[Simple Example]
[Command line function] [Segmented Fourier filter] [Interactive Fourier Filter] [Application of a narrow bandpass filter]

The Fourier filter is a type of filtering
function that is based on manipulation of specific frequency components
of a signal. It works by taking the Fourier transform of the
signal, then attenuating or amplifying specific frequencies, and
finally inverse transforming the result. In many science
measurements, such as spectroscopy and chromatography, the signals
are relatively smooth shapes that can be represented by a
surprisingly small number of Fourier components. For example, the
animation below (script)
shows in the top panel a signal of three smooth peaks, with peak
heights of 1, 2, and 3, where the x-axis is time in seconds. The
middle panel shown the first 51 frequencies of its Fourier
spectrum (zero through 50). In this case the x-axis is both the
frequency in Hz and the Fourier component number (because the
duration of the signal is exactly 1 second). The amplitude of the
Fourier components is strongest at low frequencies and drops to
near zero at 25 Hz.

The bottom panel shows in this figure the signal re-constructed
by inverse transforming the first "n" Fourier components. The
animation (visible in a web browser) shows the result of using the
frequencies between 1 through 25 progressively. The reconstructed
signal starts as a single sine wave and becomes progressively more
complex as more frequencies are added, until it is visually
indistinguishable from the original signal when 25 frequencies are
included. But notice what the reconstructed signal looks like when
it gets to 16 frequencies; the amplitude has already dropped very
low and there is relatively little amplitude in the remaining
frequencies. The three peaks are rendered well but the baseline
has a distinct ripple. That is caused by the abrupt cut-off of the
frequencies beyond that point, which can be avoided by using a
filter with an adjustable filter shape that allows the cut-off
rate to be controlled. This is illustrated in the next figure.

The figure below shows the same three-peak signal, but with the
addition of random white noise with a standard deviation equal to
the the peak height of the smallest peak (line 14 of the same
script). White noise has equal
amplitude at all frequencies, whereas most of the peak
signal is concentrated in the first 25 frequencies, so we can
expect that most but not all of the noise can be eliminated by
employing a low-pass filter with a cut-off frequency around 25 Hz.
The result (bottom panel) shows the three peaks with their
position and peak heights approximately intact. Obviously the
noise *below *25 Hz remains, which is why the signal is
imperfect.

The optimization of the Fourier filter for the
signal-to-noise (SNR) ratio of peak signals faces the same
compromise as conventional smoothing functions; namely, the
optimum SNR is achieved when the peak height is less than the
noiseless maximum. For example, the script GaussianSNRFrequencyReconstruction.m
shows that optimum SNR is reached when the peak height is about
half the true value, but the peak area is the same.

A more dramatic example is shown the figure below. In this case,
the signal (top left) seems to be only random high-frequency
noise, and its Fourier spectrum (top right) shows that
high-frequency components dominate the spectrum over most of its
frequency range (script). The
Fourier spectrum is expanded in the X and Y directions (bottom
left) to show more clearly the low-frequency region. There, the
series of relatively smooth bumps, with peaks at the 1st, 20th,
and 40th frequencies, are most likely the actual signal. Working
on the hypothesis that the components above the 40th harmonic are
increasingly dominated by noise, the Fourier filter function (FouFilter.m)
can be used to gradually reduce the higher harmonics and to
reconstruct the signal from the modified Fourier transform (red
line). The result (bottom right) shows the signal actually
contains two partly overlapping Lorentzian peaks that were totally
obscured by noise in the original signal.

`ffty=fft(y); % ffty is the fft of y`

`lfft=length(ffty); % Length of the FFT`

`ffty(n:lfft-n)=0; % All frequencies between n and lfft-n
in the fft are set to zero.`

`fy=real(ifft(ffty)); % Real part of the inverse fft`

Here, *n* is the number of frequencies that are passed; all
others are simply eliminated. The function form of this operation
is flp.m. This is the minimal essence of a
Fourier filter. The script flptest.m
demonstrates its use (figure on the right). The is not really a
practical filter, however, because its abrupt cutoff usually
results in ringing on the baseline, as shown above.

