Introduction to Equalization in General

A serial signal consists of a transmitter sending a signal over a transmission channel (examples: backplane, cable) to a receiver.

As the signal rate increases, the channel the signal travels through distorts the signal at the receiver. This can result in a partially or completely closed eye diagram where the clock/data cannot be extracted by the receiver. In order to recover a clock or data from the eye diagram, it must be re-opened. This is where equalization can help.

Look at the diagram below. You can see that an open, clean eye leaves the transmitter and is sent through the channel. As it passes through the channel, random noise, crosstalk, and inter-symbol interference (ISI) distort the signal, causing the eye to close. Equalization is then used to re-open the eye by correcting for the ISI.

Figure 1: Diagram showing the closing of the eye diagram as a result of the transmission channel and subsequent reopening of the eye via equalization

As you can see, the main purpose of equalization is to correct for the problems caused by the transmission channel. Equalization techniques provide a way to discern the original signal (the signal coming out of the transmitter) given a distorted signal at the receiver. In other words, equalization corrects for the high frequency component voltage levels and, in the process, corrects the trajectories of these components in the corresponding eye diagram (opens up the eye).

Figure 2: Screenshots showing an unequalized and equalized 6 Gb/s eye diagram

This article will discuss two types of equalization methods: Feed-Forward Equalization (FFE) and Decision Feedback Equalization (DFE).

Feed-Forward Equalization (FFE)

Feed-Forward Equalization (FFE) is an equalization technique that corrects the received waveform with information about the waveform itself and not information about the logical decisions made on the waveform. FFE basically acts like a FIR (finite impulse response) filter and uses the voltage levels of the received waveform associated with previous and current bits to correct the voltage level of the current bit. One key thing to remember when working with FFE is that the equalization is performed on the actual waveform. At no point in the FFE algorithm are logical decisions made (is this bit a 1 or a 0?). Instead, FFE is only concerned with correcting voltage levels in the waveform.

For the purpose of this discussion, assume the FFE algorithm you are using has three taps. Taps are unitless correction factors applied to voltage levels in order to correct them. One way to think of these correction factors is to view them as the ratio of the voltage the receiver should have seen to the voltage the receiver did see.

The mathematical description of a three tap FFE is as follows:

e(t) = c₀r(t – (0T_D)) + c₁r(t – (1T_D)) + c₂r(t – (2T_D))

where:

• e(t) is the corrected (or equalized) voltage waveform at time t.
• T_D is the tap delay.
• r(t-nT_D) is the uncorrected input waveform n tap delays before the present time.
• c_n is the correction coefficient (tap) multiplied by the version of the uncorrected waveform that has been time-advanced by n tap delays.

So FFE obtains the corrected (or equalized) voltage level at the location of interest on the waveform (time t) by forming a sum of the taps and voltage levels of the previous two tap-delayed locations as well as the location of interest before being equalized. Once the location of interest’s voltage level has been corrected, the algorithm moves along with the sample rate to the next location of interest and repeats the process. This continues until the entire waveform has been traversed.

Decision Feedback Equalization (DFE)

There are multiple ways to implement DFE. This section will discuss the DFE algorithm used by the Agilent N5461A Infiniium Serial Data Equalization software application (which is used on either the Agilent Infiniium 90000A Series or 9000A Series oscilloscope).

For the purpose of this article, assume the DFE algorithm you are using has two taps. Before looking at the mathematical description of DFE, it is important to understand the results of the algorithm. Generally, DFE calculates a correction value that is added to the logical decision threshold (the threshold above which the waveform is considered a logical high and below which the waveform is considered a logical low). Therefore, DFE results in the threshold shifting up or down so new logical decisions can be made on the waveform based upon this new equalized threshold level.

The mathematical description of the correction value added to the decision threshold for a two tap DFE is as follows:

V(k) = c₁s(k – 1) + c₂(k – 2)

where:
• V(k) is the correction voltage added to the decision threshold used when determining the
 logical value of bit k.
• s(k-n) is the logic value of the data bit located n bits prior to bit k.
• c_n is the correction coefficient (tap) for the bit n bits prior to the bit of interest.

So, in order for DFE to obtain the corrected voltage offset for the threshold level at the bit of interest, it first needs to be seeded with the correct bit values for the first several bits to get started. Assuming the logical decisions for the first several bits are correct, the algorithm can feed them forward to determine the logical value of the current bit. For a two tap DFE, the two bits previous to the current bit need to have their bit levels already determined. Then the algorithm multiplies their bit levels by their corresponding tap values.

The sum of these two tap/bit level products gives the amount the decision threshold should be shifted. Many DFE algorithms would then shift the voltage threshold by this amount, but the Agilent Infiniium Serial Data Equalization software does the opposite. Instead of shifting the voltage threshold, it keeps the threshold constant and shifts the corresponding voltage level by the same amount, but in the opposite direction.

The algorithm would then shift forward one index to the next bit of interest. This process repeats itself until the entire signal has been traversed.

FFE Versus DFE

FFE is the most common equalization algorithm used in today’s serial busses. As explained above FFE only corrects voltage level by removing ISI, so the equalizer chips tend to not be as complicated and require less gates than a chip designed using DFE. Under most circumstances the less expensive and easier to implement FFE will work for designers.

Now consider a design where the channel has the potential to vary from chip to chip. DFE is typically used to open the eyes in signals with more ISI than FFE can handle. Because DFE uses the current bit as part of it tap value definition it is able to dynamically open closed eyes.

Today’s oscilloscope software can be used to model both DFE and FFE to find which algorithm suits the designer’s needs best. For example Agilent’s N5461A Infiniium Serial Data Equalization Software is able to model both on a single screen to allow users to choose which equalizer they want to implement. While DFE and FFE are different equalization techniques, it is not uncommon to use both at the receiver.

Conclusion

Equalization is becoming increasingly important to today’s high speed digital designs as designers continually push material limits for increasing bus speeds. DFE and FFE are both useful equalization techniques that can be used to open closed eyes at the receiver. Today’s oscilloscopes now provide equalization software that will fully model both DFE and FFE techniques which makes using these techniques easier and more efficient than ever.
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