Fourier
transform and aliasing
Principle
In statistics, signal processing, computer
graphics and related disciplines, aliasing refers to an effect that causes
different continuous signals to become indistinguishable (or aliases of one
another) when sampled. It also refers to the distortion or artifact that
results when a signal is sampled and reconstructed as an alias of the original
signal.
Suppose we have a sine with frequency f = 0.1 Hz,
depicted in blue in Fig. 1, that is sampled with frequency fm = 1 Hz.
To our surprise we see that the samples also represent a sine with a much
higher frequency, the red sine with exactly the frequency fm - f = 0.9
Hz.
Fig. 1 Two different sinusoids that fit the
same set of samples
This effect is often called aliasing or backfolding. It
basically always happens with a discrete Fourier transform (DFT), also when
signals which are sampled with such a high frequency that the signal seems to
be well presented. However, aliasing
should preferably only be used when a signal is undersampled. Therefore,
Fig. 1 is not a good illustration of the aliasing effect. Undersampling means
that there are less than 2 sample point per period; the crosses in Fig. 1cannot
represent the sine of 0.1 Hz. Many sine frequencies can underlie the crosses. More
precisely, it implies that any signal comprising harmonics (see for harmonics Fourier analysis)
which exceed the Nyquist (from the Nyquist–Shannon sampling theorem) or folding
frequency 0.5fm, are folded backward over this frequency. Folding
means: “alias frequency” = fharmonic─0.5fm (where
0.5fm,< fharmonic<fm).
Fig. 2
Amplitude spectrum of a pure (unfiltered) symmetric
square signal of 1 Hz, amplitude 1, written as even signal (only cosine
components, see Fourier
analysis) sampled with 32 Hz. Since the phases alternate between
0 o and 180o, the phases are evaluated as a + and – sign
and attributed to the amplitude. So, the
subsequent harmonics alternate of sign. Folding frequencies are at 16, 32, etc.
Hz. The 17th harmonic is folded to 15 Hz and its amplitude is added
to the negative amplitude of the 15th harmonic, resulting in a much
too small amplitude at 15 Hz. The component 47 Hz is first folded to 17 Hz and
then to 15 Hz. This final folding result is indicated by the lower curved arrow.
Since the 47th harmonic has a 180 o phase shift, it is indicated
negative. So its amplitude subtracts from the 15th harmonic. Bar
length, dashed and solid are the theoretical amplitudes when a transform is
made with a finite sample frequency. The horizontal stripe in the solid bars in
the interval 0-16 Hz denote the final amplitude after folding.
Resuming, aliasing refers to an
effect that causes different continuous signals to become indistinguishable (or
aliases of one another) when sampled.
if a signal comprises harmonics with a frequency >0.5fm, the original signal cannot be
completely reconstructed from the outcome
of the DFT as an alias of the original signal recovered.
Aliasing can be prevented by filtering the
signal before sampling, such that all harmonics >½fm are deleted (see More
info).
It can especially occur when the signal is
periodic. Generally, biological signals are not periodic or at most
pseudo-periodic (ECG), are noisy and have a spectrum that comprises most power
in the low frequencies. Therefore, in practise filtering with a low pass Linear first order system and with fm =1.5 the cutt off frequency of the filter is sufficient.
Application
Aliasing is an unwanted effect of sampling
of any signal in the time or space domain (1D-3D).
The hardware of a signal processing
apparatus generally prevents this drawback of sampling. When computational
signal analysis software is used, combined with an analog to digital converter (ADC), generally the user himself
has to provide against the effects. How signals can be processed effectively is
discussed in Fourier transform and
signal processing.
More Info
See for more information www.biomedicalphysics.org Textbook
Medical physics the chapter Fourier transform and aliasing (section Systems and
basic concepts).
ribed as a negative frequency.