Stability Condition in Terms of the pole locations A causal Lti digital filter is BiBo stable if and only if its impulse response h[n]is absolutely summable. i.e S=∑hm]< 1=-00 We now develop a stability condition in terms of the pole locations of the transfer function H(z) Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 1 Stability Condition in Terms of the Pole Locations • A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., • We now develop a stability condition in terms of the pole locations of the transfer function H(z) = n=− S h[n]
Stability Condition in Terms of the pole locations The roc of the z-transform H(z)of the impulse response sequence hn] is defined by values of z-r for which hin]r"is absolutely summable Thus, if the roc includes the unit circle z 1. then the digital filter is stable and vice versa Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 2 Stability Condition in Terms of the Pole Locations • The ROC of the z-transform H(z) of the impulse response sequence h[n] is defined by values of |z| = r for which is absolutely summable • Thus, if the ROC includes the unit circle |z| = 1, then the digital filter is stable, and vice versa n h n r − [ ]
Stability Condition in Terms of the pole locations In addition for a stable and causal digital filter for which h[n]is a right-sided sequence, the roc will include the unit circle and entire z-plane including the point 2=0 An fir digital filter with bounded impulse response is always stable Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 3 Stability Condition in Terms of the Pole Locations • In addition, for a stable and causal digital filter for which h[n] is a right-sided sequence, the ROC will include the unit circle and entire z-plane including the point • An FIR digital filter with bounded impulse response is always stable z =
Stability Condition in Terms of the pole locations On the other hand, an IiR filter may be unstable if not designed properly In addition an originally stable iir filter characterized by infinite precision coefficients may become unstable when coefficients get quantized due to implementation Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 4 Stability Condition in Terms of the Pole Locations • On the other hand, an IIR filter may be unstable if not designed properly • In addition, an originally stable IIR filter characterized by infinite precision coefficients may become unstable when coefficients get quantized due to implementation
Stability Condition in Terms of the pole locations Example- Consider the causal iir transfer function H(z)= 1-1.845z-1+0.850586z 2 The plot of the impulse response coefficients is shown on the next slide Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 5 Stability Condition in Terms of the Pole Locations • Example - Consider the causal IIR transfer function • The plot of the impulse response coefficients is shown on the next slide 1 2 1 1 845 0 850586 1 − − − + = z z H z . . ( )