Stability Condition in Terms of the pole locations The partial sum is computed for increasing values of k until the difference between a series of consecutive values of sk is smaller than some arbitrarily chosen small number, which is typically 10 For a transfer function of very high order this approach may not be satisfactory An alternate, easy-to-test, stability condition is developed next Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 11 Stability Condition in Terms of the Pole Locations • The partial sum is computed for increasing values of K until the difference between a series of consecutive values of is smaller than some arbitrarily chosen small number, which is typically • For a transfer function of very high order this approach may not be satisfactory • An alternate, easy-to-test, stability condition is developed next S K 6 10−
Stability Condition in Terms of the pole locations Consider the causal iir digital filter with a rational transfer function H2 given by H(2)=2kW N =0 Its impulse response hn is a right-Sided sequence The roc of H(z)is exterior to a circle going through the pole farthest from 2=0 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 12 Stability Condition in Terms of the Pole Locations • Consider the causal IIR digital filter with a rational transfer function H(z) given by • Its impulse response {h[n]} is a right-sided sequence • The ROC of H(z) is exterior to a circle going through the pole farthest from z = 0 = − = − = N k k k M k k k d z p z H z 0 0 ( )
Stability Condition in Terms of the pole locations But stability requires that hn be absolutely summable This in turn implies that the dtFtH(eJo) of thin exists Now, if the roc of the z-transform H(z) includes the unit circle. then H(e0)=H(z)2=2 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 13 Stability Condition in Terms of the Pole Locations • But stability requires that {h[n]} be absolutely summable • This in turn implies that the DTFT of {h[n]} exists • Now, if the ROC of the z-transform H(z) includes the unit circle, then = = j z e j H(e ) H(z) ( ) j H e
Stability Condition in Terms of the pole locations Conclusion: All poles of a causal stable transfer function H(z) must be strictly inside the unit circle The stability region(shown shaded )in the z-plane is shown below Im stability region Re unit circle Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 14 Stability Condition in Terms of the Pole Locations • Conclusion: All poles of a causal stable transfer function H(z) must be strictly inside the unit circle • The stability region (shown shaded) in the z-plane is shown below 1 Re z j Im z − 1 − j j unit circle stability region
Stability Condition in Terms of the pole locations Example The factored form of H(z)= 1-0.845z-1+0.850586z IS H(2 (1-0.902=-)(1-0.943 which has a real pole at z=0902 and a real pole at z=0.943 Since both poles are inside the unit circle H(z)is biBO stable 15 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 15 Stability Condition in Terms of the Pole Locations • Example - The factored form of is which has a real pole at z = 0.902 and a real pole at z = 0.943 • Since both poles are inside the unit circle, H(z) is BIBO stable 1 2 1 0.845 0.850586 1 ( ) − − − + = z z H z (1 0.902 )(1 0.943 ) 1 ( ) −1 −1 − − = z z H z