EE 179, Lecture 20, Handout #35
Reducing ISI: Pulse Shaping
◮
A time-limited pulse cannot be bandlimited
◮
Linear channel distortion results in spread out, overlapping pulses
◮
Nyquist introduced three criteria for dealing with ISI.
The first criterion was that each pulse is zero at the sampling time of
other pulses.
(
1 t=0
p(t) =
0 t = ±kTb , k = ±1, ±2, . . .
Harry Nyquist, “Certain topics in telegraph transmission theory”, Trans. AIEE, Apr. 1928
EE 179, May 16, 2014
Lecture 20, Page 1
Pulse Shaping: sinc Pulse
◮
Let Rb = 1/Tb . The sinc pulse sinc(πRb t) satisfies Nyquist’s first
crierion for zero ISI:
(
1 t=0
sinc(πRb t) =
0 t = ±kTb , k = ±1, ±2, . . .
◮
This pulse is bandlimited. Its Fourier transform is
1 f P (f ) =
Π
Rb
Rb
◮
Unfortunately, this pulse has infinite width and decays slowly.
EE 179, May 16, 2014
Lecture 20, Page 2
Nyquist Pulse
Nyquist increased the width of the spectrum in order to make the pulse fall
off more rapidly.
The Nyquist pulse has spectrum width 12 (1 + r)Rb , where 0 < r < 1.
If we sample the pulse p(t) at rate Rb = 1/Tb , then
p(t) = p(t) IIITb (t) = p(t)δ(t) = δ(t) .
The Fourier transform of the sampled signal is
P (f ) = 1 =
∞
X
P (f − kRb )
k=−∞
EE 179, May 16, 2014
Lecture 20, Page 3
Nyquist Pulse (cont.)
Since we are sampling below the Nyquist rate 2Rb , the shifted transforms
overlap.
Nyquist’s criterion requires pulses whose overlaps add to 1 for all f .
For parameter r with 0 < r < 1, the resulting pulse has bandwidth
Br = 12 (Rb + rRb )
The parameter r is called roll-off factor and controls how sharply the pulse
spectrum declines above 21 Rb .
EE 179, May 16, 2014
Lecture 20, Page 4
Nyquist Pulse (cont.)
There are many pulse spectra satisfying this condition. e.g., trapezoid:


|f | < 12 (1 − r)Rb

1
1
1
b
P (f ) = 1 − |f |−(1−r)R
rRb
2 (1 − r)Rb < |f | < 2 (1 + r)Rb


0
|f | > 1 (1 − r)R
b
2
A trapezoid is the difference of two triangles. Thus the pulse with
trapezoidal Fourier transform is the difference of two sinc2 pulses.
Example: for r = 12 ,
f f − 12 Λ 1
P (f ) = 23 Λ 3
2 Rb
so the pulse is
p(t) =
9
4
2 Rb
sinc2 ( 23 Rb t) − 14 sinc2 ( 21 Rb t)
This pulse falls off as 1/t2 .
EE 179, May 16, 2014
Lecture 20, Page 5
Nyquist Pulse (cont.)
Nyquist chose a pulse with a “vestigial” raised cosine transform.
This transform is smoother than a trapezoid, so the pulse decays more
rapidly.
The Nyquist pulse is parametrized by r. Let fx = rRb /2.
EE 179, May 16, 2014
Lecture 20, Page 6
Nyquist Pulse (cont.)
Nyquist pulse spectrum is raised cosine pulse with flat porch.

1

1


!! |f | < 2 Rb − fx

1

f − 2 Rb
|f | − 21 Rb | < fx
P (f ) = 12 1 − sin π
2fx




0
|f | > 12 Rb + fx
The transform P (f ) is differentiable, so the pulse decays as 1/t2 .
EE 179, May 16, 2014
Lecture 20, Page 7
Nyquist Pulse (cont.)
Special case of Nyquist pulse is r = 1: full-cosine roll-off.
P (f ) = 12 (1 + cos πTb f )Π(f /Rb )
= cos2 ( 21 πTb f ) Π( 21 Tb f )
This transform P (f ) has a second derivative so the pulse decays as 1/t3 .
p(t) = Rb
EE 179, May 16, 2014
cos πRb t
sin(2πRb t)
sinc(πRb t) =
2
2
1 − 4Rb t
2πt(1 − 4Rb2 t2 )
Lecture 20, Page 8
Controlled ISI (Partial Response Signaling)
We can reduce bandwidth by using an even wider pulse.
This introduces ISI, which can be canceled using knowledge of the pulse
shape.
The value of y(t) at time nTb is an−2 + an−1 . Decision rule:


