- 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

## 2.9 Effect of Bandwidth on Rise Time

The term *bandwidth* is used for the highest sine-wave-frequency component that is significant in the spectrum. This is the highest sine-wave frequency we need to include to adequately approximate the important features of the time-domain waveform. All frequency components of higher frequency than the bandwidth can be ignored. In particular, as we will see, the bandwidth we choose will have a direct effect on the shortest rise time of the signal we are able to describe in the time domain.

The term *bandwidth* historically is used in the rf world to refer to the range of frequencies in a signal. In rf applications, a carrier frequency is typically modulated with some amplitude or phase pattern. The spectrum of frequency components in the signal falls within a band. The range of frequencies in the rf signal is called the bandwidth. Typical rf signals might have a carrier frequency of 1.8 GHz with a bandwidth about this frequency of 100 MHz. The bandwidth of an rf signal defines how dense different communications channels can fit.

With digital signals, bandwidth also refers to the range of frequencies in the signal's spectrum. It's just that for digital signals, the low frequency range starts at DC and extends to the highest frequency component. In the world of digital signals, since the lowest frequency will always be DC, bandwidth will always be a measure of the highest sine wave frequency component that is significant.

When we created a time-domain waveform from just the zeroth, the first, and the third harmonics included, as in Figure 2-8, the bandwidth of the resulting waveform was just up to the third harmonic, or 3 GHz in this case. By design, the highest sine-wave-frequency component in this waveform is 3 GHz. The amplitude of all other sine-wave components in this time-domain waveform is exactly 0.

When we added higher harmonics to create the waveforms in Figure 2-9, we designed their bandwidths to be 7 GHz, 19 GHz, and 31 GHz. If we were to take the shortest rise-time waveform in Figure 2-9 and transform it back into the frequency domain, its spectrum would look exactly like that shown in Figure 2-6. It would have components from the zeroth to the thirty-first harmonics. Beyond the thirty-first harmonic, all the components would be zero. The highest sine-wave-frequency component that is significant in this waveform is the thirty-first harmonic, or the waveform has a bandwidth of 31 GHz.

In each case, we created a waveform with a higher bandwidth, using the ideal-square-wave's spectrum as the starting place. And, in each case, the higher-bandwidth waveform also had a shorter 10–90 rise time. The higher the bandwidth, the shorter the rise time and the more closely the waveform approximates an ideal square wave. Likewise, if we do something to a short rise-time signal to decrease its bandwidth (i.e., eliminate high-frequency components), its rise time will increase.

For example, it is initially difficult to evaluate the time-domain response of a signal propagating down a lossy transmission line in FR4. As we will see, there are two loss mechanisms: conductor loss and dielectric loss. If each of these processes were to attenuate low-frequency components the same as they do high-frequency components, there would simply be less signal at the far end, but the pattern of the spectrum would look the same coming out as it does going in. There would be no impact on the rise time of the waveform.

However, both conductor loss and dielectric loss will attenuate the higher-frequency components more than the low-frequency components. By the time the signal has traveled through even four inches of trace, the high-frequency components, above about 8 GHz, can have lost more than 50% of their power, leaving the low-frequency terms less affected. In Figure 2-10 (top), we show the measured attenuation of sine-wave-frequency components through a four-inch length of transmission line in FR4. This transmission line happens to have a 50-Ohm characteristic impedance and was measured with a network analyzer. Frequency components below 2 GHz are not attenuated more than -1 dB, while components at 10 GHz are attenuated by -4 dB.

Figure 2-10 Top: The measured attenuation through a 4-inch length of 50-Ohm transmission line in FR4 showing the higher attenuation at higher frequencies. Bottom: The measured input and transmitted signal through a 36-inch 50-Ohm transmission line in FR4, showing the rise time to have degraded from 50 psec to more than 1.5 nsec.

This preferential attenuation of higher frequencies has the impact of decreasing the bandwidth of a signal that would propagate through the interconnect. Figure 2-10 (bottom) is an example of the measured rise time of a 50-psec signal entering a 36-inch-long trace in FR4 and this same waveform when it exits the trace. The rise time has been increased from 50 psec to nearly 1.5 nsec, due to the higher attenuation of the high-frequency components. Thirty-six inches is a typical length for a trace that travels over two 6-inch-long daughter cards and 24 inches of backplane. This rise-time degradation is the chief limitation to the use of FR4 laminate in high-speed serial links above 1 GHz.

The connection between the highest sine-wave-frequency component that is significant in a spectrum and the corresponding rise time of the waveform in the time domain is a very important property.