从其谐波中找到信号的带宽

文章：鲍勃·威特（Bob Witte）

There are several ways to evaluate the bandwidth of a signal in the time domain and frequency domain. Previously we looked at the classic relationship of rise time (t_r）和带宽（f_3DB）[参考1]，由该方程式捕获：

Eric Bogatin还提供了＃2规则，用于估算时钟频率的信号带宽[参考2]。埃里克（Eric）强调，您确实应该使用上升时间来计算信号带宽，但是您可以使用此经验法则快速获得合理的答案：

在埃里克（Eric）的文章中，他做出了一个关键的假设，即上升时间是该时期的7％。这是一个合理的假设，它使我们在上升时间进入正确的球场。但是，有些信号会更快，而有些信号会慢。

傅立叶系列
Another way to evaluate a signal’s bandwidth is through frequency domain analysis, or more specifically by using the Fourier Series. The Fourier Series for a square wave, as shown inFigure 1，是[参考3]：

该系列具有无限数量的奇谐元，它们结合在一起以代表方波。由于方程中的1/n项，每个较高的谐波的振幅都比前一个谐波小。由于我们的理想方波的上升时间为零，因此信号的带宽将是无限的。换句话说，我们需要包括所有谐波，以完美地表示方波。时间尺度是任意的。波形周期为十个单位。

Table 1lists the coefficients (zero-to-peak values) for the sinewave terms, starting with the fundamental (n =1) through the 11th harmonic.

Harmonic	系数
1	1.273
3	0.424
5	0.255
7	0.182
9	0.141
11	0.116

Let’s examine how many harmonics we need to include to have the waveform look like a decent square wave.Figure 2仅显示基本频率，纯正弦波。

Figure 3增加了第三个谐波，这开始使波形更像是方波。

Figure 4添加第五个谐波，现在我们看到波形看起来更像是方波。

每个附加的谐波产生的波形看起来更像是方波。我们不会在表1中绘制所有这些，但是Figure 5将波形显示为第11个谐波。高阶谐波使波形更加正方形，并在波形的平坦部分留下更高的频率波纹。

Adding in a specific number of harmonics is equivalent to applying a brick-wall low-pass filter in the frequency domain. The desired harmonics are included in the waveform and the higher harmonics are eliminated. This is somewhat artificial because in the real world, we would probably have a frequency response than rolls off gradually with some remnants of the higher harmonics still present.

Rise time
Qualitatively, the 5th harmonic waveform looks like a respectable square wave, so let’s take a closer look at that case.Figure 6shows this waveform with expanded horizontal axis such that we can determine the rise time using graphical techniques.

Let’s find the 10% and 90% points on the waveform and estimate the rise time. The total signal swing is 2 units so the 10% and 90% points are -0.8 and +0.8. The rise time is 2 × 0.37 = 0.74 units. Recall that the period of the waveform is 10, so this gives us a rise time that is 0.74/10 = ~7% of the period. Now isn’t that interesting? This is very close to the 7% (of period) rise time assumed in Rule of Thumb #2. Of course, our choice of including up to the fifth harmonic was an arbitrary decision based on the waveform shape.It just looked pretty good. You could argue that we need more harmonics or perhaps even less, depending on the specific application.

In summary then, we used Fourier Series analysis to determine the amplitude of the harmonics in a square wave. Then we estimated the rise time of a square wave made up of the fundamental plus the third and fifth harmonics. It matches well with Rule of Thumb #2 for estimating the bandwidth of a digital signal (five times the clock frequency).

- - - - - -鲍勃·威特is president of Signal Blue LLC, a technology consulting company.

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