Digital Signal Processing Reference
In-Depth Information
The magnitude and phase responses can be obtained from the frequency
response as follows:
()
=
(
)
j
ω
Magnitude response
:
He
2
sin
ω
N
2
180
()
(
)
(
)
⎣
⎦
=°+
⎣
⎦
−
j
ω
Phase response
:
He
90
sin
ω
N
2
ω
N
2
, degrees
π
radians,
it is sufficient to define both the magnitude and phase reponses of the system
in the interval -
Since the frequency response of the system
H
(
e
j
ω
) is periodic in 2
π
π
≤
ω
≤
π
radians.
2.1.3
Important Types of LTI Systems
The fundamental properties of LTI systems directly affect the behavior of
practical electrical components such as filters, amplifiers, oscillators, and
antennas. Some commonly used systems are described briefly below.
•
Inverse system:
As the name implies, the inverse system
H
i
(
z
) of a
given system
H
(
z
) is defined as:
1
Hz
i
()
=
(2.11)
Hz
()
Inverse systems are used in audio and video processing to recover
signals coming through noisy channels. However,
the inverse system
may not be stable, even if the original system is stable
. This is because the
zeros of the system
H
(
z
) are the poles of the system
H
i
(
z
). In order
to overcome this problem, we would require a
minimum-phase system
.
•
All-pass system:
As the name implies, an all-pass system has a
frequency response magnitude that is independent of
ω.
A stable
system function of the form:
−
1
z
−
a
az
*
Hz
ap
()
=
(2.12)
−
1
1
−
has the frequency response:
−
j
ω
()
=
e
−
a
*
j
ω
He
ap
−
j
ω
1
−
ae
(2.13)
j
ω
1
1
−
−
ae
ae
*
−
j
ω
=
e
−
j
ω
which implies that the magnitude response
H
ap
(
e
j
ω
)
= 1.
