The Fourier expansion coefficient ( in OWN) of a periodic
signal
is

and the Fourier expansion of the signal is:

which can also be written as:

where is defined as

When the period of approaches infinity , the periodic signal becomes a non-periodic signal and the following will result:

- Interval between two neighboring frequency components becomes zero:

- Discrete frequency becomes continuous frequency:

- Summation of the Fourier expansion in equation (a) becomes an integral:

the second equal sign is due to the general fact:

- Time integral over in equation (b) becomes over the entire time axis:

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

and the inverse transform (synthesis) is:

Note that is denoted by in OWN.

Comparing Fourier coefficient of a periodic signal with with
Fourier spectrum of a non-periodic signal :

we see that the dimension of is different from that of :

If represents the

**Note on notations:**

The spectrum of a time signal can be denoted by or to
emphasize the fact that the spectrum represents how the energy contained in
the signal is distributed as a function of frequency or . Moreover,
if is used, the factor in front of the inverse transform is
dropped so that the transform pair takes a more symmetric form.
On the other hand, as Fourier transform can be considered as a special case
of Laplace transform when the real part of the complex argument
is zero:

it is also natural to denote the spectrum of by (in OWN).

**Example 0:**

Consider the unit impulse function:

**Example 1:**

If the spectrum of a signal is a delta function in frequency
domain
, the signal can be found to be:

i.e.,

**Example 2:**

The spectrum is

This is the sinc function with a parameter , as shown in the figure.

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

When approaches infinity, for all , and the spectrum becomes

Recall that the Fourier coefficient of is

which represents the energy contained in the signal at (DC component at zero frequency), and the spectrum is the energy density or distribution which is infinity at zero frequency.

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

which can also be written as

Switching and in the equation above, we also have

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 where they add up to infinity, an impulse.

**Example 3:**

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 .