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Filter Stability

Definition. An LTI filter is said to be stable if the impulse response $ h(n)$ approaches zero as $ n$ goes to infinity.
In this context, we may say that an impulse response ``approaches zero'' by definition if there exists a finite integer $ n_f$, and real numbers $ A\geq 0$ and $ \alpha>0$, such that $ \left\vert h(n)\right\vert<A\exp(-\alpha
n)$ for all $ n\geq n_f$. In other terms, the impulse response is asymptotically bounded by a decaying exponential.

Every finite-order nonrecursive filter is stable. Only the feedback coefficients $ a_i$ 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