This document discusses techniques for pulse shaping to reduce inter-symbol interference (ISI) in digital communication systems. It introduces the Nyquist criteria that pulse shapes must satisfy to avoid ISI, including having zero crossings at symbol intervals, zero areas within symbol periods, and zero values at decision thresholds. Methods like raised cosine filtering are presented that trade off bandwidth for smoothness to meet the Nyquist criteria. The document also discusses partial response signaling techniques like duobinary that relax the criteria but require differential encoding to avoid error propagation.

1. Pulse Shaping
 Sy(w)=|P(w)|^2Sx(w)
 Last class:
– Sx(w) is improved by the different line codes.
– p(t) is assumed to be square
 How about improving p(t) and P(w)
– Reduce the bandwidth
– Reduce interferences to other bands
– Remove Inter-symbol-interference (ISI)
EE 541/451 Fall 2006

2. ISI Example
sequence sent 1 0 1
sequence received 1 1(!) 1
Signal received
Threshold
0 t
-3T -2T -T 0 T 2T 3T 4T 5T
Sequence of three pulses (1, 0, 1)
sent at a rate 1/T
EE 541/451 Fall 2006

3. 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)
EE 541/451 Fall 2006

4. 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.
EE 541/451 Fall 2006

5. ISI
 Nyquist three criteria
– Pulse amplitudes can be
detected correctly despite pulse
spreading or overlapping, if
there is no ISI at the decision-
making instants
x 1: At sampling points, no
ISI
x 2: At threshold, no ISI
x 3: Areas within symbol
period is zero, then no ISI
– At least 14 points in the finals
x 4 point for questions
x 10 point like the homework
EE 541/451 Fall 2006

6. 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 2t0
Equally spaced zeros,
-1 1
interval =T
2 fn
EE 541/451 Fall 2006

7. 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 = −∞
EE 541/451 Fall 2006

8. 1st Nyquist Criterion: Frequency domain
∞
∑ P( f + m T ) = T
m = −∞
f
0 fa = 2 f N 4 fN
(limited bandwidth)
EE 541/451 Fall 2006

9. Proof
∞
Fourier Transform p( t ) = ∫ P ( f ) exp( j 2πft ) df
−∞
∞
At t=T p( nT ) = ∫ P( f ) exp( j 2πfnT ) df
−∞
∞ ( 2 m +1)
p( nT ) = ∑ ∫( P( f ) exp( j 2πfnT ) df
2T
2 m −1) 2T
m = −∞
∞
∑∫ P( f + m T ) exp( j 2πfnT ) df
1 2T
=
−1 2T
m = −∞
∞
∑ P( f + m T ) exp( j 2πfnT ) df
1 2T
=∫
−1 2T
m = −∞
∞
=∫
1 2T
B( f ) exp( j 2πfnT ) df B( f ) = ∑ P( f + m T )
−1 2T m = −∞
EE 541/451 Fall 2006

10. Proof
∞ ∞
B( f ) = ∑ P( f + m T ) B( f ) = ∑ b exp( j 2πnfT )
n
m = −∞ n = −∞
B ( f ) exp( − j 2πnfT )
1 2T
bn = T ∫
−1 2T
bn = Tp ( − nT ) T ( n = 0)
bn = 
0 ( n ≠ 0)
∞
B( f ) = T ∑ P( f + m T ) = T
m = −∞
EE 541/451 Fall 2006

11. Sample rate vs. bandwidth
 W is the bandwidth of P(f)
 When 1/T > 2W, no function to satisfy Nyquist condition.
P(f)
EE 541/451 Fall 2006

12. Sample rate vs. bandwidth
 When 1/T = 2W, rectangular function satisfy Nyquist
condition
sin πt T  πt  T , ( f < W )
p( t ) = = sinc  P( f ) =  ,
πt T  0, ( otherwise )
1
0.8
0.6
0.4
Spectra
0.2
0
-0.2
-0.4
0 1 2 3 4 5 6
S bcarrier N m k
u u ber
EE 541/451 Fall 2006

