User:GabrielaV

Introduction

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Signals & Systems

Example

Find the first two orthogonormal polynomials on [-1,1]

1. What is orthogonormal? [2]

2. What is orthogonal? [3]

3. What is a polynomial? [4]

        ${\mathbf {a}}$
        ${\mathbf {b}}t+c$

4. Now we can find the values for the unknown variables.

$<a|a>=\int _{-1}^{1}aadt=1$

${\mathbf {a}}={\sqrt {\frac {1}{2}}}$

$<bt+c|a>=\int _{-1}^{1}a(bt+c)dt=0$

${\mathbf {c}}=0$

$<bt+c|bt+c>=\int _{-1}^{1}(bt+c)^{2}dt=1$

${\mathbf {b}}={\sqrt {(}}{\frac {3}{2}})$

5. Now that we know what the first two orthogonormal polynomials!

Fourier Transform

As previously discussed Fourier Series is an expansion of a periodic function so therefore we can not use it to transform a non-periodic funcitons from the time to the frequency domain. Fortunately the Fourier Transform allows for the transformation to be done for a non-periodic function.

In order to understand the relationship between a non-periodic function and it's counterpart we must go back to fourier series. Remember the complex exponential signal? [5]

$x(t)=x(t+T)=\sum _{k=-\infty }^{\infty }\alpha _{k}e^{\frac {j2\pi kt}{T}}$

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold alpha_k={1/T}\int_{-{T\over 2}}^{{T\over 2}} \x(t) e^{-j2\pi kt\over T},dt}