Sysems by Sooyong Choi 36