DT Fourier Series
The following demonstrates the passing of a complex number through an LTI system. Remember that H() is a coefficient function.
- $ \qquad \mathbf{Z}^n: \quad (\sqrt{2}\jmath)^n \rightarrow LTI \rightarrow H(\sqrt{2}\jmath)(\sqrt{2}\jmath)^n $
- $ \displaystyle \qquad \sum a_k\cdot e^{\jmath k\omega_0 n} \rightarrow LTI \rightarrow \sum a_k\cdot H(e^{\jmath K\omega_0})\cdot e^{\jmath k\omega_0n} $
Let x[n] be a periodic DT signal with fundamental period N, we can write...
$ \displaystyle \qquad \omega_0 = \frac{2\pi}{N} $
...then...
$ \displaystyle \qquad x[n] = \sum a_k \cdot e^{\jmath k\omega_0 n},\quad a_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n]\cdot e^{-\jmath k \omega_0 n} $
