User:GabrielaV

Introduction

Do you want to know how to contact me or find out some interesting things about me? [[1]]

Signals & Systems

Example

Find the first two orthogonormal polynomials on the interval [-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 therefore we can not use it to transform a non-periodic funciton from time to the frequency domain. Fortunately the Fourier transform allows for the transformation to be done on 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

${\mathbf {\alpha }}_{k}={1/T}\int _{-{T \over 2}}^{T \over 2}x(u)e^{-j2\pi ku \over T}du$

If we let

${\mathbf {T\to \infty }}$

The summation becomes integration, the harmoinic frequency becomes a continuous frequency, and the incremental spacing becomes a differential separation.

$\sum _{k=-\infty }^{\infty }\to \int _{-\infty }^{\infty }$

${{k \over T}\to f}$

${1/T}\to df$

The result is

$\lim _{T\to \infty }=\int _{-\infty }^{\infty }{\left[\int _{-\infty }^{\infty }x(u)e^{-j2\pi fu}du\right]}e^{j2\pi ft}df$

The term in the brackets is the Fourier transfrom of x(t)

${\mathcal {F}}[x(t)]=X(f)$

Inverse Fourier transform

$x(t)={\mathcal {F}}^{-1}[X(f)]$

How a CD Player Works

Today I will answer the most important question in your life! I'm sure this question has been bugging you for the last few years and it is going to be finally answered. Data is taken from the CD represented below.

$\sum _{n=-\infty }^{\infty }==HowaCDPlayerWorks==TodayIwillanswerthemostimportantquestioninyourlife!I'msurethisquestionhasbeenbuggingyouforthelastfewyearsanditisgoingtobefinallyanswered.DataistakenfromtheCDrepresentedbelow.::<math>\sum _{n=-/infty}^{\infty }$