Aliased Measurements in Feedback Control
Aliasing occurs whenever a sampled signal contains frequency components higher than half the
frequency at which sampling occurs. Half the sampling frequency is referred to as the Nyquist frequency,
named after Harry Nyquist. For a pure harmonic, f, the corresponding aliased frequency, fa, which occurs
in the sample band, is equal to:
fa = | f - mfs |
where fs is the sampling frequency and m is a positive non-zero integer. For noise components in a
signal greater than the Nyquist frequency, the noise will be aliased into a lower frequency range inside
the sample band.
So why is aliasing a concern in feedback controls ?
For feedback measurements containing noise beyond the Nyquist frequency, the noise will appear as an
aliased frequency inside the sample band. The aliased frequency or band of frequencies, seen as a
perturbation on the measurement, may be acted upon by the control as if were an in-band response of
the system. The aliased noise may then propagate actual perturbations into the controlled variable.
How can aliasing be prevented ?
Prior to conversion, a low pass analog filter should be used to attenuate input at and above the Nyquist
Why not just average or filter the converted measurements in software ?
Averaging is just a special case of filtering. Any attempts to filtering the sampled signal in software will
indeed attenuate any aliased noise that may be present, but at the expense of losing measurement
bandwidth. Also the aliased frequency may make its appearance at a very low frequency in the sample
band. Digital (software) filtering used to remove these frequencies will have a direct effect on the
measurement content as well as the noise, introduce additional phase, and make it difficult or impossible
to meet the closed loop performance requirements. It is better design practice to employ an analog filter
prior to conversion to limit signal before conversion within the Nyquist frequency.
What kind of analog filter is best ?
A filter that prevents or reduces aliasing is known as an anti-aliasing filter. The filter is generally lowpass
and attenuates noise to an acceptable level at the Nyquist frequency. The control engineer must decide
how much aliased noise can be tolerated in the sample band. If the filter cut-off frequency is chosen such
that it arbitrarily attenuates noise to an imperceptible level, the filter may contribute too much phase shift
into the measurement signal, adversely effecting loop stability, or making it difficult to achieve
performance objectives. With today's technology, the engineer is usually not limited by processor and
converter rate. So the choice of the sample rate (and processor speed) should be driven by the control
bandwidth requirements. Design the control system, pick an anti-aliasing filter with phase that will not
degrade control loop performance, then specify the required sample rate.
What if aliasing is ignored in feedback control ?
If the control system measurement is free of noise, an anti-aliasing filter is not required; but in a real
system there is always noise. The effects of aliasing on the control loop will depend on the range and
severity of the noise source. If aliasing is allowed to occur it will likely introduce roughness and inaccuracy
into the control. Noise components outside the sample band could possibly become aliased at sensitive
frequencies inside the loop bandwidth which can excite modes.
Even with the application of an anti-aliasing filter, can anything go wrong ?
Here are some precautions:
- Make sure the power supply for the analog filter does not induce any noise into the measurement
at the output stage of the amplifier. This can be insured by using a well regulated, low ripple linear
power supply or by providing proper bypass capacitors or RC filters near the supply pins to the
amplifier. Operational amplifiers do reject noise from the power supply, but only over a limited
range of (low) frequency
- If a MUX (multiplexer) is being used for multiple measurements, make sure adequate settling time is
allowed in the software for each channel after switching the MUX. Settling transients from the MUX
can appear as aliased noise in the sample data.
References on aliasing
Koenig, David M.,Control and Analysis of Noisy Processes, Prentice Hall, New Jersey 1991. pp 141-146