IMPERIAL COLLEGE LONDON DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING. COMPACT LECTURE NOTES on COMMUNICATION THEORY. Prof. Ahanassios Manikas, version Auumn 2008 Digial Communicaions: An Overview of Fundamenals Consellaion Diagram & Line Codes
6. Digial Modulaors GENERAL BLOCK STRUCTURE OF A DIGITAL COMMUNICATION SYSTEM H( f) ^ ^ ^ ^ ^ ^ Digial Communicaions - An Overview of Fundamenals 1 a poin T : =Ð>Ñ waveform. The digial modulaor akes # cs -bis a a ime a some uniform rae r cs and ransmis one of Q =2 # cs disinc waveforms = " ÐÑ,...,sQ ÐÑ i.e. we have an # cs -bi-sequence is ransmied Q-ary communicaion sysem. A new waveform corresponding o a new every seconds If # cs =" we have one bi a a ime œ 0 È= " i.e. a binary comm. sysem 1 È= # a poin T^ : noisy waveform <Ð>Ñ œ 5 =Ð>Ñ 8Ð>Ñ. The ransmied waveform =Ð>Ñ, affeced by he channel, is received a poin T^ a poin B^ : a binary sequence. based on he received signal <Ð>Ñ he digial demodulaor has o decide which of he Q waveforms = 3Ð>Ñ has been ransmied in any given ime inerval X -= Digial Communicaions - An Overview of Fundamenals 2
..01..101..00.. 0..00 s 1( )= 0..01 s 2( )= : H, Pr(H ) 1 1 : H, Pr(H ) 2 2 s ()= 1..11 s M( )= : H M, Pr(H M) s () Channel..00..101..10.. r ()=()+n() s 0..00 s 1( )= 0..01 s 2( )= 1..11 s M( )= : D 1, Pr(D 1) : D 2, Pr(D 2) : D M, Pr(D M) D Deecor wih a Decision Device (Decision Rule) r ()= Digial Communicaions - An Overview of Fundamenals 3 If Qœ# ÊBinary Digial Modulaor ÊBinary Communicaion Sysem If Q # ÊM-ary Digial Modulaor ÊM-ary Communicaion Sysem œ Ô PrÐH" ll" Ñß PrÐH" ll# Ñß ÞÞÞß PrÐH" llqñ PrÐH# ll" Ñ, PrÐH# ll# Ñ, ÞÞÞß PrÐH# llqñ Ö Ù. ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÕPrÐH ll Ñ, PrÐH ll Ñ, ÞÞÞ PrÐH ll ÑØ O " O # O Q Digial Communicaions - An Overview of Fundamenals 4
ì Binary Com. Sysems: use Qœ2 possible waveforms Ö = Ð>Ñß = Ð>Ñ à! " Q-ary Com. Sysems: use Q possible waveforms Ö = Ð>Ñß ÞÞÞß = Ð>Ñ à 1 Q ì The Q signals (or channel symbols) are characerized by heir energy I 3 X I œ ' -= # 0 = Ð>Ñ..> à a3 Ð1Ñ 3 3 Furhermore heir similariy (or dissimilariy) is characerized by heir cross-correlaion " 3 34 II 3 4 X œ ' -= * È = Ð>Ñ = Ð>Ñ.> 2 3 4 0.. ÐÑ Digial Communicaions - An Overview of Fundamenals 5 6.1. Firs Modelling of Digial Modulaors Digial Communicaions - An Overview of Fundamenals 6
Noe: How o ransform a sequence of 0's and 1's o a sequence of 1's i/p Ä o/p œ" # i/p Ä o/p Digial Communicaions - An Overview of Fundamenals 7 6.1. Examples of Binary Modulaors A binary digial modulaor maps 0's and 1's ono wo analogue symbol- È= Ð>Ñ waveforms s0ðñ and s1ðñ, ha is œ 0! 0 Ÿ Ÿ T "È= Ð>Ñ cs " The hree basic binary digial modulaors Ú Ý 0 È = 9 Ð>Ñ œ E! ÞcosÐ# 1J ->Ñ ASK ( Ampliude Shif-Keyed) Û " È = " Ð>Ñ œ E" ÞcosÐ# 1J ->Ñ Ý Ü for!ÿ>ÿx-= Ú 0 È = 9Ð>Ñ œ E-ÞcosÐ# 1J ->Ñ PSK ( Phase Shif-Keyed) Û " È = " Ð>Ñ œ E-ÞcosÐ# 1J -> ")! Ñ Ü for!ÿ>ÿx Ú 0 È = 9Ð>Ñ œ E- ÞcosÐ# 1J! >Ñ FSK ( Frequency Shif-Keyed) Û " È = " Ð>Ñ œ E- ÞcosÐ# 1J " >Ñ Ü for!ÿ>ÿx -= -= Digial Communicaions - An Overview of Fundamenals 8
