Digital Signal Processing Reference
In-Depth Information
(1) The causal implementation of an absolutely BIBO stable system must have
all of its poles in the left half of the complex s-plane.
(2) If even a single pole lies in the right half of the s-plane, the causal imple-
mentation of the system is unstable.
(3) If no pole lies in the right half of the s-plane, but one or more first-order
poles lie on the imaginary j
ω
-axis, the LTIC system is referred to as a
marginally stable system.
(4) An unstable system may be transformed into a stable system by cascading
the unstable system with an allpass system, which has zeros at the locations
of the unstable poles.
Section 6.9 described an analysis technique based on the Laplace transform to
calculate the output of an LTIC system. We showed that the Laplace-transform-
based analysis approach is suitable for studying the transient response of the
systems. The CTFT-based approach is appropriate for analyzing the steady state
response of the system.
Finally, Section 6.10 discussed the cascaded, parallel, and feedback config-
urations used to interconnect two LTIC systems. If two systems with impulse
responses
h
1
(
t
) and
h
2
(
t
) are connected, the overall impulse response and the
corresponding transfer functions are as follows:
←→
cascaded configuration
h
(
t
)
=
h
1
(
t
)
∗
h
2
(
t
)
H
(
s
)
=
H
1
(
s
)
H
2
(
s
);
←→
parallel configuration
h
(
t
)
=
h
1
(
t
)
+
h
2
(
t
)
H
(
s
)
=
H
1
(
s
)
+
H
2
(
s
);
H
1
(
s
)
1
+
H
1
(
s
)
H
2
(
s
)
.
feedback configuration
H
(
s
)
=
A practical system comprises multiple LTIC systems interconnected with a
combination of cascaded, parallel, and feedback configurations.
Problems
6.1
Using the definition in Eq. (6.5), calculate the bilateral Laplace transform
and the associated ROC for the following CT functions:
(a)
x
(
t
)
=
e
−
5
t
u
(
t
)
+
e
4
t
u
(
−
t
);
−
3
t
(d)
x
(
t
)
=
e
cos(5
t
);
−
3
t
(e)
x
(
t
)
=
e
7
t
cos(9
t
)
u
(
−
t
);
(b)
x
(
t
)
=
e
;
1
−
t
0
≤
t
≤
1
(c)
x
(
t
)
=
t
2
cos(10
t
)
u
(
−
t
);
(f)
x
(
t
)
=
0
otherwise
.
6.2
Using Eq. (6.9), calculate the unilateral Laplace transform and the associ-
ated ROC for the following CT functions:
(a)
x
(
t
)
=
t
5
u
(
t
);
−
3
t
cos(9
t
)
u
(
t
);
(d)
x
(
t
)
=
e
x
(
t
)
=
t
2
cos(10
t
)
u
(
t
);
(b)
x
(
t
)
=
sin(6
t
)
u
(
t
);
(e)
1
−
t
0
≤
t
≤
1
(c)
x
(
t
)
=
cos
2
(6
t
)
u
(
t
);
(f)
x
(
t
)
=
otherwise
.
0
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