Fourier Series: Trigonometric - Signal Processing of Random Physiological Signals - page 136

Fourier Transform representation of f ( t ). Consider the following nonperiodic signal,
f ( t ), which one would like to represent by eternal exponential functions. As a result, we
can construct a new periodic function, f T ( t ), with period T , where the function f ( t )is
represented every T seconds. The period, T , must be large enough so there is not any
overlap between pulse shapes of f ( t ). The new function is a periodic function and can
be represented with an exponential Fourier Series.

12.7 LIMITING PROCESS
What happens if the interval, T , becomes infinite for a pulse function, f ( t )? Do the
pulses repeat after an infinite interval? Such that f T ( t ) and f ( t ) are identical in the limit,
and the Fourier Series representing the periodic function, f T ( t ), will also represent f ( t )?

Discussion on Limiting Process: Let the interval between pulses become infinite, T =

∞ , in the series, so we can represent the exponential Fourier Series for f T ( t )asin

(12.14).

∞

F n e jn ω 0 t (12.14)

n =−∞

Where ω 0 is the fundamental frequency and F n is the term which represents the am-
plitude of the component of frequency, n ω 0 , the coefficient. As T becomes larger, the

fundamental frequency,

0 , becomes smaller, and the amplitude of individual compo-
nents also become smaller as shown in (12.15). The frequency spectrum becomes denser,
but its shape is unaltered.

ω

Limit T →∞ ,ω → 0 (12.15)

0

The spectrum exists for every value of ω and is no longer discrete but a continuous

function of ω . So, let us denote the fundamental frequency ω 0 as being infinite. The logic

is as follows:

π
ω

2

Since Limit T →∞ , ω → 0, then T = , o

and TF n is a function of jn ω ,or TF = F ( jn ω ).