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## Stability Revisited

As defined earlier in §5.6 (page ), a
filter is said to be *stable* if its impulse response
decays to 0 as goes to infinity.
In terms of poles and zeros, an irreducible filter transfer function
is stable if and only if all its poles are inside the unit circle in
the plane (as first discussed in §6.8.6). This is because
the transfer function is the *z* transform of the impulse response, and if
there is an observable (non-canceled) pole outside the unit circle,
then there is an exponentially increasing component of the impulse
response. To see this, consider a causal impulse response of the form

This

signal is a damped complex

sinusoid when

. It
oscillates with zero-crossing rate

zeros
per second, and it has an exponentially
decaying amplitude

envelope. If

, then the

amplitude envelope
increases exponentially as

.

The signal
has the *z* transform

where the last step holds for
, which is
true whenever
. Thus, the transfer function consists of a single pole at
, and it exists for .^{9.1}Now consider what happens when we let become greater than 1. The
pole of moves outside the unit circle, and the impulse response
has an exponentially *increasing* amplitude. (Note
.) Thus, the definition of stability is violated. Since the *z* transform
exists only for
, we see that implies that the
*z* transform no longer exists on the unit circle, so that the frequency
response becomes undefined!

The above one-pole analysis
shows that a one-pole filter is stable if
and only if its pole is inside the unit circle. In the case of an
arbitrary transfer function, inspection of its partial fraction
expansion (§6.8) shows that the behavior near any pole
approaches that of a one-pole filter consisting of only that
pole. Therefore, *all* poles must be inside the unit circle for
stability.

In summary, we can state the following:

Isolated poles

*on* the unit circle may be called

*marginally stable*. The impulse response component
corresponding to a single pole on the unit circle never decays, but
neither does it grow.

^{9.2} In

physical modeling applications, marginally stable
poles occur often in

*lossless* systems, such as

ideal vibrating string models
[

86].

**Subsections**

**Previous:** Graphical Phase Response Calculation**Next:** Computing Reflection Coefficients to
Check Filter Stability

**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.