Data Acquisition: Aliasing

Mark Colton ME 363

Spring 2011

Old Movies

Old Movies

What do these clips have in common? – The wheels appear to move backwards – The wheels appear to be moving too slowly – “Wagon wheel effect” – “Aliasing”

Why does this happen? – Sampling – (Bad selection of sampling rate)

Wagon Wheel Effect 1

Direction of Motion

True Rotation

Frame 1 Frame 2 Frame 3 Frame 4

Apparent Rotation

The wheel appears to rotate in the wrong direction!

Wagon Wheel Effect 2

Direction of Motion

True Rotation

Frame 1 Frame 2 Frame 3 Frame 4

The wheel appears to rotate too slowly!

Another Movie

Fixing the Wagon Wheel Effect

Capture at least two frames per revolution The wheel appears to rotate in the correct direction

with the correct speed

True Rotation

Frame 1 Frame 2 Frame 3

Aliasing

In instrumentation (and other fields) the wagon wheel effect is called “aliasing”

Definition: Appearance of false frequency components in a signal due to sampling

Remedy: Sample faster (we’ll see how fast)

Aliasing Example

Sample a 10 Hz sinusoid at various sample rates

Tell me when aliasing occurs

0 0.2 0.4 0.6 0.8 1-1

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t (s)

Aliasing Example

0 0.2 0.4 0.6 0.8 1-1

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t (s)

fs = 50.00fm

0 0.2 0.4 0.6 0.8 1-1

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fs = 20.00fm

0 0.2 0.4 0.6 0.8 1-1

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fs = 10.00fm

0 0.2 0.4 0.6 0.8 1-1

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fs = 5.00fm

0 0.2 0.4 0.6 0.8 1-1

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fs = 2.00fm

0 0.2 0.4 0.6 0.8 1-1

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fs = 1.50fm

0 0.2 0.4 0.6 0.8 1-1

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t (s)

fs = 1.20fm

0 0.2 0.4 0.6 0.8 1-1

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t (s)

fs = 1.05fm

0 0.2 0.4 0.6 0.8 1-1

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t (s)

fs = 1.00fm

0 0.2 0.4 0.6 0.8 1-1

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t (s)

fs = 0.55fm

Alias Frequencies

The false frequencies that appear in sampled signals are called “alias frequencies”

Once aliasing has occurred, there is no way to get back the original signal

Can we predict alias frequencies? – Yes, if we know:

• The frequency content of the measured signal • The sampling rate

Nyquist Frequency

Definition: fN = fs/2

Any frequency content in the analog signal that is above the Nyquist frequency will be aliased to frequencies below the Nyquist frequency

The “folding diagram” provides a convenient method for calculating alias frequencies

Folding Diagram

The frequencies 1.5fN, 2.5fN, 3.5fN, etc., are aliased down to 0.5fN

Example Signal frequency: f = 100 Hz Sampling rate: fs = 120 Hz Nyquist frequency: fN = fs/2 = 60 Hz Signal frequency (again): f =(100/60) fN = 1.67 fN Alias frequency (from chart): fa = 0.33fN = 20 Hz Aliased frequency will be out-of-phase to original frequency

Example

0 0.05 0.1 0.15 0.2 0.25-1

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t (s)0 0.05 0.1 0.15 0.2 0.25

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t (s)0 0.05 0.1 0.15 0.2 0.25

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t (s)0 0.05 0.1 0.15 0.2 0.25

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Summary

Problem: Aliasing occurs when the frequency content of the signal exceeds the Nyquist frequency

Solution: Select your sampling rate to be at least twice the highest frequency in the signal of interest

(This is what we already stated in the sampling theorem)

Another Problem

We showed that, if we don’t sample fast enough, our signal of interest will not be represented correctly

There is another problem due to aliasing, and it has to do with noise

Example

We want to record a 10 Hz sine wave Select a sampling rate:

fs = 80 Hz

0 0.2 0.4 0.6 0.8 1-1

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t (s)0 0.2 0.4 0.6 0.8 1

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Example

Let’s see what happens if there is noise in the signal

Assume the measuring process adds 60 Hz and 325 Hz noise

What’s the problem with this? Both of these noise components are at frequencies

above the Nyquist frequency of fN = fs/2 = 40 Hz The noise will be aliased into the frequency range

below the Nyquist frequency

Example

0 0.2 0.4 0.6 0.8 1-2

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t (s)0 0.2 0.4 0.6 0.8 1

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Example

0 10 20 30 400

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f (Hz)

M

0 10 20 30 400

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f (Hz)

M

From 60 Hz noise

From 325 Hz noise

Aliasing of Noise

Problem: Noise at higher frequencies is aliased into our frequency range of interest

Solution: Filter signal before sampling “Anti-aliasing filter”

Cutoff frequency should be no higher than the Nyquist frequency