Approximation and inverse Chebyshev order (100/140 krad/s)

Approximation in filter design

Approximation is the step of finding a realizable transfer function $T(s)$ (a ratio of polynomials with real coefficients and poles in the left half plane) whose magnitude (or phase) stays within the given specifications ($\alpha_{max}$ up to $\omega_p$, $\alpha_{min}$ beyond $\omega_s$). An ideal brick-wall response cannot be built, so it is approximated by standard functions such as Butterworth, Chebyshev, inverse Chebyshev, elliptic and Bessel-Thomson.

Order of the inverse Chebyshev low-pass filter

The inverse Chebyshev (Chebyshev Type II) response, normalized to the stopband edge ($\Omega = \omega/\omega_s$, stopband begins at $\Omega=1$), is

$$|T(j\Omega)|^2 = \frac{\varepsilon^2 C_n^2(1/\Omega)}{1+\varepsilon^2 C_n^2(1/\Omega)}$$

It is obtained by taking a Chebyshev response, subtracting it from 1 (turning it into a high-pass with equiripple stopband) and replacing $\Omega$ by $1/\Omega$.

Step 1: stopband edge. At $\Omega = 1$, $C_n(1) = 1$:

$$\alpha_{min} = 10\log_{10}\left(1+\frac{1}{\varepsilon^2}\right) \;\Rightarrow\; \varepsilon^2 = \frac{1}{10^{0.1\alpha_{min}}-1}$$

Step 2: passband edge. At $\Omega_p = \omega_p/\omega_s$, $1/\Omega_p = \omega_s/\omega_p > 1$, and the attenuation must not exceed $\alpha_{max}$:

$$\alpha_{max} \ge 10\log_{10}\left(1+\frac{1}{\varepsilon^2 C_n^2(\omega_s/\omega_p)}\right) \;\Rightarrow\; C_n^2\!\left(\frac{\omega_s}{\omega_p}\right) \ge \frac{1}{\varepsilon^2(10^{0.1\alpha_{max}}-1)} = \frac{10^{0.1\alpha_{min}}-1}{10^{0.1\alpha_{max}}-1}$$

Step 3: solve. With $C_n(x) = \cosh(n\cosh^{-1}x)$ for $x>1$:

$$n \ge \frac{\cosh^{-1}\sqrt{\dfrac{10^{0.1\alpha_{min}}-1}{10^{0.1\alpha_{max}}-1}}}{\cosh^{-1}\left(\dfrac{\omega_s}{\omega_p}\right)}$$

This is the same expression as for the Chebyshev filter: for the same specification both need the same order.

Order for the given specification

$\omega_p = 100$ krad/s, $\omega_s = 140$ krad/s, $\alpha_{max} = 0.25$ dB, $\alpha_{min} = 18$ dB, so $\omega_s/\omega_p = 1.4$.

$$\begin{aligned} 10^{0.025}-1 &= 0.05925 \\ 10^{1.8}-1 &= 62.0957 \\ \sqrt{62.0957/0.05925} &= 32.372, \quad \cosh^{-1}(32.372) = 4.1702 \\ \cosh^{-1}(1.4) &= 0.8670 \\ n &\ge \frac{4.1702}{0.8670} = 4.810 \end{aligned}$$

Answer: n = 5. Because the order formula is identical, a Chebyshev (Type I) filter for this specification also needs $n = 5$.