# Closed-Form Equation of Data Dependent Jitter in First Order Low Pass System

Authors: Sangjin Byun
Journal: The Scientific World Journal
 (2014)

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## Abstract

This paper presents a closed-form equation of data dependent jitter (DDJ) in first order low pass systems. The DDJ relates to the system bandwidth, the bit rate, the input rise/fall time, and the number of maximum consecutive identical bits of the data pattern. To confirm the derived equation, simulations have been done with a first order RC low pass circuit for various system bandwidths, bit rates, input rise/fall times, and data patterns. The simulation results agree well with the calculated DDJ values by the derived equation.


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## Body

## 1. Introduction

As bit rate increases, timing jitter becomes more critical to system performances of a high speed serial interface. Timing jitter deteriorates signal quality at the transmitter side and degrades BER performance at the receiver side [1–4]. To guarantee the satisfactory system performances of a high speed serial interface, timing jitter should be accurately predicted and carefully considered when we design system architecture, link budget, and each circuit building block.Timing jitter is composed of unbounded random jitter (RJ) and bounded deterministic jitter (DJ). The RJ is produced by Gaussian electrical noise within system components and the DJ is categorized into duty cycle distortion (DCD) jitter, data dependent jitter (DDJ), and bounded uncorrelated jitter (BUJ) [5, 6]. Among them, the DDJ is focused on in this paper. The DDJ has an impact on the high speed serial interface especially when the bit rate increases while the system bandwidth is restricted [7, 8]. As shown in Figure 1, the DDJ is generated when a certain data pattern with the bit rate, Tb, passes through a system with the limited bandwidth, fBW.Figure 1
Response of first order low pass system.So far, some papers have been published to predict the DDJ in the general transmission lines [7–9] and in the first order low pass systems [10–14]. The DDJ in the transmission line may apply to the interconnect channels such as off-chip PCB traces, off-chip cables, and on-chip interconnect lines while the DDJ in the first order low pass system may apply to the transceiver circuit building blocks such as drivers, buffers, amplifiers, and limiters. For the transmission lines, the DDJ has been predicted by using the simulated transient step response and the worst-case input bit sequence to shorten the simulation time [7–9]. On the other hand, for the first order low pass systems, the DDJ has been predicted based on the infinite number of calculated pulse or step responses of all the previous bits while the rise/fall time of the input signal was assumed to be zero ideally. However, because it is not possible to calculate the infinite number of pulse or step responses of all the previous bits, only two or four preceding bits have been considered instead for the actual DDJ prediction [10–14]. So, the calculated DDJ always underestimates the real DDJ and the prediction accuracy may degrade as the bit rate increases relatively to the system bandwidth.In this paper, a new closed-form equation of DDJ in the first order low pass system is presented. The DDJ is directly derived by solving the differential equation of the first order RC low pass circuit and by using the repetitiveness of the data pattern. Of course, this repetitiveness of the data pattern can be generalized for the real random data by increasing the pattern length to the infinity. The derived DDJ equation relates to the system bandwidth, the bit rate, the input rise/fall time, and the number of maximum consecutive identical bits of the data pattern. Contrary to the previous works, the calculated DDJ by the derived equation coincides exactly with the simulated DDJ. Additionally, the effect of nonzero input rise/fall time is also to be considered.This paper is composed of five sections. In Section2, the closed-form DDJ equation is derived by assuming zero input rise/fall time. Section 3 extends the derived DDJ equation to the data pattern with nonzero input rise/fall time. To confirm the derived DDJ equation, the simulation results are shown in Section 4 and conclusions are given in Section 5.


