Homework 3 Solution (ECE220  Fall 2007)
1.
(40 points) A bit more on transformations
(a) (10 points) Suppose you are betting at the roulette in Las Vegas  red or black. You start
with
y
[0] = $50 and every time you win you get 2
x
[
n
] of your original bet
x
[
n
] while if you
loose, you subtract the amount
x
[
n
] from your funds. Your funds amount is the output,
the game is the system and the bet you make is the input. To describe this mathematically
let
h
[
n
] =
‰
2
,
if you win at game
n
;

1
,
if you loose at game
n
.
Clearly the money you have at the
n
th game iteration is:
y
[
n
] =
y
[
n

1] +
h
[
n
]
x
[
n
]
Prove that this transformation is linear and causal.
Solution:
y
[
n
] =
y
[
n

1] +
h
[
n
]
x
[
n
] =
y
[0] +
n

1
X
k
=1
h
[
k
]
x
[
k
] +
h
[
n
]
x
[
n
] =
y
[0] +
n
X
k
=1
h
[
k
]
x
[
k
]
From that, the output at time
n
clearly only depends on the input at time
n
or before, so
the system is causal.
By strict definition the system is not linear, but with the hint posted on blackboard
αy
1
[0] +
βy
2
[0] =
y
[0] = 50.
The system can be treated as linear.
Basically what we
splitting the initial condition (the nonlinear part) between the summed signals.
This
makes for this system because if you were to have two people betting they would each
need a portion of they money to bet with.
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 Fall '05
 JOHNSON
 Signal Processing, LTI system theory, Necessary and sufficient condition

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