The Discrete-Time Fourier Series (DTFS) can be used to write N0-periodic discrete-time signals x[k] as a weighted combination of complex exponentials. In this video, we reason through the form of the DTFS, namely:
1) The DTFS must consist of exponentials whose frequencies are some multiple of the fundamental frequency of the signal. If this was not the case, then the DTFS would not be periodic with the same period as the signal x[k].
2) If x[k] is N0-periodic, only N0 terms need to be included in the weighted combination. Since discrete-time complex exponentials are non-unique, including more than N0 terms would just be adding in additional exponentials that had already been included in the summation.
At the end of this video we now know the form of the DTFS equation. In the next video, we'll derive an equation that lets us to compute the DTFS coefficients (i.e. the weights in the summation).