The Fourier expansion of a periodic signal
*x*_{T}(*t*)=*x*_{T}(*t*+*T*) is

where

If we define

the Fourier expansion becomes

When the period of

- Interval between two neighboring frequency components becomes zero:

- Discrete frequency becomes continuous frequency:

- Summation of the Fourier expansion becomes an integral:

Recall that

- Time integral over
*T*becomes over the entire time axis:

In summary, when the signal is non-periodic
,
the Fourier expansion becomes
Fourier transform. The forward transform is

and the inverse transform is

Comparing Fourier coefficient of a periodic signal

with Fourier spectrum of a non-periodic signal

we see that the dimension of is different from that of

If |

always appears in the form of , therefore can also be expressed as

it is also natural to write Fourier transform of

**Example 1:**

The spectrum is

This is the sinc function with a parameter

Note that the height of the main peak is 2*a* and it gets taller and narrower as
*a* gets larger. Also note

When

Recall that the Fourier coefficient of

which represents the energy contained in the signal at

The integral in the above transform is an important formula to be used
frequently later:

which can also be written as

Switching

representing a superposition of an infinite number of cosine functions of all frequencies, which cancel each other any where along the time axis except at

**Example 2:**

The spectrum of the cosine function is

= | |||

= |

The spectrum of the sine function

can be similarly obtained to be

Again, these spectra represent the energy density distribution of the sinusoids, while the corresponding Fourier coefficients

and

represent the energy contained at frequency .