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Undersampling Simplifies Wireless Control Systems

Wireless control involves moving accurately from the analog to the digital realm. A technique called undersampling can simplify the sesign of the conversion proscess and produce better accuracy.

RODGER H. HOSKING, PENTEK, INC.

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Wireless technology has invaded nearly every aspect of consumer, business, commercial and government electronic markets. Ranging from Bluetooth headsets for cell phones to military radios in battlefield information networks, strong growth in product demand has spawned a wealth of low-cost devices delivering sophisticated RF performance. Niche markets for embedded computing benefit by adapting these kinds of new technologies to system solutions.

One such niche market is control systems for industrial, automotive, commercial and military systems. First offering purely mechanical solutions, control system vendors have continuously injected new technology to meet increasingly tough challenges for a tremendous variety of equipment from wall thermostats to interplanetary spacecraft.

Transducers and sensors for some advanced control systems are now being equipped with wireless RF transmitters that replace traditional hard-wired harnesses. This strategy saves cable weight, minimizes the complexity of connectors, eliminates troublesome slip ring contacts in rotating equipment, enhances maintainability and simplifies installation and upgrades. During system tests, the transducer signals can be monitored remotely with a wireless receiver that replaces a custom test connector and cable assembly.

In some cases, the analog output signal from the transducer modulates the amplitude, phase or frequency of the RF carrier directly. In other systems, a digitized representation of the transducer signal drives one of many different digital modulation schemes. The modulated RF signal itself usually occupies a relatively narrow frequency band with other sensor signals allocated to adjacent frequency channels.

At the receiving end for the control computer or test system, the RF band containing the transducer channels must first be picked up using a suitable antenna. The signal is then filtered and amplified using bandpass amplifiers to enhance the desired band and reject out-of-band signals. A frequency translation stage mixes the RF band down to a lower intermediate frequency (IF) band where demodulation occurs.

Newer receiver systems now implement software defined radio solutions for the demodulation instead of older analog technology. This accommodates new complex digital modulation schemes, improves channel selectivity and eliminates component tolerance, drift, aging and calibration headaches.

In these software radio systems, the IF signal band is sampled with an A/D converter. All further processing is done using digital signal processing hardware. This IF sampling process must be implemented carefully to ensure that subsequent signal processing will be successful.

Sampling Basics

One of the most important fundamentals for correctly sampling analog signals is the concept of aliasing. The famous Nyquist criterion states that we must digitize a signal at a sampling rate of at least twice the bandwidth of the signal. For “baseband” signals with frequency components starting at DC and extending up to some maximum frequency, this means the sampling rate must be at least twice this maximum frequency. In this case, the bandwidth and the maximum frequency are the same.

But for “bandpass” signals such as the IF output signal from an RF downconverter with a 70 MHz center frequency and a 10 MHz bandwidth, the Nyquist criterion imposes constraints on bandwidth, rather than on absolute frequency. There is a major difference between sampling an IF signal at a rate twice the 10 MHz bandwidth versus twice the 75 MHz upper edge of the IF band.

By taking advantage of this subtle but profound distinction in just the right way, system designers can simplify system design and save costs. This is the basic objective of a technique known as “undersampling”, which can best be appreciated only after a basic discussion of the effects of the sampling process itself.

Modeling the Sampling Process

One tried and true technique to visualize exactly what happens when sampling occurs is the “fan-fold” paper method. Start by imagining a small stack of semitransparent fan-fold computer printer paper. Holding the paper with the folds in the vertical direction, plot the frequency axis from left to right along the bottom edge with the inward creases at multiples of the A/D sampling frequency, Fs and the outward creases at odd multiples of Fs/2, as shown in Figure 1.

The vertical axis is used to plot the spectral amplitude of the signal to be sampled, such as the wideband signal shown with energy that extends across all of the sheets. In order to see what happens after sampling, simply collapse the stack of fanfold paper, hold it up to a light and look through the stack. As shown in Figure 2, signals on all of the sheets above Fs/2 are effectively “folded” down into sheet 1 between 0 Hz and Fs/2.

