Confused between multiple representations of Fourier Series' formula

Question

I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$.

Now, suppose that I want to write the formula for complex form of the Fourier Series with the notations I am used to, it would be:

$$ f(t) = a_0 + \sum_{n=1}^\infty e^{in\pi t}(\frac{a_n-ib_n}{2}) + e^{-in\pi t}(\frac{a_n + ib_n}{2}) $$

Am I right?

Answer 1

The $w_0$ in the slides corresponds to your $\pi$. Usually, $w_0$ represents a fundamental (zeroth order) frequency, and $n w_0$ are then frequencies of its higher-order harmonics. Here, $w_0 = 2\pi f = 2 \pi/T$ so your expression is already expressed as multiples of $T/2$ which is the Nyquist minimum sampling period required for perfect signal reconstruction. Your form is more apt to digital (sampled) systems, while the form in the slides is more general and for analog systems.

The $i$ in the denominator did not go anywere; if you multiply $b_k$ by $i$ as was done, it cancels.

Answer 2

The $n$ and $k$ are identical, just a difference in index representation.

As for the $\omega$, this is angular frequency, which is equal to $2\pi{f}$, where $f$ is the cyclic frequency.

So in summary:

$$\begin{align} k&=n \\ \omega &= 2\pi{f} \end{align}$$