Inter symbol Interference

1. Digital Communications
 Inter Symbol Interference (ISI)
 Nyquist Criteria for ISI
 Pulse Shaping and Raised-Cosine Filter
 Eye Pattern
 Equalization

2. Baseband Communication System
 We have been considering the following baseband system
 The transmitted signal is created by the line coder according
to 
n
where an is the symbol mapping and g(t) is the pulse shape
Problems with Line Codes
 One big problem with the line codes is that they are not bandlimited
 The absolute bandwidth is infinite
 The power outside the 1st null bandwidth is not negligible. That
is, the power in the sidelobes can be quite high
s(t)   an g(t  nTb )

3.  If the transmission channel is bandlimited, then high frequency
components will be cut off
– Hence, the pulses will spread out
– If the pulse spread out into the adjacent symbol periods, then it is
said that intersymbol interference (ISI) has occurred
Intersymbol Interference (ISI)
 Intersymbol interference (ISI) occurs when a pulse spreads out in
such a way that it interferes with adjacent pulses at the sample instant
 Causes
– Channel induced distortion which spreads or disperses the pulses
– Multipath effects (echo)
Intersymbol Interference (ISI)

4. – Due to improper filtering (@ Tx and/or Rx), the received pulses overlap one
another thus making detection difficult
 Example of ISI
– Assume polar NRZ line code
Pulse spreading

5. – Input data stream and bit superposition
 The channel output is the sum of the contributions from each bit
Inter Symbol Interference

6. Note:
 ISI can occur whenever a non-bandlimited line code is used
over a bandlimited channel
 ISI can occur only at the sampling instants
 Overlapping pulses will not cause ISI if they have zero
amplitude at the time the signal is sampled
ISI

7. ISI Baseband Communication System Model
where hT (t)  Impulseresponseof the transmitter,
hC (t)  Impulseresponseof thechannel,
hR (t)  Impulseresponseof the receiver

s(t)   anhT (t  nT),
n

r(t)   an gT (t  nT )  n(t),
n

where g (t)  hT (t) * hC (t), T  1/ fs
where he (t)  hT (t)*hC (t)*hR (t),
ne (t)  n(t)*hC (t)*hR (t)
y(t)  anhe (t  nT)  ne (t)
n

8. or equivalently
Desired symbol scaled by
gain parameters h0
where hk  he(kT), nk  ne(kT), k  0,1,2,..
– h0 is an arbitrary constant

n
 Note that he(t) is the equivalent impulse response of the receiving filter
 To recover the information sequence {an}, the output y(t) is sampled at t = kT,
k = 0, 1, 2, …
 The sampled sequence is
y(kT)  anhe (kT  nT)  ne (kT)
 
n
 nk
yk   anhkn anhkn
n,nk
 nk  h0ak 
AWGN term
Effect of other symbols at the
sampling instants t=kT
ISI Baseband Communication System Model

9. Signal Design for Bandlimited Channel
MuhammadAli Jinnah University, Islamabad
Zero ISI
 Nyquist Criterion
– Pulse amplitudes can be detected correctly despite pulse
spreading or overlapping, if there is no ISI at the decision-
making instants

y(kT)  h0ak  anhe (kT  nT)  ne (kT)
n,nk
 To remove ISI, it is necessary and sufficient to make the term
h (kT nT)  0, for n  k and h  0
e 0

10. Nyquist Criterion: Time domain
p(t): impulse response of a transmission system (infinite length)
Suppose 1/T is the sample rate
The necessary and sufficient condition for p(t) to satisfy Nyquist
Criterion is
pnT 

  0
0, n
1,n  0

11.  Pulse shape that satisfy this criteria is Sinc(.) function, e.g.,
 The smallest value of T for which transmission with zero ISI is
possible is
 Problems with Sinc(.) function
– It is not possible to create Sinc pulses due to
– Infinite time duration
– Sharp transition band in the frequency domain
– Sinc(.) pulse shape can cause ISI in the presence of timing
errors
• If the received signal is not sampled at exactly the bit instant,
then ISI will occur
e 
T

 
h (t) or p(t)  sinc t   sinc(2Wt)
2W
T 
1
1st Nyquist Criterion: Time domain

12. 1st Nyquist Criterion: Time domain
Equally spaced zeros,
interval  T
2 fs
1
 T
2 fs
1
2t0
t0
t
0
1
p(t)
-1
 shaping function
no ISI !

