# Drop The AliasDrop The Alias

DN Staff

April 10, 2006

Previous columns described how to digitize a signal and how to ensure its stability during that measurement. Another aspect of measurements also causes engineers some difficulty. Most engineers have heard of the Nyquist Theorem and some erroneously think it means they must sample at more than twice the frequency of interest. Actually, they must sample at more than twice the frequency of any measurable frequency component in their signal.

So, if you have a pure 50-kHz sine wave, you can reconstruct it perfectly by sampling at more than 100 ksamples/sec. Some harmonics and noise may accompany the 50-kHz signal, so the reconstructed signal may include aliases due to the capture of noise and spurious signals at a sample rate too slow to reconstruct them accurately. The aliases appear as signals at frequencies that did not exist in the original signal of interest.

To avoid aliases, designers often place an anti-alias filter between a signal source and an ADC. You can specify a low-pass anti-alias filter that has the characteristics shown in the accompanying figure. This type of filter passes frequencies below a corner frequency, fc, and attenuates frequencies above it. By convention, engineers describe fc as the frequency at which the filter produces a -3 dB attenuation. The figure shows the normalized frequency response for a 4-pole and for an 8-pole Butterworth low-pass filter.

A 12-bit ADC provides a resolution of 1 part in 4096 or 1 part in 8192 for a 1/2-LSB resolution. (I'll assume the latter, better resolution.) Thus a low-pass filter must attenuate unwanted signals to below about -80 dB. The 4-pole Butterworth filter reaches a -80 dB attenuation at f/fc = 10, or 10 times the corner frequency, fc. Thus for a 50-kHz signal of interest (fc), this filter prevents a 12-bit ADC from "seeing" signals above 500 kHz - the Nyquist frequency. As a result, the 12-bit ADC must sample at more than twice this frequency, or above 1 Msamples/sec to properly capture and represent the 50-kHz signal. So, even though you sample at 1 Msample/sec, the system provides only a 50-kHz bandwidth.

To improve the bandwidth, use a "better" filter, such as an 8-pole low-pass Butterworth filter. This filter provides a faster cut off of frequencies above fc, as shown in the graph. Now, use the frequency-response graph for the 8-pole filter to locate the frequency ratio (f/fc) for a 12-bit ADC (-80 dB). The f/fc ratio at -80 dB equals 3.25, so you can calculate fc = 500 kHz/3.25, or 154 kHz. That value represents a three-fold improvement in the bandwidth for the ADC.

## Acknowledgment

Thanks go to Gregory Anderson, president of Frequency Devices, for his clear explanation of filter characteristics and sampling rates, and for supplying filter data.

You also could trade off adding an 8-pole low-pass Butterworth filter for a lower sampling rate. Again assume you have a 50-kHz signal of interest, which equates to fc on the frequency-response plot. The 8-pole filter still reaches -80 dB where f/fc = 3.25, so f must equal a Nyquist frequency of 162.5 kHz. Now, sampling can occur above 325 ksamples/sec rather than at 1 Msamples/sec.

Keep in mind you can select from a variety of fixed or programmable filter types to meet specific data-acquisition and sampling needs. When you specify or purchase equipment that includes an anti-alias filter, be sure the supplier provides filter specifications and characteristics so you can better understand the system's behavior.

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