Digital Signal Processing Reference
In-Depth Information
In the following relation,
x
(
t
) denotes the input signal and
y
(
t
) denotes the
output signal of a system.
x(t) : y(t).
(2.59)
We read this notation as “
x
(
t
) produces
y
(
t
)”; it has the same meaning as the block
diagram of Figure 2.32 and the transformation notation
[eq(2.51)]
y(t) = T[x(t)].
The following are the six properties of continuous-time systems:
Memory
A system has memory if its output at time
t
0
, y(t
0
),
depends on input values other than
x(t
0
).
Otherwise, the system is memoryless.
A system with memory is also called a
dynamic system
. An example of a system with
memory is an
integrating amplifier,
described by
t
y(t) = K
L
x(t)dt.
(2.60)
-
q
(See Section 1.2.) The output voltage
y
(
t
) depends on all past values of the input
voltage
x
(
t
), as we can see by examining the limits of integration. A capacitor also
has memory if its current is defined to be the input and its voltage the output:
t
1
C
L
v(t) =
i(t)dt.
-
q
The voltage across the capacitor at time
t
0
depends on the current
i
(
t
) for all time
before Thus, the system has memory.
A memoryless system is also called a
static system
. An example of a memoryless
system is the ideal amplifier defined earlier. With
x
(
t
) as its input and
y
(
t
) as its out-
put, the model of an ideal amplifier with (constant) gain
K
is given by
t
0
.
y(t) = Kx(t)
for all
t
. A second example is resistance, for which
v(t) = Ri(t).
A third example is
a squaring circuit, such that
y(t) = x
2
(t).
(2.61)
Clearly, a system would be memoryless, whereas a second system
has memory, because
y
1
(t) = 5x(t)
y
2
(t) = x(t + 5)
y
2
(t
0
)
depends on the value of
x(t
0
+ 5),
which is five units of time ahead of
t
0
.
Invertibility
A system is said to be invertible if distinct inputs result in distinct outputs.
