- 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

## 2.10 Bandwidth and Rise Time

The relationship between rise time and bandwidth for a re-created ideal square wave can be quantified. In each synthesized waveform in the previous example re-creating an ideal square wave, the bandwidth is explicitly known because each waveform was artificially created by including sine-wave-frequency components only up to a specified value. The rise time, defined as the time from the 10% point to the 90% point, can be measured from time-domain plots.

When we plot the measured 10–90 rise time and the known bandwidth for each waveform, we see that empirically there is a simple relationship. This is a fundamental relationship for all signals and is shown in Figure 2-11.

Figure 2-11 Empirical relationship between the bandwidth of a signal and its 10–90 rise time, as measured from a re-created ideal square wave with each harmonic added one at a time. Circles are the values extracted from the data; line is the approximation of BW = 0.35/rise time.

For the special case of a re-created square wave with only some of the higher harmonics included, the bandwidth is inversely related to the rise time. We can fit a straight-line approximation through the points and find the relationship between bandwidth and rise time as:

#### Equation 2-4

where:

- BW = the bandwidth, in GHz
- RT = the 10–90 rise time, in nsec

For example, if the rise time of a signal is 1 nsec, the bandwidth is about 0.35 GHz or 350 MHz. Likewise, if the bandwidth of a signal is 3 GHz, the rise time of the signal will be about 0.1 nsec. A signal with a rise time of 0.25 nsec, such as might be seen in a DDR3-based system, has a bandwidth of 0.35/0.25 nsec = 1.4 GHz.

There are other ways of deriving this relationship for other waveforms, such as with Gaussian or exponential edges. The approach we took here for square waves is purely empirical and makes no assumptions. It is one of the most useful rules of thumb in our toolbox.

It is important to keep the units straight. When rise time is in microseconds, the bandwidth is in MHz. For example, a very long rise time of 10 microseconds has a bandwidth of about 0.35/10 microsec = 0.035 MHz. This is equivalent to 35 kHz.

When the rise time is in nanoseconds, the bandwidth is in GHz. A 10-nsec rise time, typical of a 10-MHz clock frequency, has a bandwidth of about 0.35/10 nsec = 0.035 GHz or 35 MHz.