- 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

## 2.12 Bandwidth of Real Signals

Other than the approximation for the bandwidth of a waveform based on its rise time, there is little calculation we can do by hand. Fourier Transforms of arbitrary waveforms can only be done using numerical simulation.

For example, the spectrum of a good-quality, nearly square wave signal has a simple behavior. If a transmission line circuit is poorly terminated, the signal may develop ringing. The resulting spectrum will have peaks at the ringing frequency. The amplitudes of the ringing frequency can be more than a factor of 10 greater than the amplitudes of the signal without ringing. This is shown in Figure 2-13.

Figure 2-13 Top: The time-domain waveform of a near-square wave and one that has significant ringing due to poor termination. Bottom: the resulting DFT spectrum of these two waves, showing the effect of the ringing on the spectrum. The wide bars are for the ideal waveform while the narrow bars are for the ringing waveform.

The bandwidth of a waveform with ringing is clearly higher than one without. When ringing is present in a waveform, the bandwidth is better approximated by the ringing frequency. Just using the bandwidth to characterize a ringing signal, though, may be misleading. Rather, the whole spectrum needs to be considered.

EMI arises from each frequency component of the currents radiating. For the worst offender, the common currents, the amount of radiated emissions will increase linearly with the frequency. This means that if the current had an ideal-square-wave behavior, though the amplitude of each harmonic drops off at a rate of 1/f, the ability to radiate would increase at the rate of f, so all harmonics contribute equally to EMI. To minimize EMI, the design goal is to use the absolute lowest bandwidth possible in all signals. Above the bandwidth, the harmonic amplitudes drop off faster than 1/f, and would contribute to less radiated emissions. By keeping the bandwidth low, the radiated emissions will be kept to a minimum.

Any ringing in the circuits may increase the amplitudes of higher-frequency components and increase the magnitude of radiated emissions by a factor of 10. This is one reason why solving all signal-integrity problems is usually a starting place to minimize EMI problems.