As we shift from non-return to zero (NRZ) to PAM4, it's important to identify the distinctions between bit error ratio (BER) and symbol error ratio (SER). It sounds simple enough but after saying "BER" for your entire career, it's easy to make obvious mistakes. In most cases, if you're talking about waveforms and eye diagrams you want SER, and if you're talking about anything else, stop talking, listen carefully, and double check. As you know, PAM4 (4 level pulse amplitude modulation) encodes two bits in each of four symbol levels. The Grey coding scheme determines which pairs of bits are assigned to which symbols (see five part series links below), but Grey coding isn't relevant when you're analysing eye diagrams. That PAM4 eye diagrams come with three pupils is relevant. We can extract bathtub plots from each of PAM4's three pupils in both the horizontal jitter direction and vertical noise direction. A PAM4 jitter bathtub plot (figure 1) measures SER(x), where x is the time-delay position of the sampling point and the noise bathtub plot measures SER(V), where V is the vertical position of the sampling point—voltage for electrical systems and power for optical. It's easy to extract eye height, eye width, and total jitter defined with respect to SER from SER bathtub plots.

EDNAOL 2016JUN06 TA 01Fig1 Figure 1: Three SER bathtub plots based on a PAM4 signal.

Right now, it looks like the tardy but emerging PAM4 specifications will define the system EH (eye height) and EW (eye width) as the smallest value of the three PAM4 eye diagrams:




where LOW, MID, and UPP refer to the lowest, middle, and upper of PAM4's three eyes. Minimum allowed values for EH6 and EW6, where the 6 refers to SER = 1E-6 will be specified depending on the application. The only PAM4 serial data standard that's been released, 100 GbE's 100GBASE-KP4, as well as (rumour has it) the tardy standards, require a common time-delay centre for analysis of each eye pupil. The time-delay centre is defined to be the midpoint of the widest horizontal opening of the middle pupil. This definition accommodates receivers with symbol identifying circuits (a.k.a., voltage slicers) that have common timing. As rates grow and PAM4 evolves, expect receivers with three independent slicers and each of the three eyes to be allowed their own centres. Along with saving bandwidth by cramming two bits into one symbol period, PAM4 also introduces FEC (forward error correction) that relaxes the SER requirement, which is why they specify minimum requirements for SER < 1E-6 instead of NRZ's BER < 1E-12 or 1E-15. The higher error ratio makes it possible for oscilloscopes to acquire enough data to measure SER bathtub plots and contours. In the olden days, oscilloscopes had to extrapolate to estimate eye openings at BERs of 1E-12 or 1E-15. But be careful! Some test equipment applies Grey decoding to PAM4 signals before extracting bathtub plots and contours and that means that they report BER not SER (figure 2).

EDNAOL 2016JUN06 TA 01Fig2 Figure 2: Grey coding/decoding (graphic copyright Ransom Stephens).

The SER-BER conundrum gets worse: The FEC required by most (but not all!) PAM4 standards is applied after Grey decoding. That is, FEC corrects BER, not SER. A symbol error can mean that both bits have been misidentified, or just one of the pair. For the Grey coding schemes we'll be using, amplitude noise is more likely to cause one bit error than jitter, but there's no easy way to tell without decoding the symbols to bits and comparing to the transmitted logic. The number of bit errors that Reed-Solomon FEC can correct depends on the order of the bit errors. Since we can't decipher the errored bits from the errored symbols, it's almost impossible to predict the post-FEC BER from SER. I hope to write about the SER * FEC → BER quagmire soon. If you have results that I can report, please send me a note (ransom [at] ransomsnotes dot com ].