58 lines
2.3 KiB
Markdown
58 lines
2.3 KiB
Markdown
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# 20240221150711 極、零点とフィルタの特性
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#filter #analog #electronics
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## 周波数特性と極、零点 一般的な話
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一般にフィルタ(というか古典制御の伝達関数)は、極と零点をつかって、以下のように書ける。
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$$
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\begin{aligned}
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H(s) &= \frac{b_{ms^{m}+}b_{m-1}s^{m-1} + \dots + b_0 }{a_{n}s^{n}+ a_{n-1}s^{n-1} + \dots a_{0}}\\
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\\
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&= k\frac{(s-z_1)(s-z_{2)\dots}(s-z_m)}{(s-p_1)(s-p_2)\dots(s-p_n)}
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\end{aligned} \tag{1}
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$$
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このとき、任意の複素平面上の点$q$における振幅、位相特性は
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$$
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\begin{aligned}
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|H(s)|_{s=q} &= k\frac{(q-z_{1})(q-z_{2)} \dots (q-z_m)}
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{(q-p_1)(q-p_2)\dots(q-p_n)}\\
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\\
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&= k\frac{r_1e^{j\phi_{1}} \cdot r_2e^{j\phi_{2}} \cdots r_me^{j\phi_{m}}}
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{d_1e^{j\theta_{1}} \cdot d_2e^{j\theta_{2}} \cdots d_ne^{j\theta_{n}}}
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\end{aligned} \tag{2}
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$$
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(2) の2番目は極座標表記。
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このとき、ゲイン$|H(s)|$ は
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$$
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\begin{aligned}
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|H(s)|_{s=q} & = k\frac{r_{1}r_{2}\cdots r_m}{d_{1}d_{2}\cdots d_n}\\
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\\
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&= \frac{\prod_{t=1}^{m}r_{t}}{\prod_{t=1}^{n}d_t}
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\end{aligned} \tag{3}
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$$
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位相 $\arg H(s)$ は
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$$
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\begin{aligned}
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\arg H(s) _{s=q} & = k\frac{\phi_{1}+ \phi_{2} + \cdots + \phi_{m}}
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{\theta_{1}+ \theta_{2} + \cdots \theta_{n}}\\
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\\
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&= \frac{\sum\limits_{t=1}^{m}\phi_{t}}{\sum\limits_{t=1}^{n}\theta_t}
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\end{aligned} \tag{4}
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$$
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と表せる。
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周波数特性を気にしたい場合、 点$q$として、周波数$j\omega$ が0から$+\infty$ まで移動した時の振幅と位相を求めれば良い。つまり、虚軸を原点から上方向に進んでいく直線上で上記を求めれば良い。
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## 2次の伝達関数の場合の例
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## cf.
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- [Pole-Zero Analysis | Introduction to Digital Filters](https://www.dsprelated.com/freebooks/filters/Pole_Zero_Analysis_I.html)
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- [321 08 Poles, Zeros, and Frequency Response - YouTube](https://youtu.be/W2ubAg8RTd8?t=863)
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- [Understanding Poles and Zeros in Transfer Functions - Technical Articles](https://www.allaboutcircuits.com/technical-articles/understanding-poles-and-zeros-in-transfer-functions/)
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- [Site Unreachable](https://www.analog.com/jp/resources/technical-articles/a-filter-primer.html)
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- [Lecture 23 Filters Hung-yi Lee. - ppt download](https://slideplayer.com/slide/3990721/)
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- [Pole-Zero Analysis | Introduction to Digital Filters](https://www.dsprelated.com/freebooks/filters/Pole_Zero_Analysis_I.html)
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