## 2. Calculation of DDJ with Zero Input Rise/Fall Time

### 2.1. Differential Equation of First Order RC Low Pass Circuit

Given a data pattern, a bit rate, and a system bandwidth, the DDJ can be derived by solving the differential equation of the first order RC low pass circuit shown in Figure2. Depending on the bit transition patterns, such as 01, 00, 10, and 11, the input signal, vi(t), and the initial condition, vo(nTb), of the first order RC low pass circuit are given differently for each bit duration, Tb, as shown in Figure 3. In the figure, the input rise/fall time is zero ideally. So, vi(t) is -A or +A for nTb<t≤(n+1)Tb and a new variable, ve,n, is defined as the voltage difference between vi(t) and vo(t) at t=nTb, where n is an integer; that is, ve,n=|vi(nTb)-vo(nTb)|. Also, for the bit transition pattern of 01 or 10, another variable, τd,n, is defined as the time difference between the threshold crossing times of vi(t) and vo(t), as shown in Figures 3(a) and 3(c). The definition of the variables can be found in Table 1.Table 1
Definition of the variables.
Variable
Definition
f
BW
3 dB bandwidth of the first order low pass system
T
b
Bit duration
f
b
Bit rate, 1/Tb
Δ
T
Rise/fall time of the input signal
v
e
,
n
Voltage difference betweenvi(t) and vo(t) at t=nTb
τ
d
,
n
Time difference between the threshold crossing times ofvi(t) and vo(t) for nTb<t<(n+1)Tb
a
i
,
n
Relative bit distance of theith bit transition backwards from ve,n
N
Pattern length
K
Number of bit transitionsFigure 2
First order RC low pass circuit.Input and output waveforms when the bit transition pattern is (a) 01, (b) 00, (c) 10, and (d) 11 and the input rise/fall time is zero.
(a)
(b)
(c)
(d)Because there are four different sets of the input signal,vi(t), and the initial condition, vo(nTb), depending on the bit transition patterns, the below differential equation of the first order RC low pass circuit should be solved for each bit transition pattern:
(1)vo(t)-vi(t)R+Cdvo(t)dt=0.
First, when the bit transition pattern is 01 as shown in Figure 3(a), the initial condition and the input signal are vo(nTb)=-A+ve,n and vi(t)=A for nTb<t≤(n+1)Tb, respectively. Then, the output signal is
(2)vo(t)=vo(nTb)+{A-vo(nTb)}(1-e-(t-nTb)/RC)fornTb≤t≤(n+1)Tb.
By using that vo((n+1)Tb)=A-ve,n+1, the relationship between ve,n and ve,n+1 is obtained as
(3)ve,n+1=2Ar-ve,nr,
where r=e-(Tb/RC) and, by using that vo(nTb+τd,n)=0, τd,n is obtained as a function of ve,n:
(4)τd,n=Tblogr(A2A-ve,n).
Second, when the bit transition pattern is 00 as shown in Figure 3(b), vo(nTb)=-A+ve,n and vi(t)=-A for nTb<t≤(n+1)Tb. Then, the output signal is
(5)vo(t)=vo(nTb)+{-A-vo(nTb)}(1-e-(t-nTb)/RC)fornTb≤t≤(n+1)Tb
and, by using that vo((n+1)Tb)=-A+ve,n+1, another relationship between ve,n and ve,n+1 is obtained:
(6)ve,n+1=ve,nr.
Finally, when the bit transition pattern is 10 or 11 as shown in Figures 3(c) or 3(d), the same relationship between ve,n and ve,n+1 can be obtained as (3) or (6) and τd,n can be obtained as (4) because Figures 3(c) and 3(d) are just vertically symmetric with Figures 3(a) and 3(b). In summary, (3) and (6) describe how ve,n is updated to ve,n+1 per every Tb according to the bit transition pattern and (4) describes how τd,n relates to ve,n when a bit transition occurs.


