Data Sparse Matrix Computations - Lecture 3 Scribe :

Convolution as a Matrix/Vector multiplication Notice that (2) can be written as g = Y x where g is a column vector with elements gk = (x ∗ y)k, x is a column vector with elements xk, and Y is an NxN matrix. By examining (2), we can deduce that the elements of the first row of the matrix Y should be Y0,: = {y0, y−1, y−2, ..., y−(N−1)} Similarly, the second row should be Y1,: = {y1, y0, y−1, ..., y−(N−2)} Now, we may take advantage of the periodicity of the signal y. Namely, we note that y−1 = yN−1 y−2 = yN−2 and so on (see equation (1)). With this in mind, the rows of the matrix Y can now be written as