13. Sample rate vs. bandwidth
 W is the channel bandwidth for P(f)
 When 1/T > 2W, there is no way, we can design a
system with no ISI
P(f)

14. Sample rate vs. bandwidth
 When 1/T = 2W (The Nyquist Rate), rectangular
function satisfy Nyquist condition
,
0,otherwise


 
T,f W 

T

t
pt
sin t T
 sinct ; Pf 
Pf 
f
2W
  T rectfT ;

2W

 
1
rect
T
W

15. Sample rate vs. bandwidth
 When 1/T < 2W, numbers of choices to satisfy Nyquist
condition
– Raised Cosine Filter
– Duobinary Signaling (Partial Response Signals)
– Gaussian Filter Approximation
 The most typical one is the raised cosine function

16. Raised Cosine Pulse
 The following pulse shape satisfies Nyquist’s method for zero ISI
 The Fourier Transform of this pulse shape is
 where r is the roll-off factor that determines the bandwidth
T 2
4r 2
t 2
T 2
4r 2
t 2
1 
1 
p(t) 
T
  
 T 
 t 
  sinc

T
 
T
 
T

  
rt
sin rt  cos rt  cos rt 






1r
| f |
2T 2T
1r




0,

P(f ) T/21cos
r
| f |
2T
,
T  1r
2T
| f |
1r
2T
0| f |
1r
T ,

17. Raised cosine shaping
 W
W ω
P(ω)
r=0
r = 0.25
r = 0.50
r = 0.75
r = 1.00
π
W

0
t
0
p(t)
 π
W
2w
 Tradeoff: higher r, higher bandwidth, but smoother in
time.

18.  Bandwidth occupied beyond 1/2T is called the excess bandwidth (EB)
 EB is usually expressed as a %tage of the Nyquist frequency, e.g.,
– Rolloff factor, r = 1/2 ===> excess bandwidth is 50 %
– Rolloff factor, r = 1 ===> excess bandwidth is 100 %
 RC filter is used to realized Nyquist filter since the transition band can be
changed using the roll-off factor
 The sharpness of the filter is controlled by the parameter r
 When r = 0 this corresponds to an ideal rectangular function
 Bandwidth B occupied by a RC filtered signal is increased from its
minimum value
 So the bandwidth becomes:
s
2T
Bmin 
1
B  B 1 r
min
Rolloff and bandwidth

19.  Benefits of large roll off factor
– Simpler filter – fewer stages (taps) hence easier to
implement with less processing delay
– Less signal overshoot, resulting in lower peak to mean
excursions of the transmitted signal
– Less sensitivity to symbol timing accuracy – wider eye
opening
 r = 0 corresponds to Sinc(.) function
MuhammadAli Jinnah University, Islamabad
Rolloff and bandwidth

20. Partial Response Signals
 To improve the bandwidth efficiency
– Widen the pulse, the smaller the bandwidth.
– But there is ISI. For binary case with two symbols, there is
only few possible interference patterns.
– By adding ISI in a controlled manner, it is possible to
achieve a signaling rate equal to the Nyquist rate
i.e.
Duobinary and Polibinary Signaling

21. Eye Patterns
 An eye pattern is obtained by superimposing the actual waveforms for large
numbers of transmitted or received symbols
– Perfect eye pattern for noise-free, bandwidth-limited transmission of an
alphabet of two digital waveforms encoding a binary signal (1’s and 0’s)
– Actual eye patterns are used to estimate the bit error rate and the
signal to- noise ratio

22. Concept of the eye pattern
Eye Patterns

23. Concept of Eye diagram Mask. Waveform must not intrude into the shaded regions.
Eye Patterns

24. Cosine rolloff filter: Eye pattern
2nd Nyquist
1st Nyquist:🗸
2nd Nyquist:🗸
1st Nyquist
1st Nyquist:🗸
2nd Nyquist:🗴
🗸
2nd Nyquist:🗴
1st Nyquist:
1st Nyquist:🗸
2nd Nyquist:🗴

25. -1
0 0.2 0.4 0.6 0.8 1
Time (sec)
1.2 1.4 1.6 1.8 2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
EYE DIAGRAM
Eye Diagram Examples

26. -1
-1.5
0 0.2 0.4 0.6 0.8 1
Time (sec)
1.2 1.4 1.6 1.8 2
-0.5
0
0.5
1
1.5
EYE DIAGRAM WITH NOISE (Variance =0.1)
Eye Diagram Examples

27. -2
-3
0 0.2 0.4 0.6 0.8 1
Time (sec)
1.2 1.4 1.6 1.8 2
-1
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EYE DIAGRAM WITH NOISE (Variance =0.5)
Eye Diagram Examples