The Exponential Form Fourier Series
ECE4330 Lecture 15 The Fourier Series (continued) Prof. Mohamad Hassoun
The Exponential Form Fourier Series
Recall that the compact trigonometric Fourier series of a periodic, real signal () with frequency 0 is expressed as
() = 0 + cos(0 + )
=1
Employing the Euler's formula-based representation cos() =
1 2
(
+
- )
we
may
express
the
th
term
in
the
above
formulation
as
cos(0
+
)
=
1 2
[ (0 + )
+
-(0+)]
= (2 ) 0 + (2 -) -0
Let us define the following two complex constants, which will be referred to as the complex Fourier series coefficients,
=
2
and
its
complex
conjugate
=
-
2
We will use the following notation to refer to the magnitude and angle of those constants:
=
||
=
2
||
=
2
and
=
= 1,2,3, ...
-
= |-|-
=
2
-
|-|
=
2
and -
= -
Note
that
for
any
real
()
we
would
have
||=
|-|
=
2
and
=
-- = . Also, the dc component (signal average) is 0 = 0 which
is a real-valued constant.
Therefore, we may express the exponential Fourier series representation of () as (this is also known as the Fourier synthesis equation)
+
+
() = 0 + 0 = 0
=- 0
=-
Alternatively, the coefficients can be computed directly using the integral (which is also known as the Fourier analysis equation)
=
1 0
0
() -0
= 0, ?1, ?2, ?3, ...
In some situations, the 0 term must be computed as
0
=
1
lim
0
0
0
() -0
The above solution for the coefficients can be shown to be the one that is optimal in the sense of minimizing the total squared-error objective function,
0
+
2
() = [() - 0]
0
=-
where 0 is the frequency of the real, periodic signal ().
The above optimization problem is equivalent to minimizing the following objective function [based on the compact Fourier series expansion]
2
(, ) = [() - cos(0 + )]
0
=0
(Refer to the last few slides of Lecture 14 for an example)
The following table lists the coefficients associated with the three Fourier series representations.
Trigonometric Series 0, , = 1,2,3, ...
Compact Series 0, , = 1,2,3, ...
Exponential Series 0,
= ?1, ?2, ...
For a real and periodic (), we have the following formulas for the complex coefficients in terms of the standard trigonometric Fourier series coefficients:
= || = {} + {}
||
=
1 2
=
1 2
2
+
2
= 1,2, ...
0 = 0 = 0 = = tan-1 (-)
= 1,2,3, ...
|-| = || and - = -
The trigonometric Fourier series coefficients can be determined from the complex coefficients as follows,
0 = 0 = 2|| cos() = 2{} = -2|| sin() = -2{}
Similarly, the compact coefficients can be determined from the trigonometric (or complex) coefficients as follows,
= 2 + 2 = 2|| = 1, 2, 3, ... = tan-1 (-) =
The following table summarizes all transformations between the three Fourier series representations.
Exponential Fourier Series Spectra
The exponential Fourier series spectra of a periodic signal () are the plots of the magnitude and angle of the complex Fourier series coefficients.
Let () be a real, periodic signal (with frequency 0).
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