fourier analysis for gps asen5190 p. axelrad october 2003
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Fourier Analysis for GPS
ASEN5190
P. Axelrad
October 2003
Periodic Functions
• A periodic function of period Tp = 1/fp
• Can be expressed by a Fourier Trigonometric Series as:
pf t nT f t
01
cos 2 sin 2n p n pn
f t a a n f t b n f t
Change of variable
01
Let 2 ,
so for , 2
The function is then expressed as:
cos sin
p
p
n nn
x f t
t T x
f
f x a a nx b nx
Fourier Coefficients
0
1
2
1cos
1sin
n
n
a f x dx
a f x nx dx
b f x nx dx
Examples
2
1 1 1Constant ( ) cos 0 sin 0
2
1 1 1cos cos 0 cos 1 cos sin 0
2
1 1 1cos ( ) cos 0 cos cos 0 cos sin 0
2
sin
A Adx A A nx dx f x nx dx
nx nx dx nx dx nx nx dx
mx m n mx dx mx nx dx mx nx dx
n
21 1 1sin 0 sin cos 0 sin 1
2x nx dx nx nx dx nx dx
a0 an bn
Square Wave
if - 0
if 0
A xf x
A x
0
0
0
0
0
0
0
1 2 3 4
1 13 5
1 10
2 2
1 1cos cos 0 for all
1 1 2sin sin 1 cos
4 4 , 0 , , 0 , .
34
so sin sin 3 sin 5 ...
n
n
a Adx Adx
a A nx dx A nx dx n
Ab A nx dx A nx dx n
n
A Ab b b b etc
Af x x x x
x2
A
4
Fourier Transform of a Pulse
0
1 1
111
11
0 if 1
1 if 1
( ) ( ) cos ( )sin
( ) ( ) cos ( ) ( ) cos
sin 2sin 2( ) cos sinc
xf x
x
f x A x B x d
A f d B f d
A d
Sinc and Sinc2
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.5
0
0.5
1
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Fourier series for a signal that is periodic in P
01
/ 21
0
/ 2
/ 21/ 2
/ 2
/ 21/ 2
/ 2
cos 2 sin 2
cos 2
sin 2
n p n pn
P
PP
P
n pPP
P
n pPP
f t a a n f t b n f t
a f t dt
a f t n f t dt
b f t n f t dt
2T
2
/ 2 / 2
1 10
/ 2 / 2
/ 2/ 2 / 2
1 1/ 2 / 2
/ 2 / 2 / 2
0 if NT=P f
A if
sin 2cos 2 cos 2
2 / 2
2 sin 2 sin / 2sin
/
T
PT
P T
P PP T
TP T
pn p pP P
pP T T
pn
tf t Nf
t
AT Aa f t dt Adt
P N
A n f ta f t n f t dt A n f t dt
n f P
A n f T A n N Aa
n N n N N
cn
N
x
A
T/2T/2 P=NT
Periodic rectangular pulse
Fourier representation of a periodic rectangular pulse
1
2sinc cos 2 p
n
A A nf t n f t
N N N
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 106
-5
0
5
10
15
20x 10
-4
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x 104
0
0.5
1
1.5
2x 10
-3
Zoomed in to show lines
Series for 1/T=1.023e6
and fp=1e3