DFT coefficients meaning?
At it's most fundamental, the DFT is about fitting a set of basis functions to a given set of sampled data.  The basis functions are all sinusoidal functions, expressed as the complex exponential with a purely imaginary exponent.  Using the most common scaling convention each basis function, without its scaling coefficient, is:
$$ g_k[n] \triangleq \tfrac1N e^{j \omega_k n} $$
For the DFT:
$$ \omega_k = \frac{2 \pi k}{N} $$
and the basis functions add up as
$$\begin{align}
 x[n] &amp;= \sum\limits_{k=0}^{N-1} X[k] \ g_k[n]\\
\\
  &amp;= \sum\limits_{k=0}^{N-1} X[k] \ \tfrac1N e^{j \omega_k n} \\
\\
  &amp;=  \tfrac1N \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2 \pi k}{N} n} \\
\end{align}$$
It's real easy to solve for the coefficients:
$$\begin{align}
 X[k] &amp;= \sum\limits_{n=0}^{N-1} x[n] \ e^{ -j \frac{2 \pi k}{N} n} \\
\end{align}$$