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# Cours 6

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Communications Part I3. Nonlinear Modulation Methods0. Introduction: Stochastic Processes4. Spectral Characteristics1. Principles of Digital Transmission3. Demodulation of Dig. Waveforms1. Structure of Data Transmission1. Coherent Demodulation2. Spectra of Data Signals2. Noncoherent3. Nyquist Criterion3. Carrier Recovery4. Partial Response Transmission4. Timing Synchronisation5. Matched Filter6. Bit Error Probability7. Time Multiplex (PCM-Hierachy)2. Digital Modulation1. Lowpass – Bandpass Transform.2. Linear Modulation Methods1-11.1 Structure of Data Transmission Systemsobjective: transmitting discrete values d(i) across an analog channel¥Td(i)d()t- iT? 0di() xt()impulse i =-¥Source gt()Txgeneratordiscrete time continuous time Channel.i Tyt()% xi%()di()Desti- gt()Rxnation? weighting time-shifted analog impulses g (t-iT) with d(i)Tx¥¥Ø ø ?g (t -iT)= T ? ?d (t -iT) g (t) = T ? d (i)x(t) d (i) * Tx ? Tx? 0Œ œi=-¥º i =-¥ ß1-21.2 Spectrum of a Data SignalWhat are the spectral characteristics S (jw ) of x(t) ?XX*r (t +t,t) = E{X (t)X (t +t )}XX¥ ¥2r t -lT - l T g ( )(t +t ,t) = T ? r ( l ) ? g ( ) t -lT +tTxXX ? DD ? Txl =-¥ l=-¥r (t +t,t) = r (t + T +t ,t + T ) ? cyclostationaryXX XXeliminate time dependence by time-averagingT/ 21 E tr (t ) = r (t ,t) dt = T ? r (l) r ( + l )+t TXX ? gg?XX DDT l- T/ 21-3Spectrum of a Data Signalr (t )S ( jw)= { }F XX XX2T E E jwl Tr (t )r (t ) t = T ? r (l) S ( jw )e= r (l) r ( + lT ) F { }gg ? ggXX ? ...

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Communications Part I
0. Introduction: Stochastic Processes 3. Nonlinear Modulation Methods 4. Spectral Characteristics 1. Principles of Digital Transmission 1. Structure of Data Transmission 3. Demodulation of Dig. Waveforms 2. Spectra of Data Signals 1. Coherent Demodulation 3. Nyquist Criterion 2. Noncoherent Demodulation 4. Partial Response Transmission 3. Carrier Recovery 5. Matched Filter 4. Timing Synchronisation 6. Bit Error Probability 7. Time Multiplex (PCM-Hierachy)
2. Digital Modulation 1. Lowpass Bandpass Transform. 2. Linear Modulation Methods
1.1 Structure of Data Transmission Systems
objective: transmitting discrete values d(i) across an analog channel
¥ Source d ( i )impulse T i = å d ( i ) d 0 ( t -iT ) g Tx ( t ) x ( t ) generator
discrete time continuous time Channel i.T Desti-d ( i ) y ( t ) g Rx ( t ) ( ) nation
ð weighting time-shifted analog impulses g Tx (t-iT) with d(i) ¥ ¥ x ( t ) =éê T × å d ( i ) × d 0 ( t -iT ) úù* g Tx ( t ) = T × å d ( i ) g Tx ( t -iT ) ë i =-¥ û i =-¥
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1.2 Spectrum of a Data Signal What are the spectral characteristics S XX ( j w ) of x(t) ? r XX ( t + t , t ) = E X * ( t ) X ( t + t ) ¥ ¥ r XX ( t + t , t ) = T 2 × å r DD l × å g Tx t -l T -l T g Tx t -l T + t l =-¥ l =-¥ r XX ( t + t , t ) = r XX ( t + + t , t + ) ð cyclostationary eliminate time dependence by time-averaging T / 2 r XX ( t ) = 1 ò r XX ( t + t , t ) dt = T × å r DD ( l ) r g E g ( t + l ) -T / 2 l
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Spectrum of a Data Signal S XX ( j w ) = F r XX ( t ) r XX ( t ) = T 2 å r DD ( l ) r g E g ( t + l ) F r XX ( t ) = T × å r DD ( l ) S g E g ( j w ) e j wl T l l S XX ( j w ) G Tx ( j w ) 2 × T × å r DD ( l ) e j wl T with S g E g ( j w ) = G Tx ( j w ) 2 l transmission correlation of filter source data Conclusion The spectrum of the transmitted signal is determined by the correlation of the source data and the transmission filter. Spectral shaping is possible by design of transmission filter or by manipulation of the source data correlation ( ð Partial Response). 1-4
Spectrum of a Data Signal Example for uncorrelated, zero mean d(i) r DD ( l ) = s D 2 d ( l ) with d ( l ) = 01       o l  t h e= r0wise
r XX ( t ) = T × s D 2 × r g E g ( t )
S X ( j w ) = T × s D 2 × G Tx ( j w ) 2 X
In the case of uncorrelated, zero mean source data the spectrum is solely determined by the transmission filter.