### 2.2. DDJ Calculation

By using (3), (6), and (4), the DDJ can be derived through the following steps.(i)
Calculateve,n by using (3) and (6) for the repeated data pattern with the finite pattern length of N.(ii)
Find the maximum and minimum values,ve,max⁡ and ve,min⁡, among the set of the calculated ve,n values.(iii)
Calculateτd,min⁡ and τd,max⁡ corresponding to ve,max⁡ and ve,min⁡ by using (4). Note that τd,n is inversely proportional to ve,n as shown in Figure 4.(iv)
Finally,DDJ=τd,max⁡-τd,min⁡. For random data with the infinite pattern length, the DDJ equation should be modified appropriately.Figure 4
Relationship betweenve,n and τd,n.If a data pattern has the finite pattern length ofN and passes through a first order RC low pass circuit in a steady state, there should exist N different values of ve,n in the output waveform. For example, if a data pattern is PRBS3, the pattern length is 7 and ve,n is always mapped to one of {ve,1, ve,2, ve,3, ve,4, ve,5, ve,6, ve,7} as shown in Figure 5. However, all of ve,n do not need to be considered for calculation of the DDJ. Among {ve,1, ve,2, ve,3, ve,4, ve,5, ve,6, ve,7}, ve,1, ve,4, ve,6, and ve,7 are needed because only they are at the bit transition edges of PRBS3. Thus, if the number of bit transitions within a data pattern is K, only K values of ve,n need to be considered for calculation of the DDJ. On the other hand, ve,n has Markov property [15, 16]. A variable is said to have Markov property if the future value depends only on the present value and not on the past values. As seen from (3) and (6), ve,n+1 depends only on ve,n and not on the preceding values of ve,n such as ve,n-1 and ve,n-2. Thus,
(7)ve,2=2Ar-ve,1r,ve,3=ve,2r,ve,4=ve,3r,ve,5=2Ar-ve,4r,ve,6=ve,5r,ve,7=2Ar-ve,6r,ve,8=2Ar-ve,7r.
By using the repetitiveness of PRBS3, ve,8=ve,1 and, thus, ve,1, ve,4, ve,6, and ve,7 are obtained as follows:
(8)ve,1=2Ar1-r+r3-r61-r7(9)ve,4=2Arr2-r3+r4-r61-r7(10)ve,6=2Arr-r4+r5-r61-r7(11)ve,7=2Ar1-r2+r5-r61-r7.
Before finding ve,max⁡ and ve,min⁡ from (8) to (11), ve,n can be generalized to
(12)ve,n=2Ar∑i=1K(-1)i+1rai,n1-rN
for any data pattern with the finite pattern length of N. Here, K is the number of bit transitions and ai,n is an integer variable defined as the relative bit distance of the ith bit transition backwards from ve,n, where ai,n∈{0,1,…,N-1}. In (12), ai,n is determined by the relative bit transition positions within the data pattern because the relationship between ve,n+1 and ve,n is determined by (3) whenever a bit transition occurs like 01 or 10 and by (6) whenever a bit holds like 00 or 11. Figure 6 shows that a1,1=0, a2,1=1, a3,1=3, and a4,1=6 for ve,1 and a1,4=2, a2,4=3, a3,4=4, and a4,4=6 for ve,4, respectively, as an example. The obtained values of ai,n in Figure 6 agree well with (8) and (9). Thus, ve,n can be generally represented as (12) for any data pattern with the finite pattern length of N if the data pattern is known.Figure 5
Output waveform when the input data pattern is PRBS3 as an example.How to determine (a)ai,1 of ve,1 and (b) ai,4 of ve,4 when the input data pattern is PRBS3 as an example.
(a)
(b)Now,ve,max⁡ and ve,min⁡ can be found among {ve,1,ve,2,…,ve,N}. If ve,m=2Ar(∑i=1K(-1)i+1rai,m/(1-rN)) and ve,n=2Ar(∑i=1K(-1)i+1rai,n/(1-rN)), where m≠n, ve,m and ve,n can be compared by using the following theorems, of which proofs are given in Appendix A.Theorem 1.
Ifa1,m>a1,n, then ve,m<ve,n.Theorem 2.
Ifai,m=ai,n  for all i=1,…,2k-1, where k is an integer and a2k,m>a2k,n, then ve,m>ve,n.Theorem 3.
Ifai,m=ai,n for all i=1,…,2k, where k is an integer and a2k+1,m>a2k+1,n, then ve,m<ve,n.For PRBS3,ve,min⁡=ve,4 by Theorem 1 because a1,1=a1,7=0<a1,6=1<a1,4=2 and ve,max⁡=ve,7 by Theorem 2 because a2,1=1<a2,7=2 as seen from (8)~(11). These theorems can be generally applied to any data pattern with the finite pattern length. However, if a data pattern is random and has the infinite pattern length, there are infinite numbers of ve,n and so ve,max⁡ and ve,min⁡ are obtained in the different way. In that case, the number of maximum consecutive identical bits is also infinite so that ve,min⁡=0 from (12) and ve,max⁡=2Ar from (3).Afterve,max⁡ and ve,min⁡ are found, τd,max⁡ and τd,min⁡ are obtained by (4) as
(13)τd,max⁡=Tblogr(A2A-ve,min⁡)τd,min⁡=Tblogr(A2A-ve,max⁡)
and the DDJ is finally derived as
(14)DDJ=τd,max⁡-τd,min⁡=Tblogr(2A-ve,max⁡2A-ve,min⁡).
Thus, for PRBS3,
(15)DDJ=Tblogr(1-r+r3-r61-r3+r4-r5)
and, for random data,
(16)DDJ=Tblogr(1-r).
Additionally, for other PRBS data patterns like PRBS4 and PRBS5, the DDJ can be derived as shown in Appendix B. Carefully observing (15), (16), (B.1), and (B.2), the DDJ can be generally approximated to
(17)DDJ≈Tblogr(1-r1-rM)
for any data pattern by using the number of maximum consecutive identical bits, M. For any PRBS M data patterns, the number of maximum consecutive identical bits equals M. Finally, the calculated ve,max⁡, ve,min⁡, τd,max⁡, τd,min⁡, and DDJ are compared for various data patterns in Table 2. The DDJ of PRBS M approaches the DDJ of random data as the number of maximum consecutive identical bits, M, increases to the infinity.Table 2
Calculatedve,max⁡, ve,min⁡, τd,max⁡, τd,min⁡, and DDJ values for various data patterns. The input rise/fall time is zero. Tb = 100 ps and fBW = 2 GHz.
Data pattern
PRBS3
PRBS4
PRBS5
Random
v
e
,
max
⁡ (V)
0.52
0.56
0.57
0.57
v
e
,
min
⁡ (V)
0.04
0.01
0.00
0.00
τ
d
,
max
⁡ (ps)
53.70
54.75
55.02
55.16
τ
d
,
min
⁡ (ps)
30.99
29.23
28.67
28.51
DDJ (ps)
22.71
25.52
26.35
26.65


