Digital Signal Processing Reference
In-Depth Information
3. time-invariance
A system is causal when it uses only current and/or previous inputs to calculate
the current output. That is, if the equation describing the inputs to the outputs is
given with indexes of n on the left side of the equation, and n, n1, n2, etc., on
the righthand side. Similarly, we say that a signal is causal when it has only zeros
for index values less than zero. In other words, it starts at or after index 0.
Example:
Which of the following lters are causal? (K is positive)
P
K
y[n] =
k=0
b
k
x[nk]
P
K
y[n] =
k=0
b
k
x[n + k]
P
0
k=K
b
k
x[nk]
y[n1] = x[n]
y[n] = 0:1x[n]
y[n] = 0:1x[n] + 0:4x[n1] + 0:8x[n2]
y[n] = 0:1x[n]0:4x[n1]0:8x[n2]
y[n] = 0:1x[n] + 0:4x[n + 1] + 0:8x[n1]
y[n] = 0:1x[n] + 0:4x[n + 1] + 0:8x[n + 2]
Answer:
yes, no, no, no, yes, yes, yes, no, no
y[n] =
When a system has an output expressed as a sum of inputs multiplied by con-
stants, as with FIR lters, then this system is linear. A linear system has scaling and
additivity properties. Scaling means that we get the same result whether we multiply
the input by a constant or the output by that constant, as in Figure 3.13. For a lin-
ear system, y
1
= y
2
. For example, suppose we have a system y[n] = 2x[n] +x[n1].
If we replaced each value of x with ax, then we would calculate 2ax[n] +ax[n1] to
nd our output. This is the same as a(2x[n] + x[n1]), the same as if our output
were multiplied by a, which is what scaling means.
Additivity means that we can add two signals together before applying a linear
lter, or we can apply the linear lter to each signal and add their results, as in
Figure 3.14, i.e., y
3
= System(x
1
+ x
2
) = y
1
+ y
2
. Either way, we end up with the
same answer.
These properties hold true as long as the inputs are multiplied by constants (i.e.,
not multiplied by each other). That is, the system described by y[n] = c
0
x[n] +
c
1
x[n1] + c
2
x[n2] + ::: + c
K
x[nK] is linear (y[n] is the current output, x[n]
