Digital Signal Processing Reference
In-Depth Information
those in the examples above. Once the steady-state value has been found, check it against the formula's
prediction. In this case, (for
p
= 0.9*j), both methods produce, after a certain number of samples of output,
the steady-state value of 0.5525 +
j
0.4973.
Stability
Figure 4.32 shows a single pole filter with borderline stability. In this case, the magnitude of the pole is
exactly 1.0, and the impulse response does not decay away. This is analogous to an oscillator, in which a
small initial disturbance creates a continuous output which does not decay away. The seriousness of the
situation can be seen by using a unit step as the test signal. Figure 4.33 shows the result: a ramp which
theoretically would simply continue to increase to infinity if the filter were allowed to run forever.
1
0
1
−1
0
0
10
20
30
40
50
60
−1
(b) Sample Number of Real Output
1
0
20
40
60
(a) Sample Number of Input
0
+
−1
Input (Real)
0
10
20
30
40
50
60
(c) Sample Number of Imaginary Output
1
1
1
1
0.5
0
1
−0.5
−1
−1
0
1
Magnitude
Angle(Degrees)
(d) Real
Figure 4.32:
(a) Input sequence, a unit impulse sequence; (b) Real part of output sequence, a unit step;
(c) Imaginary part of output sequence, identically zero; (d) The pole, plotted in the complex plane.
An accumulator is a digital register having a feedback arrangement that adds the output of the
register to the current input, which is then stored in the register. The input signal is thus accumulated or
integrated. The single pole IIR with pole value equal to 1.0 functions as an
Integrator
,
which we see is
not a stable system.
As another demonstration of the visual effect of instability, let's feed a chirp into a filter having a
magnitude 1.0 pole (in this case at
π/
4 radians, or 45
◦
). Figure 4.34 shows the result: the filter ”rings,”
that is to say, once the chirp frequency gets near the pole's resonant frequency, the output begins to
