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# From Fourier Series to Fourier Transform

The Fourier expansion of a periodic signal xT(t)=xT(t+T) is

where X[k] is the Fourier coefficient

If we define

the Fourier expansion becomes

When the period of xT(t) approaches infinity , the periodic signal xT(t) becomes a non-periodic signal x(t) and the following will result:

• 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 xT(t)

with Fourier spectrum of a non-periodic signal x(t)

we see that the dimension of is different from that of X[k]:

If |X[k]|2 represents the energy contained in the kth frequency component of a periodic signal xT(t), then represents the energy density of a non-periodic signal x(t) distributed along the frequency axis. We can only speak of the energy contained in a particular frequency band :

Note: The argument of a continuous time signal's spectrum

always appears in the form of , therefore can also be expressed as X(f), , or . The expressions or X(f) emphasizes the fact that this is the spectrum of the signal representing how the energy contained in the signal is distributed as a function of frequency or f (instead of or ). Moreover, if X(f) is used (instead of ), the factor in front of the inverse transform is dropped so that the transform pair looks more symmetric. However, as Fourier transform can be considered as a special case of Laplace transform when (i.e., the real part of s is zero, ):

it is also natural to write Fourier transform of x(t) as .

Example 1:

The spectrum is

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

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

When a approaches infinity, x(t)=1 for all t, and the spectrum becomes

Recall that the Fourier coefficient of x(t)=1 is

which represents the energy contained in the signal at k=0 (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 t and f 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 t=0 where they add up to infinity, an impulse.

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 .

Next: Fourier Transform of Periodic Up: No Title Previous: No Title
Ruye Wang
2001-11-07