Aliasing and sampling

Aliasing is a phenomenia which comes to live when you are sampling an analog signal to slow. This will disturb your digitized analog signal.

The Nyquist Shannon theorem states that you must sample with more the twice the highest frequency in the signal to void aliasing.

The phenoma can be viewed on figure below. Red dot indicates sampling and by trying to reconstruct the sampled signal a red curve connects the sample points. But the orgiginal blue signal do not look like the orginal blus analog signal.

Something is wrong

The problem

The basic problem is that we are sampling to slow.
Nyquist-Shannons sampling theorem states that Fs (sampling frequency) must be above 2* Fsignal-max.
So if maximum frequency in the signal to be sampled is 100 Hz sampling mst be carried out with more than 200 Hz.

See https://www.youtube.com/watch?v=Jv5FU8oUWEY for a more in depth explanation.

Three other links for in depth explanation

Conclusion 1 - sampling rate

Given highest frequency in the signal is Fs we shall with a frequency higher than 2 * Fs

https://en.wikipedia.org/wiki/Nyquist\\-Shannon_sampling_theorem the theory behind

Theory sets a lower limit at 2 * Fmax. But to be able to handle small remains of high frequency in the signal sampling with at least 4 times stipulated max signal frequency is normally choosen.

Take this as a engineering approach just 2 b sure

As shown in the menu above we cant reconstruct signal if we do not obey to this rule https://en.wikipedia.org/wiki/Nyquist\-Shannon_sampling_theorem

Problem solving location

So high frequencies must be removed before we sample the signal with an ADC - at analog level

So what we need is an analog anti aliasing filter in front of the ADC

Ideally it should be like

As the figure above indicate the lowpass filter (the antialiasing filter) should/should be of very high order - but this is in most /many cases unrealistic.

We need a implementable solution:

From https://www.sciencedirect.com/topics/computer-science/butterworth-filter:

A Butterworth filter is defined as a maximally flat filter that provides the sharpest roll-off possible without inducing peaking in the Bode plot. It is commonly used in control systems due to its lack of peaking.

Butterworth filters are often used allthough there exist a range of very high filters. Those are not the scope here.

Butterworth filter

A Butterworth filter is describes as:

The Butterworth filter is a type of signal processing filter
designed to have a frequency response that is as flat as possible in the passband.
It is also referred to as a maximally flat magnitude filter
wikipedia

Some things to consider

To be used as anti aliasing filter we have to select a comprimise of

cutoff frequency
1. must at least be less than the half of the samplings frequency (Nyquist-Shannon criteria)
2. Prefer to be at max half or on third of samplings frequency (Jens rule)
Order of filter

Order of antialiasing filter

Increasing order of filter increase the slope or rate of damping as seen in figure below

first order filter is no often seen as antialiasing filter.
second or third order(or even higher) are preferred
As order goes up phase shift goes up as well (which also can be seen as delaying)

Order of filter - the figure below

About the figure - can be used for simple scaling to other frequencies for discussing and design.

The figure is normalised to 1 rad/sec in cut off frequency,

x axis logaritmic
y axis in dB ( so log(signal/nom signal)
at cut off frequency all WB filters has 3 dB damping = 0.707 down in signal

first order	6 dB /octave	20 dB/decade
second order	12 dB /octave	40 dB/decade
third order	18 dB /octave	60 dB/decade
fourth order	24 dB /octave	80 dB/decade

Steepness is orderOfFilter * 20dB/decade or 6 dB/octave

An example From the figure

Fs (samplings freq) == 4 rad/sec
Anti aliasing filter cutt off freq is set to 1 rad/sec
At max alowable frequency ( 2 [rad/sec]) (Nyquist freq) one octave
At 2 times the cut off frequency(2 rad/sec) (one octave above) the signal is roughly reduced to
- 1,2,3,4,5 order - in dB 6,12,18,24,30 dB or 0.5, 0.25, 0.13, 0.06, 0.003
At 10 times the cut off frequency(10 rad/sec) (one decade above) the signal is roughly reduced to
- 1,2,3,4,5 order - in dB 20,40,60,80,100 or 0.10, 0.01, 0.001, 0.0001

Please note 10 rad/sec is 2.5 times sampling frequency - this is just for illustarting purpoes

My more or less subjective selections criteria

Look at sensor signal on an oscilloscope
is noise present ?
If possible use the scope too see a frequency plot for the sensor signal
1. you should be able to see dominant noise frequencies
decide for sampling frequency, cut off frequency and order for anti aliasing filter

Out of scope here - adc resolution

We should use an anti aliasing filter
But consider the following

If you are using a high resolution adc (12,13,14,15,16,… bit) noise might reduce your real resolution on the sampled signal.
14 bit equals 2^14 or 16384 step (16 bit 65536)
So given 14 bit with max level at 1V a single bit represent 1/16384 mV or 0.6 mV.
So is average noise is at 0.6mV or even higher you will loose at least 1 bit of out the 14 bit.

Implemention of BW lowpass filter to be used for anti aliasing

We are now dealing with analog electronics.

In the links below there are suifficent mth to calculator BW filters, so no need to repeat it here.

second order bw
- local copy
higher order bw
- local copy
- it is explained a design methology to calculate order of your bw filter based on how much damping you need.
- use the table below to find the appropriate second order filters which then must be set in series.