Digital Signal Processing Reference
In-Depth Information
show that the finite-difference representation of the differential equa-
tion is given by
(1
+
5
t
+
6(
t
)
2
)
y
[
k
]
+
(
−
2
−
5
t
)
y
[
k
−
1]
+
y
[
k
−
2]
=
0
.
(ii) Show that the ancillary conditions for the finite-difference scheme
are given by
y
[0]
=
3 and
y
[
−
1]
=
3
+
7
t
.
(iii) By iteratively computing the finite-difference scheme for
t
=
0
.
02 s, show that the computed result from the finite-difference equa-
tion is the same as the result of the differential equation.
2.7
Assume that the delta modulation scheme, presented in Section 2.1.7,
uses the following design parameters:
sampling period
T
=
0
.
1 s and quantile interval
=
0
.
1V
.
Sketch the output of the receiver for the following binary signal:
11111011111100000000
.
Assume that the initial value
x
(0) of the transmitted signal
x
(
t
)at
t
=
0
is
x
(0)
=
0V.
2.8
Determine if the digital filter specified in Eq. (2.27) is an invertible system.
If yes, derive the difference equation modeling the inverse system. If no,
explain why.
2.9
The following CT systems are described using their input-output relation-
ships between input
x
(
t
) and output
y
(
t
). Determine if the CT systems are
(a) linear, (b) time-invariant, (c) stable, and (d) causal. For the non-linear
systems, determine if they are incrementally linear systems.
(i)
y
(
t
)
=
x
(
t
−
2);
(ii)
y
(
t
)
=
x
(2
t
−
5);
(iii)
y
(
t
)
=
x
(2
t
)
−
5;
(iv)
y
(
t
)
=
tx
(
t
+
10);
2
x
(
t
)
≥
0
(v)
y
(
t
)
=
0
x
(
t
)
<
0;
0
t
<
0
x
(
t
)
−
x
(
t
−
5)
t
≥
0;
(vii)
y
(
t
)
=
7
x
2
(
t
)
+
5
x
(
t
)
+
3;
(viii)
y
(
t
)
=
sgn(
x
(
t
));
(vi)
y
(
t
)
=
t
0
(ix)
y
(
t
)
=
x
(
λ
)d
λ +
2
x
(
t
);
−
t
0
t
0
d
x
d
t
(x)
y
(
t
)
=
x
(
λ
)d
λ +
;
−∞
d
4
y
d
t
4
+
3
d
3
y
d
t
3
+
5
d
2
y
d
t
2
d
2
x
d
t
2
+
3
d
y
d
t
(xi)
+
y
(
t
)
=
+
2
x
(
t
)
+
1
.
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