## 2.1. Differential Equation of First Order RC Low Pass Circuit

Given a data pattern, a bit rate, and a system bandwidth, the DDJ can be derived by solving the differential equation of the first order RC low pass circuit shown in Figure2. Depending on the bit transition patterns, such as 01, 00, 10, and 11, the input signal, vi(t), and the initial condition, vo(nTb), of the first order RC low pass circuit are given differently for each bit duration, Tb, as shown in Figure 3. In the figure, the input rise/fall time is zero ideally. So, vi(t) is -A or +A for nTb<t≤(n+1)Tb and a new variable, ve,n, is defined as the voltage difference between vi(t) and vo(t) at t=nTb, where n is an integer; that is, ve,n=|vi(nTb)-vo(nTb)|. Also, for the bit transition pattern of 01 or 10, another variable, τd,n, is defined as the time difference between the threshold crossing times of vi(t) and vo(t), as shown in Figures 3(a) and 3(c). The definition of the variables can be found in Table 1.Table 1
Definition of the variables.
Variable
Definition
f
BW
3 dB bandwidth of the first order low pass system
T
b
Bit duration
f
b
Bit rate, 1/Tb
Δ
T
Rise/fall time of the input signal
v
e
,
n
Voltage difference betweenvi(t) and vo(t) at t=nTb
τ
d
,
n
Time difference between the threshold crossing times ofvi(t) and vo(t) for nTb<t<(n+1)Tb
a
i
,
n
Relative bit distance of theith bit transition backwards from ve,n
N
Pattern length
K
Number of bit transitionsFigure 2
First order RC low pass circuit.Input and output waveforms when the bit transition pattern is (a) 01, (b) 00, (c) 10, and (d) 11 and the input rise/fall time is zero.
(a)
(b)
(c)
(d)Because there are four different sets of the input signal,vi(t), and the initial condition, vo(nTb), depending on the bit transition patterns, the below differential equation of the first order RC low pass circuit should be solved for each bit transition pattern:
(1)vo(t)-vi(t)R+Cdvo(t)dt=0.
First, when the bit transition pattern is 01 as shown in Figure 3(a), the initial condition and the input signal are vo(nTb)=-A+ve,n and vi(t)=A for nTb<t≤(n+1)Tb, respectively. Then, the output signal is
(2)vo(t)=vo(nTb)+{A-vo(nTb)}(1-e-(t-nTb)/RC)fornTb≤t≤(n+1)Tb.
By using that vo((n+1)Tb)=A-ve,n+1, the relationship between ve,n and ve,n+1 is obtained as
(3)ve,n+1=2Ar-ve,nr,
where r=e-(Tb/RC) and, by using that vo(nTb+τd,n)=0, τd,n is obtained as a function of ve,n:
(4)τd,n=Tblogr(A2A-ve,n).
Second, when the bit transition pattern is 00 as shown in Figure 3(b), vo(nTb)=-A+ve,n and vi(t)=-A for nTb<t≤(n+1)Tb. Then, the output signal is
(5)vo(t)=vo(nTb)+{-A-vo(nTb)}(1-e-(t-nTb)/RC)fornTb≤t≤(n+1)Tb
and, by using that vo((n+1)Tb)=-A+ve,n+1, another relationship between ve,n and ve,n+1 is obtained:
(6)ve,n+1=ve,nr.
Finally, when the bit transition pattern is 10 or 11 as shown in Figures 3(c) or 3(d), the same relationship between ve,n and ve,n+1 can be obtained as (3) or (6) and τd,n can be obtained as (4) because Figures 3(c) and 3(d) are just vertically symmetric with Figures 3(a) and 3(b). In summary, (3) and (6) describe how ve,n is updated to ve,n+1 per every Tb according to the bit transition pattern and (4) describes how τd,n relates to ve,n when a bit transition occurs.


## 2.2. DDJ Calculation

By using (3), (6), and (4), the DDJ can be derived through the following steps.(i)
Calculateve,n by using (3) and (6) for the repeated data pattern with the finite pattern length of N.(ii)
Find the maximum and minimum values,ve,max⁡ and ve,min⁡, among the set of the calculated ve,n values.(iii)
Calculateτd,min⁡ and τd,max⁡ corresponding to ve,max⁡ and ve,min⁡ by using (4). Note that τd,n is inversely proportional to ve,n as shown in Figure 4.(iv)
Finally,DDJ=τd,max⁡-τd,min⁡. For random data with the infinite pattern length, the DDJ equation should be modified appropriately.Figure 4
Relationship betweenve,n and τd,n.If a data pattern has the finite pattern length ofN and passes through a first order RC low pass circuit in a steady state, there should exist N different values of ve,n in the output waveform. For example, if a data pattern is PRBS3, the pattern length is 7 and ve,n is always mapped to one of {ve,1, ve,2, ve,3, ve,4, ve,5, ve,6, ve,7} as shown in Figure 5. However, all of ve,n do not need to be considered for calculation of the DDJ. Among {ve,1, ve,2, ve,3, ve,4, ve,5, ve,6, ve,7}, ve,1, ve,4, ve,6, and ve,7 are needed because only they are at the bit transition edges of PRBS3. Thus, if the number of bit transitions within a data pattern is K, only K values of ve,n need to be considered for calculation of the DDJ. On the other hand, ve,n has Markov property [15, 16]. A variable is said to have Markov property if the future value depends only on the present value and not on the past values. As seen from (3) and (6), ve,n+1 depends only on ve,n and not on the preceding values of ve,n such as ve,n-1 and ve,n-2. Thus,
