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Aliasing

by Siemens Experimenter Siemens Experimenter ‎09-29-2016 04:08 PM - edited ‎04-14-2017 09:24 AM

 

When converting signals from their true analog form into digital form, frequency errors can be induced due to “aliasing”.

 

Aliasing is an effect that causes distortion in the spectrum of a sampled signal due to the sampling rate being too low to capture the frequency content properly. Aliasing causes high frequency data to appear at a lower frequency than it actually is (see Figure 1 below): thus assuming a “false identity” frequency or “alias” frequency.

 

Figure 1: TOP: The red sine wave is the original signal. The blue dots represent how often the signal is being sampled. MIDDLE: The blue line is how the signal will appear due to the low sampling rate. BOTTOM: What the user will see in the time domain. Notice the acquired frequency is much lower than the actual frequency.Figure 1: TOP: The red sine wave is the original signal. The blue dots represent how often the signal is being sampled. MIDDLE: The blue line is how the signal will appear due to the low sampling rate. BOTTOM: What the user will see in the time domain. Notice the acquired frequency is much lower than the actual frequency.

 Some essential terms to know when talking about aliasing:

 

  • Sampling frequency (Hz): The number of samples per second being acquired of an incoming frequency. The sampling frequency is two times the bandwidth.
  • Bandwidth (Hz): The frequency range over which measurements will be taken. Bandwidth is defined as half of the sampling frequency.
  • Span (Hz): The frequency range over which measurements will be taken and not be effected by the anti-aliasing low-pass filters (i.e. the alias-free region of the bandwidth). The span is 80% of the bandwidth.
  • Nyquist rate (Hz): Minimum frequency at which a signal can be sampled without introducing frequency errors. The Nyquist rate is twice the highest frequency of interest in the sample.

 

To properly sample all the desired frequency content of an incoming signal, and thereby avoid aliasing, one must sample at (or above) the Nyquist rate. In data acquisition, the sampling frequency is twice as high as the specified bandwidth. So, all frequency content below the specified bandwidth will be sampled at a rate sufficient to accurately capture the frequency content. However, if the incoming signal contains frequency content above the specified bandwidth, the sampling frequency (2x bandwidth) will violate the Nyquist theorem for this higher frequency content.

 

Figure 2: fs represents the sampling frequency, fsin¬e represents the frequency of the sine wave. a) When sampling at the same frequency as the incoming signal, the observed frequency is 0Hz. b) When sampling at twice the frequency of the sine wave, the observed frequency is fsine, the true frequency of the sine wave.Figure 2: fs represents the sampling frequency, fsin¬e represents the frequency of the sine wave. a) When sampling at the same frequency as the incoming signal, the observed frequency is 0Hz. b) When sampling at twice the frequency of the sine wave, the observed frequency is fsine, the true frequency of the sine wave.

 

When the Nyquist theorem is violated, spectral content above the bandwidth is mirrored about the bandwidth frequency. This means that frequency content X Hz above the bandwidth will then appear X Hz below the bandwidth. Watch the video at the top of this article to see mirroring in action.

 

Thus, higher frequency content appears to be at a lower frequency, or an “alias” frequency.

 

Figure 3: Aliasing causes frequency above the bandwidth to be mirrored across the bandwidth. Here, the bandwidth is 1000Hz. The actual frequency component in the signal is at 1300Hz. The frequency is 300Hz over the bandwidth. It will be mirrored 300Hz below the bandwidth at 700Hz.Figure 3: Aliasing causes frequency above the bandwidth to be mirrored across the bandwidth. Here, the bandwidth is 1000Hz. The actual frequency component in the signal is at 1300Hz. The frequency is 300Hz over the bandwidth. It will be mirrored 300Hz below the bandwidth at 700Hz.

 

Figure 4: This table shows the actual frequency being acquired by the system vs the observed frequency after sampling. For all frequencies being acquired, the bandwidth is 100Hz.Figure 4: This table shows the actual frequency being acquired by the system vs the observed frequency after sampling. For all frequencies being acquired, the bandwidth is 100Hz.

 

Preventing Aliasing

 

An anti-aliasing filter is a low-pass filter that removes spectral content that violates the Nyquist criteria (aka spectral content above the specified bandwidth). The ideal anti-aliasing filter would be shaped like a “brick wall”, completely attenuating all signals beyond the specified bandwidth (see Figure 5).

 

Figure 5: The ideal anti-aliasing filter would be shaped like a wall: cutting off all frequencies beyond the specified bandwidth (fs/2).Figure 5: The ideal anti-aliasing filter would be shaped like a wall: cutting off all frequencies beyond the specified bandwidth (fs/2).

In the real world, it is impossible to have this “wall shaped” filter. Instead, a very sharp analog filter is used that has a -3dB roll off at the bandwidth and attenuates all frequencies 20% beyond the bandwidth to zero.

 

Figure 6: The anti-aliasing filter has a -3dB roll off point at the bandwidth.Figure 6: The anti-aliasing filter has a -3dB roll off point at the bandwidth.

This is why the “trustable”, alias-free region of the spectrum is from zero Hz to 80% of the bandwidth. This alias-free range is called the frequency span.

 

Equation 1: Span is 80% of the bandwidthEquation 1: Span is 80% of the bandwidth

If the bandwidth was set at 1000Hz, the span would be 800Hz.

 

Figure 7: The spectral content is being mirrored about the bandwidth. All mirrored content is between 80% of the bandwidth and the full bandwidth. The alias free frequency range is from 0Hz to 80% of the bandwidth, also known as the span.Figure 7: The spectral content is being mirrored about the bandwidth. All mirrored content is between 80% of the bandwidth and the full bandwidth. The alias free frequency range is from 0Hz to 80% of the bandwidth, also known as the span.

The LMS SCADAS hardware has an anti-aliasing filter built into it. The video at the top of this article demonstrates how this anti-aliasing filter works.

 

TIP: Remember to set the bandwidth at least 20% higher than the highest frequency of interest.

In Test.Lab, it is possible to specify the span instead of the bandwidth. This way, you can be sure all data up to that frequency value will be alias free. See Video 2 below for instructions on how to set the default view in LMS Test.Lab as the span instead of bandwidth (Tools -> Options -> General tab -> Frequency: Span/Bandwidth/Sampling Rate).

 

 

Conclusion:

 

Aliasing can cause spectral content to be mirrored about the bandwidth thus causing false representation of the frequency content. To prevent this an anti-aliasing filter is implemented. The alias-free portion of the bandwidth is called the span. The span is the first 80% of the bandwidth. Remember, always set your bandwidth 20% higher than the highest frequency of interest to avoid aliasing.

 

Related Digital Signal Processing Links: 

 

 

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