Dale Lankford, Daytronic Corp.
(As seen in Sensors' 1998 Data Acquisition Special Issue)
To be sure that the sensor data you acquire are valid,
you must understand aliasing and apply the appropriate remedies.
(Hint: The problem must be solved in the analog domain)
The concept of aliasing can be illustrated by means of two analogies. The spokes on the wagon wheels in a western movie appear to be turning backwards at a slow rate. A spinning shaft seems to stop when illuminated by a strobe lights. In the movie, the frame rate of the film is much slower than the rotational frequency of the wheel spokes. The result is an apparent low-frequency movement that is not in phase with the actual rotation of the wheel. With the strobe, the sampling rate is deliberately slowed to freeze the action of the rotating part. Here, a high frequency event has been reduced to a constant DC value. Both are examples of signal aliasing (from the Latin alius, meaning other), a phenomenon that occurs when the sampling rate is too slow to accurately represent the physical event being measured. (See "The DA Time Bomb," Strether Smith, www.sensorsmag.com)
In real-world measurement and control of analog phenomena, signal aliasing can and does have disastrous implications. When analog signals are sampled and digitized, the data are converted into a series of points. Because all the data between samples are lost, once aliasing is introduced into the measurement it is impossible to remove. The collected data are worthless. It is therefore essential that aliasing be prevented in the first place. And since the damage occurs in the conversion from analog to digital, digital filters will not eliminate the problem (see Figure 1). Aliasing must be dealt with in the analog world.
Figure 1. Typical high-frequency aliasing arises when data are being sampled at a 1 ms rate. The apparent data have a shape significantly different from that of the actual force-time profile. Even filtering does not solve the problem, and conclusions based on the sampled data could be substantially wrong.
How Can Aliasing Be Eliminated?
The primary considerations involved in anti-aliasing are the sampling rate of the A/D system and the frequencies present in the sampled data. Most engineers are familiar with the basic sampling theory (Shannon's theorem), which states that to reconstruct the data, the sampling rate (Nyquist frequency) must be at least twice that of the component with the highest frequency. What often eludes the practitioner, however, are the tasks of recognizing the presence of higher-than-expected frequencies in the sensor output, differentiating between high-frequency signal and noise, and using practical methods to eliminate the higher frequencies that cause aliasing. To successfully contend Keith the problem, you must:
- Establish the useful bandwidth of the measurement
- Select a sensor with sufficient bandwidth
- Select the low-pass anti-aliasing analog filter that can eliminate all frequencies exceeding this bandwidth
- Sample the data at a rate at least twice that of the filter's upper cutoff frequency
Useful Measurement Bandwidth
Aboard large, tandem-rotor three-blades-per-rotor helicopters (e.g., the Army CH-46 or the Navy CH-47), a frequency component produced by the rotor vibration is usually found in the raw data. This 3-per-rev vibration, unless properly filtered, causes serious aliasing of sensor data when measuring parameters such as hydraulic pressure, airspeed, motor torque, and other low frequency components. Data outside the useful measurement bandwidth come in the form of electrical noise from contact closures, electrical machinery, and other sources, as well as from unwanted dynamic content.
In every physical measurement and control application, it is possible to establish a practical bandwidth that eliminates both the electrical noise and the unwanted dynamic content. But you must be careful not to overly limit this bandwidth. Clearly, the pulsations caused by a pump are outside the range of frequencies of interest for a flow control system. Structural analysis, however, would require many multiples of this fundamental frequency. The message should be clear- physical measurements must be engineered, and a critical part of this work is to define as accurately as possible the bandwidth of interest.
Once the measurement bandwidth has been determined, it is necessary to select a sensor with the appropriate response characteristics. It is not unreasonable (within practical budgetary boundaries) to use sensors with the widest possible bandwidth when making any physical measurement. This is the one way to ensure that the basic measurement system is capable of responding linearly over the full range of interest. The wider the bandwidth of the sensor, however, the more you must be concerned with eliminating sensor response to higher-than-desired frequencies. This is where a properly designed low-pass filter comes in.
