As I am sure some of you have noticed, there have been a few changes on the editorial staff of RF Design in the last few months. I am the latest addition to this venerable magazine. Therefore, I will take this opportunity to introduce myself and the new column to everyone.

First, a little about me: my education and career have been a little atypical when compared to most engineers. My formal education consists of a B.S.M.E. from the University of Texas in 1994 and a M.S.E.E. (May 2003) from University of Colorado at Colorado Springs. My M.S.E.E. education consisted of microwave circuits, DSP and communications. It is not often one sees a gear head and sparky rolled up into one, but that would be me. I guess you could say I am a sparking gear falling from a disintegrating car of a highway chase — I feel like my career thus far consisted of trying to avoid the hatchet.

My career looks like that of a journeyman. I started with Applied Materials soon after my graduation working as a manufacturing and later, field service engineer. With Applied I went through four layoffs and a semiconductor fabrication plant closure. In 2000, I joined Agilent Technologies hoping for more stability. However, I subsequently went through five layoffs with the last one terminating my employment as an EMC Engineer. They are both great companies trying to weather what everyone is telling me is a horrendous high tech downturn. Personally, I have only known employment instability, having had many bosses and seen many friends leave, so it looks the same to me.

After my layoff I was contacted by Todd Erickson, editor of RF Design, for technical editing work. Naturally I jumped the opportunity to contribute to RF Design. It is an excellent opportunity to keep abreast of industry developments.

Now, what is this column about? It about a few things: returning to the fundamentals of RF design, allowing engineers to keep their knowledge fresh. The fundamentals are necessary so when one encounters statements like, “10-bit, 3 tap, twice oversampled 16K-point,” it makes sense. In addition, there will be some commentary on the historical development of RF topics. As the column name suggest, the topics will be an examination of the fundamental areas related to RF design.

My first column will focus on something very simple as a good starter. Hopefully, I will get it right.

What is Nyquist frequency? Well it is rooted in the effects of sampling an analog or continuous signal. It turns out that when one looks at the frequency spectrum of a signal (in this case arbitrary) it can be represented as shown in figure 1.

The continuous signal, c(t), is sampled at a frequency of

where T is the sampling period. Mathematically, this is expressed as x[n] = c(nT), where n is all the integers from negative to positive infinity (assuming an infinite in time signal). This sampling essentially duplicates the frequency signal of the continuous signal offset by the sampling frequency as shown in figure 2.

Note that it becomes apparent that if the sampling frequency is not at least two times the highest spectral content of the continuous signal, then something called aliasing occurs. This is when the original signal content (and spurious) spectral content overlay each others. As can be imagined, this is bad news.

I suspect that most engineers already know all of the above. Personally, I like to see the math that shows the origins of the aliasing. I can visualize that sampling moves the spectral content around, but I still scratch my head and ask why does the signal get duplicated along the spectrum? Well, the below series of equations explains why.

Assuming an impulse sampling train, s(t), that is multiplied with the original continuous signal is denoted by

The term δ (t) is the Dirac delta function. Well, one can look at this equation and say, hey, I can apply the Fourier transform to this and what do you know, one arrives at a spectrum of the sampled signal that is a series of the spectrum of the original continuous signal. This is expressed as:

The large upper case letters denote the Fourier transform. What about the term ψ_s This is the time axis transform of the sampling impulse train. In effect,

So as one can see from the last equation, the spectral frequency of the original continuous signal is, in addition to being scaled by

duplicated along the spectrum at interval lengths of ψ_s. As we pointed out, if this frequency is too small then aliasing will occur.

If there are any comments or questions, please don't hesitate to contact me. I am more than happy to correspond with anyone.

Keith Vick
Technology Editor
RF Design
keith_vick@adelphia.net