Chebyshev Type I Filter Theory

The Prototype Lowpass Filter Concept

Filter design begins by designing a prototype lowpass filter with cutoff angular frequency $\omega_c = 1$.

Suppose there exists a function $f(\omega)$ that is small for $\omega$ in the range from $0$ to $1$ and increases rapidly for $\omega > 1$.

Using this $f(\omega)$, we construct

which yields a lowpass filter amplitude response.

Chebyshev Polynomials

The Chebyshev filter uses the square of the Chebyshev polynomial of the first kind $T_n(\omega)$ as $f(\omega)$.

$T_n(\omega)$ is defined by the following recurrence relation:

The first several Chebyshev polynomials are listed below:

Closed-Form Expressions

Although $T_n(\omega)$ defined by the recurrence is a polynomial of degree $n$ in $\omega$, it can be written in closed form using trigonometric or hyperbolic functions depending on the range of $\omega$:

These are different representations of the same polynomial. At $\omega = 1$, $\cos^{-1}1 = \cosh^{-1}1 = 0$, so both give $T_n(1) = 1$.

Properties of Chebyshev Polynomials

• For $|\omega| \leq 1$, $|T_n(\omega)| \leq 1$ (oscillates between $-1$ and $1$)
• For $|\omega| > 1$, $|T_n(\omega)|$ grows rapidly
• $T_n(1) = 1$ (for all $n$)
• The leading coefficient is $2^{n-1}$ (for $n \geq 1$)
• $T_n(-\omega) = (-1)^n T_n(\omega)$ (parity property)

For $|\omega| > 1$, Eq.\eqref{eq:hyp} gives $T_n(\omega) = \cosh(n\cosh^{-1}\omega)$. Figure 5 shows the shape of $\cosh(\alpha)$, where $\alpha = \cosh^{-1}\omega$ and $\omega = 1$ corresponds to $\alpha = 0$ (Eqs. (4) and (5) both yield $T_n(1) = 1$ at $\omega = 1$). Since $\cosh$ is flat near the origin but grows exponentially away from it, the growth of $T_n(\omega)$ becomes steeper as $n$ increases.

Best Approximation (Minimax Property)

The Chebyshev polynomial $T_n(\omega)$ possesses the following best-approximation property on the interval $[-1, 1]$:

Among all polynomials $p_n(\omega)$ of degree $n$ with leading coefficient $2^{n-1}$, $T_n(\omega)$ is the one that minimizes $\displaystyle\max_{|\omega| \leq 1} |p_n(\omega)|$.

This property provides the mathematical foundation for the equiripple characteristic.

Amplitude-Squared Response

The amplitude-squared response of the Chebyshev Type I filter is defined as:

Here $\varepsilon$ ($0 < \varepsilon < 1$) is the parameter that determines the passband ripple. Let us examine how $T_n^2(\omega)$ in the denominator behaves.

For $|\omega| \leq 1$, $T_n^2(\omega) \in [0, 1]$ oscillates, so $|H(\omega)|^2$ ripples within $[1/(1+\varepsilon^2),\; 1]$. For $|\omega| > 1$, $T_n^2(\omega)$ increases rapidly so $|H(\omega)|^2 \to 0$, and the transition becomes steeper as $n$ increases.

Designing the Ripple Parameter $\varepsilon$

For the ripple to be $-r$ dB at the passband edge ($\omega = 1$), since $T_n(1) = 1$:

Setting this equal to $-r$ dB:

Solving for $\varepsilon$:

Representative ripple values and corresponding $\varepsilon$:

DC Gain

Pole Locations

Eq.\eqref{eq:H2} gives the amplitude response on the imaginary axis ($s = j\omega$, $\omega$ real), but to find the filter poles we must extend this to the entire $s$-plane.

Since the coefficients of the transfer function $H(s)$ are all real, the complex conjugate of $H(j\omega)$ equals $H(-j\omega)$:

On the imaginary axis $s = j\omega$, so we can write $$\omega = s/j = -js$$ This substitution makes the right-hand side a rational function of $s$, defined over the entire $s$-plane:

Since $T_n(\omega)$ is a polynomial of degree $n$ in $\omega$, $T_n^2(-js)$ is a polynomial of degree $2n$ in $s$. Therefore the denominator $1 + \varepsilon^2 T_n^2(-js)$ is also a polynomial of degree $2n$ in $s$, and by the fundamental theorem of algebra, it has exactly $2n$ zeros (counting multiplicity), which are the poles of $H(s)\,H(-s)$. These poles are distributed symmetrically with respect to both the real and imaginary axes, and to obtain a stable filter $H(s)$, we must select only the $n$ poles in the left half-plane.

