Communication Technologies

Transcription

1 Communication Tchnologis. Principls of Digital Transmission. Structur of Data Transmission.2 Spctrum of a Data Signal 2. Digital Modulation 2. Linar Modulation Mthods 2.2 Nonlinar Modulations (CPM-Signals) 2.3 Spctral Charactristics 3. Rcivr Structurs 3. Cohrnt Dmodulation 3.2 MSK-Dmodulator 3.3 Carrir Rcovry 3.4 Noncohrnt Dmodulation 4. AWGN Transmission 4. Matchd Filtr (Whit Nois) 4.2 S/N MF in Basband for Ral and Complx Data 4.3 Bit Error Probability 4.4 Optimal Rcivr (MAP/ML) 5. Linar Equalization 5. Multipath Channl / Frquncy Slctiv Channl 5.2 T-Equalizr: LS/MMSE-Solution 5.3 T/2-Equalizr: LS-Solution 5.4 Dcision-Fdback Equalizr

2 Why do w nd to qualiz? Frquncy-Slctiv Channls mobil scattrr transmittr mobil rcivr fixd scattrr Multipath radio channls causd.g. from rflctions from objcts v 2

3 Multipath Channls (Frquncy-Slctiv Channls) Tim domain n h( t) xp( j ) ( t ) (unlimitd channl bandwidth) Dlayd vrsions of transmittd signal at rcivr intrsymbol intrfrnc (ISI) ow much ISI is a function of th signal duration T (or signal bandwidth /T) and th maximum dlay max of channl: If T >> max ) ngligibl ISI Frquncy domain n ( j) xp( j )xp( j ) channl varis ovr frquncy frquncy-slctiv channl If signal bandwidth B << cohrnc bandwidth (¼ / max ) ) ngligibl ISI 3

4 Linar Equalization Digital transmission systm (symbol rat /T) Channl n(t) T k w g Tx (t) h c (t) g Rx (t) Equalizr Downsampling w Impuls rspons of ntir channl : h t g t h t g t Sampld impuls rspons: hkt / w Tx c Rx hk / w for k w othrwis w: Ovrsampling factor finit lngth impuls rspons no ISI! Our task is to dsign an qualizr (k) such that a Nyquist-impuls (no ISI at tim instancs it) rsults at its output aftr down-sampling 4

5 n+l y( k) h( k) x( k) h( ) x( k ) n Convolutional Matrix causal input x( k) [ x(), x(),, x( L )] y() h() x() y() h() x() h() x() Full quation systm y(2) h(2) x() h() x() h() x(2) y( n) h( n) x() h( n ) x() h() x( n) y( n ) h( n) x() h() x( n ) T Toplitz structur Equation systm in matrix notation with convolutional matrix L h() y() h() h() y() h(2) h() h() y(2) x() h(2) h() x() h() x(2) hn ( ) hn ( ) xl ( ) hn ( ) h( n) y( L n) : causality : finit impuls rspons 5

6 6 Equalization: Dfinitions Vctor algbra i : Elmnt of vctor,, n n n m n m mn m m m m M m n m n mn m m m m M n n T,, n n T : Transpos of vctor : rmitian of vctor Matrix M M : rmitian of matrix M * i: Conjugat lmnt of vctor * * * * n n

7 Communication Tchnologis. Principls of Digital Transmission. Structur of Data Transmission.2 Spctrum of a Data Signal 2. Digital Modulation 2. Linar Modulation Mthods 2.2 Nonlinar Modulations (CPM-Signals) 2.3 Spctral Charactristics 3. Rcivr Structurs 3. Cohrnt Dmodulation 3.2 MSK-Dmodulator 3.3 Carrir Rcovry 3.4 Noncohrnt Dmodulation 4. AWGN Transmission 4. Matchd Filtr (Whit Nois) 4.2 S/N MF in Basband for Ral and Complx Data 4.3 Bit Error Probability 4.4 Optimal Rcivr (MAP/ML) 5. Linar Equalization 5. Multipath Channl / Frquncy Slctiv Channl 5.2 T-Equalizr: LS/MMSE-Solution 5.3 T/2-Equalizr: LS-Solution 5.4 Dcision-Fdback Equalizr 7

