# Same convolution

A same convolution is a type of convolution where the output matrix is of the same dimension as the input matrix.

For a $$n \times n$$ input matrix $$A$$ and a $$f \times f$$ filter matrix $$F$$, the output of the convolution $$A * F$$ is of dimension $$\left \lfloor \frac{n + 2p - f}{s} \right \rfloor + 1 \times \left \lfloor \frac{n + 2p - f}{s} \right \rfloor + 1$$ where $$s$$ represents the stride length and $$p$$ represents the padding.

In a same convolution:

• $$s$$ is typically set to $$1$$
• $$p$$ is set to $$\frac{f - 1}{2}$$
• $$f$$ is an odd number

The result is that $$A$$ is padded to be $$n + p \times n + p$$ and $$A * F$$ becomes $$n \times n$$ – the same as the original dimensions of $$A$$.