Digital Signal Processing Reference
In-Depth Information
2.18.
(a)
x
1
1
t
2
= 2t[u
1
t
2
- u
1
t - 1
2
] +
1
-2t + 4
2
[u
1
t - 1
2
- u
1
t - 2
2
]
= 2tu
1
t
2
- 4
1
t - 1
2
u
1
t - 1
2
+ 2
1
t - 2
2
u
1
t - 2
2
=
a
k=-
q
x
1
=
a
k=-
q
x
1
(c)
x
1
t
2
1
t - kT
0
2
1
t - 2k
2
2.20.
(c) (i)
1
(iii)
1
2.21.
(a)
u
1
t + 3
2
2.22.
(a)
(c)
1 - u
1
t
2
t[1 - u
1
t
2
]
2.23.
(a)
y
2
1
t
2
= T
2
[T
1
[x
1
t
2
]], y
3
1
t
2
= T
3
[T
1
[x
1
t
2
]],
y
1
t
2
= T
2
[T
1
[x
1
t
2
]] + T
4
[T
3
[T
1
[x
1
t
2
]] + T
5
[x
1
t
2
]]
(b)
(c)
(d)
y
1
t
2
= T
3
[T
2
[T
1
[x
1
t
2
]]] + T
4
[T
2
[T
1
[x
1
t
2
]]] + T
5
[T
1
[x
1
t
2
]]
y
1
t
2
= T
2
[T
1
[x
1
t
2
] + T
4
[T
3
[T
1
[x
1
t
2
]] * T
5
[x
1
t
2
]]
y
1
t
2
= T
3
[T
2
[T
1
[x
1
t
2
]]] * T
4
[T
2
[T
1
[x
1
t
2
]]] * T
5
[T
1
[x
1
t
2
]]
2.26.
(a)
(i) has memory; (ii) not invertible; (iii) stable; (iv) Time invariant;
(v) Linear;
(b)
Causal for
a Ú 1
2.28.
(a)
2y
1
1
t + 1
2
+ y
1
1
t
2
2.29.
(i) not memoryless unless (ii) invertible; (iii) causal if
otherwise not; (iv) stable; (v) time invariant; (vi) linear
t
0
= 0;
t
0
Ú 0,
2.30.
2.31.
h
1
t
2
= 2tu
1
t
2
- 2
1
t - 1
2
u
1
t - 1
2
(a)
(i) memoryless; (ii) not invertible; (iii) causal; (iv) stable; (v) time
invariant; (vi) not linear;
(c)
(i) memoryless; (ii) not invertible; (iii) causal; (iv) stable; (v) time in-
variant; (vi) not linear;
CHAPTER 3
3.1.
(a)
1
t - 2
2
u
1
t - 2
2
3.3.
3.4.
1
t - t
0
- t
1
2
u
1
t - t
0
- t
1
2
(a)
y
1
t
2
= 4
1
t - 1
2
[u
1
t - 1
2
- u
1
t - 2
2
] + 4[u(t - 2) - u(t - 3)]
] + (-t
2
+ 4
1
4 - t
2
[u
1
t - 3
2
- u
1
t - 4
2
+ 12t - 32)[u(t - 4)
t
2
- u(t - 6)] +
1
- 16t + 64
2
[u
1
t - 6
2
- u
1
t - 8
2
]
(b)
y
1
t
2
=-4
1
t - 1
2
[u
1
t - 1
2
- u
1
t - 2
2
] +
1
4t - 12
2
[u(t - 2)
- u(t - 4)] + (-4t + 20)[u(t - 4) - u(t - 5)]
3.5.
(a)
y
1
0
2
= 0,
y
1
1
2
= 4,
y
1
2
2
=-8,
y
1
2.667
2
= 0
1 - e
t - 2
2 - e
t - 1
- e
t - 2
3.7.
(a)
1
2
[u
1
t - 1
2
- u
1
t - 2
2
] +
1
2
[u(t) - u(t - 1)] + (2e
t
- e
t - 1
- e
t - 2
)u(-t)
e
-1
u
+ e
-
1
t - 1
2
(c)
(e)
1
2 - t
2
u
1
t - 2
2
1 - e
- 400
e
-t
- e
-400
1
2
u
1
-t
2
+
1
2
[u
1
t
2
- u
1
t - 400
2
]
