- 2.1 The Time Domain
- 2.2 Sine Waves in the Frequency Domain
- 2.3 Shorter Time to a Solution in the Frequency Domain
- 2.4 Sine Wave Features
- 2.5 The Fourier Transform
- 2.6 The Spectrum of a Repetitive Signal
- 2.7 The Spectrum of an Ideal Square Wave
- 2.8 From the Frequency Domain to the Time Domain
- 2.9 Effect of Bandwidth on Rise Time
- 2.10 Bandwidth and Rise Time
- 2.11 What Does Significant Mean?
- 2.12 Bandwidth of Real Signals
- 2.13 Bandwidth and Clock Frequency
- 2.14 Bandwidth of a Measurement
- 2.15 Bandwidth of a Model
- 2.16 Bandwidth of an Interconnect
- 2.17 The Bottom Line

## 2.13 Bandwidth and Clock Frequency

As we have seen, bandwidth relates to the rise time of a signal. It is possible to have two different waveforms, with exactly the same clock frequency but different rise times and different bandwidths. Just knowing the clock frequency cannot tell us what the bandwidth is. Figure 2-14 shows four different waveforms, each with exactly the same clock frequency of 1 GHz. However, they have different rise times and hence different bandwidths.

Figure 2-14 Four different waveforms, each with exactly the same 1-GHz clock frequency. Each of them has a different rise time, as a fraction of the period, and hence different bandwidths.

Sometimes, we don't always know the rise time of a signal but need an idea of its bandwidth anyway. Using a simplifying assumption, we can estimate the bandwidth of a clock wave from just its clock frequency. Still, it is important to keep in mind that it is not the clock frequency that determines the bandwidth, it is the rise time. If all we know about the waveform is the clock frequency, we can't know the bandwidth for sure; we can only guess.

To evaluate the bandwidth of a signal from just its clock frequency, we have to make a very important assumption. We need to estimate what a typical rise time might be for a clock wave.

How is the rise time related to the clock period in a real clock waveform? In principle, the only relationship is that the rise time must be less than 50% of the period. Other than this, there is no restriction, and the rise time can be any arbitrary fraction of the period. It could be 25% of the period, as in cases where the clock frequency is pushing the limits of the device technology, such as in 1-GHz clocks. It could be 10% of the period, which is typical of many microprocessor-based products. It could be 5% of the period, which is found in high-end FPGAs driving external low-clock-frequency memory buses. It could even be 1% if the board-level bus is a legacy system.

If we don't know what fraction of the period the rise time is, a reasonable generalization is that the rise time is 7% of the clock period. This approximates many typical microprocessor-based boards and ASICs driving board-level buses. From this, we can estimate the bandwidth of the clock waveform.

It should be kept in mind that this assumption of the rise time being 7% of the period is a bit aggressive. Most systems are probably closer to 10%, so we are assuming a rise time slightly shorter than might typically be found. Likewise, if we are underestimating the rise time, we will be overestimating the bandwidth, which is safer than underestimating it.

If the rise time is 7% of the period, then the period is 1/0.07 or 15 times the rise time. We have an approximation for the bandwidth as 0.35/rise time. We can relate the clock frequency to the clock period, because they are each the inverse of the other. Replacing the clock period for the clock frequency results in the final relationship; the bandwidth is five times the clock frequency:

#### Equation 2-5

where:

- BW
_{clock}= the approximate bandwidth of the clock, in GHz - F
_{clock}= the clock repeat frequency, in GHz

For example, if the clock frequency is 100 MHz, the bandwidth of the signal is about 500 MHz. If the clock frequency is 1 GHz, the bandwidth of the signal is about 5 GHz.

This is a generalization and an approximation, based on the assumption that the rise time is 7% of the clock period. Given this assumption, it is a very powerful rule of thumb, which can give an estimate of bandwidth with very little effort. It says that the highest sine-wave-frequency component in a clock wave is typically the fifth harmonic!

It's obvious, but bears repeating, that we always want to use the rise time to evaluate the bandwidth. Unfortunately, we do not always have the luxury of knowing the rise time for a waveform. And yet, we need an answer *now*!