When We Create a Sprite, We Usually Completely Replace Pixel $1-\alpha$

Essay

When we create a sprite, we usually completely replace pixel values by those in the sprite. Suppose, however, we wish to replace pixel values by a combination with weights $1-\alpha$ from the original background, plus $\alpha$ from the sprite, where $\alpha$ is in $[0,1]$ .
State how you could do this, using only LOGICAL operators and an ADD.
Assume you already have a sprite $S_{\alpha}$ which has the character's colors all multiplied by the factor $\alpha$ , and black outside the character;
and also suppose you already have a second image of the (entire) background, already multiplied by $(1-\alpha)$ , called $B_{1-\alpha}$ .