Digital Signal Processing Reference
In-Depth Information
Time domain
Spectr um
Fr eq- domain
Step funct.
GAUSS- wi ndow
5,0
4,5
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
15,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
4,5
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
7
6
5
4
3
2
1
0
That´s not a step function ! This "step " ends
after 1 second at the end of the pictur e /block
with an inverse step . The computer thinks it
sees a periodic r ectangular signal of
1 Hz
and therefor e pr oduces such a line
spectr um . Only the even harm onics
are to be seen .
The LAPLACE tr ick : by multiplying the "step "
by a GAUSS window , the signal r eturns ver y
gently to zer o . So we have a r eal step which
only causes a sm all uncer tainty of fr equency
measurement ! After one second the next
step ar ises . So the step is periodic and the
distance of the lines in the spectr um is
between 1 Hz. The envelope of the step
12,5
10,0
7,5
5,0
-
spectrum is now in har mony with theory
.
2,5
0,0
15,0
12,5
GAUSS window causes an
"uncertainty of bandwidth
10,0
"
of near ly 1 Hz
7,5
5,0
2,5
0,0
0
100 200 300 400 500 600 700 800 900
5 10
20
30
40
50
60
70
ms
Hz
Illustration 114:
The trick with the LAPLACE Transformation
The step function contains mathematical and measuring technology problems because it is unclear what
happens after the steps. The end of the measuring procedure implies the end or the “jumping back” of the
step function. Thus, (in retrospect) we have not used the step function as a test signal but rather a rectan-
gular of the width t. This problem is got rid of by means of a trick. The step function is ended as gently as
possible as it fades exponentially or according to a GAUSSian function. The more slowly this takes place,
the more this test signal has the features of the (theoretical) step function. This trick in connection with the
subsequent FT is called LAPLACE Transformation.
Carrying out the step function of an FT is not easy without a trick because it is neither
periodic nor is its duration defined. (What happens after the "step" - a leap backwards?).
Thinking about this takes us a step further - the step function is a kind of rectangle in
Illustration 33 - a measure for the zero positions of the rectangular amplitude spectrum.
The greater
, the smaller the distance between the zero positions of the spectrum. The
second "step" (jump back) is practically where the measurement procedure is ended.
τ
