 Signal and Power Integrity: Time and Frequency Domains

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2.8 From the Frequency Domain to the Time Domain

The spectrum, in the frequency domain, represents all the sine-wave-frequency amplitudes of the time-domain waveform. If we have a spectrum and want to look at the time-domain waveform, we simply take each frequency component, convert it into its time-domain sine wave, then add it to all the rest. This process is called the Inverse Fourier Transform. It is illustrated in Figure 2-7. Figure 2-7 Convert the frequency-domain spectrum into the time-domain waveform by adding up each sine-wave component.

Each component in the frequency domain is a sine wave in the time domain, defined from t = - infinity to t = + infinity. To re-create the time-domain waveform, we take each of the sine waves described in the spectrum and add them up in the time domain at each time-interval point. We start at the low-frequency end and add each harmonic based on the spectrum.

For a 1-GHz ideal-square-wave spectrum, the first term in the frequency domain is the zeroth harmonic, with amplitude of 0.5 v. This component describes a constant DC value in the time domain.

The next component is the first harmonic, which is a sine wave in the time domain with a frequency of 1 GHz and an amplitude of 0.63 v. When this is added to the previous term, the result in the time domain is a sine wave, offset to 0.5 v. It is not a very good approximation to the ideal square wave. This is shown in Figure 2-8. Figure 2-8 The time-domain waveform is created by adding together the zeroth harmonic and first harmonic and then the third harmonic, for a 1-GHz ideal square wave.

The next term is the third harmonic. The amplitude of the 3-GHz sine-wave-frequency component is 0.21 v. When we add this to the existing time-domain waveform, we see that it changes the shape of the new waveform slightly. The top is a bit more flat, better approximating a square wave, and the rise time is a little sharper. As we go through this process, adding each successive higher harmonic to re-create the ideal square wave, the resulting waveform begins to look more and more like a square wave. In particular, the rise time of the resulting time-domain waveform changes as we add higher harmonics.

To illustrate this in more detail, we can zoom in on the rise time of the waveform, centered about the beginning of a cycle. As we add all the harmonics up to the seventh harmonic, and then all the way up to the nineteenth, and finally, all the way up to the thirty-first harmonic, we see that the rise time of the resulting waveform in the time domain continually gets shorter. This is shown in Figure 2-9. Figure 2-9 The time domain waveform created by adding together the zeroth harmonic and first harmonic, then the third harmonic and then up to the seventh harmonic, then up to the nineteenth harmonic, and then all harmonics up to the thirty-first harmonic, for a 1-GHz ideal square wave.

Depending on how the DFT was set up, there could be over 100 different harmonics listed in the spectrum. The logical question to ask is, do we have to include all of them, or can we still re-create a "good enough" representation of the original time-domain waveform with just a limited number of harmonics? What really is the impact of limiting the highest harmonic included in the re-created time-domain waveform? Is there a highest sine-wave-frequency component at which we can stop?