Pulse Shaping and ISI (Proakis: chapter 10.1, 10.3) EEE3012 Spring 2018
Digital Communication System
Introduction Bandlimited channels distort signals the result is smeared pulses intersymol interference (ISI) efore after ISI reduces the noise margin
Baseand Pulse Transmission Pulse shaping can control the ISI Î k { 0,1} pulse shaping transmit filter channel xt () G( jw ) H ( jw ) H ( jw) T c () yt H jw decision R ( ) receive filter sample ì+ a if k = 1 xt ( ) = å Agt k ( - kt) ; A ï k =í k ïî ï- a if k = 0 polar signaling
å () = ( - ) xt Agt kt k k Î k { 0,1} pulse shaping transmit filter channel x() t G ( jw ) H ( jw ) H ( jw ) T c ( ) a ( ) = å k - k y t A p t kt ( ) ( ) ( ) ( ) ( ) a P f = G f H f H f H f ( ) p 0 = 1 T R C () yt H jw decision R ( ) receive filter sample y(t) is sampled at t=nt ( ) a a ( ) ynt = A + å ApnT- kt contriution from the nth it n k k¹ n this is the ISI
to have zero ISI, we need ( ) a é ( ) ISI = a å A a é kp nt- kt = åakp n k T ù ë - û = 0 k¹ n k¹ n this happens with many pulse shapes consider a sinc( ) pulse ( ) = sinc ( t/ T ) pt / ( ) P jw t/ T - /2 /2 w o w o
A series of sinc pulses corresponding to the sequence 1011010.
Is this a good pulse shape? No! A small amount of timing jitter can result in a great deal of ISI. Also ifiit infinite duration non-causal
Nyquist s First Criterion for Zero ISI To have zero ISI, we need ( ) pt ì 1 t = 0 = ï í ï ïî t= nt 0 if p(t) is sampled y an impulse train, then ( t ) d ( t- T ) = d ( t ) å ptåd t nt d t n known as the zero-forcing criterion often a goal of equalizers
( ) d ( - ) = d ( ) å pt t nt t n taking the Fourier transform gives ( ) P f 1 n * åd æ f ö - = 1 T ç è T ø k P( jw) ( ) å P f - nf = T f = 1/ k Nyquist s First Criterion T w T - w w /2 w w
assuming that P(f)=0, f > f = 1/T gives ( ) ( ) 0 P f + P f - f = T < f < f P( w ) ( - ) P w w this happens when w /2 P( w) fold over and add w /2
P( w) cont /2 w form an equivalent pulse y folding over at f /2 zero ISI if the result is rectangular Peq ( w) w /2 the minimum andwidth Nyquist-1 pulse is clearly æ ö sinc t sinc ç èt ø ( ) = i = i [ f t ] p t
Roll-Off Factor Letting the pulse BW grow gives flexiility how does the pulse BW affect the eye? narrower andwidth more ringing in the time domain narrower eyes (in general) the roll-offoff factor parameterizes the BW ( ) P f g g = = f /2 excess andwidth /2 2g f f 2 f f [Hz] also commonly called a
Raised Cosine Pulses A family of pulse shapes with different s invented y Nyquist ~ 1928 cos ( pt/ T ) ( ) = 2 1- ( 2 t / T ) p t 1 2 / sinc / ( t T )
In the frequency domain 1- ìï T f 2 T T P( f) = í ï é1+ cos é f - ùù < f 2 ê ë ê ë ú û ûú 1+ ï 0 f 2T ïî 1-1- 1+ 2T 2T 2T
Eye Diagram Effectively shows the effects of ISI extract each symol period plot on top of each other
Easy to show on a scope f efore and after pre-distortion ti
Interpretation width wasted power slope indicates sensitivity to timing errors noise tolerance zero crossing variation
Works for multi-level systems consider a 4-level system where should the slicing levels go?
Need pulse shapes that give OPEN eyes the sinc pulse is not one of these!
Examples (1) What is the eye diagram RC 1 + j w RC pt ( ) RC= T we have full width pulses and polar signaling strategy? find the filter pulse response & superimpose
the step response is -t/ RC -t/ T ( ) = 1 - = 1 -, ³ 0 st e e t the pulse response is 1 ( ) ( - / ) ( - ( - )/ ) p t = 1 -e u( t) - 1 -e u( t-t ) t T t T T T 2T 3T assume zero after 3 time constants
the filtered data look at all the ISI! T 2T 3T find the resultant
T 2T 3T T 2T 3T eye the est sampling time is clearly at t = nt
- / ( ) ( ) compute the eye opening worst case ISI all destructive ( ) ( - - / ) pt= 1 -e ut ( )- 1 -e ut ( -T ) t T t T T 1 V 2 V 3 T 2T 3T ( -2 T / T ) ( -T / T ) ( -3 T / T ) ( -2 T / T -e - - e + -e - -e ) max ISI = 1 1 1 1 1 = e - = 0.368
The worst case opening is thus é ë -e - e ù û = V -1-1 eye opening = 2 1 0.53 +ve and ve pulses peak ISI wanted signal proaility of error is dominated y the worst case
Examples (2) Effect of channel noise - quaternary system (M = 4) (M-1) = 3 eye openings! - raised cosine pulse shaping ( =05) 0.5) No noise - symol time T = T log 2 M = 2T - no andlimiting SNR = 20 db SNR = 10 db
f o = 0.975 Hz cont Effect of andlimiting - low-pass Butterworth filter 2 1 H( f) = 2 1 + ( f / f ) N o ( N = 25) - signal andwidth BT = W (1 + a) = 0.75 Hz ( W = 0.5 Hz, a = 0.5) f o = 05H 0.5 Hz