Stability Condition in Terms of the Pole locations 010203040 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 Terns 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 ith 2 digits after the decimal point H(z)= 1-1.85x-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 a plot of the impulse response of hn is shown below 10203040506070 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 Terns of the pole locations In this case the impulse response coefficient hIn] increases rapidly 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 Terns 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 <OO n=-0 For a causal iir transfer function the sum s can be computed approximately as Sx=∑ =0 h[n] 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