This paper resolves the jitter impairment of non-return-to-zero data in transmission lines. The limited bandwidth of the transmission line introduces data-dependent jitter. Crosstalk between neighbouring lines results in bounded uncorrelated jitter in the data eye. An analytical approach to representing data-dependent jitter and crosstalk-induced bounded uncorrelated jitter is presented. Comparison with jitter measurements of microstrip lines on FR4 board demonstrated accuracy to within 15% of the predictions for deterministic jitter

Analui, Behnam

Buckwalter, James

Hajimiri, Ali

English

Caltech Authors - Main

Data-dependent jitter and crosstalk-induced bounded uncorrelated jitter in copper interconnects

TH2C-1 
Data-Dependent Jitter and Crosstalk-Induced Bounded 
Uncorrelated Jitter in Copper Interconnects 
James Buckwalter, Behnam Analui, and Ali Hajimiri 
California Institute of Technology, Pasadena, CA 9 1 125 
Abstract - This paper resolves the jitter impairment of 
non-return-to-zero data in transmission lines. The limited 
bandwidth of the transmission line introduces data- 
dependent jitter. Crosstalk between neighboring Lines results 
in bounded uncorrelated jitter in the data eye. An analytical 
approach to representing data-dependent jitter and crosstalk- 
induced bounded uncorrelated jitter is presented. 
Comparison with jitter measurements of microstrip lines on 
FR4 board demonstrates accuracy to within 15% of the 
predictions for deterministic jitter. 
Index Terms - Jitter, Data-dependent Jitter, Bounded 
Uncorrelated Jitter, Interconnect, Crosstalk 
I. INTRODUCTION 
Exceptional effort is focused on extending the data rate 
capabilities of economical copper backplanes. While VLSI 
circuit speed is steadily increasing, the wires between 
chips and even on-chip are becoming bottlenecks [1][2]. 
To avoid costly optical interconnects, circuit designers are 
forced to equalize the channel to increase the data rate, 
implement alternative channel coding schemes, or simply 
use more parallel channels [3][4]. Ultimately, space and 
power dissipation limits the number of parallel channels. 
Increasing the interconnect density in controlled imped- 
ance environments is a problem common to copper back- 
plane and VLSI. Electromagnetic coupling between 
transmission lines is overwhelming as trace separation 
reduces. To prevent an aggressive neighboring signal from 
generating errors in a victim signal, circuit designers match 
line lengths such that the strongest coupling fkom the 
aggressor occurs not in the center of the data eye but dur- 
ing data transitions. While this prevents errors at the sam- 
pling point, .the aggressor generates jitter on the victim 
line. 
This paper studies the jitter penalty associated with cou- 
pling between neighboring microstrip lines. Jitter consists 
of deterministic and random components. In this paper, the 
important deterministic components are data-dependent jit- 
ter and bounded-uncorrelated jitter. First, the data-depen- 
dent jitter (DDJ) due to the transmission line is analyzed 
and a DDJ probability density knction (PDF) is intro- 
duced. Fig. 1 illustrates the data transition ambiguity 
Fig. 1. The data-dependent jitter of a interconnect on FR4 
board with various line separations. The illustration shows the 
threshold crossing time deviation through the transmission line 
and the capacitive coupling between neighboring lines. 
caused by the microstrip line. Second, an aggressive neigh- 
boring signal causes bounded uncorrelated jitter (BUJ) and 
a PDF is formulated that accounts for the coupling. These 
two jitter PDFs dominate jitter in interconnects. The PDFs 
are predicted from the step response and S-parameters and 
matched to measurements on an FR4 board with microstrip 
lines of various separation. 
11. DATA-DEPENDENT JI TER IN COPPER INTERCONNECTS 
The properties of DDJ are determined analytically h m  
a characterization of the transmission line. The received 
non-return-to-zero (NU) data signal is 
r( t )  = 2 a#( t -"T) .  (1) 
,=-m 
where a, is 
response with 
the binary value and p ( t )  is the pulse 
period T. The pulse response is related to 
the step response, s( t) ; 
p ( t )  = s ( t ) - s ( t - T ) .  (2) 
Substituting (2) into (l), we have 
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0-7803-8331-1/04/$20.00 0 2004 IEEE 2004 IEEE MTT-S Digest 
m 
r ( t )  = c a , [ s ( t - n r ) - s ( t - ( n +  l)T)]. (3) 
n = - m  