**General-purpose Fourier filter
function.** To make the Fourier filter more generally useful,
we must add code to include not only low-bass, but also high-pass,
band pass, and band reject filter modes, plus a provision for more
gentle and variable cut-off rates. This, and more, is done in the
in the following section. The custom Matlab/Octave function **FouFilter.m** is a more flexible
Fourier filter that can serve as a bandpass or bandreject (notch)
filter with variable cut-off rate. (Version 2, March, 2019,
correction thanks to Dr. Raphael Attié, NASA/Goddard Space Flight
Center). Has the form` [ry,fy,ffilter,ffy] = FouFilter(y,
samplingtime, centerfrequency, frequencywidth, shape, mode)`,
where y is the time-series signal vector, 'samplingtime' is the
total duration of sampled signal in sec, millisec, or microsec;
'centerfrequency' and 'frequencywidth' are the center frequency
and width of the filter in Hz, KHz, or MHz, respectively; 'Shape'
determines the sharpness of the cut-off. If shape = 1, the filter
is Gaussian; as shape increases the filter shape becomes more and
more rectangular. Set mode = 0 for band-pass filter, mode = 1 for
band-reject (notch) filter. Set centerfrequency to zero for a
low-pass filter. FouFilter returns the filtered signal in 'ry'. It
can handle signals of virtually any length, limited only by the
memory in your computer. Here are two examples of its application:
TestFouFilter.m demonstrates a
Fourier bandpass filter applied to a noisy 100 Hz sine wave which
appears in the middle third of the signal record. TestFouFilter2.m is an animated
demonstration of the Fourier bandpass filter applied to a noisy
100 Hz sine wave signal, with the filter center frequency swept
from 50 to 150 Hz. Both requires the FouFilter.m function in the
Matlab/Octave path.

iFilter 4.3, an Interactive Fourier Filter for Matlab, allows you to select from six filter modes ('band-pass', 'low-pass', 'high-pass', 'band-reject (notch), 'comb pass', and 'comb notch'). (In the

MorseCode.m
is a script that uses iFilter to demonstrate the abilities and
limitations of Fourier filtering. It creates a pulsed *fixed
**frequency* (0.05) sine wave
that spells out “SOS” in Morse code
(dit-dit-dit/dah-dah-dah/dit-dit-dit), adds random white noise so
that the SNR is extremely poor
(about 0.1 in this example). The white noise has a frequency
spectrum that is spread out over
the entire range of frequencies; the signal itself is
concentrated mostly at a fixed frequency (0.05) but the presence
of the Morse Code pulses spreads out its spectrum over a narrow frequency range of
about 0.0004. This suggests that a Fourier bandpass filter
tuned to the signal frequency might be able to isolate the signal
from the noise. As the bandwidth
is reduced, the signal-to-noise ratio improves and the signal begins to emerges from the
noise until it
becomes clear, but if the bandwidth is *too *narrow,
the *step response time* is too slow to give distinct “dits”
and “dahs”. The step response time is inversely proportional to
the bandwidth. (Use the ? and " keys to adjust the bandwidth.
Press '**P**' or the Spacebar to hear the sound). You can
actually *hear *that sine wave component better than you
can *see *it in the waveform plot (upper panel), because the ear
works like a spectrum analyzer, with separate nerve endings
assigned to to specific
frequencies, whereas the eye analyzes the graph spatially, looking
at the overall amplitude and not at individual frequencies.

Watch an mp4 video of this script in
operation, with sound. Also on YouTube.

The script RealTimeFourierFilter.m
is a demonstration of a *real-time* Fourier filter which is
discussed in Appendix
Y.

An older version of this page is also available in French.

Last updated December, 2019. This page is part of "

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