Express Circular Convolution in Terms of Linear Convolution
$h[n]_N$ is the periodic extension of $h[n]$.

$$ h[n]_N = h[n + mN] \quad \text{for } 0 \le n+mN &lt; N \text{ and } m \in \mathbb{Z}$$

so, given an $n$ and knowing $N$, you have to choose the correct integer $m$ so that

$$ 0 \le n+mN &lt; N $$
or
$$ -n \le mN &lt; N-n $$
or
$$ -\frac{n}{N} \le m &lt; \frac{N-n}{N} = 1 - \frac{n}{N} $$.

Or, for the term $h[n-k]_N$, 

$$ h[n-k]_N = h[n -k + mN] \quad \text{for } 0 \le n-k+mN &lt; N \text{ and } m \in \mathbb{Z}$$

and

$$ -\frac{n-k}{N} \le m &lt; 1 - \frac{n-k}{N} $$

so, you have to split your summation into two summations.