Digital Signal Processing Reference
In-Depth Information
x t
()
⎯⎯ →
FFT
X
( )
f
x(t)
∗
IFFT
y(t)
•
Yf
()
⎯⎯ →
yt
()
FFT
h(t)
ht
()
⎯⎯ →
H f
( )
Illustration 317:
Two ways to process convolution
Convolution in the time range often faces considerable mathematical difficulties in the solution of the con-
volution integral. Computer programs almost always select the route via the frequency range.
FFT
and
IFFT
are rapid standard procedures and multiplication does not cause any difficulties. DASYLab also fol-
lows the latter route.
for unlimited duration. For limited duration the following applies
T
1
2
φ
=
³
x
() ()
tx tdt
1
2
TT
−
2
1
T
1
The standardized term
φ
ϕ
=
whereby -1
≤
ϕ
≤
1
φ
max
is called the
correlation coefficient
. The following notation is also practicable:
∞
φ
³
()
τ
=
kxtmt
()
⋅
( )
+
τ
t
1
−∞
Here,
m(t)
is the pattern function. By defining the value of
k
, several signal forms are
classified:
-
°
1 with aperiodic signals
1 2 with periodic signals (integral limits -T bis +T)
lim 1 2
k
=
T
®
°
¯
T
with random signals
T
→∞
τ
φ
(τ)
max
results. The similarity of this formula
φ
(τ)
At a definite value of
to the convo-
lution is apparent. Initially a
fixed
value of
(integral dt instead of dIJ) is used in the cal-
culation. By contrast, during the convolution one signal is shifted past the other, and
τ
τ
is
thus altered constantly.
m(t) is initially an arbitrary pattern function. It decides what signal process is involved:
