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