(7)ve,2=2Ar-ve,1r,ve,3=ve,2r,ve,4=ve,3r,ve,5=2Ar-ve,4r,ve,6=ve,5r,ve,7=2Ar-ve,6r,ve,8=2Ar-ve,7r.
By using the repetitiveness of PRBS3, ve,8=ve,1 and, thus, ve,1, ve,4, ve,6, and ve,7 are obtained as follows:
(8)ve,1=2Ar1-r+r3-r61-r7(9)ve,4=2Arr2-r3+r4-r61-r7(10)ve,6=2Arr-r4+r5-r61-r7(11)ve,7=2Ar1-r2+r5-r61-r7.
Before finding ve,max⁡ and ve,min⁡ from (8) to (11), ve,n can be generalized to
(12)ve,n=2Ar∑i=1K(-1)i+1rai,n1-rN
for any data pattern with the finite pattern length of N. Here, K is the number of bit transitions and ai,n is an integer variable defined as the relative bit distance of the ith bit transition backwards from ve,n, where ai,n∈{0,1,…,N-1}. In (12), ai,n is determined by the relative bit transition positions within the data pattern because the relationship between ve,n+1 and ve,n is determined by (3) whenever a bit transition occurs like 01 or 10 and by (6) whenever a bit holds like 00 or 11. Figure 6 shows that a1,1=0, a2,1=1, a3,1=3, and a4,1=6 for ve,1 and a1,4=2, a2,4=3, a3,4=4, and a4,4=6 for ve,4, respectively, as an example. The obtained values of ai,n in Figure 6 agree well with (8) and (9). Thus, ve,n can be generally represented as (12) for any data pattern with the finite pattern length of N if the data pattern is known.Figure 5
Output waveform when the input data pattern is PRBS3 as an example.How to determine (a)ai,1 of ve,1 and (b) ai,4 of ve,4 when the input data pattern is PRBS3 as an example.
(a)
(b)Now,ve,max⁡ and ve,min⁡ can be found among {ve,1,ve,2,…,ve,N}. If ve,m=2Ar(∑i=1K(-1)i+1rai,m/(1-rN)) and ve,n=2Ar(∑i=1K(-1)i+1rai,n/(1-rN)), where m≠n, ve,m and ve,n can be compared by using the following theorems, of which proofs are given in Appendix A.Theorem 1.
Ifa1,m>a1,n, then ve,m<ve,n.Theorem 2.
Ifai,m=ai,n  for all i=1,…,2k-1, where k is an integer and a2k,m>a2k,n, then ve,m>ve,n.Theorem 3.
Ifai,m=ai,n for all i=1,…,2k, where k is an integer and a2k+1,m>a2k+1,n, then ve,m<ve,n.For PRBS3,ve,min⁡=ve,4 by Theorem 1 because a1,1=a1,7=0<a1,6=1<a1,4=2 and ve,max⁡=ve,7 by Theorem 2 because a2,1=1<a2,7=2 as seen from (8)~(11). These theorems can be generally applied to any data pattern with the finite pattern length. However, if a data pattern is random and has the infinite pattern length, there are infinite numbers of ve,n and so ve,max⁡ and ve,min⁡ are obtained in the different way. In that case, the number of maximum consecutive identical bits is also infinite so that ve,min⁡=0 from (12) and ve,max⁡=2Ar from (3).Afterve,max⁡ and ve,min⁡ are found, τd,max⁡ and τd,min⁡ are obtained by (4) as
(13)τd,max⁡=Tblogr(A2A-ve,min⁡)τd,min⁡=Tblogr(A2A-ve,max⁡)
and the DDJ is finally derived as
(14)DDJ=τd,max⁡-τd,min⁡=Tblogr(2A-ve,max⁡2A-ve,min⁡).
Thus, for PRBS3,
(15)DDJ=Tblogr(1-r+r3-r61-r3+r4-r5)
and, for random data,
(16)DDJ=Tblogr(1-r).
Additionally, for other PRBS data patterns like PRBS4 and PRBS5, the DDJ can be derived as shown in Appendix B. Carefully observing (15), (16), (B.1), and (B.2), the DDJ can be generally approximated to
(17)DDJ≈Tblogr(1-r1-rM)
for any data pattern by using the number of maximum consecutive identical bits, M. For any PRBS M data patterns, the number of maximum consecutive identical bits equals M. Finally, the calculated ve,max⁡, ve,min⁡, τd,max⁡, τd,min⁡, and DDJ are compared for various data patterns in Table 2. The DDJ of PRBS M approaches the DDJ of random data as the number of maximum consecutive identical bits, M, increases to the infinity.Table 2
Calculatedve,max⁡, ve,min⁡, τd,max⁡, τd,min⁡, and DDJ values for various data patterns. The input rise/fall time is zero. Tb = 100 ps and fBW = 2 GHz.
Data pattern
PRBS3
PRBS4
PRBS5
Random
v
e
,
max
⁡ (V)
0.52
0.56
0.57
0.57
v
e
,
min
⁡ (V)
0.04
0.01
0.00
0.00
τ
d
,
max
⁡ (ps)
53.70
54.75
55.02
55.16
τ
d
,
min
⁡ (ps)
30.99
29.23
28.67
28.51
DDJ (ps)
22.71
25.52
26.35
26.65