The above binary digial modulaors using complex represenaion: Ú 0 È= 9 Ð>ÑœEÞ! expð# j 1J>Ñ - ASK ( Ampliude Shif-Keyed) Û "È=Ð>ÑœEÞ " " expð# j 1J>Ñ - Ü for!ÿ>ÿx -= Ú œ E- èëëéëëê Ý 0 È= 9Ð>Ñœ E-expÐ!ÑÞ j expð# j 1J>Ñ - PSK ( Phase Shif-Keyed) Û "È=Ð>Ñœ " ðóóóóñóóóóò EÞ - expð")!ñ j expð# j 1J>Ñ - Ý œ E- Ü for!ÿ>ÿx -= Ú 0 È= 9Ð>ÑœEÞ - expð# j 1J>Ñ! FSK ( Frequency Shif-Keyed) Û "È=Ð>ÑœEÞ " - expð# j 1J>Ñ " Ü for!ÿ>ÿx -= Digial Communicaions - An Overview of Fundamenals 9 6.2. Examples of M-ary Modulaors Digial Communicaions - An Overview of Fundamenals 10
6.3. An Imporan M-ary Modulaor ( Q œ '%Ñ A 64-ary Modulaor: Based on Walsh-Hadamard marix " " ì 1= c1 d # œ marix " " Ð# #Ñ 64 œ ðóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóò # Œ # Œ # Œ # Œ # Œ # 6-imes where Œ denoes he Kronecker produc of wo marices e.g. for wo marices and if œ E "" E "# hen E E #" ## Œ œ E"" E"# E E #" ## Digial Communicaions - An Overview of Fundamenals 11 ì Properies: ˆ T 64 œ '% ˆ 64 '% ˆ Le H 3 denoe he 3-h column of 64 i.e. 64 œ ch ß H ß ÞÞÞß H " # '% Then he following example illusraes an applicaion of "Walsh Hadamard" as a 64-ary Modulaor demodulaor d Digial Communicaions - An Overview of Fundamenals 12
Digial Communicaions - An Overview of Fundamenals 13
Sysems: 1. Coheren (if demodulaor is coheren) 2. Non-coheren (if demodulaor is non-coheren) Noe: Opimum demodulaors are coheren. Digial Communicaions - An Overview of Fundamenals 14 6.4. Second Modelling of Digial Modulaors: Signal Consellaion ì We may represen he signal = Ð>Ñ 3 by a poin ÐA =3 Ñin a H-dimensional Euclidean space wih H ŸQÞ ì The se of poins (vecors) specified by he columns of he marix œ ca ß A ß ÞÞÞß A = = = " # Q d is known as "signal consellaion". H I 3 œa A œ¼a ¼ ÐÑ 3 = = = 3 3 3 # " 3 34 II = H. = œ A A œ È 3 4 3 4 A = H. A 3 = 4 ½A ½Þ½A ½ = 3 = 4 ÐÑ 4 Digial Communicaions - An Overview of Fundamenals 15
6.5. Disance beween wo M-ary signals ì The disance beween wo signals = 3Ð>Ñ and = 4Ð>Ñ is he Euclidean disance beween heir associae vecors A and A i.e.. 34 = = 3 4 œ ½A A œ ÊŠ A A Š A A œ œ ÉAHA AH A #AHA œ ÉI I 23 ÈI I Ê = = 3 4 ½ H = = = = 3 4 3 4 = = = = = = 3 3 4 4 3 4 3 4 34 3 4. # œ I I 23 ÈI I 34 3 4 34 3 4 ì I is clear from he above ha he Euclidean disance. 