Chapter 2: Principles of High-Speed Communications 10
2.6 Overall Impact of Jitter and ISI on the BER
2.6.3 ISI and Jitter Trade-off
If the BER caused by the timing jitter in equation (2.23) is plotted vs. the sampling point in a unit interval, i.e., when , a curve is achieved that resembles the shape of a bathtub and is thus called a bathtub curve. It graphically demonstrates that as the sampling point approaches the edges of the data eye diagram, the BER significantly increases. An example bathtub curve is shown in Figure 2.27, when the total jitter distribution is Gaussian with zero mean and standard deviation, σj=0.05 UI. A unit interval (UI) is a unit of time that equals the time normalized to a bit period. The bathtub curve is a useful tool for characterization of high-speed links. It is used to define an eye diagram opening for a given BER. For instance, in Figure 2.27, the eye diagram opening at the BER=10-12 is about 0.3 UI. The eye diagram opening corresponds to the available timing margin for the location of the sampling clock in the eye diagram that can achieve
0≤Ts≤Tb
0 0.2 0.4 0.6 0.8 1
10-12 10-10 10-8 10-6 10-4 10-2 100
Figure 2.27: Bathtub curve for σj=0.05 UI
Log[BER]
Ts [UI]
the target BER. Therefore, the bathtub curve can be used as a measure for the trade-off between the link data jitter budget, σj, and the clock jitter budget, the eye opening.
2.6.3.2 The BER Contours: 3D Bathtub Curve
We can generalize the concept of the bathtub curve to a data link with noise and ISI. If we use (2.30) to calculate the BER, we can plot a three-dimensional bathtub curve as a function of the sampling time and the system bandwidth that represents the ISI in the case of a first-order system. Consequently, we obtain an insight about the trade-offs between the data link’s jitter and ISI budget and the sampling clock timing margin. Such trade-offs are important in determining the specifications of the pre-amplifier response and the clock and data recovery characteristics for achieving minimum BER.
Figure 2.28(a) shows the 3D bathtub curve when the link is modeled with a first-order system. The BER is calculated for various sampling points and normalized 3dB bandwidths, when σj=0.05UI and N0=4e-3V/Hz2. If N0=0, the cross section of the plot, when bandwidth approaches infinity, becomes the conventional bathtub curve.
Figure 2.28(b) shows the contours of the BER as a function of the sampling point and the bandwidth, which is equivalent to the top view of the 3D bathtub curve. The contours
0.1 0.3 0.5 0.7 0.9
0.4 0.6 0.8 1 1.2 1.4
-12
-10
-10
-10-10 -8
-8
-8
-8-8 -8
-6
-6
-6
-6-6 -6
-4
-4
-4
-4-4 -4
-2
-2
-2
-2-2 -2
Figure 2.28: (a) Three dimensional bathtub curve for a first-order system for various normalized bandwidths; σj=0.05UI and N0=4e-3v2/Hz (b) Contours of BER from top view of plot (a)
log10[BER]
Ts [UI]
Ts [UI] f-3d
/(BBit Rate)
f-3dB/(Bit Rate)
(a) (b)
show that the BER is independent of bandwidth as the sampling point approaches the data edges because the timing jitter dominates the BER. Moreover, the optimum bandwidth that minimizes BER is about 75% of the bit rate. At this bandwidth, the optimum sampling point is neither in the center of the eye nor at Ts=Tb as we saw in a first-order system.
Finally, we can see that the timing margin for the sampling clock reduces drastically at smaller system bandwidths.
We can use (2.30) to find the contours of the BER for any given σj and N0. The noise standard deviation, σn, is a function of the bandwidth and N0. Figure 2.29 shows the BER contours for two more cases. In Figure 2.29(a), the BER is plotted when the σj=0.025 UI, half the value in Figure 2.28(b)’s plot. All the other parameters are the same. The jitter reduction lowers the minimum achievable BER. It also moves the optimum sampling point to the left, closer to Ts=Tb, which is the optimum sampling point for first-order response in the absence of jitter. The optimum bandwidth is shifted to lower values, as the sampling point is closer to Ts=Tb. This is because the ISI terms from (2.16) and (2.17) decrease, and amplitude noise dominates BERISI. Therefore, a smaller system bandwidth that filters more of the noise power can achieve a lower BER. Figure 2.29(b) shows the
0.1 0.3 0.5 0.7 0.9
0.4 0.6 0.8 1 1.2 1.4
-8 -9 -8
-8
-7 -7
-7 -7
-6 -6
-6
-6
-6 -6
-5
-5
-5
-5-5 -5
-4
-4
-4
-4-4 -4
-3
-3
-3
-3-3 -3
-2
-2
-2
-2-2-2
-1
-1
-1
-1-1-1
0.1 0.3 0.5 0.7 0.9
0.4 0.6 0.8 1 1.2 1.4
-16
-14
-14
-12 -12
-12
-10
-10
-10
-10-10
-8
-8
-8
-8-8-8
-6
-6
-6
-6-6-6
-4
-4
-4
-4-4-4
-2
-2
-2
-2-2-2
Ts [UI]
(a) (b)
Ts [UI]
Figure 2.29: BER contours (a) σj=0.025UI and N0=4e-3V2/Hz (b)σj=0.05UI and N0=5e-3V2/Hz
f-3dB/(Bit Rate) f-3dB/(Bit Rate)
BER contours when N0=5e-3V/Hz and all the other parameters are the same as Figure 2.28(b). The increase in amplitude noise degrades the overall BER by 3-4 orders of magnitude. The optimum system bandwidth-to-bit rate ratio is smaller compared to Figure 2.28(b), as we discussed above. The sampling timing margin for achieving BER of 10-12 has significantly decreased.
The y-axis in Figure 2.29 is the normalized bandwidth only for a first-order system.
For a general LTI system the y-axis is related to the size of the pulse response at each subsequent sampling point, which is in turn associated with the received pulse response as a result of the combination of the channel response and pre-amplifier transfer function.
Therefore, the designer can use the BER contours to determine the optimum front end time response shape for achieving a target BER. In addition, the optimum sampling point and its associated timing margin can be obtained from the BER contours with a target BER and is used to design the parameters for the clock recovery circuit.
In the next chapter, we introduce the data-dependent jitter (DDJ) phenomenon that is the impact of the ISI on the threshold-crossing time of the data. The DDJ modifies the jitter distribution by effectively increasing the jitter variance. In this case, (2.30) is not sufficient for finding the total BER. We complete the equation for computing the BER by including the impact of DDJ in our calculations.