What is the pdf of a signal y = x + n where x takes the values {-1,0,1} with equal probability and n is independent of x and has a pdf
$$ fn(n)=\begin{cases}\frac{1}{L}(1-\frac{|n|}{L}) \,,~~~|n|\leq L\\ 0 \,,~~~|n|>L \end{cases}$$
What is the probability of error?
What happens for $$L \rightarrow \infty $$
$$\begin{aligned}F_{Y}\left(y\right) & =\frac{1}{3}P\left(1+N\leq y\mid X=1\right)+\frac{1}{3}P\left(N\leq y\mid X=0\right)+\frac{1}{3}P\left(-1+N\leq y\mid X=-1\right)\\ & =\frac{1}{3}P\left(1+N\leq y\right)+\frac{1}{3}P\left(N\leq y\right)+\frac{1}{3}P\left(-1+N\leq y\right)\\ & =\frac{1}{3}F_{N}\left(y-1\right)+\frac{1}{3}F_{N}\left(y\right)+\frac{1}{3}F_{N}\left(y+1\right) \end{aligned} $$
Taking the derivative we find:
$$f_{Y}\left(y\right)=\frac{1}{3}f_{N}\left(y-1\right)+\frac{1}{3}f_{N}\left(y\right)+\frac{1}{3}f_{N}\left(y+1\right)$$