34 associaed wih wo signals = 3Ð>Ñ and = 4Ð>Ñ indicaes, like he cross-correlaion coefficien, he similariy or dissimilariy of he signals. Digial Communicaions - An Overview of Fundamenals 16 6.6. Examples of Consellaion Diagram Consider an M-ary Sysem having he following signals Ö = 1Ð>Ñ, = 2Ð>Ñ,..., = Q Ð>Ñ wih! Ÿ>ŸX -= ì M-ary ASK ˆ channel symbols: = 3Ð>Ñ œ E3ÞcosÐ2 J >Ñ 9< = 3Ð>Ñ œ E3Þexp 4# 1J - > ˆdimensionaliy of signal space œ Hœ" ˆ if Qœ% hen 1 - where E3 œ given= #3 " Q (say) Es 1 Es 2 Es 3 Es 4 Origin ˆ if Qœ) hen 00 01 11 10 s1( ) s2( ) s3( ) s4( ) Es 1 Es 2 Es 3 Es 4 Origin Es 5 Es 6 Es 7 Es 8 000 s1( ) 001 s2( ) 011 s3( ) 010 s4( ) 110 s5( ) s 6( ) 101 s7( ) 100 s8( ) Digial Communicaions - An Overview of Fundamenals 17
ì M-ary PSK # ˆ channel symbols: = 3Ð>Ñœ EÞ cosˆ # 1J > 1 - Q Þ3 " 9 3 œ "ß #ß ÞÞß # 9< = 3Ð>Ñ œeþexpˆ 4ˆ 1 Q Þ 3 " 9 exp 4# 1J> - ˆdimensionaliy of signal-space œ Hœ2 ˆ if Qœ% & 9 œ! hen if Qœ% & 9 œ45 hen 01 Es 2 s 2( ) Es 2 01 s2( ) Es1 s1( ) 00 Es 3 11 s 3( ) Origin s 1( ) Es 1 00 Origin 45 0 s4( ) 10 11 s 3( ) Es 3 s 4( ) Es 4 10 Es 4 Digial Communicaions - An Overview of Fundamenals 18 ˆ if Qœ8 & 9 œ! hen Es 4 010 s4( ) Es 3 011 s3( ) Es 2 001 s2( ) 110 Es 5 s5( ) Origin 000 s1( ) Es 1 s6( ) 101 Es s 6 7( ) 100 s8( ) Es 8 Es 7 Digial Communicaions - An Overview of Fundamenals 19
ì M-ary FSK À very difficul o be represened using consellaion diagram Digial Communicaions - An Overview of Fundamenals 20 ì M-ary Ampliude & Phase - M-ary QAM À ˆ channel symbols: = Ð>ÑœE 3 Þ cos # 1J - > : 9 3œ"ß#ßÞÞßQ 3 3 9< = Ð>Ñ œ E 8 Þ cos # 1J - > : 9 87 7 ˆdimensionaliy of signal-space œ Hœ2 PAM-PSK QAM Ú 8 œ "ß #ß ÞÞß Q1 Û 7 œ "ß #ß ÞÞß Q Ü QœQ Q # " # Digial Communicaions - An Overview of Fundamenals 21
ì Two Represenaive Examples:..01..101..00.. 0..00 s 1( )= 0..01 s 2( )= : H, Pr(H ) 1 1 : H, Pr(H ) 2 2 s ()= 1..11 s M( )= : H M, Pr(H M) s () Channel..00..101..10.. r ()=()+n() s 0..00 s 1( )= 0..01 s 2( )= 1..11 s M( )= : D 1, Pr(D 1) : D 2, Pr(D 2) : D M, Pr(D M) D Deecor wih a Decision Device (Decision Rule) r ()= For an ASK M=4 and a QPSK Communicaion Sysem he figure above is equivalen o: Digial Communicaions - An Overview of Fundamenals 22 Equivalen Model of an ASK (M=4) Communicaion Sysem Digial Communicaions - An Overview of Fundamenals 23
Equivalen Model of a QPSK Communicaion Sysem Digial Communicaions - An Overview of Fundamenals 24 Remember: A QPSK modulaor has four "channel-symbols" which are described by he following equaion: # 1 = 3( > ) = Ecos(2 1J -> + Q ( 3 " ) 9) for 3=1,2,3,4 ÐÑ 5 wih Q =4 and!ÿ>ÿx -= and he modulaion process is described by he so called "consellaion diagram". The previous figure shows he consellaion diagram for 9 œ 45. From his figure i is clear ha he consellaion diagram shows he mapping of binary digis o QPSK channel symbols (consellaion poins) as well as he square roo of he energy ÈI = of he channel symbols. The diagram also indicaes a Gray code mapping from binary digis o channel symbols (consellaion poins). Digial Communicaions - An Overview of Fundamenals 25