8 Linar Equalization: Conditions Zro forcing solution: i! h k k i i,,,,,,,, k i w st Nyquist condition N B n w w Numbr of sampls aftr downsampling N B conditions for calculating n qualizr cofficints: N B n w w n qualizr ordr: n w w w n! 8

9 Linar Equalization: T-Equalizr w = : No xact solution (with n < ) for mting st Nyquist Critrion possibl Last Squars approximation: Introduc a rsidual (rror i) and minimiz its nrgy Vctor algbra: n +l - = i + i h i h h i h2 h h i h3 h2 h h i 2 h3 h2 h h i 3 h3 h2 h i 4 h3 h 2 i h 3 i n with i ; h h it it i 9

10 Linar Equalization: T-Equalizr LS solution: minimiz th nrgy of th rror i (rsidual): min i i i Δi Δi Δi Normal quations: LS i ) Last squars (LS) solution for qualizr cofficint vctor : LS i if ( ) - xists Enrgy of rsidual (rror vctor) for LS: Δi Δi i LS min (= iff - xists, i 2 of columns spac of )

11 Linar Equalization: T-Equalizr Problms with LS T-qualization: LS qualizr is adaptd to channl only, not on nois n LS qualizr amplifis nois ) high nois amplification, if qualizr cofficints ar larg (if zros of channls ar clos to unit circl) LS qualization assums prfct knowldg of channl ) channl stimation is rquird

12 Moor-Pnros-Psudoinvrs min Ax b 2 Th solution of th gnral problm x x A b A + is th (nxm)-psudo invrs of th (mxn) matrix A. is givn by For an ovrdtrmind systm, A + givs us th shortst vctor x. It is givn as follows dpnding on its rank: A A A A A A A A AA m n, rank m m n, rank n m n, rank A n A A - 2

13 Moor-Pnros-Psudoinvrs (cont.) Singular valu dcomposition of A: A UΣV Psudo-invrs in trms of SVD: A VΣ U V is unitary (mxm)-matrix of Eign vctors of A A U is unitary (nxn)-matrix of Eign vctors of AA / for Σ with for m 3

14 Linar Equalization: T-Equalizr MMSE (Minimum Man Squar Error) - solution which considrs nois Equalizr input at tim sampl i: xi hid i ni Minimizing th powr of (qualizr output original data): qualizr cofficint vctor: 2 ˆ FMMSE =E d i d i i min rror: i n qualizr input vctor: x() i xi xi n di : Data ni : Nois qualizr output: ˆ d i x() i 4

15 ACF-matrix: Linar Equalization: T-Equalizr E xx * * * rxx rxx rxx n r r r n! xx xx xx R xx rxx n r CCF-vctor of data and qualizr input: In vctor notation: r xx * * du to r E x i x i E x i x i r xx ( ) * xd E xd i i rmitian Toplitz matrix F MMSE R r r 2 xx xd xd d min 5

16 Linar Equalization: T-Equalizr MMSE - solution for qualizr cofficint vctor : R r MMSE xx xd rror nrgy: 2 Error nrgy includs channl and nois (both in x) ) MMSE balancs nois and ISI to minimiz rror nrgy R xx and r xd ndd for calculation of R xx : stimatd from rcivd data min r F MMSE d xd MMSE r xd : contains original data trainings squnc (pilots) ncssary no nois, zro man uncorrlatd data and prfct knowldg of : LS solution = MMSE-solution 6

17 Linar Equalization: Exampls (w = ) E s /N =5dB Last Squars (ZF) powr of ISI nois rror,98,4487,4585 MMSE powr of ISI nois rror,73,78,5 7

