# Complex Exponential Fourier Series

Inner product is just Ravi's fancy math way of saying dot product for functions, and orthogonal is his fancy math way of saying perpendicular.

#### Definition: Orthonormal
$V_1$ and $V_2$ are orthonormal if:
1. $V_1 \cdot V_2 = 0$
2. $\|V_1\| = \|V_2\| = 1$

If $\psi_l$ and $\psi_k$ are orthonormal on $[a,b]$:
$$\int\limits_a^b \psi_k(t) \psi_l^*(t)dt = \begin{cases} 1 & k=l \\ 0 & k \ne l \end{cases}$$

### Example
Consider the interval $[0,1]$:
$$\text{Let } \psi_1(t) = 1; t \in [0,1]$$
$$\text{Let } \psi_2(t) = \begin{cases} \end{cases}$$

### Fourier Series in Exponential Form
$$x(t) = \sum_{k=-\infty}^\infty X_k e^{jk\omega_0 t}$$

$$T_0 = 1$$
$$\omega_0 = 2\pi$$

$$x(t) = 1 + \sum_{k=-\infty, k\ne0}^\infty {2 \sin\left({ \pi k \over 2}\right) \over \pi k}e^{2\pi k t}$$

$$x(t) = 1 + \sum_{k=-\infty, k\ne0}^\infty {2 \sin\left({\pi k \over 2}\right) \over \pi k} e^{jk2\pi t}$$

$$x(t - 0.1) = 1 + \sum_{k=-\infty, k\ne0}^\infty {2 \sin\left({\pi k \over 2}\right) \over \pi k} e^{jk2\pi (t - 0.1)}$$

### Fourier Series in Trigonometric Form
$$x(t) = c_0 + 2\sum_{k=1}^\infty \left[c_k \cos(k \omega_0 t) + d_k \sin(k \omega_0 t)\right]$$

### Odd Signal Example
$A = 1$, $T = 2$, $T_0 = 2$, $\omega_0 = \pi$

$$x(t) = t$$
$$x(t) = \sum_k X_k e^{j\pi k t}$$
$$X_k = {1\over2} \int\limits_{-1}^1 t e^{-j \pi k t} dt$$
This gives an imaginary $X_k$.
$$X_0 = {1\over2} \int\limits_{-1}^1 t dt = 0$$
$$x(t) = c_0 + 2\sum_{k=1}^\infty c_k \cos(\pi k t) + d_k \sin(\pi k t)$$
Since it's odd:
$$c_k = 0$$
$$d_k = {1\over2} \int\limits_{-1}^1 t \sin(\pi k t) dt$$
$$x_t = 2 \sum_{k=1}^\infty {(-1)^{k+1} \over \pi k} \sin(\pi k t)$$