For insanceß if he bi-pair a he inpu of he QPSK modulaor is "01" hen he oupu is he waveform (channel symbol) =Ð>Ñ. # Overall, using complex number represenaion, is is clear from he consellaion diagram ha!! È = " Ð>Ñ œ ðóóóóñóóóóò ÈI= exp(j45 ) exp 4# 1J -> œ7"! 1 È = # Ð>Ñ œ ðóóóóóñóóóóóò ÈI= exp(j135 ) exp 4# 1J -> œ7# 11 È = $ Ð>Ñ œ ðóóóóóñóóóóóò ÈI= exp(j225 ) exp 4# 1J -> œ7$ 1!È =Ð>Ñœ % ðóóóóñóóóóò ÈI= exp(j45 ) exp 4# 1J -> œ7 % ÐÑ 6 Digial Communicaions - An Overview of Fundamenals 26 In oher words, =Ð>Ñœ7 3 3exp 4# 1J - > for 3œ1,2,3 and 4 ÐÑ 7 I is imporan o poin ou ha he four symbols 7ß7ß7 " # $ and 7% are known as baseband QPSK "channel symbols" and are used by he "QPSK consellaion symbol mapping" block shown in he previous figure. The erm exp 4# 1J - > in Equaions 6 and 7 is shown seperaely in he previous figure, in order o indicae he up-conversion from baseband o bandpass. In a similar fashion he down-conversion from baseband o bandpass is shown a he receiver's fron-end using he complex conjugae of he ransmier's carrier, i.e. using exp 4# J >. 1 - Thus, overall, we have exp 4# 1J > exp 4# 1J > =1. - - Digial Communicaions - An Overview of Fundamenals 27
Based on he above discussion i is clear ha he presence of he carrier does no affec he analysis of he sysem. Therefore, i is common pracice o ignore he carrier when analysing communicaion sysems, by working on he baseband. For he res of his explanaory noe he carrier erm will be ignored from boh Tx and Rx. Digial Communicaions - An Overview of Fundamenals 28 Summarising, a QPSK modulaor/demodulaor is represened by is consellaion diagram and he QPSK symbol mapper ransforms he binary sequence o a sequence of QPSK complex channel symbols 7 3, forming he baseband QPSK message signal 7Ð>Ñ of bandwidh F 3Þ/Þ 7ß7ß7 " # $ 9<7% å 7Ð>Ñ aò8ó. -Ð> n. X-= Ñà nt-= Ÿ Ðn "Ñ.T-= 8 where Ú Ý Û Ý Ü > -Ð>Ñ œ recš X -= ÖÒ8Ó a œ sequ. of independen daa symbols Ðmi Ñ Fœ " X -= Digial Communicaions - An Overview of Fundamenals 29
Thus, wih reference o he binary sequence of bis "001001" (message), by looking a he QPSK consellaion diagram i is clear ha he we have he following mapping 00 7" œ E exp(j %& ) "! 7% œ E exp(j $"& )!" 7# œ E exp(j "$& ) where Eœ I È = Noe ha, for a binary sysem mð>ñ å " aò8ó. -Ð> n. X-= Ñà nt Ÿ Ðn "Ñ.T where Ú Ý Û Ý Ü n > -= -= -Ð>Ñ œ recš X -= ÖÒ8Ó a œ sequ. of independen daa bis Ð 1sÑ Fœ " X -= Digial Communicaions - An Overview of Fundamenals 30 Examples of Decision Variables (QPSK- Receiver) 1.5 8 x 106 1 6 4 0.5 2 Im 0 Im 0-0.5-2 -4-1 -6-1.5-1.5-1 -0.5 0 0.5 1 1.5 Re 'good' -8-8 -6-4 -2 0 2 4 6 8 Re x 10 6 'bad' Digial Communicaions - An Overview of Fundamenals 31
