Rise times and bandwidths: as rise times keep dropping, bandwidths will keep heading through the roof.

WE ARE ACCUSTOMED to thinking about high-speed digital signals in the time domain, where signals are sequential voltages that change in time. After all, it is in the time domain where performance such as propagation delay, jitter, skew, rise time and clock period are measured.

But even though the time domain is the real world, and ultimately we must be able to return to the time domain to make the final evaluation of a high-speed digital circuit, it is certainly not the only domain.

For some signal integrity problems we can get to an answer faster by taking a short cut through the frequency domain. The frequency domain is a mathematical construct; it is not the real world. As a mathematical construct, there are certain very well defined rules that must be followed in the frequency domain. One such rule is that the only sort of waveform we are allowed to talk about is a sine wave. When we describe signals in the frequency domain, we may only use combinations of sine waves, each with a specific frequency, amplitude and phase.

The process we use to translate a time domain waveform into its frequency domain description is the Fourier transform. This tool converts a voltage as a function of time into its frequency components. Luckily, all versions of SPICE simulators have built-in Fourier transform tools, so we never have to perform this calculation by hand. If we can simulate the time domain waveform of a signal, we can easily generate its frequency domain description in SPICE.

FIGURE 1 depicts an example of a time domain clock signal with a clock frequency of 1 GHz and a rise time of 1 psec. This is a very short rise time signal, and a good approximation to an ideal square wave. Its Fourier transform, showing its frequency components or harmonics, is also shown.

[FIGURE 1 OMITTED]

Even though this is nearly an ideal square wave, the amplitudes of the harmonics drop off pretty quickly. In fact, for an ideal square wave, with a voltage transition from 0 v to 1 v, the amplitudes of each harmonic are A(n) = 2/([pie] x n) with n = the harmonic number, for odd values of n.

To translate this frequency domain signal into the time domain, we could use an inverse Fourier transform. But we can gain more insight by just converting each harmonic component in the spectrum into its corresponding sine wave in the time domain, and adding them all up. This can also be done in SPICE.

How many harmonics do we have to include in order to re-create the ideal square wave? What is the highest sine wave frequency component we must include in the resulting time domain waveform? We call this highest sine wave frequency component the bandwidth of the signal. When we convert a frequency domain spectrum into its time domain signal, what is the impact on the time domain signal with ever-increasing bandwidth?

We can use SPICE to explore the impact of the bandwidth on a time domain signal. An AC voltage source can be used to generate a sine wave. A different AC source is used for each harmonic. Its amplitude is adjusted to match the harmonic amplitudes of each harmonic in an ideal square wave. Selected voltages can be added up to create the resulting time domain waveform when the bandwidth is 3 GHz, 5 GHz and even 11 GHz. FIGURE 2 shows the resulting time domain waveforms with increasing bandwidth from 1 GHz to 11 GHz. These waveforms clearly demonstrate that the higher the bandwidth, the shorter the rise time.

[FIGURE 2 OMITTED]

This is a very important general principle for all signals: the shorter the rise time of the signal, the higher the bandwidth of the signal. Using almost any version of SPICE, you can explore for yourself how the highest sine wave frequency in a signal influences its rise time. Using SPICE you can verify that the 10% to 90% rise time, RT, of a signal is roughly related to the bandwidth, BW, by RT = 0.35/BW.

Remember, it is absolutely inevitable that as clock frequencies increase, rise times will decrease and bandwidths will only be headed onward and upward.

REFERENCES

This topic is reviewed in Bogatin's book Signal Integrity--Simplified (Prentice Hall) and in online lectures available at www.BeTheSignal.com.

DR. ERIC BOGATIN is currently the CTO of IDI in Kansas City, KS. He has written four books on signal integrity and interconnect design, as well as over 200 papers. He can be reached at [e-mail].