Transmitter FFE makes the channel do the work

-June 30, 2016

At high data rates, around and above 10 Gbits/s, we have to face the reality that conducting traces glued to dielectric (a.k.a., printed circuit boards) are truly horrible waveguides. They attenuate the signal, whether it's NRZ/PAM-2 or PAM-4, mess up the relationships between the Fourier components' amplitudes, frequencies, and phases—everything that makes the waveform a signal.

Figure 1 shows an example of a PCB channel response. You can see how the signal is 60 dB down at 28 MHz, just where you want to be for 56 Gbit/s NRZ.

Figure 1. This ADS simulation of a Wild River Technology channel-modeling platform shows its frequency response. Vertical scale: dB, horizontal scale: frequency in GHz.

We must do a lot of work to help the receiver recognize the resulting waveform as a signal. In addition to careful layout and use of quality components—all in a cost-optimized way, of course—equalization does a lot of work.

Equalization effectively removes the channel response by applying the inverse transfer function of the channel to the signal. That is, if the transfer function of the channel is G(s), where

s = + α,

then we're after an equalizer whose transfer function is G-1(s). If we call the transmitter signal Tx(s) and the unequalized received signal Rx(s), then

G(s)Tx(s) = Noise(s).

Apply equalization at the receiver and you get

G-1(s)Rx(s) = G-1(s)G(s)Tx(s) + G-1(s)Noise(s) = Tx(s) + G-1(s)Noise(s)

which, except for the noise, is exactly what we want.

Here's another way to think of it that I find more intuitive: the channel distorts the signal, so why not pre-distort the signal in such a way that the channel itself removes that distortion? In other words, pre-distort the transmitted signal in a way that includes the inverse channel frequency response so that the channel cancels the pre-distortion.

Lets call Symbols(s) the signal that we want. It has the wide open eye that we’d like the receiver to see after the transmitted signal Tx(s) has been through the channel. Then the pre-distorted signal should be as close to G-1(s)Symbols(s) as we can get. Now go back to the simple relationship between the transmitted signal, channel response, and received signal:

G(s)Tx(s) + Noise(s) = Rx(s),

plug in the pre-distorted wave, Tx(s) = G-1(s)Symbols(s), and get

G(s)G-1(s)Symbols(s) + Noise(s) = Symbols(s) + Noise(s) = Rx(s)

and, voila, except for the inevitable noise, the receiver sees a perfect waveform!

We call the predistorted transmitted signal transmitter FFE (feed forward equalization):

TxFFE(s) = G-1(s)Symbols(s)

Well, not so fast. Because the transmitter, in most cases (every case I've ever seen), can only modify a pulse once per bit period, we have to use a discrete form of transmitter FFE rather than the continuous ideal:

where the ki are called taps and the symbol voltage levels, VTxSymbol, are called cursors. The expression amounts to distorting the voltages of the symbol we're receiving and those surrounding it so that the channel compensates for the distortion.

A great way to get immediate gratification from your equalization genius is to see how your equalization scheme affects a single bit. Enter the pulse or single-bit response. Figure 2 shows the pulse shapes with no equalization, DFE (decision-feedback equalization), and TxFFE+DFE.

Figure 2. Comparison of single bit response with and without Tx FFE and DFE.
Image courtesy Wild River Technology and Keysight Technologies.

You can even see how the TxFFE and receiver DFE (decision feedback equalizer) work together; even the nonlinear nature of the DFE jumps out.

So, nothing really new here, but a pretty cool way to think of Tx FFE.

Also see:

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