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Fourier Convolution

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. Fourier convolution is used in this way to correct the analytical curve non-linearity caused by spectrometer resolution, in the “Tfit” method for absorption spectroscopy.

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.

Fourier convolution can also be used as a very general algorithm for the smoothing and differentiation of digital signals, by convoluting the signal with a (usually) small set of numbers representing the convolution vector. Smoothing is performed by convolution with sets of positive numbers, e.g. [1 1 1] for a 3-point boxcar. Convolution with [–1 1] computes a first derivative; [1 -2 1] computes a second derivative. Successive convolutions by Conv1 and then Conv2 is equivalent to one convolution with the convolution of Conv1 and Conv2.  First differentiation with smoothing is done by using a convolution vector in which the first half of the coefficients are negative and the second half are positive (e.g.[-1 -2 0 2 1]).

Simple whole-number convolution vectors:
Smoothing
[1 1 1]     = 3 point boxcar (sliding average) smooth
[1 1 1 1]     = 4 point boxcar (sliding average) smooth
[1 2 1]         = 3 point triangular smooth
[1 2 3 2 1]       = 5 point triangular smooth
[1 4 6 4 1]         = 5 point Gaussian smooth 
[1 4 8 10 8 4 1]     = 7 point Gaussian smooth 
[1 4 9 14 17 14 9 4 1] =
9 point Gaussian smooth 

Differentiation:
[-1 1]         First derivative
[1 -2 1]       Second derivative
[1 -2 1 -1]    Third derivative
[1 -4 6 -4 1]  Fourth derivative

Results of successive convolution by two vectors Conv1 and Conv2:
Conv1                    Conv2                      
[1 1 1]     x   [1 1 1]  = [1 2 3 2 1]     Triangular smooth           
[1 2 1]     x   [1 2 1]  = [1 4 6 4 1]     Pseudo-Gaussian smooth           
[-1 1]      x   [-1 1]   = [1 -2 1]        2nd derivative           
[-1 1]      x   [1 -2 1] = [1 -3 3 -1]     3rd derivative           
[-1 1]      x   [1 1 1]  = [1 0 0 -1)      1st derivative with gap-segment           
[-1 1]      x   [1 2 1]  = [1 1 -1 -1)     Smoothed 1st derivative           
[1 1 -1 -1] x   [1 2 1]  = [1 3 2 -2 -3 -1]    1st derivative with more smoothing       
[1 -2 1]    x   [1 2 1]  = [1 0 -2 0 1]    2nd derivative with gap-segment   


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.xlsx for 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 1],'same')./5;

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.

c=exp(-(1:length(y))./30);  
yc=conv(y,c,'full')./sum(c);

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.  

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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 toh@umd.edu.
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