Chebyshev poles lie on an ellipse

The poles of a Chebyshev low-pass filter lie on an ellipse whose semi-minor axis is $\sinh a$ (along the real axis) and whose semi-major axis is $\cosh a$ (along the imaginary axis), where $a = \frac{1}{n}\sinh^{-1}(1/\varepsilon)$.

Finding the poles

The normalized Chebyshev response is $|T(j\omega)|^2 = \dfrac{1}{1+\varepsilon^2 C_n^2(\omega)}$ with $C_n(\omega) = \cos(n\cos^{-1}\omega)$. Replacing $\omega$ by $s/j$, the poles satisfy

$$1 + \varepsilon^2 C_n^2(s/j) = 0 \;\Rightarrow\; \cos\left[n\cos^{-1}(s/j)\right] = \pm\frac{j}{\varepsilon}$$

Let $\cos^{-1}(s/j) = u + jv$. Then

$$\cos(nu)\cosh(nv) - j\sin(nu)\sinh(nv) = \pm\frac{j}{\varepsilon}$$

• Real part: $\cos(nu)\cosh(nv) = 0$. Since $\cosh(nv) \ge 1$, $\cos(nu) = 0$, so $u_k = \dfrac{(2k-1)\pi}{2n}$, $k = 1,\dots,2n$.
• Imaginary part: with $\sin(nu_k) = \pm1$, $\sinh(nv) = 1/\varepsilon$, so $v = a = \dfrac{1}{n}\sinh^{-1}\dfrac{1}{\varepsilon}$.

Now $s = j\cos(u_k + ja) = j[\cos u_k\cosh a - j\sin u_k\sinh a]$:

$$s_k = \sigma_k + j\omega_k = \sin u_k \sinh a + j\cos u_k \cosh a$$

The left-half-plane poles (for a stable filter) are

$$\sigma_k = -\sinh a\,\sin u_k, \qquad \omega_k = \cosh a\,\cos u_k, \qquad k = 1,\dots,n$$

Showing the locus is an ellipse

$$\left(\frac{\sigma_k}{\sinh a}\right)^2 + \left(\frac{\omega_k}{\cosh a}\right)^2 = \sin^2u_k + \cos^2u_k = 1$$

This is an ellipse centred at the origin.

Major and minor axes

Since $\cosh^2 a - \sinh^2 a = 1$, the foci are at $s = \pm j1$.

              jw
              |  cosh a
        x . . + . . o
      .       |       .
 ----x--------+--------o---- sigma
   -sinh a    |      sinh a
      .       |       .
        x . . + . . o
              |
   x = LHP poles used, o = RHP mirror

Geometric construction: draw two circles of radius $\sinh a$ and $\cosh a$; draw radial lines at angles $u_k$ from the $j\omega$ axis. Each pole takes its real part from the small circle and its imaginary part from the large circle (compare: Butterworth poles lie on the unit circle).