Thus far, only periodic functions have been considered though most functions are not periodic. To move from periodic functions (with period *T*) to aperiodic functions we simply let the period get very large, i.e., *T→∞*. Though this seems straightforward in concept, it fundamentally changes the nature of the transformation from time domain to frequency domain. Throughout this section we will work exclusively with the Exponential Fourier Series (which will lead to the Fourier Transform).

In the previous document, the Fourier Series of the pulse function was derived and discussed. To recap, the periodic pulse function *Π _{T}(t/T_{p})* has Fourier Series coefficients

To understand what happens as the period *T* increases (with *T _{p}=1.5*), consider the images below.

T = 2
...4
...8
...16

The graph on the left side of the image shows the periodic pulse function (a function of time). The width of the pulse is 1.5 seconds and, the period of the function is chosen with the buttons at the bottom of the image. There are two horizontal scales; the top one shows time in seconds, and the bottom one shows the location of the times *-2T*, *-T*, *-T/2*, *0*, *T/2*, *T* and *2T*. The right graph shows the Fourier Series coefficients *c _{n}*. Note that there are also two scales on the horizontal axis. The bottom scale is the integer index,

As T increases, there are two important features to note:

- The spacing of the
*c*coefficients decreases on the_{n}*ω*scale. - The magnitude of the
*c*coefficients decreases. In particular, the_{n}*c*coefficient (the average value) decreases because the time domain function is high for a smaller fraction of time._{0}

If *T* were to increase to become arbitrarily large, the magnitude of the Fourier Series coefficients goes to zero. We can keep this from happening by multiplying *c _{n}* by

The result is shown below (where the right hand graph is now *T·c _{n}*. Now a

T = 2
...4
...8
...16

Now imagine what would happen as *T* increases further. The overall magnitude of the coefficients is determined by the *sinc()* function, but the coefficients get closer and closer together (the spacing is *ω _{0}*) until, as

& T{c_n} = \int_T {x(t){e^{ - jn \cdot {\omega _0}t}}dt} \quad \quad T \to \infty \cr

& X(\omega ) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt} \cr} $$

Likewise, if we start from the Synthesis equation of the Fourier Series,

$$x(t) = \sum\limits_{n = - \infty }^{ + \infty } {{c_n}{e^{jn \cdot {\omega _0}t}}} $$Multiply and divide the right side by *T*, then replace *1/T* by *ω _{0}/(2*π)*

Make the substitutions as described in the previous paragraph (*T ·cn=X(ω)*,

In the limit as *T→∞* and *Δ ω→0 t*his becomes an integral

Using the definition given above, the Fourier Transform of the pulse function *Π(t/T _{p})* is given by

& x(t) = \Pi \left( {{t \over {{T_p}}}} \right) \cr

& X(\omega ) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt} = \int\limits_{ - {{{T_p}} \over 2}}^{ + {{{T_p}} \over 2}} {{e^{ - j\omega t}}dt} = - {1 \over {j\omega }}\left( {{e^{ - \omega {{{T_p}} \over 2}}} - {e^{ + \omega {{{T_p}} \over 2}}}} \right) \cr & \ \ \ \ \ \ \ \ \ = - {1 \over {j\omega }}\left( { - 2j\sin \left( {\omega {{{T_p}} \over 2}} \right)} \right) = {2 \over \omega }\sin \left( {\omega {{{T_p}} \over 2}} \right) \cr} $$

To get this into a form using the *sinc() *function multiply and divide by *T _{p}* and (and, later, by

& X(\omega ) = {2 \over \omega }{{{T_p}} \over {{T_p}}}\sin \left( {\omega {{{T_p}} \over 2}} \right) = {{{T_p}} \over {\omega {{{T_p}} \over 2}}}\sin \left( {\omega {{{T_p}} \over 2}} \right) = {{{T_p}} \over {\pi \omega {{{T_p}} \over {2\pi }}}}\sin \left( {\pi \omega {{{T_p}} \over {2\pi }}} \right) \cr

& X(\omega ) = {T_p}{\mathop{\rm sinc}\nolimits} \left( {\omega {{{T_p}} \over {2\pi }}} \right) \cr} $$

Thus, the pulse function, *x(t)=Π(t/T _{p})* and the

If you are interested in the Fourier Transform, go here.

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College