Discrete–Time Linear, Time Invariant Systems
and z–Transforms
Linear, time invariant systems
“Continuous–time, linear, time invariant systems” refer to circuits or processors that take one input
signal and produce one output signal with the following properties.
◦
Both the input and output are continuous–time signals.
◦
The system is linear. This means that if the input signals
x
1
(
t
) and
x
2
(
t
) generate the output signals
y
1
(
t
) and
y
2
(
t
), respectively, and if
a
1
and
a
2
are constants, then the input signal
a
1
x
1
(
t
) +
a
2
x
2
(
t
)
generates the output signal
a
1
y
1
(
t
) +
a
2
y
2
(
t
).
◦
The system is time invariant. This means that if the input signal
x
(
t
) generates the output signal
y
(
t
),
then, for each real number
s
, the time shifted input signal ˜
x
(
t
) =
x
(
t

s
) generates the time shifted
output signal ˜
y
(
t
) =
y
(
t

s
).
Example 1
A simple example of a continuous–time, linear, time invariant system is the RC lowpass filter
that is used, for example in amplifiers, to suppress the high frequency parts of signals. This is the electrical
circuit
+

x
(
t
)
R
C
i
(
t
)
+

y
(
t
)
with the voltage source
x
(
t
) viewed as the input signal and the voltage
y
(
t
) across the capacitor viewed as
the output signal. If
i
(
t
) denotes the current in the circuit, the voltage across the resistor is
Ri
(
t
). If
q
(
t
)
denotes the charge on the capacitor, then the voltage across the capacitor is
y
(
t
) =
q
(
t
)
C
. So by Kirchhoff’s
voltage law,
x
(
t
) =
Ri
(
t
) +
y
(
t
)
Since
i
(
t
) =
dq
dt
(
t
) =
C
dy
dt
(
t
), we have that
RC
dy
dt
(
t
) +
y
(
t
) =
x
(
t
)
⇒
dy
dt
(
t
) +
1
RC
y
(
t
) =
1
RC
x
(
t
)
This first order constant coefficient linear ODE is easily solved by multiplying it by the integrating factor
(1)
e
t/RC
.
e
t/RC dy
dt
(
t
) +
1
RC
e
t/RC
y
(
t
) =
1
RC
x
(
t
)
e
t/RC
⇒
d
dt
(
e
t/RC
y
(
t
)
)
=
1
RC
x
(
t
)
e
t/RC
Change
t
to
τ
in this equation and integrate both sides from
τ
=
∞
to
τ
=
t
. Assuming that
e
t/RC
y
(
t
)
tends to zero as
t
→ ∞
,
e
t/RC
y
(
t
) =
Z
t
∞
1
RC
x
(
τ
)
e
τ/RC
dτ
⇒
y
(
t
) =
Z
t
∞
1
RC
x
(
τ
)
e
(
τ

t
)
/RC
dτ
For each possible input signal
x
(
t
), this determines the corresponding output signal
y
(
t
).
This rule is
obviously linear. To see that it is also time invariant, observe that the output signal that corresponds to the
(1)
In general, multiplying the equation
y
0
(
t
) +
p
(
t
)
y
(
t
) =
g
(
t
) by the integrating factor
e
R
p
(
t
)
dt
turns
the left hand side into a perfect derivative.
April 4, 2007
Discrete–Time Linear, Time Invariant Systems and z–Transforms
1
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input signal
x
(
t

s
) is
Z
t
∞
1
RC
x
(
τ

s
)
e
(
τ

t
)
/RC
dτ
τ
0
=
τ

s
=
Z
t

s
∞
1
RC
x
(
τ
0
)
e
(
τ
0
+
s

t
)
/RC
dτ
0
=
y
(
t

s
)
as desired.
“Discrete–time, linear, time invariant systems” refer to linear, time invariant circuits or processors
that take one discrete–time input signal and produce one discrete–time output signal.
Example 2
Let
x
[
n
] denote the net deposit (i.e. the sum of all deposits minus the sum of all withdrawals)
to a bank account during month number
n
and let
y
[
n
] denote the balance in the account at the end of
month number
n
. We want to think of
x
[
n
] as the known input to the system. We wish to determine the
output
y
[
n
]. We start by setting up a equation that tells us about the time evolution of
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 Spring '10
 YILMAZ
 Digital Signal Processing, LTI system theory, Impulse response, Time invariant systems

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