- 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

## 2.3 Shorter Time to a Solution in the Frequency Domain

In some situations, if we use the naturally occurring sine waves in the frequency domain rather than in the time domain, we may arrive at a simpler description to a problem and get to a solution faster.

It is important to keep in mind that there is fundamentally no new information in the frequency domain. The time- and the frequency-domain descriptions of the same waveforms will each have exactly the same information content.

However, some problems are easier to understand and describe in the frequency domain than in the time domain. For example, the concept of bandwidth is intrinsically a frequency-domain idea. We use this term to describe the most significant sine-wave frequency components associated with a signal, a measurement, a model, or an interconnect.

Impedance is defined in both the time and the frequency domain. However, it is far easier to understand, to use, and to apply the concepts of impedance in the frequency domain. We need to understand impedance in both domains, but we will often get to an answer faster by solving an impedance problem in the frequency domain first.

Looking at the impedance of the power and ground distribution in the frequency domain will allow a simpler explanation and solution to rail-collapse problems. As we shall see, the design goal for the power-distribution system is to keep its impedance below a target value from direct current (DC) up to the bandwidth of the typical signals.

When dealing with EMI issues, both the FCC specifications and the methods of measuring the electromagnetic compliance of a product are more easily performed in the frequency domain.

With today's current capabilities of hardware and software tools, the quality of the measurements and the computation speed of the numerical-simulation tools can sometimes be better in the frequency domain.

A high signal-to-noise ratio (SNR) means higher quality measurements. The SNR of a vector-network analyzer (VNA), which operates in the frequency domain, is constant over its entire frequency range, which can be -130 dB from 10 MHz up to 50 GHz and more. For a time-domain reflectometer (TDR), the effective bandwidth may be as high as 20 GHz, but the SNR starts at -70 dB at low frequency and drops to as low as -30 dB at 20 GHz.

Many of the effects related to lossy transmission lines are more easily analyzed, measured, and simulated by using the frequency domain. The series resistance of a transmission line increases with the square root of frequency, and the shunt AC leakage current in the dielectric increases linearly with frequency. The transient (time-domain) performance of lossy transmission lines is often more easily obtained by first transforming the signal into the frequency domain, looking at how the transmission line affects each frequency component separately, and then transforming the sine-wave components back to the time domain.