Example Possible levels for the a’s : +A and -A 1 1 0 1 0 0 I a  d + 1 R ( k ) = å ( a n a n + k ) i P i n  an a n k A i = 1 (c) Polar NRZ 0 -A    R ( 0) = i å 2 = 1 ( a n a n ) i P i = A 2 21 + ( -A ) 2 12 = A 2 T b For   k ¹ 0, 4    R ( k ) = å ( a n a n + k ) P i = A 2 1 / 4 + ( -A )( A )1 / 4 + ( A )( -A )1 / 4 + ( -A ) 2 1 / 4 = 0 i = 1  ) 2 , 0   Þ R polar ( k =íìî 0 A , kk =¹ 0  n above equation Substitut alon with F f f ( t ) = C ( t / T b ) « ( f ) = T b sin ÕÕ fT b T b i P s ( f ) = F ( f ) 2 å ¥ R ( k ) e 2 p kfTs T s k =-¥ gives 2 sin ö ÷   P polar NRZ ( f ) = A 2 T b æ ç ÕÕ fTf b T b è ø
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time
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PSD for line codes
2 ( ) = 2 æ ç sin Õ fT b ÷ ö P polar NRZ f A T b è Õ fT b ø
If ‘A’ is chosen so that normalized average power of the polar NRZ signal is unity, then A=1
Bit rate: R=1/T b
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Spectral efficiency Definition : Spectral efficiency of a digital signal is given by R bit s where R - data rate  h =       B Hz  B - bandwidth If limited BW is desired, then a signaling technique that has high spectral efficiency is desired. Maximum spectral efficiency (which is limited by channel noise) is given by   h max = BC = log 2 æç 1 + SN ö÷ Shannon’s channel è ø capacity formula Spectral efficiency for multilevel signaling bit s  h = l       Hz
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Intersymbol interference Ø If the rectangular multilevel pulses are filtered improperly as they pass through a communications system, they will spread in time, and the pulse for each symbol may be smeared into adjacent time slots and cause Intersymbol Interference
How can we restrict BW and not introduce ISI? Ø 3 Techniques
Intersymbol Interference Flat-topped multilevel input signal:  w in t = å a n h t -nT s where h ( t ) = Õ æ ç Tt ö ÷ Symbol rate: D = 1   pulses/s è s ø T s Þ  w in t = å a n h t * d t -nT s n                =éêë å a n d ( t -nT s ) ûùú * h ( t ) n Output signal is given by: w out ( t ) =êëé å a n d ( t -nT s ) úûù * h e ( t ) n Equivalent impulse response: he t = h t * hT t * hC t * hR t where h e t is the pulseshape at output Equivalent transfer function: H e f = H f H T f H C f H R f H ( f ) = F é æ t öù T s æ sin p T s f ö where ëê Õ è ç T s ø ÷ ûú=è ç p T s f ø ÷ H e f Receiving filter is given by: H R ( f ) = H ( f ) H T ( f ) H C ( f ) Output signal can be rewritten as: w out t = å a n h e t -nT s n H e (f) overall filtering characteristic (chosen to minimize ISI)
