9.3. OPERATOR OF CYCLIC SUPERPOSITION

When using the cyclic superposition operator, the arrays of the input and output images, as well as the impulse response, must be periodic in spatial variables with the same period. For the sake of unity, we assume, as before, that all these arrays have finite dimensions. Assume also that the array of source samples by size placed in the upper left corner of the array containing zeros i.e. an extended array is formed

at , (9.3.1а)

at . (9.3.1b)

Similarly, an extended array of impulse response samples is formed:

at , (9.3.2а)

at . (9.3.2b)

Next, we form periodically continued arrays and , repeating (propagating) extended arrays with a period of counts. The cyclic convolution of these arrays is by definition equal to

. (9.3.3)

The similarity of this expression with equality (9.1.1), describing the superposition of finite arrays, is striking. In fact, if chosen so that then at . It should also be noted the similarity of cyclic superposition with the superposition of discretized arrays. These relationships become clearer when using the vector representation of cyclic superposition.

Assume that arrays and presented accordingly -component vectors and . Then for the operation of cyclic superposition, you can write the ratio

, (9.3.4)

Where - matrix of array elements size . The cyclic superposition operator can be conveniently expressed as a block matrix with blocks size :

, (9.3.5)

Where

, (9.3.6)

and and a and . It should be noted that each row and each column of the block matrix contain nonzero blocks. If the array of impulse response samples is spatially invariant, then

(9.3.7)

and any row (or column) can be obtained by cyclically rearranging the blocks of the first row (or the first column). In fig. 9.3.1, and the example of the cyclic convolution operator is given, when the input and output arrays have dimensions , and the size of the array of samples of the impulse response is equal to . In fig. 9.3.1, b shows the matrix structure of the same operator with and when the impulse response is Gaussian.

Fig. 9.3.1. Examples of matrices of the cyclic convolution operator.

a - the general case ; b - impulse response of the Gaussian form, .

If the impulse response is spatially invariant and separable, then

, (9.3.8)

Where and - view matrix

, (9.3.9)

which size is equal . In this case, the two-dimensional cyclic convolution is calculated in accordance with the ratio

. (9.3.10)