I've received many responses to my first “RF Fundamentals” column. There were two messages I overwhelmingly received. First, many people wrote that this column is needed and widely desired. Second, I received a number of messages to tone down the math and use more figures and examples to explain concepts.

For example, the first column dealt with aliasing. I received an e-mail from James Campbell that gave an excellent example of aliasing that helped him address it in some designs he was working on.

He said he was watching an old western on TV and noticed that the wagon wheels, as the wagon sped up, starting rotating slower and slower. Eventually the spokes rotated backwards. This is because the wheel is rotating faster than the snapshot or sampling rate of each wheel. So, as the viewer puts all of this information into his head, the wheel appears to be moving backwards instead of forward.

Also, if my memory serves me correctly, as the wagon gained further speed, the spokes would start to appear jumbled and messed up, not smoothly rotating in either direction. This is the aliasing the math describes.

Now on to the next topic of interest: harmonics and double side band modulation.

In the RF world, there are two interesting phenomenon that play a role in the design of communications circuits. The first is the generation of harmonics by nonlinear circuits from a continuous wave. That is, a sinusoidal wave such as:

Have you ever heard some rather smart engineer say, “Well, you see it is a nonlinear device so you have to filter the harmonics”? and like me, you scratch your head and ask yourself, “I know he is right, and I know the reason is simple, but for the life of me I can't remember why”?

Well, if you have a simple nonlinear system, such as:

and we plug equation 1 into equation 2, we get:

Recall the trigonometric identity:

Then equation 3 becomes:

Note that the original signal is replicated at double the original frequency. Also, keep in mind this a nonlinear system with a single nonlinear term, x². Things get very complicated once we start adding in higher order terms.

An example we can all relate to is considering any kind of wood instrument. Imagine a string that only generated a perfect tone without any harmonics. In our minds we place that string on a violin and strum it. We can easily imagine that the tone we hear sounds like a violin, even with the string generating a perfect tone.

The violin can be thought of a very complicated nonlinear system that generates harmonics from the original tone. Obviously, this is an oversimplification of the instrument, but I believe the example serves the purpose of demonstrating the effects of a nonlinear system. The input is the pure tone from the string and the output is composed of harmonics that, to our ears, sound like a violin.

The second phenomenon deals with the generation of double side band (DSB) modulation when we multiply a signal with a cosine function. If we were to modulate the original signal x(t) as shown in:

It turns out that when one does this the original signal frequency, ω, of equation 1 is offset by ±ω_m. That is the original signal is bracketed DSB signals that are a ω_m distance from the original signal. Essentially, looking at m(t) from the perspective of frequency spectrum we get:

In the RF world, mixers are used to perform this kind of modulation. However, they are nonlinear devices because they are typically comprised of diodes (a classic nonlinear device so see results of equation 5). Mixers are components that mix signal (RF) with a local oscillator (LO) to generate harmonics that are the sum and differences of the two frequencies, f_out = f_RF ± f_LO. These DSB families are duplicated over the frequency spectrum (e.g. 2 f_LO) due to the nonlinearity of the mixer. The mixer is typically followed by a filter (lowpass or bandpass depending on application) to eliminate the harmonics generated by the nonlinearity of the mixers.

Keith Vick
kvick@primediabusiness.com

1. David Pozar, Microwave Enginering, 2^nd ed., John Wiley & Sons, 1997.

2. R.E. Ziemer and W.H. Tranter, Principles of Communications, 4^th ed., John Wiley & Sons, 2001.