EECS 20N: Structure and Interpretation of Signals and Systems
Department of Electrical Engineering and Computer Sciences
U
NIVERSITY OF
C
ALIFORNIA
B
ERKELEY
Problem Set 4
SOLUTIONS
HW 4.1
(1)
(a) The system cannot be memoryless because same input values are
mapped into different output values depending if their time stamp
was positive or negative
(b) The system must be causal.
Consider
x
1
(
n
)
and
x
2
(
n
)
such that
x
1
(
n
) =
x
2
(
n
)
,
∀
n
≤
n
0
, then for the definition of the system we
will also have
y
1
(
n
) =
y
2
(
n
)
,
∀
n
≤
n
0
, hence it is causal.
(c) The system must be linear. Consider
ˆ
x
(
n
) =
ax
1
(
n
) +
bx
2
(
n
)
...
(d) The system is not time invariant.
Consider
ˆ
x
(
n
) =
x
(
n
−
N
)
, it
follows
ˆ
y
(
n
) =
braceleftBigg
−
ˆ
x
(
n
)
n <
0
+ˆ
x
(
n
)
n
≥
0
=
braceleftBigg
−
x
(
n
−
N
)
n <
0
+
x
(
n
−
N
)
n
≥
0
On the other hand:
y
(
n
−
N
) =
braceleftBigg
−
x
(
n
−
N
)
n
−
N <
0
+
x
(
n
−
N
)
n
−
N
≥
0
which is different from
ˆ
y
.
(2)
(a) The system must be memoryless and the function
f
is exactly the
ceiling function
f
(
x
) =
⌈
x
⌉
.
(b) The system must be causal, because it is memoryless and
memoryless
⇒
causal
.
(c) The system cannot be linear. Consider
ˆ
x
(
t
) =
ax
1
(
t
) +
bx
2
(
t
)
, then
F
2
(ˆ
x
)(
t
) =
⌈
ax
1
(
t
)+
bx
2
(
t
)
⌉
which is different from
a
⌈
x
1
(
t
)
⌉
+
b
⌈
x
2
(
t
)
⌉
.
(Assume for instance that
a
=
b
= 1
and that for a given time
t
0
, we
have
x
1
(
t
0
) =
x
2
(
t
0
) =
1
2
, in this case
⌈
ax
1
(
t
) +
bx
2
(
t
)
⌉ negationslash
=
a
⌈
x
1
(
t
) +
b
⌉
x
r
(
t
)
).)
(d) The system must be time invariant.
(3)
(a) The system cannot be memoryless. Consider
N < t
0
< N
+1
where
N
is a Natural number,
y
(
t
0
) =
x
(
N
)
, hence it is not memoryless.
(b) The system must be causal, because it is looking at current or past
values of the input.
(c) The system must be linear. Consider
ˆ
x
(
t
) =
ax
1
(
t
) +
bx
2
(
t
)
, then
F
3
(ˆ
x
)(
t
) =
ax
1
(
⌊
t
⌋
) +
bx
2
(
⌊
t
⌋
) =
aF
3
(
x
1
)(
t
) +
bF
3
(
x
2
)(
t
)
.
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 Fall '08
 Ayazifar
 Electrical Engineering, Fourier Series, Cos, Trigraph, Yorkshire Television, Bell X1

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