6.7. Performance Evaluaion ì In general he qualiy of a digial communicaion sysem is expressed in erms of he accuracy wih which he binary digis delivered a he oupu of he deecor represen he binary digis ha were fed ino he digial modulaor. ì I is generally aken ha i is he fracion of he binary digis ha are delivered back in error ha is a measure of he qualiy of he communicaion sysem. This fracion, or rae, is referred o as he bi error probabiliy : /, or, Bi- Error-Rae BER. Digial Communicaions - An Overview of Fundamenals 32 ì The performance of Q-ary communicaion sysems is evaluaed by means of he average probabiliy of symbol error : /ß-=, which ß for Q #ß is differen han he average probabiliy of bi error (or Bi-Error-Rae BER), : /. Tha is œ : /ß-= :/ß-= Á :/ for Q # œ : for Q œ # / (i.e. Binary Communicaion Sysems) However, because we ransmi binary daa, he probabiliy of bi error a more naural parameer for performance evaluaion han. : /ß-= : / is Digial Communicaions - An Overview of Fundamenals 33
Alhough, hese wo probabiliies are relaed i.e. : / œ f{ : /ß-= } heir relaionship depends on he encoding approach which is employed by he digial modulaor for mapping binary digis o Q-ary signals (channel symbols) Digial Modulaor bis Ä 3-h word of #-= bis èëëëëëëëëëëëëëëëëëëëéëëëëëëëëëëëëëëëëëëëê 1s bi 2nd bi...//... # -h bi cs È= 3 Ä channel symbols where # cs œ log 2 QÞ Digial Communicaions - An Overview of Fundamenals 34 >2 >2 ì An Imporan Bound involving. 34 (disance beween he 3 and 4 consellaion poin À ˆ The symbol error probabiliy is bounded as follows: : /ß=- Q Q. :/ß-s Ÿ Ð= 3Ñ 34 Pr T{ } Ð8Ñ 3œ" 4œ" 4Á3 È#R! ì definiion of he "minimum disance":. min œ mine. 34f a3ß 4 Digial Communicaions - An Overview of Fundamenals 35
6.8. Performance of BINARY Ðequiprobable Ñ DIGITAL MODUL./DEMODULATORS ì Consider a Binary Communicaions sysem 0 È= " œ or a more popular noaion: 0 È= 1 È= œ 1 È= # Consellaion diagram À 0 1 = 0 = 1 ÈI ÈI. 0 1 0 1. È#R! 3 p e œ T{ } œ ÞÞÞ=T{ ÈÐ" 3Ñ. EUE } Digial Communicaions - An Overview of Fundamenals 36 ì Thus, a he oupu of an opimum digial demodulaor he probabiliy of error can be calculaed by using he following expression: p e œ T{ ÈÐ" 3Ñ. EUE } Ð9Ñ where EUE œ E N! and PSD n ÐfÑ œ N! i # Ú " X-= E.' # # œ #! Š s! ÐÑ s" ÐÑ d Ý œ average signal energy wihû " X-= Ý 3 œ.' E! s! ÐÑ s" ÐÑ d Ü œ he ime cross-correlaion beween signals Digial Communicaions - An Overview of Fundamenals 37
