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