18 Communication Tchnologis. Principls of Digital Transmission. Structur of Data Transmission.2 Spctrum of a Data Signal 2. Digital Modulation 2. Linar Modulation Mthods 2.2 Nonlinar Modulations (CPM-Signals) 2.3 Spctral Charactristics 3. Rcivr Structurs 3. Cohrnt Dmodulation 3.2 MSK-Dmodulator 3.3 Carrir Rcovry 3.4 Noncohrnt Dmodulation 4. AWGN Transmission 4. Matchd Filtr (Whit Nois) 4.2 S/N MF in Basband for Ral and Complx Data 4.3 Bit Error Probability 4.4 Optimal Rcivr (MAP/ML) 5. Linar Equalization 5. Multipath Channl / Frquncy Slctiv Channl 5.2 T-Equalizr: LS/MMSE-Solution 5.3 T/2-Equalizr: LS-Solution 5.4 Dcision-Fdback Equalizr 8

19 Linar Equalization: T/2-Equalizr (w = 2) T zro forcing: h( kt / 2) h, h/2, h, h3/2,, h l2 /2 2,,, n 2 n 2 +l 2 2 = i h h/ 2 h h h/ 2 h h3/ 2 h h/ 2 h h2 h3/ 2 h h/ 2 h h5/ 2 h2 h3/ 2 h h/ 2 2 h3 h5/ 2 h2 h3/ 2 h 3 h3 h5/ 2 h2 h 3/ 2 4 h3 h5/ 2 h 2 h3 h5/ 2 h 3 n 2 + T st Nyquist cond. considrs symbol rat /T, not (w/t) Arbitrary valus of no intrst 9

20 Linar Equalization: T/2-Equalizr w = 2: For th spcific choic of th lngth of th qualizr n w can lav out vry 2nd row and obtain a quadratic matrix 2 : 2 2 n = i 2 h/ 2 h h h h h 3/ 2 / 2 h5 / 2 h2 h3/ 2 h h/ 2 2 h3 h5 / 2 h2 h3/ 2 3 h3 h 5 / 2 4 n 2 + zro forcing solution (xact) for qualizr cofficint vctor i : non-singular T/2-qualizr: no ISI at qualizr output (prfct qualization) Rmmbr: LS/ZF T-qualizr minimal in LS sns rmaining ISI at output 2

21 Linar Equalization: T/2-Equalizr Drawback: Equalizr cofficints may bcom vry larg high nois amplification Solution: W allow for largr qualizr ordr than ncssary ( n2 n2) and prform a LS-approach with an additional minimization of th cofficint nrgy 22 2 = i + i with 2 2 cost function : FLS i i 2 2 min I i n n Not: W solv an undrtrmind quation systm by LS with rgularization trm Rgularization trm: 22 Zro forcing solution (last squars) for qualizr cofficint vctor 2 : 2

22 Linar Equalization: Exampls (w = 2) h norm. channl impuls rspons z-plan of th channl no ISI larg nois amplification h 2 (k) 2 (k)* 2 (k) kt/2 xact solution (n=9), 2 2 = k 5 h 2 (k)* 2 (k) Im{z} R{z} ZF, last squars, n=6, 2 2 = k ISI small nois amplification many mor cofficints 22

23 Signal Spac for QPSK (no nois) w=, MMSE.5.5 Im{y(i)} Im{y(i)} w=, Zro Forcing R{y(i)} w=2, xact.5.5 Im{y(i)} Im{y(i)} R{y(i)} w=2, with SC R{y(i)} R{y(i)} 23

24 Signal Spac for QPSK (ES/N=5 db) w=, MMSE Im{y(i)} Im{y(i)} w=, Zro Forcing R{y(i)} w=2, xact Im{y(i)} Im{y(i)} R{y(i)} w=2, with SC - R{y(i)} R{y(i)}

25 S/ISI in db Influnc of Sampling Tim Offst Not that sampling is applid aftr th usd rciv filtr, but bfor qualization G( j2 f ) cos-roll-off 45 T-Equalizr T/2-Equalizr Sampling: it+ r= (roll-off) w / 2T aliasing /T f 4 35 G( j2 f ) 3 25 w 2 / 2T /T 2/T f /T T-Equalizr: dos not mt sampling thorm vry snsitiv to sampling offst T/2-Equalizr: mts sampling thorm vry robust to sampling offst 25

