Digital Signal Processing Reference
In-Depth Information
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
-1,00
0,50
0,45
0,40
0,35
0,30
0,25
0,20
0,15
0,10
0,05
0,00
0,250
0,225
0,200
0,175
0,150
0,125
0,100
0,075
0,050
0,025
0,000
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0,00
0,030
0,025
0,020
0,015
0,010
0,005
0,000
2, 8H z
0 ,375 s
6H
z
0,16s
0,
06 s
16 H z
0
,03s
33 H z
0
100
200
300
400
500
600
700
800
900
50
75
100
125
150
ms
Hz
Illustration 46:
Bandwidth
Δ
f, time length
Δ
t and limiting case of UP
Here a so-called GAUSSian oscillation pulse is more and more restricted in time. The GAUSSian function
as revealing a "sinusoidal signal restricted in time" guarantees that the oscillation pulse begins gently and
ends gently without any abrupt changes. As a result of this option selected the spectrum also develops
according to a GAUSS function; it also begins and ends gently.
The time duration
Δ
t and the bandwidth
Δ
f must now be defined as a GAUSS pulse is also theoretically
"infinitely long". If the time duration
f relate to the two threshold values at which
the maximum functional value (of the envelope) has dropped to 50%, the product of
Δ
t and the bandwidth
Δ
Δ
f
<
Δ
t is roughly 1,
i.e. the physical limiting case
Δ
f
<
Δ
t = 1.
Check this assertion using a ruler and calculation using the rule of three for the above four cases: e.g. 100
Hz on the frequency axis are x cm, the band width
f entered - marked by arrows - is y cm. Then the same
measurement and calculation for the corresponding time duration
Δ
Δ
t. The product
Δ
f
<
Δ
t ought to be
about 1 in all four cases.
Sinusoidal signal and
δ
-pulse as a limiting case of the Uncertainty Principle
In the "ideal" sinusoidal signal
Δ
t
−>
∝
(e.g. a billion) applies for the time duration. It
follows that for the band width
Δ
f
−>
0 (e.g. a billionth part) as the spectrum consists of
a line or a thin stroke or a
δ
-function. In contrast, the
δ
-pulse has the time duration
Δ
t
−>
0. In contrast to the sine the band width
Δ
f
−>
∝
(with a constant amplitude!)
applies. Sine and the
δ
-function give the limiting values 0 and
∝
in the time and frequency
