In an October column about undersampling, I explained how a data acquisition system could use aliasing to move signals within bandwidth x into a lower-frequency portion of the spectrum. That column mentioned a spreadsheet used to calculate a sample frequency based on signal bandwidth and the center frequency (f_{c}) of the bandwidth of interest. The spreadsheet for a signal at 55MHz with a bandwidth of 8MHz produces six upper- and lower-limit sampling frequencies, which are listed in the table below in Msamples/sec.

Before you choose a frequency, you should understand how each upper and lower sampling frequency affects the aliasing of signals in the bandwidth of 8MHz. The 102-Msamples/sec rate gives an f_{s}/2 value of 51MHz. That means aliasing simply folds the wanted signals along the f_{s}/2 axis to 43-51MHz. The aliasing also inverts the order of the frequencies, so those near 59MHz now appear near 43MHz, as shown in the figure below. If you sample at 59Msamples/sec, the signals fold along the f_{s}/2 axis at 29.5Hz and appear at 0-8MHz -- also inverted. The diagram also shows results when n equals 4 and 5.

After diagramming the aliasing for all six values of n, the following rules emerged. When n equals an odd number, you should use the lower sampling frequency, and you can use signal processing software to invert the frequencies back to their original form. When n equals an even number, you should use the higher sampling frequency, and the signals remain in the proper frequency order.

Reducing an undersampling rate proportionally reduces the Nyquist bandwidth and folds your signals to lower frequencies between 0 and fs/2 Hz.

As n increases, the Nyquist bandwidth, or the 0-to-f_{s}/2 frequency span, decreases and comes close to the bandwidth of the original signal. For higher values of n, the f_{s}/2 value can encroach on the signal bandwidth and lead to unwanted signal aliasing that will affect your measurements.

As I explained in a September column, you cannot create a perfect anti-alias filter that cuts off frequencies below 51MHz and above 59MHz. Some signals and noise outside this range will appear in your aliased signal band. Allow space in your Nyquist bandwidth, f_{s}/2, for these signals. In other words, do not try to choose a sampling frequency that places the Nyquist bandwidth too close to the bandwidth of your signals. Also, your bandwidth must not cross an f_{s}/2 boundary.

In my next column, I'll discuss sample rates in more detail.

Do you have a measurement question you'd like answered here? Contact me at tituslabs@comcast.net.

Reference:

Lyons, Richard G., "Understanding Digital Signal Processing," 2 ed., Pearson Education, Inc., 2004. ISBN: 978-0-13-108989-7.

Dear Jon, I am very amazed in seeing that in your article you have ignored my name in reference to the formulas for the undersampling reported in the table while you are quoting only the book by Lyons Richard. Yet in a your preceding article: Undersampling Shifts Frequency-Jon Titus, Contributing Editor-Design News, September 4, 2006, my name was quoted after I had informed you that I had published the formulas long before in: Angelo Ricotta, "Some remarks on the sampling and processing of SODAR data", Technical Report, IFA 83/11, IFA-CNR, July 1983, (pp. 4-7, in Italian). At that time I also had written to Lyons Richard who, in the first edition of his book, had attributed the formulas to: Hill G. "The Benefits of Undersampling, " Electronic Design, July 11 1994. But George Hill had simply copied them from me! Lyons Richard recognized the plagiarism but now I see that in the second edition of his book he not only doesn't cite me but describes the formulas as if he had deduced them! All of this is very incorrect! See the whole story in http://angeloricotta.altervista.org/UndersamplingAR/UndersamplingAR.htm

When considering under-sampling and its potential effects on bandwidths and spectrum, an associated question naturally occurs: 'What are the expected effects to amplitude response or sensitivity that would arise from undersampling'?

Thanks for your comments. The column should have had the headline, "Undersampling Shifts Bandwidths," but it got posted before I suggested that change.

Op-amp filters and digital filters would make for interesting columns. I have a couple of columns about sensors but can tackle filters after that. Thanks for the good suggestion. In the meantime keep in mind the digital filters also represent sampling devices, which adds another "wrinkle" to the sample-rate selection. More later.

Thanks for the link to your Filter Wizard articles. These webpage is awesome! This webpage will serve as a great resource for the Design News community, my ITT Tech Electronics Technology students, and myself as well.

Hi mrdon - if you're interested in comparing the performance of digital and analog filters, my Filter Wizard articles on that topic might be of interest. You can access them all at http://www.electronics-eetimes.com/en/News/filter-wizard.html#. Undersampling is, by definition, a sampled domain activity. But mixing, in other words multiplication by a carrier waveform, has very similar behaviour in the continuous time world. moving signals around between different frequency bands is essentially what a radio does. best / Kendall

Hi naperlou, Thanks for the comment and your vote on the op-amp. I'm somewhat gravitating towards the op-amp as well. I agree, Jon should do a column on the performance of op-amps(Active Filters) versus digital filtering techniques via DSP ICs and microcontrollers as well as the signal management benefits of each as it relates to Controls theory and analysis.

mrdon, I vote for the op-amp based filter. I looked it up after reading your comment. An op-amp was the first IC I played with when I was young (about 12 years old).

Jon, perhaps you could address that in your next column. It would be interesting to compare some of the operational parameters and performance of the two. Filtering is an important concern in many types of applications.

Hi Jon, Nice article as usual. The question I have relates to the best prototyping method to see the signal behavior in Undersampling Changes related to bandwiidth. Although digital filters seems to be the choice technique in doing controls analysis for Mechatronics of this subject matter, is it possible to obtain good analysis results using op-amp based Active Filters?

Suppose you wanted to create a FIR filter with your own requirements. How would you find the necessary coefficients, and how many of them would you need?

Switched-capacitor filters have a few disadvantages. They exhibit greater sensitivity to noise than their op-amp-based filter siblings, and they have low-amplitude clock-signal artifacts -- clock feedthrough -- on their outputs.

The Machinist Calc Pro computes speeds and feed rates for milling, turning, and drilling: cutting speed, spindle speed, feed rate (inches/minute), cutting feed, etc.

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