## 3. Calculation of DDJ with Nonzero Input Rise/Fall Time

### 3.1. Differential Equation of First Order RC Low Pass Circuit

Now, the effect of the nonzero input rise/fall time on the DDJ can be considered. The differential equation of (1) should be solved again for four different bit transition patterns, such as 01, 00, 10, and 11, when the input rise/fall time is ΔT as shown in Figure 7. Although there are more accurate models for the rising/falling edges of the input signal, vi(t), the first order model is adopted for simplicity of calculation to derive the closed-form DDJ equations in this paper.Input and output waveforms when the bit transition pattern is (a) 01, (b) 00, (c) 10, and (d) 11 and the input rise/fall time is nonzero.
(a)
(b)
(c)
(d)First, when the bit transition pattern is 01 as shown in Figure7(a), the initial condition is vo(nTb)=-A+ve,n and the input signal is vi(t)=-A+(2A/ΔT)(t-nTb) for nTb<t≤nTb+ΔT and vi(t)=A for nTb+ΔT<t≤(n+1)Tb, respectively. Then, the output signal is
(18)vo(t)=-A+2AΔT(t-nTb-RC)+(2AΔTRC+ve,n)e-(t-nTb)/RCmmmmfornTb<t≤nTb+ΔT,(19)vo(t)=vo(nTb+ΔT)+{A-vo(nTb+ΔT)}(1-e-(t-nTb-ΔT)/RC)mmmmiimmmfornTb+ΔT<t≤(n+1)Tb.
By using that vo((n+1)Tb)=A-ve,n+1, the relationship between ve,n and ve,n+1 is obtained as
(20)ve,n+1=2ASr-ve,nr,
where r=e-(Tb/RC) and S=(RC/ΔT)(eΔT/RC-1). Also, by using that vo(nTb+(ΔT/2)+τd,n)=0, τd,n can be obtained as a function of ve,n. However, to solve vo(nTb+(ΔT/2)+τd,n)=0, (18) should be used if τd,n<ΔT/2 and (19) should be used if τd,n>ΔT/2. Because vo(nTb+ΔT)<0 is equivalent to τd,n>ΔT/2 in Figure 7(a), we can say that (18) should be used if vo(nTb+ΔT)>0 and (19) should be used if vo(nTb+ΔT)<0. Figure 8 shows the sufficient condition for τd,n>ΔT/2regardless ofve,n as region 1 and τd,n<ΔT/2regardless ofve,n as region 2, respectively. Region 3 is located between region 1 and region 2, in which τd,n can be larger or less than ΔT/2depending onve,n. The regions 1, 2, and 3 of Figure 8 can be obtained by solving vo(nTb+ΔT)=0. Here, fBW is the 3 dB bandwidth of the first order low pass system which is defined as fBW=1/2πRC. Thus, τd,n is obtained as
(21)τd,n=Tblogr(A2AS-ve,n)-ΔT2
from (19) in region 1 and
(22)τd,n=Tblogr{(2A/ΔT)(RC-τd,n)(2A/ΔT)RC+ve,n}-ΔT2
from (18) in region 2, respectively. In region 3, either (21) or (22) should be appropriately chosen for τd,n but after both (21) and (22) are evaluated and compared with ΔT/2 because (21) is used if τd,n>ΔT/2 and (22) is used if τd,n<ΔT/2. Second, when the bit transition pattern is 00 as shown in Figure 7(b), the initial condition and the input signal are equal to those of Figure 3(b) so that the same relationship between ve,n and ve,n+1 is obtained:
(23)ve,n+1=ve,nr.
Finally, when the bit transition pattern is 10 or 11 as shown in Figures 7(c) or 7(d), the same relationship between ve,n and ve,n+1 can be obtained as (20) or (23) and τd,n can be obtained from (21) or (22) because Figures 7(c) and 7(d) are just vertically symmetric with Figures 7(a) and 7(b), respectively.Figure 8
Sufficient conditions forτd,n>ΔT/2 and τd,n<ΔT/2 regardless of ve,n.


### 3.2. DDJ Calculation

As seen from (20) and (23), ve,n has Markov property also when the input rise/fall time is ΔT. Thus, ve,n can be generally represented as
(24)ve,n=2ASr∑i=1K(-1)i+1rai,n1-rN
for any data pattern with the finite pattern length of N by following the same steps explained in Section 2.2. Here, K is the number of bit transitions within the data pattern and ai,n is an integer variable defined as the relative bit distance of the ith bit transition backwards from ve,n, where ai,n∈{0,1,…,N-1}. The only difference between (24) and (12) is the additional multiplication factor, S, in (24). So, ve,max⁡ and ve,min⁡ can be found among {ve,1,ve,2,…,ve,N} by comparaing only ai,n values based on the same theorems stated in Section 2.2. Of course, if a data pattern is random and has the infinite pattern length, ve,min⁡=0 and ve,max⁡=2ASr. For any data pattern, τd,max⁡ and τd,min⁡ can be obtained from either (21) or (22) depending on fBW/fb and ΔT/Tb. If fBW/fb and ΔT/Tb are in region 1 of Figure 8,
(25)τd,max⁡=Tblogr(A2AS-ve,min⁡)-ΔT2τd,min⁡=Tblogr(A2AS-ve,max⁡)-ΔT2
from (21) and if fBW/fb and ΔT/Tb are in region 2,
(26)τd,max⁡=Tblogr{(2A/ΔT)(RC-τd,max⁡)(2A/ΔT)RC+ve,min⁡}-ΔT2τd,min⁡=Tblogr{(2A/ΔT)(RC-τd,min⁡)(2A/ΔT)RC+ve,max⁡}-ΔT2
from (22). Otherwise, if fBW/fb and ΔT/Tb are in region 3, the appropriate equations should be chosen from (25) and (26) for calculation of τd,max⁡ and τd,min⁡ by comparing the calculated values of τd,max⁡ and τd,min⁡ with ΔT/2. Consequently, the DDJ is derived as
(27)DDJ=τd,max⁡-τd,min⁡=Tblogr(2AS-ve,max⁡2AS-ve,min⁡)
in region 1 and the DDJ is calculated by using (26) in region 2. Although (27) and (14) look a bit different, the DDJ equation of (27) equals the DDJ equation of (14) since ve,min⁡ and ve,max⁡ are linearly proportional to the multiplication factor, S, as shown in (24) and thus S is cancelled out from (27). This means that the DDJ value does not depend on ΔT in region 1 and equals the DDJ value when ΔT=0. On the other hand, the DDJ value in region 2 is slightly larger than the DDJ value when ΔT=0. However, the difference is quite small and acceptable because the system bandwidth is relatively large compared to the bit rate in region 2 as shown in Figure 8 and so the DDJ value is very small in itself.


