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## Frequency Response for Small

As can be seen from the pole-zero plot, when , the vector length is minimized to . If is small (), and is dominant in , and is maximized when . In this case, the system behaves like a band-pass filter which peaks at approximately . Due to the second vector , the actual peak frequency is slightly lower than . The precise peak frequency can be found by solving

to be

which is very close to for small . The bandwidth of the passing band is defined as the frequency range

where and are the cutoff frequencies on either side of ( ) at which

or

i.e., when , only half of the power contained in the signal can pass through the filter. To find , consider the two frequencies in the pole-zero plot

When is at either or , the vector in the denominator satisfies

and we have

Here we assumed as the second pole (in the 3rd quadrant) is farther away from both and than in the 2nd quadrant. We can now find approximately the bandwidth as

Next: Higher Order Systems Up: Evaluation of Fourier Transform Previous: Root locus of Second
Ruye Wang 2012-01-28