- 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

## 2.11 What Does *Significant* Mean?

We defined the bandwidth of a signal as the highest sine-wave-frequency component that is significant. In the example, where we started with an ideal square wave and limited the high-frequency components, there was absolutely no ambiguity about what *significant* meant. We explicitly cut off all higher frequency sine-wave components in the frequency domain so that the highest significant component was the last harmonic in the spectrum.

We simply showed that if we include 100% of all the frequency components of an ideal square wave, up to the bandwidth, we would be able to re-create a square wave with a limited rise time, where the relationship of rise time = 0.35/BW. But what is the impact from adding only a fraction of the next component?

For example, if we take an ideal-square-wave clock signal with clock frequency of 1 GHz, its first harmonic will be a 1-GHz sine-wave frequency. If we were to include 100% of every component up to the twenty-first harmonic, the bandwidth would be 21 GHz and the resulting rise time of the re-created signal would be 0.35/21 GHz = 0.0167 nsec or 16.7 psec.

How would the rise time change if we added the twenty-third harmonic? The rise time would be 0.35/23 GHz = 0.0152 nsec or 15.2 psec. The rise time dropped by 1.5 psec. This is about 10% of the rise time, which is consistent, because we increased the bandwidth by 10%. The magnitude of the component we added was just 0.028 v, compared with the first harmonic of 0.63 v. Even though this amplitude is a small amount, less than 5% of the first harmonic amplitude and less than 3% of the peak value of the original square wave, it had the impact of dropping the rise time by 10%.

The spectrum of an ideal square wave has components that extend to infinite frequency. In order to achieve the zero rise time of an ideal square wave, each of these components is needed and is significant.

For a real time-domain waveform, the spectral components will almost always drop off in frequency faster than those of an ideal square wave of the same repeat frequency. The question of significance is really about the frequency at which amplitudes of the higher harmonics become small compared to the corresponding amplitudes of an ideal square wave.

By "small," we usually mean when the power in the component is less than 50% of the power in an ideal square wave's amplitude. A drop of 50% in power is the same as a drop to 70% in amplitude. This is really the definition of significant. Significant is when the amplitude is still above 70% of an ideal square wave's amplitude of the same harmonic.

In a slightly different view, we can define *significant* as the frequency at which the harmonic components of the real waveform begin to drop off faster than 1/f. The frequency at which this happens is sometimes referred to as the *knee frequency*. The harmonic amplitudes of an ideal square wave will initially drop off similarly as 1/f. The frequency at which the harmonic amplitudes of a real waveform begin to significantly deviate from an ideal square wave is the knee frequency.

To evaluate the bandwidth of a time-domain waveform, we are really asking what is the highest frequency component that is just barely above 70% of the same harmonic of an equivalent ideal square wave. When the harmonic amplitudes of the real waveform are significantly less than an ideal square wave's, these lower amplitude harmonics will not contribute significantly to decreasing the rise time and we can ignore them.

For example, we can compare, in the time-domain waveform, two clock waves with a repeat frequency of 1 GHz: an ideal square wave and an ideal trapezoidal waveform, which is a non-ideal square wave with a long rise time. In this example, the 10–90 rise time is about 0.08 nsec, which is a rise time of about 8% of the period, typical of many clock waveforms. These two waveforms are shown in Figure 2-12.

Figure 2-12 Top: Time domain waveforms of 1-GHz repeat frequency: an ideal square wave and an ideal trapezoidal wave with 0.08-nsec rise time. Bottom: Frequency-domain spectra of these waveforms showing the drop-off of the trapezoidal wave's higher harmonics, compared to the square wave's.

If we compare the frequency components of these waveforms, at what frequency will the trapezoid's spectrum start to differ significantly from the ideal square wave's? We would expect the trapezoid's higher frequency components to begin to become insignificant at about 0.35/0.08 nsec = about 5 GHz. This is the fifth harmonic. After all, we could create a non-ideal square wave with this rise time if we were to take the ideal-square-wave spectrum and drop all components above the fifth harmonic, as we saw earlier.

When we look at the actual spectrum of the trapezoid compared to the square wave, we see that the first and third harmonics are about the same for each. The trapezoid's fifth harmonic is about 70% of the square wave's, which is still a large fraction. However, the trapezoid's seventh harmonic is only about 30% of the ideal square wave's. This is illustrated in Figure 2-12.

We would conclude, by simply looking at the spectra of the trapezoid, that harmonics above the fifth harmonic (i.e., the seventh and beyond) are contributing only a very small fraction of the amount of voltage as in the ideal square wave. Thus, their ability to further affect the rise time is going to be minimal. From the spectrum, we would say that the highest sine-wave-frequency component that is significant in the trapezoid, *compared to that in the ideal square wave*, is the fifth harmonic, which is what our approximation gave us.

There are higher harmonics in the trapezoid's spectrum than the fifth harmonic. However, the largest amplitude is 30% of the square wave's and then only a few percent after this. Their magnitude is such a small fraction of the amplitude of the ideal square wave's that they will contribute very little to the decrease of the rise time and can be ignored.

The bandwidth of any waveform is always the highest sine-wave-frequency component in its spectrum that is comparable in magnitude to a corresponding ideal square wave. We can find out the bandwidth of any waveform by using a DFT to calculate its spectrum and compare it to an ideal square wave. We identify the frequency component of the waveform that is less than 70% of the ideal square wave, or we can use the rule of thumb developed earlier, that the BW is 0.35/rise time.

If you have a problem where it is important to know whether the bandwidth of a waveform is 900 MHz or 950 MHz, you should not use this term *bandwidth*. Rather, you should use the whole spectrum. The entire spectrum is always an accurate representation of the time-domain waveform.