The discrete periodic image $x(m, n)$ is periodic with period 4 samples in both the directions. Express the image in terms of complex exponentials and, thereby, find its DFT coefficients $X(k, l)$. Find the $4 \times 4$ samples from both the expressions and check that they are the same. Find the least-squares errors, if $x(m, n)$ is represented by its DC component alone with the values $X(0,0), 0.9 X(0,0)$, and $1.1 X(0,0)$.
*(i)
$$
x(m, n)=1+2 \cos \left(\frac{2 \pi}{4}(m+n)-\frac{\pi}{3}\right)+\cos \left(2 \frac{2 \pi}{4}(m+n)\right)
$$
(ii)
$$
x(m, n)=2+2 \cos \left(\frac{2 \pi}{4}(m+2 n)+\frac{\pi}{3}\right)-\cos \left(2 \frac{2 \pi}{4}(m+n)\right)
$$
(iii)
$$
x(m, n)=-1+2 \cos \left(\frac{2 \pi}{4}(2 m+n)-\frac{\pi}{6}\right)+2 \cos \left(2 \frac{2 \pi}{4}(m+n)\right)
$$
.