Influence of higher order poles on waveform

Jane Medium ： In this paper, the order of poles in the system function is increased , Analysis of the influence of system unit reset and corresponding waveform . Help students understand the influence of pole position on the time-domain characteristics of the system （ stability ） Influence .

key word ： Signals and systems , pole , Order

 

§01 high Order pole

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One 、 Questions mentioned in the Q & A

   today （2022-06-10） In the morning, I was answering questions in the course of signal and system , Some students mentioned the influence of the unit impulse response of the system corresponding to the higher-order poles in the system function .  Usually , The pole of the system function determines the waveform characteristics of the corresponding impulse response ,  For example, whether the index increases or decreases , Oscillation speed and other parameters .  So the order of the poles , Do you have any effect on the time domain waveform ？

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Two 、 Influence of pole order on time domain waveform

   The influence of pole order at several typical positions on time domain waveform is discussed below .

  

1、 The higher order pole at the origin

   For the pole at the origin .   If it is a first-order pole , Corresponding unit step signal .  If the order of poles increases , It corresponds to t Power form of .  This is the relation between the pole order at the origin and the time domain signal .  So we can know , The higher-order pole at the origin causes the signal to show a polynomial increasing trend .

  

2、 Higher order poles on the imaginary axis

   If the pole is on the imaginary axis ,  Then as the order of poles increases ,  The time domain waveform will also change from the original equal amplitude oscillation signal ,  Gradually evolved into the form of increasing amplitude polynomial .  The higher the pole order ,  The higher the order of the polynomial with the corresponding increase in amplitude .  From this point of view , The influence of pole order on the amplitude of waveform is similar .
  

3、 Positive real axis and first quadrant higher order pole

   When the pole is on the positive real axis and the first quadrant ,  The corresponding high-order pole is also in front of the original signal , multiply t The polynomial .  The corresponding signal waveform is based on the original exponential increase ,  The superposition has a polynomial increasing trend .  From the outside , Only the value of waveform at zero point is changed ,  It's from 0 Beginning to evolve .

  

The negative real axis and the pole of the second quadrant

   When the pole is on the negative real axis and in the second quadrant ,  For the first order pole , The corresponding signal amplitudes are all exponentially attenuated signals .  When the order of poles increases ,  The amplitude of the signal will be more polynomial .  But because of the exponential decay in t Towards infinity , Will be much smaller than a polynomial ,  So the amplitude of the final signal decays exponentially .  From the outside , The waveform of this part shows , First increase when the index decreases .  Here, the corresponding higher-order pole waveform on the negative real axis is given .  For the third 、 Fourth quadrant high order pole case , And second , The first quadrant is similar .

  

3、 ... and 、 Pole order pairs Waveform effect

   The influence of the order of poles at different positions on the waveform is discussed above .  On the whole , Pole order increases , The polynomial part will be added to the signal amplitude , Showing an increasing trend .  This change has the greatest impact on the poles on the imaginary axis ,  This makes it change from a constant amplitude signal to a polynomial signal .  For the poles on the right and left of the imaginary axis ,  The order of the poles does not change t Towards infinity ,  Signal changes .

  

 

※ total    junction ※

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   In this paper, the order of poles in the system function is increased , Analysis of the influence of system unit reset and corresponding waveform . Help students understand the influence of pole position on the time-domain characteristics of the system （ stability ） Influence .

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