Digital Signal Processing Reference
In-Depth Information
Ti me domai n
Noisy signal
Sum
Autoc or r el at
4, 0
3, 5
3, 0
2, 5
2, 0
1, 5
1, 0
0, 5
0, 0
2
-
-
-3
-4
10, 0
7, 5
5, 0
2, 5
0, 0
-2,5
-5,0
1, 00
0, 75
0, 50
0, 25
0, 00
-0,25
Signal 1: periodic sawtooth
Signal 2: noise
Signal 3 = Signal 1 + Signal 2: very noisy sawtooth
Autokorrelation function of Signal 3
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
ms
Abbildung 318:
Several time signals and their autocorrelation function
Start the DASYLab process to see the phase information generally getting lost due to averaging. The
comparison between the “clean” and the noisy signals shows how to filter out the “sustaining tendency”
respectively the exact frequency/ cycle duration by autocorrelation.
Autocorrelation of pure noise doesn´t show any tendency to sustain. It also shows noise respectively pure
random.
The WIENER-KHINTSHINE- theorem
How can the autocorrelation function be calculated by a computer in the easiest possible
way? As with convolution, this can be carried out with the help of
FFT
and
IFFT
.
In the case of real time signals the average value of the product of a time signal is deter-
mined by the autocorrelation function. As we are dealing here mainly with signals u(t) or
i(t) that came into being by the measurement of physical values, the electrical output can
be determined via the autocorrelation function even in more complicated cases, for exam-
ple, with noisy signals on electronic components:
2
ut
()
2
pt
()
=⋅
ut it
() ()
=
it R
()
=
R
The WIENER- KHINTSHINE theorem states that the average value of the spectral noise
output is this autocorrelation function, having been FOURIER- transformed, or that the
auto correla
tion function is the
IFT
of the spectral power density:
T
∞
1
1
jt
ω
³
³
φ
=
x
D
x
=
lim
x
( )
τ
x
∗
(
τ
−
t d
)
τ
=
S(
ω
)
e
d
ω
xx
1
1
1
1
2
T
2
π
11
T
→∞
−
T
−∞
