How do you analyze the stability of a digital filter?

A digital system is BIBO (bounded-input, bounded-output) stable if all poles of H(z) lie inside the unit circle |z| < 1. Analysis methods: Pole location - find roots of denominator polynomial, check magnitudes. Jury stability test - algebraic test without finding roots (analog of Routh-Hurwitz). For IIR filters, check each second-order section (biquad) independently. FIR filters are always stable (no poles except at origin). Marginally stable if poles on unit circle. Coefficient quantization can move poles, potentially causing instability in fixed-point implementations. Use scaled structures and proper word lengths.