Understanding the Fourier Transform



The projection of vector a onto vector b is defined: $$\operatorname{proj}_{\mathbf{v}} \mathbf{a} = \frac{\langle \mathbf{a}, \mathbf{b} \rangle}{|\mathbf{a}|^{2}} \mathbf{a}$$

Thus, if ${\mathbf{e_1}, \cdots, \mathbf{e_n}}$ is an orthogonal basis of a vector space, then

$$\mathbf{v}=\sum_{i=1}^n \operatorname{proj}_{\mathbf{e_i}}(\mathbf{v})$$

As it turns out, functions are vectors. This is because vectors are simply mathematical objects that can be added together and multiplied by scalars. That is to say, because you can take linear combinations of factors, they are vectors.

We also define the inner product of two periodic functions s1(t), s2(t) as:

$$\langle s_1(t),s_2(t)\rangle = \frac{1}{T}\int_T s_1(t)s_2^{*}(t) dt $$

$$ |s(t)|^2 = \frac{1}{T}\int_T |s(t)|^2 dt$$

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