Computes the convolution of the input sequences X and Y. Wire data to the X and Y inputs to determine the polymorphic instance to use or manually select the instance.


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1D Convolution

The linear convolution of the signals x(t) and y(t) is defined as:

where the symbol * denotes linear convolution.

When algorithm is direct, this VI uses the following equation to perform the discrete implementation of the linear convolution and obtain the elements of X * Y.

for i = 0, 1, 2, … , M+N–2

where h is X * Y

N is the number of elements in X, M is the number of elements in Y, the indexed elements outside the ranges of X and Y are equal to zero, as shown in the following relationships:

xj = 0, j < 0, or j ≥ N

and

yj = 0, j < 0, or j ≥ M.

When algorithm is frequency domain, this VI completes the following steps, in order, to compute the linear convolution:

  • First, this VI pads the end of X and Y with zeros to make their lengths M + N – 1, as shown in the following equations.
  • Second, this VI calculates the Fourier transform of X' and Y' according to the following equations.
X'(f) = FFT(x') Y'(f) = FFT(y')
  • Third, this VI multiplies X'(f) by Y'(f) and calculates the inverse Fourier transform of the product. The result is the linear convolution of X and Y, as shown in the following equation.
X * Y = IFFT(X'(f) · Y'(f))

Thus, this VI computes the linear convolution, not the circular convolution. However, because x(t) * y(t)N ⇔ X(f)Y(f) is a Fourier transform pair, where x(t) * y(t)N is the circular convolution of x(t) and y(t), you can create a circular version of the convolution. To compute the circular convolution, you can use a block diagram similar to the block diagram shown in the following illustration.

2D Convolution

When algorithm is direct, this VI uses the following equation to compute the two-dimensional convolution of the input matrices X and Y.

for i = 0, 1, 2, … , M1+M2–2 and j = 0, 1, 2, … , N1+N2–2

where h is X * Y,

M1 is the number of rows of matrix X, N1 is the number of columns of matrix X, M2 is the number of rows of matrix Y, N2 is the number of columns of matrix Y, The indexed elements outside the ranges of X and Y are equal to zero, as shown in the following relationships:

x(m,n), m < 0 or m ≥ M1 or n < 0 or n ≥ N1

and

y(m,n) , m < 0 or m ≥ M2 or n < 0 or n ≥ N2.

When algorithm is frequency domain, this VI completes the following steps, in order, to compute the two-dimensional convolution:

  • First, this VI pads the end of X and Y with zeros to make their sizes (M1 + M2 – 1)-by-(N1 + N2 – 2), as shown in the following equations.
  • Second, this VI calculates the Fourier transform of X' and Y' according to the following equations.
X'(f) = FFT(x') Y'(f) = FFT(y')
  • Third, this VI multiplies X'(f) by Y'(f) and calculates the inverse Fourier transform of the product. The result is the two-dimensional convolution of X and Y, as shown in the following equation.
X * Y = IFFT(X'(f) · Y'(f))

The output size determines the size of the output matrix X * Y as shown in the following illustration.

  1. full

    The output matrix X * Y is (M1+M2–1)-by-(N1+N2–1).

  2. size X

    This is useful in image processing. If X is the image you want to filter, Y is a small matrix called the convolution kernel. X * Y is the filtered image whose size is the same as that of image X. The output M1-by-N1 matrix is the central part of the output matrix when the output size is full.

  3. compact

    Computing the edge elements of X * Y requires zero-padding if the output size is full or size X. LabVIEW removes these edge elements if the output size is compact. The output (M1–M2+1)-by-(N1–N2+1) matrix is the central part of the output matrix when the output size is size X.

Examples

Refer to the following example files included with LabVIEW.

  • labview\examples\Signal Processing\Signal Operation\Edge Detection with 2D Convolution.vi