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Closed-form equation of data dependent jitter in first order low pass system.

Byun S - ScientificWorldJournal (2014)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Division of Electronics and Electrical Engineering, Dongguk University-Seoul, 26 Pil-dong 3-ga, Jung-gu, Seoul 100-715, Republic of Korea.

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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Output waveform when the input data pattern is PRBS3 as an example.
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fig5: Output waveform when the input data pattern is PRBS3 as an example.

Mentions: If a data pattern has the finite pattern length of N 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−rNfor 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.


Closed-form equation of data dependent jitter in first order low pass system.

Byun S - ScientificWorldJournal (2014)

Output waveform when the input data pattern is PRBS3 as an example.
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4217351&req=5

fig5: Output waveform when the input data pattern is PRBS3 as an example.
Mentions: If a data pattern has the finite pattern length of N 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−rNfor 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.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Division of Electronics and Electrical Engineering, Dongguk University-Seoul, 26 Pil-dong 3-ga, Jung-gu, Seoul 100-715, Republic of Korea.

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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