## 3.1. Differential Equation of First Order RC Low Pass Circuit

Now, the effect of the nonzero input rise/fall time on the DDJ can be considered. The differential equation of (1) should be solved again for four different bit transition patterns, such as 01, 00, 10, and 11, when the input rise/fall time is ΔT as shown in Figure 7. Although there are more accurate models for the rising/falling edges of the input signal, vi(t), the first order model is adopted for simplicity of calculation to derive the closed-form DDJ equations in this paper.Input and output waveforms when the bit transition pattern is (a) 01, (b) 00, (c) 10, and (d) 11 and the input rise/fall time is nonzero.
(a)
(b)
(c)
(d)First, when the bit transition pattern is 01 as shown in Figure7(a), the initial condition is vo(nTb)=-A+ve,n and the input signal is vi(t)=-A+(2A/ΔT)(t-nTb) for nTb<t≤nTb+ΔT and vi(t)=A for nTb+ΔT<t≤(n+1)Tb, respectively. Then, the output signal is
(18)vo(t)=-A+2AΔT(t-nTb-RC)+(2AΔTRC+ve,n)e-(t-nTb)/RCmmmmfornTb<t≤nTb+ΔT,(19)vo(t)=vo(nTb+ΔT)+{A-vo(nTb+ΔT)}(1-e-(t-nTb-ΔT)/RC)mmmmiimmmfornTb+ΔT<t≤(n+1)Tb.
By using that vo((n+1)Tb)=A-ve,n+1, the relationship between ve,n and ve,n+1 is obtained as
(20)ve,n+1=2ASr-ve,nr,
where r=e-(Tb/RC) and S=(RC/ΔT)(eΔT/RC-1). Also, by using that vo(nTb+(ΔT/2)+τd,n)=0, τd,n can be obtained as a function of ve,n. However, to solve vo(nTb+(ΔT/2)+τd,n)=0, (18) should be used if τd,n<ΔT/2 and (19) should be used if τd,n>ΔT/2. Because vo(nTb+ΔT)<0 is equivalent to τd,n>ΔT/2 in Figure 7(a), we can say that (18) should be used if vo(nTb+ΔT)>0 and (19) should be used if vo(nTb+ΔT)<0. Figure 8 shows the sufficient condition for τd,n>ΔT/2regardless ofve,n as region 1 and τd,n<ΔT/2regardless ofve,n as region 2, respectively. Region 3 is located between region 1 and region 2, in which τd,n can be larger or less than ΔT/2depending onve,n. The regions 1, 2, and 3 of Figure 8 can be obtained by solving vo(nTb+ΔT)=0. Here, fBW is the 3 dB bandwidth of the first order low pass system which is defined as fBW=1/2πRC. Thus, τd,n is obtained as
(21)τd,n=Tblogr(A2AS-ve,n)-ΔT2
from (19) in region 1 and
(22)τd,n=Tblogr{(2A/ΔT)(RC-τd,n)(2A/ΔT)RC+ve,n}-ΔT2
from (18) in region 2, respectively. In region 3, either (21) or (22) should be appropriately chosen for τd,n but after both (21) and (22) are evaluated and compared with ΔT/2 because (21) is used if τd,n>ΔT/2 and (22) is used if τd,n<ΔT/2. Second, when the bit transition pattern is 00 as shown in Figure 7(b), the initial condition and the input signal are equal to those of Figure 3(b) so that the same relationship between ve,n and ve,n+1 is obtained:
(23)ve,n+1=ve,nr.
Finally, when the bit transition pattern is 10 or 11 as shown in Figures 7(c) or 7(d), the same relationship between ve,n and ve,n+1 can be obtained as (20) or (23) and τd,n can be obtained from (21) or (22) because Figures 7(c) and 7(d) are just vertically symmetric with Figures 7(a) and 7(b), respectively.Figure 8
Sufficient conditions forτd,n>ΔT/2 and τd,n<ΔT/2 regardless of ve,n.


## 3.2. DDJ Calculation

As seen from (20) and (23), ve,n has Markov property also when the input rise/fall time is ΔT. Thus, ve,n can be generally represented as
(24)ve,n=2ASr∑i=1K(-1)i+1rai,n1-rN
for any data pattern with the finite pattern length of N by following the same steps explained in Section 2.2. Here, K is the number of bit transitions within the data pattern and ai,n is an integer variable defined as the relative bit distance of the ith bit transition backwards from ve,n, where ai,n∈{0,1,…,N-1}. The only difference between (24) and (12) is the additional multiplication factor, S, in (24). So, ve,max⁡ and ve,min⁡ can be found among {ve,1,ve,2,…,ve,N} by comparaing only ai,n values based on the same theorems stated in Section 2.2. Of course, if a data pattern is random and has the infinite pattern length, ve,min⁡=0 and ve,max⁡=2ASr. For any data pattern, τd,max⁡ and τd,min⁡ can be obtained from either (21) or (22) depending on fBW/fb and ΔT/Tb. If fBW/fb and ΔT/Tb are in region 1 of Figure 8,
(25)τd,max⁡=Tblogr(A2AS-ve,min⁡)-ΔT2τd,min⁡=Tblogr(A2AS-ve,max⁡)-ΔT2
from (21) and if fBW/fb and ΔT/Tb are in region 2,
(26)τd,max⁡=Tblogr{(2A/ΔT)(RC-τd,max⁡)(2A/ΔT)RC+ve,min⁡}-ΔT2τd,min⁡=Tblogr{(2A/ΔT)(RC-τd,min⁡)(2A/ΔT)RC+ve,max⁡}-ΔT2
from (22). Otherwise, if fBW/fb and ΔT/Tb are in region 3, the appropriate equations should be chosen from (25) and (26) for calculation of τd,max⁡ and τd,min⁡ by comparing the calculated values of τd,max⁡ and τd,min⁡ with ΔT/2. Consequently, the DDJ is derived as
(27)DDJ=τd,max⁡-τd,min⁡=Tblogr(2AS-ve,max⁡2AS-ve,min⁡)
in region 1 and the DDJ is calculated by using (26) in region 2. Although (27) and (14) look a bit different, the DDJ equation of (27) equals the DDJ equation of (14) since ve,min⁡ and ve,max⁡ are linearly proportional to the multiplication factor, S, as shown in (24) and thus S is cancelled out from (27). This means that the DDJ value does not depend on ΔT in region 1 and equals the DDJ value when ΔT=0. On the other hand, the DDJ value in region 2 is slightly larger than the DDJ value when ΔT=0. However, the difference is quite small and acceptable because the system bandwidth is relatively large compared to the bit rate in region 2 as shown in Figure 8 and so the DDJ value is very small in itself.


