Fourier transform: Difference between revisions

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<br><br>
Now let's make a periodic function <math> \gamma(t) </math> by repeating <math> \beta(t) </math> with a fundamental period <math> T_\zeta </math>.
Now let's make a periodic function <math> \gamma(t) </math> by repeating <math> \beta(t) </math> with a fundamental period <math> T_\zeta </math>.
Note that <math> \lim_{T_\zeta \to \infty}\gamma(t)=\beta(t) </math>
<br>
<br>
The Fourier Series representation of <math> \gamma(t) </math> is
The Fourier Series representation of <math> \gamma(t) </math> is
<br>
<br>
<math> \gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt} </math> where <math> f={1\over T_\alpha}
<math> \gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt} </math> where <math> f={1\over T_\zeta}
</math> <br>and <math> \alpha_k={1\over T_\alpha}\int_{-{T_\alpha\over 2}}^{{T_\alpha\over 2}} \gamma(t) e^{-j2\pi kt}\,dt</math>
</math> <br>and <math> \alpha_k={1\over T_\zeta}\int_{-{T_\zeta\over 2}}^{{T_\zeta\over 2}} \gamma(t) e^{-j2\pi kt}\,dt</math>
<br>
<math> \alpha_k </math> can now be rewritten as <math> \alpha_k={1\over T_\zeta}\int_{-\infty}^{\infty} \beta(t) e^{-j2\pi kt}\,dt </math>
<br>From our initial identity then, we can write <math> \alpha_k </math> as
<math>
\alpha_k={1\over T_\zeta}\Beta(kf)
</math>
<br> and
<math>
\gamma(t)
</math>
becomes
<math>
\gamma(t)=\sum_{k=-\infty}^\infty {1\over T_\zeta}\Beta(kf) e^{j2\pi fkt}
</math>

Revision as of 10:02, 8 December 2004

An initially identity that is useful:

Suppose that we have some function, say , that is nonperiodic and finite in duration.
This means that for some

Now let's make a periodic function by repeating with a fundamental period . Note that
The Fourier Series representation of is
where
and
can now be rewritten as
From our initial identity then, we can write as
and becomes