Question 2014.S3.33: A continuous-time LTI system with system function H(ω) has t......
A continuous-time LTI system with system function H(ω) has the following pole-zero plot. For this system, which of the alternatives is TRUE?
(a) |H(0)| > |H(ω)|;|ω| > 0
(b) |H(ω)| has multiple maxima, at {{\omega}}_{1}\mathop{\mathrm{and}}\;{{\omega}}_{2}
(c) |H(0)| < |H(ω)|; |ω| > 0
(d) |H(ω)| = constant; -∞ < ω < ∞
The given plot can be redrawn as
From the given plot, the transfer function is
H(s)=K \frac{\left(s-z_1\right)\left(s-z_1^*\right)\left(s-z_2\right)\left(s-z_2^*\right)}{\left(s-p_1\right)\left(s-p_1^*\right)\left(s-p_2\right)\left(s-p_2^*\right)}
Put s = jω,
H(j \omega)=K \frac{\sqrt{\omega^2+\left|z_1\right|^2} \sqrt{\omega^2+\left|z_1\right|^2} \sqrt{\omega^2+\left|z_2\right|^2} \sqrt{\omega^2+\left|z_2\right|^2}}{\sqrt{\omega^2+\left|p_1\right|^2} \sqrt{\omega^2+\left|p_1\right|^2} \sqrt{\omega^2+\left|p_2\right|^2} \sqrt{\omega^2+\left|p_2\right|^2}}
From the plot \left|z_1\right|=\left|p_2\right| \text { and }\left|z_2\right|=\left|p_1\right|
Therefore, H(jω) = K.
