Appendix B: Fourier Series
This appendix summarises the basic facts of Fourier series and interelates
its various forms so that the reader understands how to
recover the amplitude and phase of a Fourier component in each
representation.
The most general form of the Fourier series in one dimension describing
a function f(x) is
|
fs(x) = |
¥
å
n=0
|
Dn cos( nx + Fn ) |
| (1) |
so the function f
s(x) represents the object f(x) on the range
[
-p,+
p] and is periodic with period 2
p. The meaning of
D
n and
Fn is explained in appendix one.
The equation for f
s(x) may be expanded to give
|
|
|
|
|
¥
å
n=0
|
Dn cos( nx ) cos( Fn ) - Dn sin( nx ) sin( Fn ) |
| |
|
| |
A0
2
|
+ |
¥
å
n=1
|
An cos( nx ) + Bn sin( nx ) |
| (2) |
|
where A
0 is often referred to as the constant or `DC' term in the
series and
The amplitude Dn and phase Fn of each Fourier
component may be found from
and the specific values of A
n and B
n for a particular object
function f(x) are found from the integrations
|
|
|
|
|
1
p
|
|
ó
õ
|
+p
-p
|
f(x) cos( nx ) dx |
| (7) | |
|
| |
1
p
|
|
ó
õ
|
+p
-p
|
f(x) sin( nx ) dx |
| (8) |
|
(The factor of one half in front of the DC term allows these equations
to be generally applicable for all `n'.)
A complex Fourier series may be formed by setting
to form a series representation of f(x) described by
Alternatively one may calculate C
n directly via the equation
|
Cn = |
1
2p
|
|
ó
õ
|
+p
-p
|
f(x) e-i nx dx |
| (11) |
The complex form of the series thus incorporates A
n and B
n
into the REAL and IMAGINARY parts of one complex amplitude
C
n. Knowing this, one may easily recover the amplitude and phase
of the Fourier component from the complex representation. In fact
with the factor of 2 arising due to the summation over
-¥ to
+
¥ in the complex series but only from 0 to +
¥ in the
'standard' series representation.
From equation 2.10 one sees that the phase of C
n is identical to
Fn, and an advantage of the complex Fourier series is this clear
identification of the phase of the component and the linear relationship
between D
n and
|C
n|.
Two Dimensional Series
Fourier's theorem may be extended into two dimensions to express a two
dimensional function as a linear combination of 2-D sinusoidal basis
functions. The series representation of f(x,y) is then
|
fs(x,y) = |
1
2
|
A0 + |
+¥
å
m=-¥
|
|
+¥
å
n=-¥
|
Amn cos( mx + ny + Fmn ) |
| (13) |
which may be expanded as in the 1-D case, but this is not explicitly
done here. The amplitude and phase of each component are found by
performing the integrations
|
|
|
|
|
1
4p2
|
|
ó
õ
|
|
ó
õ
|
+p
-p
|
f(x,y) cos( mx + ny ) dx dy |
| (14) | |
|
|
- |
1
4p2
|
|
ó
õ
|
|
ó
õ
|
+p
-p
|
f(x,y) sin( mx + ny ) dx dy |
| (15) | |
|
| |
1
4p2
|
|
ó
õ
|
|
ó
õ
|
+p
-p
|
f(x,y) dx dy |
| (16) |
|
from which A
mn and
Fmn may be found.
Square Wave series: Evaluation
Functions may be
represented on other ranges by the appropriate scaling of the cosine
argument to be [(n
px)/L] where f
s(x) is now periodic with
period 2L, so that (in 1-D),
|
fs(x) = |
+¥
å
n=-¥
|
Cn e+i [(npx)/L] |
| (17) |
where
|
Cn = |
1
2L
|
|
ó
õ
|
+L
-L
|
f(x) e-i [(npx)/L] dx |
| (18) |
A square wave with period 2L has, in general, pulses of width `a'
separated by a distance `s', so that 2L=a+s. Figure B.1 shows a section
of the square wave.
Figure B.1: One dimensional square wave function
The Fourier coefficients Cn are found from the integration
|
Cn = |
1
2L
|
|
ó
õ
|
+[ a/2]
-[ a/2]
|
H e-i [(npx)/L] dx |
| (19) |
where H represents the height of the pulse. This integration reduces to
and, by dividing and multiplying by [ 2L/a], can be written as
|
Cn = H |
a
2L
|
sinc( |
na
2L
|
) |
| (21) |
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On 27 Oct 2001, 23:42.