26 Influnc on whit nois Assum: root-cosin matchd filtring and n(t) zro man whit nois T-Equalizr: Aliasing nois is still whit aftr matchd filtr (qualizr input) T/2-Equalizr: no aliasing nois is colord aftr matchd filtr (qualizr input) 26

27 T-Equalizr Summary of Linar Equalizrs T/2-Equalizr Last Squars Solution LS i Last Squars Solution (Zro Forcing) i 2 2 MMSE Solution R r MMSE XX XD 2 N MMSE I i (for uncorrlatd data and whit nois) LS Solution (n largr than ncssary and minimal cofficint nrgy) I i 27

28 Communication Tchnologis. Principls of Digital Transmission. Structur of Data Transmission.2 Spctrum of a Data Signal 2. Digital Modulation 2. Linar Modulation Mthods 2.2 Nonlinar Modulations (CPM-Signals) 2.3 Spctral Charactristics 3. Rcivr Structurs 3. Cohrnt Dmodulation 3.2 MSK-Dmodulator 3.3 Carrir Rcovry 3.4 Noncohrnt Dmodulation 4. AWGN Transmission 4. Matchd Filtr (Whit Nois) 4.2 S/N MF in Basband for Ral and Complx Data 4.3 Bit Error Probability 4.4 Optimal Rcivr (MAP/ML) 5. Linar Equalization 5. Multipath Channl / Frquncy Slctiv Channl 5.2 T-Equalizr: LS/MMSE-Solution 5.3 T/2-Equalizr: LS-Solution 5.4 Dcision-Fdback Equalizr 28

29 Nonlinar Equalization: Dcision Fdback Rcivd signal, sampld with symbol rat /T: x i h d( i ) n( i) h d( i) h d( i ) n( i) d: Data h ; : Impuls rspons of channl and filtr (g TX, g RX ) symbol of intrst, i.. i = Ida: At sampling instant it w alrady hav dcidd symbols If w assum prfct prcding dcisions,! subtract prcding dcisions from data d(i) and tak channl h v into account signal aftr subtraction: h h x i h d( i ) d( i) n( i) Symbol d(i) + (amplifid) nois AWGN (no ISI) 29

30 Dcidd symbol Nonlinar Equalization: Dcision Fdback ˆ d( i) Q x i b d( i ) ; b h ˆ h h dcision oprator prviously dcidd data 3

31 Nonlinar Equalization: Dcision Fdback xi () yi () FIR pr-qualizr,,, n Without pr-qualizr: n b y i x i b dˆ i d i q h T /h + - yq () i dcision b b 2 b nb z z z dcision: dˆ i d ˆ( i i ) n b : # of fdback cofficints (mmory of fdback structur) 3

32 Problm: Nonlinar Equalization: Dcision Fdback Error propagation du to dcision rrors. If first impuls h of channl is small ) high amplification of nois (high SNR loss) )high probability of dcision rror )First impuls h should b larg (not that hr: i =) Solution for any i : FIR pr-qualizr () kills sampls of h(k) from to tim (i -)T Subsquntly, DF qualizr (b) kills sampls from (i +)T to nd Exampls of dsign approachs: sparatly ( : ZF or MMSE, b : MMSE) or jointly (, b : MMSE) Joint dsign for and b basd on MMSE-critria,,, n T T b b, b2,, b nb 2 yq i d i i x d b d i i E E min 2 b, y () q i 32

33 Nonlinar Equalization: Dcision Fdback T * d d( i i ),, d( i i n b ) rxd E xd i i R x d xd E 2 MMSE-solution for uncorrlatd data, r MMSE R xx 2 xdr xd rxd d R bmmse R 2 xd MMSE d dd : d 33

34 h(i)*(i) h(i)*(i) h(i) DF-Equalization 4 crucial zros n b = 4 aftr prqualizr 34

35 DF-Equalization : ZF or MMSE, b : ZF (sparat dsign) b(k) quals xactly th last 4 sampls of h(k) * (k) b(k) quals xactly th last 4 sampls of h(k) * (k) 35