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Rise Times and Harmonics: Introducing Mr. Fourier

In a recent article (Note 1) I discussed rise times and frequencies and pointed out that (in general) there is no necessary relationship between them. In very general terms, frequency relates to information and rise time relates to how quickly we can process that information. A circuit only needs to have a rise time fast enough (and not faster) to process the information flow.

Rise time is usually the driving factor behind system bandwidth. Bandwidth refers to how wide a frequency range a circuit (or PCB) needs to handle without distortion. For example, a 1.0 MHz sine wave needs a circuit bandwidth of 1.0 MHz to pass undistorted. But a 1.0 MHz square wave needs a much wider bandwidth. That’s because the square wave has a much faster rise time than does the sine wave, and the wider bandwidth is required to accommodate that faster rise time.

Which all leads to a very interesting question: How do I know how wide a bandwidth I need to pass my signals?

If I had a short, easy answer to that question, I’d be more famous! But there is an analytical, and intuitive, way we can look at that question. It relies on some work done by Joseph Fourier (Note 2), that resulted in an area of mathematics we call “Fourier Series.” The fundamental principle behind a Fourier series is this (paraphrased):

Every signal or curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of individual sine (and or cosine) waveforms of different frequencies (harmonics) and different phase shifts. (Note 3.)

Consider the application of this to a square wave clock signal. The principle behind the Fourier series tells us that we can decompose that square wave signal into a series of sine or cosine terms of different frequencies, amplitudes, and phases. If we can do this, then the bandwidth only needs to be wide enough to pass the highest frequency on the series. Pretty slick.

There are only two practical problems (well, maybe even more than that!). One is that the mathematics behind a Fourier analysis is pretty rigorous and difficult, well beyond the scope of this article. Another is that many “typical” Fourier series are infinite in length, leading to the consequence that our bandwidth would have to be infinite to pass the entire series without distortion. But these practical limitations don’t weaken the concept. Let’s look into a couple of examples.

A square wave is a typical type of computer (clock or signal) waveform. At least we’d like our signals to resemble a square wave, with its fast rising edge and flat top. A Fourier series representing a square wave of fundamental frequency Ɵ  is (see Note 4):

    [Equation 1]

Note some of its characteristics:

1. The series is an infinite series.
2. The series consists entirely of odd harmonics (a harmonic is a multiple of the fundamental frequency).
3. Each term has a smaller amplitude than the term before it, divided by the value of the harmonic.
4. The signs of the individual terms alternate between plus and minus.

Since each term has a smaller amplitude, and in the limit the amplitude goes to zero (because the denominator goes to a very large number), there is probably a point where we can terminate this series without causing very much distortion. The question is, where is this point? Again, I apologize that that type of discussion is beyond the scope of the article! However, I can provide a partial answer that may be of some use.

UltraCAD has created a small tool that makes looking at this type of question pretty easy (Note 5).  It is shown in Figure 1. It provides a graphical representation of a Fourier series for four types of waveforms: square, triangular, trapezoid, and sawtooth. The user can set the number of harmonics to be used in the series and get a visual representation of how close the resulting series reflects the ideal.

For example, Figure 1 shows the results of using the first 11 harmonics of the series for a square wave (and it also shows the fundamental cosine wave for the frequency of the signal). Someone may feel that this is close enough for practical purposes, and that therefore the bandwidth need only be 11 times the fundamental square wave frequency.

Such a bandwidth will pass a square wave with this degree of distortion. Another user, however, may feel that a wider bandwidth is necessary. Setting the “Total Harmonics” entry on the tool (and looking at the results) can be a useful way of estimating the necessary bandwidth of the system.

Figure 1. UltraCAD’s Fourier Simulation Tool.

Another fun use of this tool is to eliminate one of the harmonics. Such a thing might happen, for example, if there were an inadvertent filter in the system that shorted out one particular frequency (and that particular frequency happened to be one of the harmonics of the fundamental.)

For example, Figure 2 illustrates what happens when we short out the 5th harmonic from the series shown in Figure 1. The difference in the distortion is notable.

Figure 2. Removing the 5th harmonic from the series in Figure 1.

Another way of looking at the issue of bandwidth was alluded to in the article referenced in Note 1.  I made the comment that, for a sine wave, “…the rise time is approximately 1/3 of the period of the waveform (the period being the inverse of the frequency).” Thus, if we have a trapezoidal clock frequency (with, say, a rise time of 1.0 ns) as opposed to a square wave clock signal (with effectively 0.0 ns rise time) there will be (for all practical purposes) an upper limit to the harmonic frequencies required to represent that waveform. That upper limit is 1/(3xrise time). For example, a 1.0 ns rise time signal has an upper harmonic frequency limit of:

Max harmonic = 1/(3 x 1.0 ns) = 333 MHz           [Equation 2]

This frequency represents the bandwidth required to pass the trapezoidal waveform without distortion.

The ideas of frequency harmonics and Fourier Series can be helpful in visualizing the frequency and bandwidth requirements of our circuits and boards.

Notes

1. Rise Time vs Frequency, What’s the Relationship, June 2011, PCBDesign007. 
2. Jean Baptiste Joseph Fourier (21 March 1768 – 16 May 1830), see Wikipedia entry. 
3. In theory, we usually require that the signal or curve be repetitive. It practice, we often ignore this theoretical requirement.
4. There is also a sine series representing a square wave. So there is not necessarily just one unique Fourier series that can represent a waveform.
5. See www.ultracad.com/fourier_tool.htm. This tool has a nominal licensing fee associated with it.

Douglas Brooks has an MS/EE from Stanford University and a Ph.D. from the University of Washington. He has spent most of his career in the electronics industry in positions of engineering, marketing, general management, and as CEO of several companies. He has owned UltraCAD Design Inc. since 1992. He is the author of numerous articles in several disciplines, and has written articles and given seminars all over the world on Signal Integrity issues since founding UltraCAD. His book, Printed Circuit Board Design and Signal Integrity Issues was published by Prentice Hall in 2003. Visit his Web site at www.ultracad.com.