The spectra from all sheets are superimposed on top of each other. This truly represents the exact frequency content of the resulting signal at the output of the A/D converter. Such a situation obviously results in a hopelessly corrupted signal. Now you can easily see why anti-aliasing filters are normally required to eliminate all baseband signals above Fs/2 before sampling takes place.

Before proceeding, note that in Figure 1 signals on every odd numbered sheet are translated down in frequency by a multiple of Fs. Signals on even numbered sheets are translated down in frequency by odd multiples of Fs/2.

Also worth noting is that odd numbered sheets fold down to sheet 1 with no spectral reversal. Specifically, signals at the left edge of the original sheet fold to 0 Hz and signals at the right edge fold to Fs/2. For even numbered sheets, the frequency scale is reversed with the highest frequency folded to 0 Hz and the lowest to Fs/2.

Most digital signal processing systems that follow the A/D converter can easily invert the spectrum so this normally causes no problem. Again, all of these folding effects are much easier to follow by visualizing the fan-fold model.

Defining an Undersampling Strategy

Now that the effects of sampling are squared away, we can see how to exploit this model for sampling of the IF output bandpass signals from the RF downconverter. This can be accomplished by a judicious choice of the sampling rate based on the frequencies present in the IF band. Suppose all of the frequencies in the IF bandpass signal fall on a single sheet of fan-fold paper as shown in Figure 3.

In this case, after sampling, all of the signal energy on sheet 5 will fold down onto sheet 1 and be represented in the output sample stream just as if it were a baseband signal between 0 and Fs/2. In the resulting spectrum shown in Figure 4, the sampling process causes a downward frequency translation of all IF signals on sheet 5 (an odd numbered sheet) by 2Fs, with no frequency reversal.

Because we are intentionally aliasing the IF bandpass signal by using a carefully chosen sampling rate lower than twice the absolute input frequency, this technique is called undersampling.

Undersampling: Rules and Guidelines

The key rule for undersampling is obvious from the fan-fold paper model: choose the sampling rate, Fs, so that the entire band of the bandpass signal falls on a single sheet. Depending on the odd or even number of that sheet, the frequency axis of the resulting sampled spectrum will either be normal or reversed, respectively.

There are a multitude of different sample clock frequencies that will work for undersampling. While this fan-fold model will greatly help you develop several correct frequency plans, the best choice will usually be determined by several other important practical considerations:

• Some A/D converters are specifically characterized for undersampling applications, while others are designed only for baseband sampling. Check the manufacturer’s specifications carefully.

• The analog signal path of the A/D converter must handle the input frequencies of the bandpass signal with minimum distortion and noise. In this case, a transformer-coupled input stage is often the best answer.

• The quality of the sample-and-hold amplifier at the front end of the A/D converter becomes more critical at higher bandpass input frequencies. Often, an additional external high-performance sample-and-hold will be necessary.

• Any out-of-band signals or noise must be kept to a minimum because they will fold down into the output spectrum, exactly as shown in Figure 2. An extra input bandpass filter can help reduce these components.

• Jitter and phase noise of the sample clock signal can seriously degrade undersampling performance. Use a high-quality crystal oscillator with a simple, direct connection to the A/D converter for maximum performance.

When properly implemented, undersampling can save an extra stage of analog frequency translation from the IF band down to baseband because this translation is inherent in the process. Since undersampling results in a lower sampling rate, the A/D converter will be less expensive, draw less power and usually have more bits of resolution. In addition, the lower sample rate relaxes the complexity and speed required for the subsequent stage of digital signal processing. As RF wireless technology migrates to new applications, a wider range of A/D converters specifically characterized for undersampling will appear, improving performance and expanding design options for control systems.

Pentek
Upper Saddle River, NJ.
(201) 818-5900.
[www.pentek.com].