ECE 4371, Fall, 2016
Introduction to Telecommunication
Engineering/Telecommunication Laboratory
Zhu Han
Department of Electrical and Computer Engineering
Class 12
Oct. 3rd, 2016
Outline

Midterm review

Inter-Symbol-Interference

Nyguist Three Criteria

Eye Diagram
ISI Example
sequence sent
1
sequence received 1
0
1
1(!)
1
Signal received
Threshold
t
0
-3T
-2T
-T
0
T
2T
3T
4T
Sequence of three pulses (1, 0, 1)
sent at a rate 1/T
5T
Baseband binary data transmission system.

ISI arises when the channel is dispersive

Frequency limited -> time unlimited -> ISI

Time limited -> bandwidth unlimited -> bandpass channel ->
time unlimited -> ISI
p(t)
ISI


First term : contribution of the i-th transmitted bit.
Second term : ISI – residual effect of all other transmitted bits.

We wish to design transmit and receiver filters to minimize the
ISI.

When the signal-to-noise ratio is high, as is the case in a
telephone system, the operation of the system is largely limited
by ISI rather than noise.
ISI

Nyquist three criteria
– Pulse amplitudes can be
detected correctly despite pulse
spreading or overlapping, if
there is no ISI at the decisionmaking instants



1: At sampling points, no
ISI
2: At threshold, no ISI
3: Areas within symbol
period is zero, then no ISI
– At least 14 points in the finals


4 point for questions
10 point like the homework
1st Nyquist Criterion: Time domain
p(t): impulse response of a transmission system (infinite length)
p(t)
1
 shaping function
0
no ISI !
t
1
T
2 fN
t0
Equally spaced zeros,
-1
interval
1
T
2 fn
2t0
1st Nyquist Criterion: Time domain
Suppose 1/T is the sample rate
The necessary and sufficient condition for p(t) to satisfy
1, n  0
pnT   
0, n  0
Is that its Fourier transform P(f) satisfy

 P f  m T   T
m  
1st Nyquist Criterion: Frequency domain

 P f  m T   T
m  
0
fa  2 f N
f
4 fN
(limited bandwidth)
Proof
Fourier Transform

pt    P f exp  j 2ft df


pnT    P f exp  j 2fnT df
At t=T


pnT  
 2 m 1 2T
 
m  



m  
1 2T

1 2T
1 2T

1 2T
1 2T
1 2T
2 m 1 2T
P f  exp  j 2fnT df
P f  m T  exp  j 2fnT df

 P f  m T exp  j 2fnT df
m  
B f  exp  j 2fnT df
B f  

 P f  m T 
m  
Proof
B f  

 P f  m T 

B f  
b
n
n  
m  
bn  T 
1 2T
1 2T
bn  Tp nT 
B f   T
exp  j 2nfT 
T
bn  
0

B  f exp  j 2nfT 
n  0
n  0
 P f  m T   T
m  
Sample rate vs. bandwidth

W is the bandwidth of P(f)

When 1/T > 2W, no function to satisfy Nyquist condition.
P(f)
Sample rate vs. bandwidth
When 1/T = 2W, rectangular function satisfy Nyquist
condition

T ,  f  W 
sin t T
 t 
pt  
 sinc   P f   
,
t
T 
0, otherwise 
1
0.8
Spectra
0.6
0.4
0.2
0
-0.2
-0.4
0
1
2
3
4
Subcarrier Number k
5
6
Sample rate vs. bandwidth

When 1/T < 2W, numbers of choices to satisfy Nyquist
condition

A typical one is the raised cosine function
Cosine rolloff/Raised cosine filter

Slightly notation different from the book. But it is the same
sin(  Tt ) cos( r Tt )
prc0 (t ) 

t
T
1  (2 r Tt ) 2
r : rolloff factor
0  r 1
1
Prc0 ( j 2f ) 
1
2
1  cos(
0
f
(
2 r T  r  1))


f  (1  r ) 21T
if
1
2T
(1  r )  f 
f 
1
2T
(1  r )
1
2T
(1  r )
Raised cosine shaping

Tradeoff: higher r, higher bandwidth, but smoother in time.
W
P(ω)
r=0
r = 0.25
r = 0.50
r = 0.75
r = 1.00

