2.5: The Chebyshev Lowpass Approximation
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The maximally flat approximation to the ideal lowpass filter response is best near the origin but not so good near the band edge. Chebyshev filters have better responses near the band edge, with lower insertion loss near the edges, but at the cost of ripples in the passband. Example reflection and transmission responses are shown in Figure 2.4.2 for a seventh-order and a sixth-order Chebyshev lowpass filter.
2.5.1 Chebyshev Filter Design
The general form of the Chebyshev transmission coefficient is
\[\label{eq:1}|T(s)|^{2}=\frac{1}{1+\varepsilon^{2}|K(s)|^{2}} \]
where \(\varepsilon\) is the ripple factor and defines the passband ripple (PBR):
\[\label{eq:2}\text{PBR}=1+\varepsilon^{2},\quad\text{or in decibels}\quad R_{\text{dB}}=\text{PBR}|_{\text{dB}}=10\log (1+\varepsilon^{2}) \]
The PBR can be seen in the transmission response, \(|T(s)|^{2}\), in Figure 2.4.2. In the passband the peaks of the lossless filter response have \(|T(s)|^{2} = 1\) and the minimums of the ripple response all have \(|T(s)|^{2} = 1/(1 +\varepsilon^{2}) = 1/\text{PBR}\). Consequently Chebyshev filters are also known as equiripple all-pole lowpass filters. Also note that the corner radian frequency, \(\omega = 1\) for the lowpass filter prototype, has a transmission response (i.e., insertion loss \(\text{IL}\)) of \(|T(s)|^{2} = 1/(1 + \varepsilon^{2})\), whereas the Butterworth transmission response was at half power at the corner frequency. For the Chebyshev filter, the insertion loss at the corner frequency is the ripple:
\[\label{eq:3}\text{IL}=1R_{\text{dB}}=10\log(1+\varepsilon^{2}) \]
For the \(n\)th-order Chebyshev (lowpass filter) approximation, the square of the characteristic function is
\[\label{eq:4}|K_{n}(\omega)|^{2}=\left\{\begin{array}{ll}{\cos^{2}[n\:\cos^{-1}(\omega)]}&{-1\leq\omega\leq 1} \\ {\cosh^{2}[n\cosh^{-1}(|\omega|)]}&{\omega\leq -1,\:\omega\geq 1}\end{array}\right. \]
which can be expressed as a polynomial. For example, with \(n = 3\),
\[\label{eq:5}K_{3}(\omega)=4\omega^{3}-3\omega,\quad\text{for all }\omega \]
(This equivalence was derived by Pafnuty Chebyshev.) It is surprising that the trigonometric expression has such a simple polynomial equivalence. From Equation (2.2.11) the transmission coefficient is (for \(−1 ≤ \omega ≤ 1\))
\[\label{eq:6}|T(\omega)|^{2}=\frac{1}{1+\varepsilon^{2}\cos^{2}[n\cos^{-1}(\omega)]} \]
and the reflection coefficient is
\[\label{eq:7}|\Gamma_{1}(\omega)|^{2}=\frac{\varepsilon^{2}\cos^{2}[n\cos^{-1}(\omega)]}{1+\varepsilon^{2}\cos^{2}[n\cos^{-1}(\omega)]} \]
Factorizing the denominator of either Equation \(\eqref{eq:6}\) or Equation \(\eqref{eq:7}\) yields the following roots (of the denominators of \(\Gamma_{1}(s)\) and \(T(s)\)):
\[\begin{align} s_{i}&=\sin\left[\frac{(2i-1)\pi}{2n}\right]\sinh\left[\frac{1}{n}\sinh^{-1}\left(\frac{1}{\varepsilon}\right)\right] \nonumber \\ \label{eq:8}&\quad +\jmath\cos\left[\frac{(2i-1)\pi}{2n}\right]\cosh\left[\frac{1}{n}\sinh^{-1}\left(\frac{1}{\varepsilon}\right)\right]\quad i=1,2,\ldots ,n\end{align} \]
The roots of the numerator of \(\Gamma_{1}(s)\) in the \(s\) plane are
\[\label{eq:9}s_{k}=\jmath\cos\frac{(2k-1)\pi}{2n}\quad k=1,2,\ldots ,n \]
Equations \(\eqref{eq:8}\) and \(\eqref{eq:9}\) can be used to obtain the reflection and transmission coefficients directly in the \(s\) domain.
