$z_i={}_1{x}_{i-1}+{}_2{x}_i+{}_3{x}_{i+1}$

$(10\mathrm{.}3)$

Problem $10\mathrm{.}2$ Equation $10\mathrm{.}3$ defines $1$D convolution with a kernel size of three, stride of one, and dilation one. Write out the equivalent equation for the $1$D convolution with a kernel size of three and a stride of two as pictured in figure $10\mathrm{.}3$a$-$b$.$

a$)$

b$)$

$(x_8$

c$)$

d$)$

Figure $10\mathrm{.}3$ Stride, kernel size, and dilation. a$)$ With a stride of two, we evaluate the kernel at every other position, so the first output $z_1$ is computed from a weighted sum centered at $x_1,$ and $b)$ the second output $z_2$ is computed from a weighted sum centered at $x_3$ and so on$.$ c$)$ The kernel size can also be changed. With a kernel size of five, we take a weighted sum of the nearest five inputs. d$)$ In dilated or atrous convolution $($from the French $"$ trous" $-$ with holes$),$ we intersperse zeros in the weight vector to allow us to combine information over a large area using fewer weights.