华南理工大学电子与信息学院：《数字信号与处理》（英文版）Lecture 11 Stability Condition in Terms of the pole locations

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 . . ( )

Stability Condition in Terms of the pole locations 三 1020304050 070 Time index n as can be seen from the above plot the impulse response coefficient h[n] decays rapidly to zero value as n increases Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 6 Stability Condition in Terms of the Pole Locations • As can be seen from the above plot, the impulse response coefficient h[n] decays rapidly to zero value as n increases 0 10 20 30 40 50 60 70 0 2 4 6 Time index n Amplitude h[n]

Stability Condition in Terms of the pole locations The absolute summability condition of hn is satisfied Hence, H(zis a stable transfer function Now. consider the case when the transfer function coefficients are rounded to values with 2 digits after the decimal point A( 1-1.85z-1+0.85z 2 Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 7 Stability Condition in Terms of the Pole Locations • The absolute summability condition of h[n] is satisfied • Hence, H(z) is a stable transfer function • Now, consider the case when the transfer function coefficients are rounded to values with 2 digits after the decimal point: 1 2 1 1 85 0 85 1 − − − + = z z H z . . ( ) ^

Stability Condition in Terms of the pole locations 入 a plot of the impulse response of hn] is shown below 3x06010-1600309040004000P4A A 3504AAMAPMMAMMAED h[n]6 总4 02030405060 8 Time index n Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 8 Stability Condition in Terms of the Pole Locations • A plot of the impulse response of is shown below h[n] ^ 0 10 20 30 40 50 60 70 0 2 4 6 Time index n Amplitude h[n] ^

Stability Condition in Terms of the pole locations In this case, the impulse response coefficient hn Increases rapi idly to a constant value as n increases Hence the absolute summability condition of is violated Thus, H(z)is an unstable transfer function Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 9 Stability Condition in Terms of the Pole Locations • In this case, the impulse response coefficient increases rapidly to a constant value as n increases • Hence, the absolute summability condition of is violated • Thus, is an unstable transfer function h[n] ^ H(z) ^

Stability Condition in Terms of the pole locations The stability testing of a IiR transfer function is therefore an important problem In most cases it is difficult to compute the infinite sum n<oo n=-0 For a causal iir transfer function the sum s can be computed approximately as K SK=∑=01n Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 10 Stability Condition in Terms of the Pole Locations • The stability testing of a IIR transfer function is therefore an important problem • In most cases it is difficult to compute the infinite sum • For a causal IIR transfer function, the sum S can be computed approximately as =     n=− S h[n]  − = = 1 0 K n S h[n] K