N.B.: Ð" Ñ œ Å Êp œ Æ 3 E N e! if E N! œ fixed hen he opimum sysem is ha for which he correlaion coeff is " i.e. 3 œ 1 Ê s!ðñœ s"ðñ Ð10Ñ This is known as opimum, or ideal binary Communicaion Sysem p e ρ=1 ρ= -1 ρ=0 EUE Digial Communicaions - An Overview of Fundamenals 38 Examples - Baseband MODEMS: ˆAnipodal Ä 0 È=Ð>Ñœ E œ "È=Ð>ÑœE! - " - 0 Ÿ Ÿ ˆ =Ð>Ñ œ E-7Ð>Ñ Ð ÖÒ8Ó a œsequ. of independen daa bis Ð 1sÑ A - ˆ p= / T{ } 5 COHERENT MODEMS: 1. Ampliude Shif-Keyed ÐASK Ñ or On-Off Keying ÐOOKÑ 0 ˆ ÐASK or OOK Ñ Ä È=Ð>Ñœ!! œ 0 Ÿ Ÿ T "È=Ð>ÑœE " -cos # 1J> - ˆ sðñœe -.mðñ. cosð# 1F - Ñ É E N ˆ p= / T{ # } = 1 0 cs Digial Communicaions - An Overview of Fundamenals 39
2. Biphase Shif-Keyed: ìgeneral Ä 0 È=Ð>ÑœE! -cos # 1J> -?) œ "È=Ð>ÑœE cos # 1J>?) " - - 0 Ÿ Ÿ p = T{ È2.EUE. sin# Ð?) Ñ} / ì Phase-Reversal Keying ÐRSKÑ 0 È=Ð>Ñœ! E -sin # 1J> - ÐRSK Ñ Ä œ "È=Ð>ÑœE sin # 1J> " - - 0 Ÿ Ÿ p / = T{ È 2.EUE } Digial Communicaions - An Overview of Fundamenals 40 ì N.B.: for?)= 1 2 hen general=rsk and i is called BPSK BPSK s ÐÑœE-. sinð# 1J - > 7Ð>Ñ. # Ñ Ð12Ñ Equaion Ð11 Ñ can be rewrien as follows: 1 BPSK s ÐÑœE-.m ÐÑ. cosð# 1F - Ñ Ð13Ñ... BPSK can be considered as PM œ AM The PSD ÐÑ f 's of mðñ and =Ð>Ñare shown below: Digial Communicaions - An Overview of Fundamenals 41
Digial Communicaions - An Overview of Fundamenals 42 3. Frequency Shif-Keyed ÐFSKÑ ÐFSK Ñ À 0 È=Ð>ÑœE! -cos # 1J> - œ "È=Ð>ÑœE cos # 1 J? 0 > " - - m 0 Ÿ Ÿ,? f= #Xcs Digial Communicaions - An Overview of Fundamenals 43
Digial Communicaions - An Overview of Fundamenals 44 1 ASK : œ / TœÉ " # EUE # non-coheren ASK ÐNASK Ñ p= 0.5 expš 0.5TœÉ $ FSK : œ Tš ÈEUE % FSK (non-coheren) : œ E= E= / % N # N / / exp " " # # 1 1 0 0 EUE & BPSK :/ œ Tš È2 EUE " ' BPSK (differenial) :/ œ # exp e EUEf ( QPSK : œ Tš È2 EUE ) MSK : œ Tš È1.7 EUE * Gaussian MSK : Tš È1.36 EUE / / / "! M-ary PSK (coheren) : 2Tš È4 EUE sinˆ 1 / #Q " "" M-ary QAM :/ 4 Š 1 TœÉ ÈQ $ Q " EUE Digial Communicaions - An Overview of Fundamenals 45
Digial Communicaions - An Overview of Fundamenals 46 6.9. LINE CODES Digial Communicaions - An Overview of Fundamenals 47
Digial Communicaions - An Overview of Fundamenals 48?????? Digial Communicaions - An Overview of Fundamenals 49
Digial Communicaions - An Overview of Fundamenals 50 Digial Communicaions - An Overview of Fundamenals 51
Digial Communicaions - An Overview of Fundamenals 52 EXAMPLE Show ha for a bipolar line code he auocorrelaion funcion of he code sequence Öa[ 8 ] is as follows: Ú "Î# if 5 œ! VaaÐ5Ñ œ Û "Î% if 5 œ " Ü! if 5 # > X -= If PrÐ!Ñ œ PrÐ "Ñ œ!þ& and -Ð>Ñ œ recš ß derive an expression for he power specral densiy for he bipolar line code waveform 8= 7Ð Ñ œ aò8ó. -Ð> 8X Ñ -= Digial Communicaions - An Overview of Fundamenals 53
Digial Communicaions - An Overview of Fundamenals 54 Digial Communicaions - An Overview of Fundamenals 55