Chebyshev order (0.5 dB, 20 dB, 2000 rad/s)

Derivation of the order relation

The Chebyshev low-pass magnitude response (normalized so that the passband edge is $\Omega = \omega/\omega_p = 1$) is

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

where $C_n(\Omega)$ is the Chebyshev polynomial of order $n$:

$$C_n(\Omega) = \begin{cases} \cos(n\cos^{-1}\Omega), & |\Omega| \le 1 \\ \cosh(n\cosh^{-1}\Omega), & |\Omega| > 1 \end{cases}$$

The attenuation in dB is $\alpha(\Omega) = 10\log_{10}\left[1+\varepsilon^2 C_n^2(\Omega)\right]$.

Step 1: ripple factor from the passband edge. At $\Omega = 1$, $C_n(1) = 1$ for every $n$, so the attenuation is exactly $\alpha_{max}$:

$$\alpha_{max} = 10\log_{10}(1+\varepsilon^2) \;\Rightarrow\; \varepsilon^2 = 10^{0.1\alpha_{max}} - 1$$

Step 2: stopband condition. At $\Omega_s = \omega_s/\omega_p\ (>1)$ the attenuation must be at least $\alpha_{min}$:

$$10\log_{10}\left[1+\varepsilon^2 C_n^2(\Omega_s)\right] \ge \alpha_{min} \;\Rightarrow\; C_n^2(\Omega_s) \ge \frac{10^{0.1\alpha_{min}}-1}{\varepsilon^2}$$

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

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

The order is the next whole number above this value. (Useful identity: $\cosh^{-1}x = \ln\left(x+\sqrt{x^2-1}\right)$.)

Required order

Given $\alpha_{max} = 0.5$ dB, $\alpha_{min} = 20$ dB, $\frac{\omega_s}{\omega_p} = \frac{2000}{1000} = 2$.

$$\begin{aligned} 10^{0.1\alpha_{max}} - 1 &= 10^{0.05} - 1 = 0.12202 \\ 10^{0.1\alpha_{min}} - 1 &= 10^{2} - 1 = 99.0000 \\ \sqrt{\frac{99.0000}{0.12202}} &= 28.4843 \\ \cosh^{-1}(28.4843) &= 4.0422 \\ \cosh^{-1}(2) &= 1.3170 \\ n &\ge \frac{4.0422}{1.3170} = 3.069 \end{aligned}$$

Answer: n = 4 (next integer above 3.069).

Note that $n = 3.07$ is only slightly above 3, but the order must be rounded up; a 3rd-order filter would give only about 19.2 dB at 2000 rad/s, which fails the 20 dB requirement. (For comparison, a Butterworth filter for the same specification needs $n = 5$.)