As I mentioned earlier, we should pay attention to the signal rise time. Many signal integrity problems are caused by the short signal rise time. This paper talks about a basic concept: the relationship between signal rise time and signal bandwidth.

For digital circuits, the output is usually a square wave signal. The rising edge of the square wave is very steep. According to Fourier analysis, any signal can be decomposed into a series of sinusoidal signals with different frequencies. The square wave contains very rich spectral components.

Aside from the boring theoretical analysis, we use experiments to intuitively analyze the frequency components of square waves and see how sinusoidal signals of different frequencies are superimposed into square waves. Firstly, we superimpose a 1.65v DC and a 100MHz sinusoidal waveform to obtain a single frequency sine wave with a DC offset of 1.65v. We superimpose this signal with a sinusoidal signal of integer multiple frequency, which is commonly referred to as harmonic. The frequency of the third harmonic is 300MHz, the frequency of the fifth harmonic is 500MHz, and so on. The higher harmonic is an integral multiple of 100MHz. Figure 1 shows the comparison before and after superposition of different harmonics. In the upper left corner is the 100MHz fundamental frequency waveform of DC bias. In the upper right corner, the waveform after the fundamental frequency is superimposed with the third harmonic is a bit similar to the square wave. The lower left corner is the waveform of fundamental frequency + 3rd harmonic + 5th harmonic, and the lower right corner is the waveform of fundamental frequency + 3rd harmonic + 5th harmonic + 7th harmonic. It can be seen intuitively that the more harmonic components are superimposed, the more the waveform is like a square wave.

Figure 1

Analysis of the relationship between signal rise time and signal bandwidth in signal integrity problem

Therefore, if enough harmonics are superimposed, we can approximately synthesize the square wave. Fig. 2 is a waveform superimposed to 217 harmonics. It is already very similar to the square wave. Don’t care about the burrs on the corner. It is the famous Gibbs phenomenon. This kind of simulation is bound to exist, but it doesn’t affect the understanding of the problem. Here, the highest frequency of our superimposed harmonic reaches 21.7ghz.

Figure 2

Analysis of the relationship between signal rise time and signal bandwidth in signal integrity problem

The above experiments are very helpful for us to understand the essential characteristics of square wave waveform. The ideal square wave signal contains infinite harmonic components, which can be said that the bandwidth is infinite. There is a gap between the actual square wave signal and the ideal square wave signal, but there is one thing in common, that is, the spectrum component with high frequency.

Now let’s look at the effect of superposition of different spectral components on the rising edge. Figure 3 is a comparative display. Blue is the rising edge of fundamental frequency signal, green is the rising edge of waveform superimposed with 3rd harmonic, red is the rising edge of fundamental frequency + 3rd harmonic + 5th harmonic + 7th harmonic, and black is the rising edge of waveform superimposed with 217th harmonic.

Figure 3

Analysis of the relationship between signal rise time and signal bandwidth in signal integrity problem

Through this experiment, we can intuitively see that the more harmonic components, the steeper the rising edge. Or from another point of view, if the rising edge of the signal is very steep and the rising time is very short, the bandwidth of the signal is very wide. The shorter the rise time, the wider the bandwidth of the signal. This is a very important concept. You must have an intuitive understanding and engrave it in your mind, which is very good for you to learn signal integrity.

Here, the waveform repetition frequency of the final synthesized square wave is 100MHz. Superimposed harmonics only change the signal rise time. The signal rise time has nothing to do with the frequency of 100MHz. It is the same rule to change to 50MHz. If the output data signal of your circuit board is only tens of MHz, you may not care about the signal integrity. But then you think about the impact of those high-frequency harmonics in the spectrum due to the short rise time of the signal? Remember an important conclusion: it is not the repetition frequency of the waveform that affects the integrity of the signal, but the rise time of the signal.

The simulation code in this paper is very simple. I post the code here. You can run it on MATLAB.

clc;     clear   all;     pack;

Fs = 10e9;

Nsamp = 2e4;

t = [0:Nsamp-1].*(1/Fs);

f1 = 1e6;

x0 = 3.3/2;

x1 = x0 + 1.65*sin(2*pi*f1*t);

x3 = x0;

for   n=1:2:3

x3 = x3 + 3.3*2/(pi*n) * sin(2*pi*n*f1*t);

end

x5 = x0;

for   n=1:2:5

x5 = x5 + 3.3*2/(pi*n) * sin(2*pi*n*f1*t);

end

x7 = x0;

for n=1:2:7

x7 = x7 + 3.3*2/(pi*n) * sin(2*pi*n*f1*t);

end

figure

subplot(221)

plot(x1)

subplot(222)

plot(x3)

subplot(223)

plot(x5)

subplot(224)

plot(x7)

x217 = x0;

for   n=1:2:217

x217 = x217 + 3.3*2/(pi*n) * sin(2*pi*n*f1*t);

end

figure

plot(x217)

figure

plot(x217,’k’)

hold   on

plot(x1,’b’)

plot(x3,’g’)

plot(x7,’r’)

hold   off

axis([8000 12000 -0.5 4])

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