The pole positions are found by setting the denominator to zero:

That is:

Derivation of Pole Positions

Identity \eqref{eq:Tn-cos} was derived solely from the addition formula for $\cos$ and the recurrence, so it holds for complex $\theta$ as well (defining $\cos\theta = (e^{j\theta} + e^{-j\theta})/2$, the addition formula follows from $e^{a+b} = e^a e^b$, which holds for complex numbers).

We wish to apply this identity to $T_n(-js)$ in Eq.\eqref{eq:pole-condition}. Let $\theta = x + jy$ ($x, y$ real) be the complex number such that $\cos\theta = -js$:

That is:

To expand the left-hand side, we first determine $\cos$ and $\sin$ for purely imaginary arguments. From Euler's formula definitions:

substituting $\theta = j\alpha$ ($\alpha$ real) gives:

Using these results and the addition formula for $\cos$, we expand the left-hand side:

Therefore:

Equating real and imaginary parts:

Since $\cosh(ny) > 0$, we need $\cos(nx) = 0$. Therefore:

In this case $\sin(nx) = \pm 1$, so:

Pole Coordinates

We now find the pole position $s$. Since we set $\cos\theta = -js$:

Expanding with the addition formula for $\cos$ as before:

Substituting into Eq.\eqref{eq:s-from-theta}:

Denoting the poles of $H(s)\,H(-s)$ by $p$:

The poles are distributed on an ellipse in the $s$-plane. The semi-axes (distances from center to edge) of the ellipse are:

• Real-axis direction (semi-minor axis): $a = \sinh(y)$
• Imaginary-axis direction (semi-major axis): $b = \cosh(y)$

where $y = \dfrac{1}{n}\sinh^{-1}\left(\dfrac{1}{\varepsilon}\right)$.

Selecting the Stable Poles

The denominator of $H(s)\,H(-s)$ is a polynomial of degree $2n$ in $s$, giving $2n$ poles. These are distributed symmetrically about both the real and imaginary axes of the $s$-plane.

To obtain a stable filter, we select only the left half-plane poles ($\text{Re}(p) < 0$). This reduces the filter order from $2n$ to $n$.

Dark-colored markers are the left half-plane poles used for the filter; light-colored markers are the unstable right half-plane poles that are discarded. The poles are equally spaced on the ellipse. For odd orders, one pole lies on the real axis.

Comparison with Butterworth

Butterworth filter poles are equally spaced on a circle of radius 1. In a Chebyshev filter, this circle deforms into an ellipse, and the poles move closer to the imaginary axis (yielding a steeper roll-off).

As the ripple $\varepsilon \to 0$, $y \to \infty$ and $\sinh(y) \approx \cosh(y)$, so the ellipse approaches a circle. This represents the asymptotic convergence to the Butterworth response.

Constructing the Transfer Function

From Poles to Transfer Function

Once the $n$ left half-plane poles $p_1, p_2, \ldots, p_n$ are found, the transfer function is:

The numerator constant $K$ is adjusted so that the DC gain or the gain at the passband edge is correct.

Decomposition into First- and Second-Order Sections

Real-axis pole (odd order)

For odd-order filters, when $m = (n-1)/2$, $x = \pi/2$ and $\cos(x) = 0$, so the pole lies on the real axis:

The corresponding first-order section:

Complex conjugate poles (second-order sections)

The remaining poles appear as complex conjugate pairs $p, p^*$:

Each pair is combined into a second-order section:

where:

Correspondence with the Standard Form

The standard form of a second-order section is:

We compare coefficients with the denominator $s^2 - 2\text{Re}(p)\,s + |p|^2$ from Eq.\eqref{eq:H2-section}.

Comparing constant terms:

Comparing coefficients of $s$:

Solving for $Q$:

Substituting $\text{Re}(p) = -\sin(x)\sinh(y)$:

Related Topics

• Theory (Type II): Inverse Chebyshev Filter
• Frequency Response: Magnitude, Phase, and Group Delay
• Design: Order Determination and Design Procedure
• Remez Theorem (Mathematical basis of Chebyshev approximation)