A first-order Taylor series approximates the, step response. 
s ( t - n T )  = s ( t , - n T ) + ( t - t , ) s  ( 1 )  ( t , - n T )  (4) 
The superscript denotes the derivative order and to is the 
time at which s ( t )  crosses a voltage threshold, v t h  . Substi- 
tuting (4) into (3), the threshold crossing time, t, , is' 
v t h - ~ ' n [ ~ ( t o  - - s( to  - ( n  + )T)l  
tc = t ,  + * ( 5 )  
~ u , [ s " ) ( t o - n T ) - S ( l ) ( t o - ( ( n  + 1)T)] 
Jitter is the deviation of the signal fiom a reference such 
as to (i.e., Atc = t c - t o ) .  Clearly, t, depends on earlier 
bits. Therefore, At, is the DDJ. If two previous bits are 
considered, there are four possible sequences with transi- 
tions at the current bit: 001, 101, 110, and 010. If we 
assume V i h  = 0.5, there are two different t, for the 001 
and 101 sequences as illustrated in Fig. 1. For 001 and 110 
sequences, t ,  = to since, by definition, s ( t , )  = V t h .  For 
the 101 and 010 sequences, 
Notably, the denominator contains the slope implying 
that slow waveforms suffer fiom more DDJ. The first- 
order DDJ is a two-valued PDF of Atc as modelled in [SI, 
1 
PdfDDJ(Atc) = 2 [ 8 ( A t c )  + 8 ( A z c - A f c , D D J ) l  ' (7) 
If additional bits are considered, the PDF extends to 
more peaks. Both victim and aggressor signals suffer from 
the DDJ impairment because the transmission lines are 
identical. 
111. BOUNDED UNCORRELATED JI-ITER 
Capacitive Coupling introduces energy from an aggres- 
sor to a victim signal as illustrated in Fig. 1. Inductive 
effects can also be considerable in certain cases. We 
assume capacitive coupling but inductive coupling can be 
treated similarly. The nature of the coupling capacitance, 
C,, between microstrip lines has been studied (e.g. [6]).  
1. Causality restricts the summation from -a to 0. These limits are 
assumed in the following summations. 
Even and odd modes experience C, with relative strength 
of zero or two. For equiprobable NRZ data, the excitatiaas 
are uncorrelated and both modes occur equally. Hence, the 
C, has a relative strength of one. Notably, the: mocle- 
dependent capacitance ultimately disperses the signal on 
the microstrip line. 
Although the capacitance is distributed, the coupled sig- 
nal travels through the same amount of transmission line 
irrespective of the coupling point, as shown in Fig. 1. 
Therefore, the coupling between the victim and aggressor 
line is approximated as a lumped capacitance: 
IS4, (0)l = 0ccz0/2 . (8) 
A differentiated version of the aggressor contributes to (3), 
The timing jitter resulting fiom the uncorrelated &ita 
sequence can now be calculated as 
z o c c ~ ( ~ " - b , - l ) - s  (1) ( t 0 - C )  
Atc, BUJ = -- 9 (10) 
where b, is the binary value of the aggressor signal. Equa- 
tion (10) features only the slope of the step response. Since 
a data transition occurs in the victim signal, a. - is 
always non-zero. Under certain cases, (1 0) simplifies fiu- 
ther. If the aggressor and victim are identical, 
t,, BuJ = -Z,Cc/2.  However, C, = 0 because the mode 
is even. If the aggressor is a differential signal, a, = 4, 
and t,,BuJ = ZoC, since the mode is odd. Therefore, the 
coupling delays the signal. For two uncorrelated bits, the 
role of the slope cancels and there are three values for (1 0). 
Now, the PDF for the BUJ can be approximated. 
Convolving the PDF of the DDJ and BUJ determines the 
total deterministic jitter impairment of the received signal; 
d f J i t t e A A t c )  = ~ d f J d A t c )  @ , P ~ B u J ( A ~ , )  . (12) 
Since the first-order DDJ has two terms and the BUJ has 
three terms, the total deterministic jitter is a PDF with at 
most six delta fimctions corresponding to jitter peaks. 
w. EXPERIMENTAL RESULTS 
The FR4 board shown in Fig. 1 has 9" coupled micros- 
trip lines separated by 0.25", OS', and 1". Network analy- 
1628 
sis of the lines is plotted in Fig. 2. SZl indicates that the 
3dJ3 bandwidth is 2.5GHz for the 0.25” lines and 3.8GHz 
for the 0.5” and 1” lines. 
The measured step responses are plotted in Fig. 3 and are 
used to calculate the DDJ in Table 1. The step response 
value is normalized to one in the table. The 0.25” lines fea- 
ture a frequency null near 9GHz and the step response is 
slowest. The negative value for the DDJ indicates that the 
101 sequence is faster than the 001 sequence. Interestingly, 
while the 0.25” line response in Fig. 3 is slowest, the DDJ 
