Master of your domain: everything you ever wanted to know about frequency and time domain--and maybe more.

THERE IS NO fundamental difference between the information contained in the frequency domain description and the time domain description of the behavior of an interconnect. The two terms possess everything you could ever want to know about the properties of an interconnect. In the frequency domain, we refer to the insertion loss and the return loss. In the time domain, we refer to the time domain reflected signal and the time domain transmitted signal.

The insertion and return loss is a measure of how sine wave signals reflect and are transmitted by the entire interconnect. The time domain signals measure how a Gaussian edge reflects from or transmits through the interconnect. If we measure the response of an interconnection in one of these domains--be it a connector, an IC package, a circuit board trace or combinations of these components--we can mathematically transform the data to be displayed in the other domain.

This means signal integrity engineers, who typically have better intuition in the time domain, can borrow the techniques from the well-established frequency domain, the usual territory of the RF engineer.

In the frequency domain, the insertion and return loss are special names for two of the four scattering parameters, or S parameters, for an interconnect. When there is only one interconnect, the two ends (or ports) are labeled with the index numbers port 1 and port 2.

The insertion loss is the ratio of the signal coming out of port 2 to the signal going into port 1, labeled as S21. The return loss is the ratio of the signal coming out of port 1 to the signal going into port 1, or S11. Of course, each S parameter will vary with frequency, and they have a magnitude and phase. These are two of the four S parameters. The others are generated by looking from the other side of the interconnect. While S12 will always equal S21, a property of all linear systems, any asymmetry in the interconnect will make S11 not equal to S22.

Everything you ever wanted to know about a single-ended interconnect is contained in these three unique S parameters, whether they are displayed in the time or the frequency domain. But what if you have two interconnects that might have some coupling? How do we use S parameters to describe the two lines?

The formalism for using S parameters starts by assigning index labels to the ends of the transmission lines. We usually use port 1 as the front of one line and port 2 as the other end of that line. Port 3 is the end of the line adjacent to port 1 and port 4 is the other end of the line, near port 2. This is illustrated in FIGURE 1. With 4 ports, there are 16 S parameters, but only ten of them are unique. Everything you ever wanted to know about the behavior of these two lines is contained in these 10 S parameters.

[FIGURE 1 OMITTED]

Of course, the insertion and return loss of the first interconnect is the same S21 and S11. The insertion and return loss of the second line is just S43 and S33. There are comparable values looking from the other side of the lines.

We can describe the near-end crosstalk between the two lines by looking at the signal coming out of port 3 when we send a signal into port 1. This would be S31. The far-end crosstalk would be S41. This is the ratio of the signal out of port 4 when we send a signal into port 1.

All of these combinations of S parameters can be stored in a simple, logical matrix known as, naturally, the S parameter matrix. For two single-ended coupled transmission lines, the S parameter matrix is a 4 x 4 matrix. Whether the terms are displayed in the frequency domain or the time domain, absolutely everything you ever wanted to know about the two lines can be found in this matrix.

Of course, these two lines, with some coupling, can also be described as a single differential pair. Next month, we look at the S parameter matrix for a single differential pair.

DR. ERIC BOGATIN is the CTO of IDI, and president of Bogatin Enterprises. Many of his papers are available on his Web site, www.BeTheSignal.com. He can be reached at [e-mail].