## Filter Stability

In this context, we may say that an impulse response ``approaches zero'' by definition if there exists a finite integer , and real numbers and , such that for all . In other terms, the impulse response is

Definition.An LTI filter is said to bestableif the impulse response approaches zero as goes to infinity.

*asymptotically bounded*by a

*decaying*exponential.

Every finite-order nonrecursive filter is stable. Only the feedback
coefficients in Eq.(5.1) can cause instability. Filter
stability will be discussed further in §8.4 after poles
and zeros have been introduced. Suffice it to say for now that, for
stability, the feedback coefficients must be restricted so that the
feedback gain is less than 1 at every frequency. (We'll learn in
§8.4 that stability is guaranteed when all filter poles
have magnitude less than 1.) In practice, the stability of a
recursive filter is usually checked by computing the filter
*reflection coefficients*, as described in §8.4.1.

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Impulse Response Example

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Impulse-Response Representation