0
π
W

0
ECE 4371 Fall 2008
ω
2w
W
p(t)
π
W
t
Figure 4.10 Responses for different rolloff factors.
(a) Frequency response. (b) Time response.
Cosine rolloff filter: Bandwidth efficiency

Vestigial spectrum
data rate
1/ T
2 bit/s
 rc 


bandwidth (1  r ) / 2T 1  r Hz
bit/s
1
Hz

2
(1  r )
bit/s
 2
Hz


2nd Nyquist (r=1)
r=0
2nd Nyquist Criterion

Values at the pulse edge are distortionless

p(t) =0.5, when t= -T/2 or T/2; p(t)=0, when t=(2k-1)T/2, k≠0,1
-1/T ≤ f ≤ 1/T
Pr ( f )  Re[
PI ( f )  Im[

 (1)
n
P ( f  n / T )]  T cos( fT / 2)
n
P ( f  n / T )]  0
n  

 (1)
n  
Example
3rd Nyquist Criterion

Within each symbol period, the integration of signal (area) is
proportional to the integration of the transmit signal (area)

 ( wt ) / 2
,w

 sin( wT / 2)
T
P ( w)  

 0,
w 

T

1
p(t ) 
2
2 n1T
2
A  2 n1
2
T
 /T
( wt / 2)
jwt
e
dw

sin( wT / 2)
 / T
1,
p(t )dt  
0,
n0
n0
Cosine rolloff filter: Eye pattern
2nd Nyquist
1st Nyquist:
1st Nyquist:
2nd Nyquist:
2nd Nyquist:
1st Nyquist
1st Nyquist:
2nd Nyquist:

2nd Nyquist:
1st Nyquist:
Eye Diagram

The eye diagram is created by taking the time domain signal and
overlapping the traces for a certain number of symbols.

The open part of the signal represents the time that we can
safely sample the signal with fidelity
Vertical and Horizontal Eye Openings

The vertical eye opening or noise
margin is related to the SNR, and
thus the BER
– A large eye opening corresponds
to a low BER

The horizontal eye opening relates
the jitter and the sensitivity of the
sampling instant to jitter
– The red brace indicates the range
of sample instants with good eye
opening
– At other sample instants, the eye
opening is greatly reduced, as
governed by the indicated slope
Interpretation of Eye Diagram
Jitter in Circuit design

Circuit design
Raised Cosine Eye Diagram

The larger , the wider the
opening.

The larger , the larger
bandwidth (1+ )/Tb

But smaller  will lead to larger
errors if not sampled at the best
sampling time which occurs at
the center of the eye.
Eye Diagram Setup

Eye diagram is a retrace display of
data waveform
– Data waveform is applied to
input channel
– Scope is triggered by data
clock
– Horizontal span is set to cover
2-3 symbol intervals

Measurement of eye opening is
performed to estimate BER
– BER is reduced because of
additive interference and noise
– Sampling also impacted by
jitter
Eye Diagram

Eye diagram is a means of evaluating the quality of a received
“digital waveform”
– By quality is meant the ability to correctly recover symbols and
timing
– The received signal could be examined at the input to a digital
receiver or at some stage within the receiver before the decision
stage





Eye diagrams reveal the impact of ISI and noise
Two major issues are 1) sample value variation, and 2) jitter and
sensitivity of sampling instant
Eye diagram reveals issues of both
Eye diagram can also give an estimate of achievable BER
Check eye diagrams at the end of class for participation
Figure 4.34 (a) Eye diagram for noiseless quaternary system. (b) Eye diagram for quaternary system
with SNR  20 dB. (c) Eye diagram for quaternary system with SNR  10 dB.
Figure 4.35 (a) Eye diagram for noiseless band-limited quaternary system:
cutoff frequency fo  0.975 Hz. (b) Eye diagram for noiseless band-limited
quaternary system: cutoff frequency fo  0.5 Hz.
Eye Diagram In Phase
Linear Modulation with Nyquist Impulse Shaping
QPSK diagram under limited bandwidth conditions
 if system (tx and rx filter) meets 1st Nyquist : 4 sharp signal points (right diagram)