For Bounded Input, Bounded Output stability (BIBO), there is a very nice interpretation. All it means is that if for any finite signal the system produces a finite output signal, then the system (cellphone, radio, lamp, etc...) is BIBO stable.

But what about asymptotic stability? Asymptotic stability states that without an input signal, any initial internal state of the system will lead to the internal state decay to zero.

Can someone come up with an example that illustrate this effect?

More interestingly is the case that a system can be BIBO stable without being asymptotically stable. Are there any realistic system that can achieve this?



IIR digital filters can be BIBO stable but not asymptotically stable.

When quantization is neglected, IIR filters can be designed to be stable in both a BIBO sense and asymptotically. However, when a Quantizer is introduced into the feedback path the IIR filter can exhibit small scale limit cycles, and is therefore not asymptotically stable. (A.K.A Zero-input small scale limit cycles, which makes clear these are happening even though the input is zero). This limit cycle will persist for infinite time. It may also be a D.C. value instead of an oscillation.

This is illustrated below (source). The impulse response (dotted line) is both BIBO and asymptotically stable without a rounding quantizer. Including the rounding quantizer results in a limit cycle (squares).

Whether IIR filters will exhibit a small scale limit cycle such as this is governed by several factors, including: the location of the filter poles (closer to unit circle is worse); the type of quantizer; the location and number of quantizers (i.e. 1Q or 2Q). This limit cycling is caused by an apparent movement of the poles onto the unit circle by virtue of the inclusion of the quantizer.

Incidentally, any quantized feedback system exhibits this phenomenon, and it is also commonly observed in Delta-Sigma ADCs and DACs. For Delta-Sigmas in general, DC input values translate into a "dither" between several discrete output values, causing residual tones in the output.

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