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1.3 1st Nyquist Criterion: Time domain g(t) : impulse response of a transmission system (infinite length) g(t) 1 ß shaping function
0
1 = T 2 f N t 0 equally spaced zeros, -1 interval 1 = T 2 f n
2 t 0
no ISI ! t
1st Nyquist Criterion: Time domain
limitation of length (0 £ t £ 2 t 0 ) by multiplying with a shaping function and sampling (rate f a = T 1 = 2 f N ), t 0 = i 0 T
ð 1st Nyquist Criterion in time domain =íì 1 for i = i 0 g ( iT ) î 0 for i ¹ i 0
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1st Nyquist Criterion: Frequency domain ¥ å G ( j ( w -2 p i T 1 )) = e -w i 0 T i =-¥ G ( j w ) × e + j w i 0 T
f = 2 f N a 4 f N 0 (limited bandwidth)
å = 1
f
1st Nyquist Criterion: Frequency domain G ( j w ) with linear phase: G ( j w ) = G 0 ( j w ) × e -j w i 0 T ( G 0 ( j w ) Î Â ) G 0 ( j w )
D D 1 b a 0,5 a b = f f N 21 T symmetry to f N : Nyquist rolloff
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Definition:
Cosine rolloff filter
ì 1, f < B ï Transfer Function:    H e ( f ) =ïíï 12 ïîïìí 1 + cos éëê p ( 2 ff -D f 1 ) ûúùýïþïü , f 1 < f < B B- Absolute BW ïî 0, f > B   f D = B -f 0 where f o is the 6 - dB bandwidth of the filter  f 1 º f 0 -f D Rolloff factor:  r = f D f 0 é ù Impulse response is given by:  h e t () = F -1 [ H e ( f ) ]= 2 f 0 æ ç è sin2 p 2 f p 0 tf 0 t ÷ öê 1cos ( 42 p f D f t D ) t 2 ú øëê-ûú
Cosine rolloff filter
= × 0 ( ) sin( p tT )1cos((2 r p tT )) 2 g rc t p tT -r tT r : rolloff factor 0 r 1
w 1 2 T £ 1 -r G rc 0 ( j w ) = 21 [1 + cos( 2 p r ( 2 w T + r -1))] if 1 -r £ 2 w T £ 1 + r w 0 2 T ³ 1 + r
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Cosine
rolloff
filter: Examples
Demonstration: Eye pattern
(w=4)
(r=0,5)
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Distorted polar NRZ waveform and corresponding eye pattern Received line code
Effect of channel filtering & channel noise
2nd 1 N st y N q y ui q s u t ist: P 2nd Nyquist: P 1st Nyquist
1st Nyquist: P 2nd Nyquist: O
Resemble human eye Oscilloscope presentations Normal- Eye open Noise - Eye close Information from the eye pattern: Ø Timing error à eye opening Ø Sensitivity à slope of the open eye Ø noise margin à height of the eye openin 1 g -19
Cosine rolloff filter: Eye pattern
1st Nyquist: P 2nd Nyquist: O
1st Nyquist: P 2nd Nyquist: O
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Cosine rolloff filter: Bandwidth efficiency
d(i)
E data rate 1/ T 2 bit/s ff rc = bandwidth = (1 + r ) / 2 T = 1 + r Hz
1bit/s £ 2 < 2bit/s Hz (1 + r ) Hz ò ò 2nd Nyquist (r=1) r=0
1.5 Matched Filter
Noise n a (t) i T g Tx (t) g Rx (t) ?
task: design a g Rx (t) that maximizes the SN -Ratio
d ( i ) + n ( i ) S ® max
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