y(nTb ) > 0
1
ân−1 = 0
y(nTb ) < 0


′
(ân−2 ) y(nTb ) = 0
A related approach is decision feedback equalization: once a bit has been
detected, its contribution to the received signal is subtracted.
EE 179, May 16, 2014
Lecture 20, Page 9
Partial Response Signaling (cont.)
The ideal duobinary pulse is
p(t) =
sin πRb t
πRb t(1 − Rb t)
The Fourier transform of p(t) is
P (f ) =
πf f 2
Π
e−jπf /Rb
cos
Rb
Rb
Rb
The spectrum is confined to the theoretical minimum of Rb /2.
EE 179, May 16, 2014
Lecture 20, Page 10
Zero-ISI, Duobinary, Modified Duobinary Pulses
Suppose pa (t) satisfies Nyquist’s first criterion (zero ISI). Then
pb (t) = pa (t) + pa (t − Tb )
is a duobinary pulse with controlled ISI. By shift theorem,
Pb (f ) = Pa (1 + e−j2πTb f )
Since Pb (Rb /2) = 0, most (or all) of the pulse energy is below Rb /2.
We can eliminate unwanted DC component using modified duobinary,
where pc (−Tb ) = 1, pc (Tb ) = −1, and pc (nTB ) = 0 for other integers n.
pc (t) = pa (t + Tb ) − pa (t − Tb ) ⇒ Pc (f ) = 2jPa (f ) sin 2πTb f
The transform of pc (t) has nulls at 0 and ±Rb /2.
EE 179, May 16, 2014
Lecture 20, Page 11
Zero-ISI, Duobinary, Modified Duobinary Pulses (cont.)
Zero−ISI
1
0.5
0
−5
−4
−3
−2
−1
0
1
2
3
4
1
2
3
4
2
3
4
Duobinary
1.5
1
0.5
0
−5
−4
−3
−2
−1
0
Modified Duobinary
1
0.5
0
−0.5
−1
−5
EE 179, May 16, 2014
−4
−3
−2
−1
0
1
Lecture 20, Page 12
Zero-ISI, Duobinary, Modified Duobinary Pulses (cont.)
150
100
50
0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−1.5
−1
−0.5
0
0.5
1
1.5
2
200
150
100
50
0
−2
200
150
100
50
0
−2
EE 179, May 16, 2014
Lecture 20, Page 13
Partial Response Signaling Detection
Suppose that sequence 0010110 is transmitted (first bit is startup digit).
Digit xk
0
Bipolar amplitude
-1
Combined amplitude
Decoded values
Decode sequence
Partial response signaling is susceptible
0 1
-1 1
-2 0
-2 0
0 1
to error
0 1 1 0
-1 1 1 -1
0 0 2 0
2 0 0 2
0 1 1 0
propagation.
If a nonzero value is misdetected, zeros will be misdetected until the next
nonzero value.
Error propagation is eliminated by precoding the data: pk = xk ⊕ pk−1 .
EE 179, May 16, 2014
Lecture 20, Page 14
Eye Diagrams
Polar Signaling with Raised Cosine Transform (r = 0.5)
P (f ) =
EE 179, May 16, 2014


1




1
2



0
1 − sin π
f − 12 Rb
Rb
!!
|f | < 14 Rb
||f | − 21 Rb | < 12 Rb
|f | > 34 Rb
Lecture 20, Page 15
Eye Diagrams (cont.)
Polar Signaling with Raised Cosine Transform (r = 0.5)
The pulse corresponding to P (f ) is
p(t) = sinc(πRb t)
cos(πrRb t)
1 − 4r 2 Rb2 t2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−1.5
EE 179, May 16, 2014
−1
−0.5
0
0.5
1
1.5
2
Lecture 20, Page 16
Eye Diagram Measurements
◮
Maximum opening affects noise margin
◮
Slope of signal determines sensitivity to timing jitter
◮
Level crossing timing jitter affects clock extraction
◮
Area of opening is also related to noise margin
EE 179, May 16, 2014
Lecture 20, Page 17