The Anti-Aliasing Filter
The ideal low-pass filter would faithfully pass the analog signal over the entire passband without distortion or delay and completely eliminate all frequencies exceeding the defined cutoff. As can be seen in Figure 2, this would mean that the filter's gain would be 1 in the passband and 0 in the stopband. While such ideal filters do not exist, when it comes to most physical measurements a top-quality Butterworth filter is a good approximation.
Figure 2. Comparing Butterworth and Chebychev filters with the same 3 dB cutoff frequency, we see that the Chevychev's sharper transition from passband to stopband is achieved at the cost of significant passband ripple. Therefore, data accuracy near the cutoff can be poor. With an essentially flat passband performance, the Butterworth filter yields a stopband slope of ~20 dB/decade/pole.
While there are other filter types available (see Figure 3), the Butterworth design is appropriate for most physical measurements because it exhibits optimal flatness over the passband with a relatively sharp roll-off at the cutoff frequency. At Daytronic, we have modified the basic Butterworth filter design to eliminate the overshoot and ringing that occur when there is a step change in input. This is particularly important in peak-capture S/H applications. The overshoot of the standard Butterworth filter can easily lead to errors in these peak measurements.
Figure 3. A comparison of the response of five different 3-pole filters to a 5 V step demonstrates that most filters introduce not only error into the frequency domain, but overshoot and/or settling-time problems as well. In contrast to most other anti-aliasing filters, the Daytronic filter produces virtually no overshoot.
One Size Does Not Fit All
As with most things, not all filters are created equal. It is important that the filter meet the standards of accuracy, stability and reliability your measurement system requires. And since measurement criteria can change from one data gathering or control task to another, the filter should be designed for programmability over a wide range of cutoff frequencies. For temperature or other slowly changing parameters, a cutoff frequency of 0.2 Hz might be needed to eliminate noise spikes that would otherwise corrupt the measurement. The same filter might have a cutoff as high as 200 Hz for vibration and shock measurements. In short, your filter must:
- Exhibit linear (flat) amplitude response over the passband
- Exhibit uniform or near-uniform frequency response over the measurement bandwidth
- Provide reliable, stable performance
- Be programmable over a wide range of cutoff frequencies
Data Sampling Rate
The ultimate goal of the analog signal conditioning system is to present a signal to the A/D system that when digitized will faithfully represent the physical measurement. We referred earlier to the Nyquist frequency, which is a sampling rate at least twice the upper cutoff frequency. The question now arises: How much higher than twice the upper cutoff frequency of the anti-aliasing filter should the sampling rate be? You would be tempted to say that it need not be any higher at all. And if the measurements you are taking are in the frequency domain, this is true- in theory. (For frequency domain measurements, engineering practice typically recommends a sampling rate of 5 X the highest frequency.)
For time domain measurements, the normal practice is to sample at 10 X the upper cutoff frequency. And since most control applications are time domain in nature, this is a good rule to follow. But even 10 X the cutoff frequency can result in a relatively low number of data points. With an anti-aliasing filter set with a cutoff frequency of 0.2 Hz, this is only 2 sps. This fact can be used to allow a DA system to process more variables or store data over a longer period of time, or it could allow the user to select a lower speed A/D converter. Once again, there are tradeoffs that must be engineered.
Analog signals acquired in the real world usually come accompanied by noise and/or unwanted dynamic content. A thermocouple measurement may have an occasional noise spike induced into the signal from a nearby contactor. A squirrel-cage blower or piston compressor will add impulses to flow and pressure measurements. Gear train torque measurements will include impulses caused by each tooth as it engages its opposite member.
In all these cases, and in virtually every physical measurement, noise and unwanted dynamic content must be identified and filtered out before the A/D conversion takes place. Anti-aliasing is not complicated but it does take engineering. And engineering requires careful analysis of the physical measurement's useful bandwidth and selection of all aspects of the measurement system to faithfully record the data.
Dale Lankford is Principal Engineer at Daytronic Corporation, Dayton, OH , www.daytronic.com