Fourier Series: Trigonometric - Signal Processing of Random Physiological Signals - page 134

If you continue this process for other integers of n , you will obtain b n = 0 for all values

of n .

Before moving into Discrete and Fast Fourier transformations and frequency-
domain analysis, let me regress into perhaps the most important function to engineers,
physicists, and mathematicians—the exponential function —which has the property
that the derivative and integral yield a function proportional to itself as in (12.10).
d
dt e st

e st

1
s

se st e st

= and dt = (12.10)

It turns out that every function or waveform encountered in practice can always be
expressed as a sum of various exponential functions.

12.5 EULER EXPANSION
Euler showed that the expression, s =− j ω , of a complex frequency represents signals

oscillating at angular frequency, ω , and that the cos ω t could thereby then be represented

as (12.11).

e j ω t e − j ω t +

cos( ω

t )

= (12.11)

2

In a way, the Discrete Fourier Series is a method of representing periodic sig-
nals by exponentials whose frequencies lie along the j axis. Expressing the Fourier

Trigonometric Series in an equivalent form in terms of exponentials, or e ± jn ω t :

ω

∞

f ( t ) = a 0 +

( a n cos n

ω 0 t + b n sin n ω 0 t )

n = 1

2 e jn ω 0 t

e − jn ω 0 t

1

cos n ω 0 t = +

2 j e jn ω 0 t

e − jn ω 0 t

1

sin n ω

0 t

= −

a n e jn ω 0 t

∞

e − jn ω 0 t

b n e jn ω 0 t

e − jn ω 0 t + −

then f ( t ) = + +

a 0

2 2 j

n = 1

1

Grouping terms, − j =

j :

a n e jn ω 0 t a n e − jn ω 0 t

∞

− jb n + jb n

f ( t ) = a 0 + +

2 2

n = 1