Digital Signal Processing Reference
In-Depth Information
(c)
the system's natural response does not contain a damped sinusoid;
(d)
the system's natural response contains a damped sinusoid;
(e)
the system's natural response contains an undamped sinusoid;
(f)
the system's frequency response approaches a constant at very high frequencies.
(g)
the system's frequency response approaches zero at very high frequencies;
7.19.
For each of the systems of Problem 7.17, determine
(a)
stability (use MATLAB as required);
(b)
the system modes;
(c)
the inverse system's transfer function.
7.20.
(a)
You are given a linear, time-invariant (LTI) system that produces an output
to an input
y(t) = e
-bt
u(t)
x(t) = e
-at
u(t)
, where
a 7 0
and
b 7 0.
Find the im-
pulse response of the system.
(b)
You are given a linear, time-invariant (LTI) system that produces an output
to an input
h(t)
y(t) = e
-at
cos(bt)u(t)
x(t) = u(t),
where
a 7 0
and
b 7 0.
Find the
impulse response
h(t)
of the system.
7.21.
Find the bilateral Laplace transforms of the following functions, giving the ROCs:
e
-2t
u(t)
(a)
(b)
(c)
(d)
(e)
(f)
e
-2t
u(t - 1)
-e
2t
u(-t)
e
2t
u(-t - 1)
e
-2t
u(t + 4)
e
-2t
u(-t + 1)
7.22.
Sketch each waveform, and find its bilateral Laplace transform and its ROC for each of
the signals given. If the transform does not exist, simply state that.
f(t) = e
-10t
u(t) + e
5t
u(-t)
(a)
(b)
(c)
(d)
f(t) = e
10t
u(t) + e
-5t
u(-t)
f(t) = e
10t
u(t) + e
5t
u(-t)
f(t) = e
-10t
u(t) + e
-5t
u(-t)
7.23.
Consider the function
e
5t
,
-1
F
t
F
2
b
f(t) =
.
0,
otherwise
(a)
Calculate the bilateral Laplace transform of this function, using definition (7.1),
and give its ROC.
(b)
This function can be expressed as
f(t) = e
5t
[u(t + 1) - u(t - 2)].
Use tables to find its bilateral transform and its ROC.