2.5.2 Chebyshev Approximation and Recursion
The characteristic function of the Chebyshev approximation can be obtained from the recursion formula,
\[\label{eq:10}K_{n}(\omega)=2\omega K_{n-1}(\omega)-K_{n-2}(\omega) \]
| Response \(1\text{ dB}\) down | ||||
|---|---|---|---|---|
| Ripple | \(n=3\) | \(n=5\) | \(n=7\) | \(n=9\) |
| \(0.01\text{ dB}\) | \(1.564\) | \(1.192\) | \(1.097\) | \(1.058\) |
| \(0.1\text{ dB}\) | \(1.202\) | \(1.071\) | \(1.036\) | \(1.022\) |
| \(0.2\text{ dB}\) | \(1.127\) | \(1.045\) | \(1.023\) | \(1.014\) |
| \(1\text{ dB}\) | \(1.000\) | \(1.000\) | \(1.000\) | \(1.000\) |
| \(3\text{ dB}\) | \(-\) | \(-\) | \(-\) | \(-\) |
| Response \(3\text{ dB}\) down | ||||
| Ripple | \(n=3\) | \(n=5\) | \(n=7\) | \(n=9\) |
| \(0.01\text{ dB}\) | \(1.877\) | \(1.291\) | \(1.145\) | \(1.087\) |
| \(0.1\text{ dB}\) | \(1.389\) | \(1.134\) | \(1.068\) | \(1.041\) |
| \(0.2\text{ dB}\) | \(1.284\) | \(1.099\) | \(1.050\) | \(1.030\) |
| \(1\text{ dB}\) | \(1.095\) | \(1.0338\) | \(1.017\) | \(1.010\) |
| \(3\text{ dB}\) | \(1.000\) | \(1.000\) | \(1.000\) | \(1.000\) |
Table \(\PageIndex{1}\): Radian frequencies at which the transmission response of an \(n\)th Chebyshev filter is down \(1\text{ dB}\) and \(3\text{ dB}\) for a corner frequency \(\omega_{0} = 1\text{ rad/s}\). (Note that \(\omega_{0}\) is the radian frequency at which the transmission response of a Chebyshev filter is down by the ripple, see Figure 2.4.2.)
with
\[\label{eq:11}K_{1}(\omega)=\omega;\qquad K_{2}(\omega)=2\omega^{2}-1 \]
For example, with \(n = 3\),
\[\begin{align}\label{eq:12}K_{3}(\omega)&=2\omega K_{3-1}(\omega)-K_{3-2}(\omega) \\ &=2\omega(2\omega^{2}-1)-\omega=4\omega^{3}-2\omega-\omega=4\omega^{3}-3\omega\end{align} \nonumber \]
2.5.3 Bandwidth Consideration
At the corner frequency of the Chebyshev filter the transmission response is down by the amount of the ripple. This can be seen in Figure 2.4.2. However, the bandwidth of a filter is usually specified in terms of its \(1\text{ dB}\) or \(3\text{ dB}\) bandwidth at which the transmission response is down \(1\text{ dB}\) or \(3\text{ dB}\), respectively, from its maximum response. The radian frequencies at which the responses of various orders of Chebyshev filters are \(1\text{ dB}\) down and \(3\text{ dB}\) down are given in Table \(\PageIndex{1}\). By frequency scaling the Chebyshev response, the filter can be designed for a specified \(1\text{ dB}\) or \(3\text{ dB}\) bandwidth.