46 lines
1.6 KiB
Markdown
46 lines
1.6 KiB
Markdown
# Complex Exponential Fourier Series
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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.
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#### Definition: Orthonormal
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$V_1$ and $V_2$ are orthonormal if:
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1. $V_1 \cdot V_2 = 0$
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2. $\|V_1\| = \|V_2\| = 1$
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If $\psi_l$ and $\psi_k$ are orthonormal on $[a,b]$:
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$$\int\limits_a^b \psi_k(t) \psi_l^*(t)dt = \begin{cases} 1 & k=l \\ 0 & k \ne l \end{cases}$$
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### Example
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Consider the interval $[0,1]$:
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$$\text{Let } \psi_1(t) = 1; t \in [0,1]$$
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$$\text{Let } \psi_2(t) = \begin{cases} \end{cases}$$
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### Fourier Series in Exponential Form
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$$x(t) = \sum_{k=-\infty}^\infty X_k e^{jk\omega_0 t}$$
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$$T_0 = 1$$
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$$\omega_0 = 2\pi$$
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$$x(t) = 1 + \sum_{k=-\infty, k\ne0}^\infty {2 \sin\left({ \pi k \over 2}\right) \over \pi k}e^{2\pi k t}$$
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$$x(t) = 1 + \sum_{k=-\infty, k\ne0}^\infty {2 \sin\left({\pi k \over 2}\right) \over \pi k} e^{jk2\pi t}$$
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$$x(t - 0.1) = 1 + \sum_{k=-\infty, k\ne0}^\infty {2 \sin\left({\pi k \over 2}\right) \over \pi k} e^{jk2\pi (t - 0.1)}$$
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### Fourier Series in Trigonometric Form
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$$x(t) = c_0 + 2\sum_{k=1}^\infty \left[c_k \cos(k \omega_0 t) + d_k \sin(k \omega_0 t)\right]$$
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### Odd Signal Example
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$A = 1$, $T = 2$, $T_0 = 2$, $\omega_0 = \pi$
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$$x(t) = t$$
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$$x(t) = \sum_k X_k e^{j\pi k t}$$
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$$X_k = {1\over2} \int\limits_{-1}^1 t e^{-j \pi k t} dt$$
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This gives an imaginary $X_k$.
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$$X_0 = {1\over2} \int\limits_{-1}^1 t dt = 0$$
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$$x(t) = c_0 + 2\sum_{k=1}^\infty c_k \cos(\pi k t) + d_k \sin(\pi k t)$$
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Since it's odd:
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$$c_k = 0$$
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$$d_k = {1\over2} \int\limits_{-1}^1 t \sin(\pi k t) dt$$
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$$x_t = 2 \sum_{k=1}^\infty {(-1)^{k+1} \over \pi k} \sin(\pi k t)$$
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