## 4. Simulation Results

Figure9 shows the eye diagrams of the simulated input and output waveforms when the data pattern is (a) PRBS3 and (b) random, respectively. The input rise/fall time is zero, the bit rate is 10 Gb/s, and the system bandwidth is 2 GHz. All the waveforms were obtained by running Cadence Spectre. In Figure 9(a), among {ve,1, ve,4, ve,6, ve,7}, ve,max⁡=ve,7 and ve,min⁡=ve,4, as discussed in Section 2.2. So, τd,max⁡=τd,4, τd,min⁡=τd,7, and DDJ=τd,4-τd,7. In Figure 9(b), ve,min⁡=0 and ve,max⁡=2Ar, and τd,max⁡ and τd,min⁡ correspond to ve,min⁡ and ve,max⁡, respectively. All the simulated ve,max⁡, ve,min⁡, τd,max⁡, τd,min⁡, and DDJ for various data patterns agree exactly with the calculated values in Table 2.Eye diagrams of the simulated input and output waveforms when the input data pattern is (a) PRBS3 and (b) random. The input rise/fall time is zero.Tb = 100ps and fBW=2 GHz.
(a)
(b)Figure10 compares the eye diagrams of the simulated output waveforms for ΔT=0 and ΔT=0.4Tb. The data pattern is random, the bit rate is 10 Gb/s, and the system bandwidth is 2 GHz. The input signal, of which rise/fall time is modeled by ΔT as first order approximation, is applied to the first order low pass system. As shown in Figure 10, two output waveforms slightly differ from each other only around the transition edges of the input signals; however, they exactly coincide with each other around the transition edges of the output signals. Thus, the simulated DDJ when ΔT=0.4Tb equals the simulated DDJ when ΔT=0. This coincidence between ΔT=0 and ΔT=0.4Tb is due to the fact that fBW/fb and ΔT/Tb are in region 1.Figure 10
Comparison of eye diagrams of the output waveforms whenΔT=0 and ΔT=0.4Tb. Tb=100ps and fBW=2GHz.Figure11(a) shows the comparison of the simulated DDJ with the calculated DDJ by (16) in this work and the calculated DDJ by (10) in [10]. The calculated DDJ in this work better estimates the simulated DDJ. As the system bandwidth, fBW, decreases, the calculated DDJ by (10) in [10] underestimates the simulated DDJ because (10) in [10] was derived by using only two preceding bits as discussed in Appendix C. Figure 11(b) shows the simulated DDJ for various input rise/fall times. ΔT varies from 0 to 0.75Tb. If fBW/fb and ΔT/Tb are in region 1, the simulated DDJ does not depend on ΔT; however, in region 2, the simulated DDJ starts to deviate as ΔT increases. Even in this case, the deviation of the simulated DDJ is smaller than 0.008 UI because the absolute DDJ value is very small in itself, that is, less than 0.01 UI, in region 2. The system bandwidth, fBW, is relatively large compared to the bit rate, fb, in region 2 as shown in Figure 8.(a) Comparison of the DDJ values calculated by (16) in this work and by (10) in [10] and (b) the simulated DDJ values versus various input rise/fall times.
(a)
(b)Additionally, the simulated DDJ values when the low pass system has the additional second pole,fp2, are summarized in Table 3. As shown in the table, the calculated DDJ values by the derived equations agree well with the simulated results with the accuracy of less than 0.021 UI when the second pole is larger than 5 times of the first pole, that is, 10 GHz. However, if the second pole approaches the first pole, the system bandwidth now decreases less than the first pole so that the simulated results start to deviate from the calculated DDJ values.Table 3
Simulated DDJ values when the low pass system has the additional second pole,fp2. The input rise/fall time is zero. Tb = 100 ps and fp1 = 2 GHz.
Second pole
Data pattern
PRBS3
PRBS4
PRBS5
Random
No second pole
22.71
25.52
26.35
26.65
f
p
2 = 20 GHz
22.81
25.68
26.52
26.83
f
p
2 = 10 GHz
24.35
27.48
28.41
28.75
f
p
2 = 5 GHz
32.23
36.86
38.48
38.81


## 5. Conclusion

The closed-form equation of DDJ in a first order low pass system has been derived. If the bit rate, the system bandwidth, the input rise/fall time, and the number of maximum consecutive identical bits are given, the DDJ can be calculated exactly in region 1 and accurately in regions 2 and 3. The simulated DDJ agrees well with the calculated DDJ. Because the DDJ in the transmission line may apply to the interconnect channels such as off-chip PCB traces, off-chip cables, and on-chip interconnect lines and the DDJ in the first order low pass system may apply to the transceiver circuit building blocks such as drivers, buffers, amplifiers, and limiters, the derived equation can be generally used for a high speed serial interface when we design system architecture, link budget, and each circuit building block which can be modeled as a first order low pass system.



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*Source: 694178-2014-10-16.xml*