Convolution is an
operation performed on two signals which involves multiplying one
signal by a delayed or shifted version of another signal,
integrating or averaging the product, and repeating the process for
different delays. Convolution is a useful process because it
accurately describes some effects that occur widely in scientific
measurements, such as the influence of a low-pass filter on an
electrical signal or of the spectral bandpass of a spectrometer on
the shape of a spectrum.
Figure 11. Fourier convolution is used here to
determine how the atomic line spectrum in Window 1 (top left)
will appear when scanned with a spectrometer whose slit function
(spectral resolution) is described by the Gaussian function in
Window 2 (top right). The Gaussian function has already been
rotated so that its maximum falls at x=0. The resulting
convoluted spectrum (bottom center) shows that the two lines
near x=110 and 120 will not be resolved but the line at x=40
will be partly resolved.
In practice, the calculation is usually performed by
point-by-point multiplication of the two signals in the Fourier
domain. First, the Fourier transform of each signal is obtained.
Then the two Fourier transforms are multiplied point-by-point by
the rules for complex multiplication and the result is then
inverse Fourier transformed. Fourier transforms are usually
expressed in terms of complex numbers, with real and imaginary
parts; if the Fourier transform of the first signal is a + ib, and the Fourier transform of the second
signal is c + id, then the product of the
two Fourier transforms is (a
+ ib)(c + id) = (ac
- bd) + i(bc + ad).
Although this seems to be a round-about method, it turns out
to be faster then the shift-and-multiply algorithm when the number
of points in the signal is large. Convolution can be used as a
powerful and general algorithm for smoothing and differentiation.
The example of Figure 11 shows how it can be used to predict the
broadening effect of a spectrometer on an atomic line spectrum. In
the “Tfit” method for absorption spectroscopy (TFit.html),
Fourier convolution is used to correct the analytical curve
non-linearity caused by spectrometer resolution.
SPECTRUM, the freeware
signal-processing application that accompanies this tutorial,
includes convolution and auto-correlation (self-convolution)
functions. Spreadsheets can be used to
perform "shift-and-multiply" convolution for small data sets
(for example,MultipleConvolution.xls or MultipleConvolution.xlsxfor Excel and MultipleConvolutionOO.ods
for Calc), but for larger data sets the performance is much slower
that Fourier convolution (which is much easier done in Matlab or
Octave than in spreadsheets). Matlab and Octave have a built-in
function for convolution of two vectors: conv.
This function can be used to create very general type of filters and
smoothing functions, such as sliding-average
and triangular smooths. For example,
ysmoothed=conv(y,[1 1 1 1
smooths the vector y with a 5-point unweighted sliding average
(boxcar) smooth, and
ysmoothed=conv(y,[1 2 3 2 1],'same')./9;
smooths the vector y with a 5-point triangular smooth. The optional
argument 'same' returns the central part of the convolution that is
the same size as y. The keyword 'same' returns the central part of
the convolution that is the same size as y.
Differentiation is carried out with smoothing by using
a convolution vector in which the first half of the
coefficients are negative and the second half are positive (e.g.[-1 0 1],[-2 -1 0 1 2], or
[-3 -2 -1 0 1 2 3]) to compute a
first derivative with increasing amounts of smoothing.
The next example creates an exponential trailing transfer function
(c), which has an effect similar to a simple RC low-pass filter, and
applies it to y.
In each of the above three examples, the result of the convolution
is divided by the sum of the convolution transfer function, in order
to insure that the convolution has a net gain of 1.000 and thus does
not effect the area under the curve of the signal. This page is maintained by Prof. Tom O'Haver ,
Department of Chemistry and Biochemistry, The University of Maryland
at College Park. Comments, suggestions and questions should be
directed to Prof. O'Haver at firstname.lastname@example.org.
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