- 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

## 2.6 The Spectrum of a Repetitive Signal

In practice, the DFT or FFT is used to translate a real waveform from the time domain to the frequency domain. It is possible to take a DFT of any arbitrary, measured waveform. A key requirement of the waveform is that it be repetitive. We usually designate the repeat frequency of the time-domain waveform with the capital letter F.

For example, an ideal square wave might go from 0 v to 1 v, with a repeat time of 1 nsec and a 50% duty cycle. As an ideal square wave, the rise time to transition from 0 v to 1 v is precisely 0 sec. The repeat frequency would be 1/1 nsec = 1 GHz.

If a signal in the time domain is some arbitrary waveform over a time interval from t = 0 to t = T, it may not look repetitive. However, it can be turned into a repetitive signal by just repeating the interval every T seconds. In this case, the repeat frequency would be F = 1/T. Any arbitrary waveform can be made repetitive and the DFT used to convert it to the frequency domain. This is illustrated in Figure 2-5.

Figure 2-5 Any arbitrary waveform can be made to look repetitive. A DFT can be performed only on a repetitive waveform.

For a DFT, only certain frequency values exist in the spectrum. These values are determined by the choice of the time interval or the repeat frequency. When using an automated DFT tool, such as in SPICE, it is recommended to choose a value for the period equal to the clock period. This will simplify the interpretation of the results.

The only sine-wave frequency values that will exist in the spectrum will be multiples of the repeat frequency. If the clock frequency is 1 GHz, for example, the DFT will only have sine wave components at 1 GHz, 2 GHz, 3 GHz, etc.

The first sine-wave frequency is called the first harmonic. The second sine-wave frequency is called the second harmonic, and so on. Each harmonic will have a different amplitude and phase associated with it. The collection of all the harmonics and their amplitudes is called the spectrum.

The actual amplitudes of each harmonic will be determined by the values calculated by the DFT. Every specific waveform will have its own spectrum.