Sampling a signal causes the original signal spectrum (blue) to create sum (purple) and difference (red) frequencies around the sampling frequency, fS. When the difference signals fall into the original signal spectrum below half the sample rate (fMAX), these are reconstructed as if they are part of the original signal. These false signals are called aliased signals.
Many years ago, when digital oscilloscopes were just becoming popular, I was in the lab with a new engineer who was troubleshooting a circuit and seemed very frustrated. When I asked what the trouble was, he explained that he was seeing a signal with the shape he expected, but it was the wrong frequency. It should have been around 105 kHz, but it was only at 5 kHz, and he spent several hours trying to figure out why. I looked at the scope and smiled.
“Look at your sample rate. It’s only 100 kHz,” I said. He was very relieved when I turned the sample rate up to 10 MHz and, suddenly, his signal was at the right frequency!
What this young engineer missed was that the signal he was looking at on the scope was not really his signal, but an alias of the signal. Any time you sample a signal, your system has the original signal frequencies as well as the sum and difference of those frequencies with the sample frequency (see the figure).
In your data, then, you may see the difference signals come back into your bandwidth of interest. These signals are called aliased signals because, while they look like part of your original signal, they aren’t really there. They’re aliases of the actual frequency content in the signal.
Aliasing, In Theory
The Nyquist-Shannon sampling theorem states that, when sampling a signal at discrete intervals, the sampling frequency must be greater than twice the highest frequency of the input signal to be able to reconstruct the original perfectly from the sampled version. If the sampling frequency is less than this limit, then frequencies in the original signal above half the sampling rate will be aliased and appear in the resulting signal as lower frequencies.
To avoid this, you must sample at a rate of fS that is at least twice that of your highest input frequency (fMAX), i.e., fMAX < fS/2. As defined here, fMAX is often called the Nyquist frequency. But noise and interference signals may be present in your system, and those may have frequency content much higher than the signal you are interested in.
Sampling at a rate higher than these figures may not be possible or practical. The alternative is to band-limit the signal going into your sampling system so you know what the maximum input frequency is, and then you can be assured that the sampling rate is at least twice that frequency.
Now look again at the figure and observe that the input signal is attenuated above some frequency. (Perhaps these signals aren’t even your signal, but simply noise or some interfering signal.) So, the aliased signals are attenuated somewhat as well. If the level of the aliased signals is below the resolution of the data converter, then these aliased signals are no cause for concern.
The key to a good data acquisition system design is to ensure that whatever aliased signals are present in your system are attenuated below the noise floor of the data converter. It also follows that, like any low-noise system design, you should band-limit the system to only use the bandwidth that you actually need. This means that you should put some kind of band-limiting filter before the data converter. These filters are often called anti-aliasing filters.
Preventing Aliases
To keep the aliased signals or noise below the resolution of the converter, we must ensure that the aliased signals are attenuated by 6.02(N + 1) + 1.76 dB so they are below half the system’s least significant bit (LSB), where N is the number of bits of resolution of the converter. For example, a 16-bit converter requires aliased signals to be attenuated by 6.02(17) + 1.76 dB, or 104.1 dB, at fS/2.
System designers often believe they should be able to use all the bandwidth of their data converter up to half the sample rate. Let’s say you have a 16-bit converter sampling at 500 kHz. You’ll need a filter with a –104.1-dB response at 250 kHz to prevent aliasing. But if you want to have a 250-kHz input bandwidth, your filter is attenuating everything at 250 kHz. So, what corner (–3 dB) frequency does your filter have?
Using a Butterworth response as a means to estimate the filter order required, we can use this approximate equation:
where n is the order of the filter required, A is the attenuation (in dB) required at the Nyquist frequency, fMAX, and fC is the corner frequency of the filter.
In our example, if you could accept 5% less bandwidth, your corner frequency will be at 237.5 kHz. This means the filter you need must transition approximately –100 dB in 12.5 kHz, which requires well over a two-hundredth-order filter! This clearly is not practical.
Filter Reality Check
Most people can stomach building no more than an eighth-order filter. It’s easy to design one with most active filter design software (such as FilterPro from Texas Instruments), and it can be realized utilizing a quad operational amplifier (op amp). If all you can handle putting in your system is an eighth-order filter, what does this mean to your system bandwidth? Solving the equation above for fC where n = 8, your corner frequency would be around 56 kHz. That’s almost one-tenth of your sample rate!
The good news is that, for most data acquisition systems, the actual signal you’re measuring generally isn’t close to the maximum frequency you can sample. Some signals that alias aren’t the signal of interest but, rather, are noise or some interference signal. Those troublesome signals may already be attenuated, perhaps by the response of the input sensor itself, and not present at full scale.
In our example system, assume that we’re concerned about a signal on our board that’s around 300 kHz, perhaps from a switching power regulator also on the board. The 300 kHz probably isn’t coming into our signal chain at full scale, but perhaps it’s down –40 dB.
This means we need an additional 60 dB of attenuation of this signal, so we need our antialiasing filter to have –60-dB response at 300 kHz. The corner frequency of our eighth-order filter then can be extended to about 125 kHz and still be adequate for our design.
Alternative To Filtering
Another approach you may see to prevent aliasing is to increase the sample rate. Delta-sigma converters use this approach, called oversampling, to reduce the antialiasing filter requirements. By sampling at a rate much higher than 2fMAX (for example, at rates up to 256 times fMAX), the antialiasing filter can be very simple.
The antialiasing filter cannot be eliminated, regardless of the rate at which you sample. You always need to band-limit the input signal, even if it’s only a simple RC filter that’s required. Aliasing will always occur in any sampling system, but now you know how to mitigate its effects.
You often can take advantage of your system knowledge, such as the frequency responses of your input sensors, to help you design the right amount of filtering. But don’t overlook this important part of your data acquisition system design!
References
1. Nyquist, H., “Certain topics in telegraph transmission theory,” AIEE Trans., vol. 47, pp. 617-644, 1928.
2. Shannon, Claude E., “Communications in the Presence of Noise,” Proceedings of the Institute of Radio Engineers, Vol. 37, pp. 10-21, 1949.
3. Download your free FilterPro software from Texas Instruments at www.ti.com/filterpro-ca.