Digital Signal Processing Reference
In-Depth Information
• If
m(t) =
e
-j
ω
t
, the
FOURIER- transformation
FT
results
• If
m(t) = x
1
(t)
, the
autocorrelation function
ACF
results
• If
m(t) = x
2
(t)
, the cross
correlation function
CCF
(
x
2
(t)
unequal
x
1
(t)
)
This multiplicative structure in the integral apparently includes
the most important forms of signal processes such as the analysis
and synthesis of signals as well as statistical characteristics of
patterns and the extraction of certain information from noise.
This clue will be followed up by taking a closer look at
ACF
und
CCF
and by deriving
further connections. Here are once more the definitions:
Autocorrelation function
:
T
1
φ
³
∗
=
x
D
x
=
lim
x
( )
τ
x
(
τ
−
t d
)
τ
xx
1
1
1
1
2
T
11
T
→∞
−
T
Crosscorrelation function:
T
1
φ
³
∗
=
x
D
x
=
lim
x
( )
τ
x
(
τ
−
t d
)
τ
xx
1
2
1
2
2
T
12
T
→∞
−
T
NB:
• If we have complex input signals, in their multiplication one signal has to
be
complex conjugate
to the other, i.e. both signals must counter-rotate in
the GAUSSian plane. Only in this way, the maximum value /
(x
1
)
2
/ or
/
(x
1
)(x
2
)
/, be reached which is valid according the multiplication rule (see
Illustration 303. Generally, the following is valid for
complex
signals in
symbolic notation:
xt x t
()
⋅
∗
()
=
xt x t
() ( ) bzw.
⋅
−
X
()
ωω ω ω
⋅
X
∗
()
=
X
()
⋅
X
( )
−
• The
ACF
shows the correlation of time function
x
1
(t)
and the temporally
shifted function
x
1
(t+
).
an. Thus the
ACF
is a measure of the analogy of
later and earlier parts of a signal, a measure of its inherent tendency to
sustain itself, for the prediction of the time curve as well as the interference
ability with itself.
τ
• In the sine oscillation of the
ACF
(see Illustration 317), the amplitude and
frequency of the time function reappear, but not the zero phase angle. As a
result of the
ACF
the phase information is generally lost due to averaging.