13. Sample rate vs. bandwidth
 When 1/T < 2W, numbers of choices to satisfy Nyquist
condition
 A typical one is the raised cosine function
EE 541/451 Fall 2006

14. Cosine rolloff/Raised cosine filter
 Slightly notation different from the book. But it is the same
sin(π T ) cos(rπ T )
t t
prc 0 (t ) = ⋅
π Tt
1 − (2 r T ) 2
t
r : rolloff factor 0 ≤ r ≤1
1 f ≤ (1 − r ) 21T
Prc 0 ( j 2πf ) = 1
2
[1 + cos( π
2r ( πTf + r − 1)) ] if 1
2T (1 − r ) ≤ f ≤ 1
2T (1 + r )
0 f ≥ 1
2T (1 + r )
EE 541/451 Fall 2006

15. Raised cosine shaping
 Tradeoff: higher r, higher bandwidth, but smoother in time.
P(ω)
π W r=0
r = 0.25
r = 0.50
r = 0.75
r = 1.00
p(t)
W 2w ω
π π
− +
W W
0
0 t
EE 541/451 Fall 2006

16. Cosine rolloff filter: Bandwidth efficiency
 Vestigial spectrum
 Example 7.1
data rate 1/ T 2 bit/s
β rc = = =
bandwidth (1 + r ) / 2T 1 + r Hz
bit/s 2 bit/s
1 ≤ < 2
Hz (1 + r ) Hz
 
2nd Nyquist (r=1) r=0
EE 541/451 Fall 2006

17. 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[ ∑( −1) n P ( f + n / T )] = T cos( fT / 2)
n =−∞
∞
PI ( f ) = Im[ ∑( −1) n P ( f + n / T )] = 0
n =−∞
EE 541/451 Fall 2006

18. Example
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19. 3rd Nyquist Criterion
 Within each symbol period, the integration of signal (area) is
proportional to the integration of the transmit signal (area)
 ( wt ) / 2 π
sin( wT / 2) , w ≤ T

P ( w) = 
 0, π
 w >
 T
π /T
1 ( wt / 2)
p (t ) = ∫/ T sin( wT / 2) e dw
jwt
2π −π
2 n +1T
1, n=0
A = ∫2 n−1 p(t )dt = 
2
2
T
0, n≠0
EE 541/451 Fall 2006

20. Cosine rolloff filter: Eye pattern
2nd Nyquist
1st Nyquist: 1st Nyquist:

2nd Nyquist: 
2nd Nyquist:
1st Nyquist
1st Nyquist: 
1st Nyquist:

2nd Nyquist: 
2nd Nyquist:
EE 541/451 Fall 2006

21. Example
 Duobinary Pulse
– p(nTb)=1, n=0,1
– p(nTb)=1, otherwise
 Interpretation of received signal
– 2: 11
– -2: 00
– 0: 01 or 10 depends on the previous transmission
EE 541/451 Fall 2006

22. Duobinary signaling
 Duobinary signaling (class I partial response)
EE 541/451 Fall 2006

23. Duobinary signal and Nyguist Criteria
 Nyguist second criteria: but twice the bandwidth
EE 541/451 Fall 2006

24. Differential Coding
 The response of a pulse is spread over more than one signaling
interval.
 The response is partial in any signaling interval.
 Detection :
– Major drawback : error propagation.
 To avoid error propagation, need deferential coding (precoding).
EE 541/451 Fall 2006

25. Modified duobinary signaling
 Modified duobinary signaling
– In duobinary signaling, H(f) is nonzero at the origin.
– We can correct this deficiency by using the class IV partial
response.
EE 541/451 Fall 2006

26. Modified duobinary signaling
 Spectrum
EE 541/451 Fall 2006

27. Modified duobinary signaling
 Time Sequency: interpretation of receiving 2, 0, and -2?
EE 541/451 Fall 2006

28. Pulse Generation
 Generalized form of
correlative-level
coding
(partial response signaling)
Figure 7.18
EE 541/451 Fall 2006