calculated in Table I is smallest. This emphasizes the 
importance of the slope after one period. Fig. 4,5, and 6 
show the data eyes at 5Gb/s for the 0.25”, O S ’ ,  and 1.0’’ 
lines without the impact of the aggressor signal. The DDJ 
values measured from these eyes are 6ps, lops, and lops. 
The DDJ measured from the data eyes agrees with the pre- 
dicted DDJ value in (6). 
TABLE I 
Calculated and Measured DDJ for FR4 Coupled Microstrip 
In Fig. 2, S41 shows the coupling is capacitive to 6GHz. 
From ADS simulations, the coupling capacitance is 1.4pF 
for the 0.25” line, 400fF for the 0.5” line, and 120fF for the 
1 .O” line. The time constants that determine the PDF of the 
BUJ are therefore 35ps, lops, and 3ps for these line separa- 
tions, respectively, since the characteristic line impedance 
0.1 1 10 
Frequency (GHz) 
Fig. 2. 
with varying microstrip separations. 
Measurement of Szl and S41 for coupled microstrip line 
330 340 350 360 370’ 
Separation: Dark- 0.25” 
Medium- 0.5 
I inht- 1 n‘‘ 
0 50 100 150 200 250 300 350 400 
Time (ps) 
Fig. 3. 
graphs show slope at t ,  and at t, + T. 
Step response for different line separations. Inset 
is 50Q The accuracy of the line impedance can have a 
drastic impact on the accuracy of the BUJ prediction. 
Finally, the predicted values for the DDJ and the BUJ 
determine the location and probability of the jitter peaks 
given in (7) and (1 1) in the presence of an aggressive sig- 
nal. A 5GB/s PRBS is introduced to the victim line. The 
differential line is delayed by four periods and introduced 
to the aggressor line. For the 0.25” separation, the total 
deterministic jitter PDF is calculated in Table 11. In this 
case, BUJ > DDJ. Fig. 4 exhibits a central Atc consisting 
of the -5.lps and Ops terms. The random jitter limits the 
Fig. 4. 
5GbIs for 0.25” line separation. 
Victim data eye with and without the aggressor signal at 
1629 
Fig. 5. Victim data eye for 0.5” line separation. 
resolution of the peaks. The probability indicates that the 
central threshold crossing occurs twice as often as the outer 
threshold crossings. The increased contrast of the central 
threshold crossing time in Fig. 4 confirms this. The peaks 
are separated by 31 ps indicating that the overall error for 
the deterministic jitter prediction is about 11%. 
For the 0.5” separation, the BUJ is comparable to the 
DDJ. More At, are obvious but the total range of the jitter 
is smaller. Again, random jitter prevents resolving the - 
l lps  and -10 ps crossings and the -Ips and Ops crossings. 
Clearly, these peaks are more probable than the outer peaks 
in the Fig. 4 because they occur three times as often. The 
At, are separated by 9ps. The predicted BUJ is lops so the 
error is about 10%. 
Finally, the 1” separation illustrates the case that the BUJ 
DDJ. The BUJ obscures the distinct At, of the DDJ and 
behaves like random jitter. As a result, verifying the pre- 
dicted At, for the total deterministic jitter is difficult. 
TABLE 11 
Predicted Location and Probability of Deterministic Jitter Peaks 
m e  
1.25” 
1.5” 
1.0” 
A t ,  DDJ A t e ,  BUJ Ate, DDJB BUJ 
-5.lps 35ps -4Ops (Pr= 1/8), -35ps (Pr = Mi), 
-5.lps (R = 1/4), Ops (Pr = 1/4), 
3Ops (R = l/8), 35ps (p’= 1/8) 
-1lps l O p ~  -21pS (R= 1/8), -1lpS (Pr= 1/4), 
-Ips (pr= 1/8), -1%~ (Pr= 1/8), 
Ops (R = 1/4), lOps (PI = 118) 
- 8 . 7 ~ ~  3ps - 1 2 ~ ~  (R= 1/8), - 8 . 7 ~ ~  (Pr = 1/4), 
- 5 . 7 ~ ~  (PI= 1/8), -3ps (PI 1/8), 
ODs Pr = 1/4),30s P r  = 118) 
Fig. 6. Victim data eye for 1.0” line separation 
v. CONCLUSIONS 
This paper develops an analysis of DDJ and BUJ in a 
coupled microstrip environment. The analysis of the DDJ 
emphasized the role of the step response slope in determil- 
ing the DDJ impairment. The BUJ analysis stressed the 
coupling capacitance in determining the BUJ impairment. 
Predicted values for the data-dependent and bound uncor- 
related jitter were calculated from the step response and !$- 
parameter measurement of microstrip traces on an FR.4 
board. The observed data eyes agree to within 15% of the 
predicted DDJ and to 11% of the predicted BUJ. This gives 
a reasonable design rule for managing jitter degradation 
resulting from capacitive coupling. 
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http://authors.library.caltech.edu/3156/1/BUCmttsims04.pdf

Data-dependent jitter and crosstalk-induced bounded uncorrelated jitter in copper interconnects

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