Elliptic Filter Approximation and Equiripple Analysis

1. The Elliptic Filter's Magnitude Response

Just as Chebyshev polynomials were the key to Chebyshev filters, the magnitude response of an elliptic filter is defined by a special class of functions called elliptic rational functions.

The magnitude-squared response of an Nth-order low-pass elliptic filter is given by:

$|G_N(j\omega ){|}^2=\frac{1}{1+{\epsilon }^2{R}_n^2(\xi ,\omega /{\omega }_0)}$

Let's break this down:

• $R_n(\xi ,x)$ is the Nth-order elliptic rational function. This function is the heart of the filter.
• $\epsilon$ is the ripple factor, a parameter that controls the amount of passband ripple, just like in the Chebyshev filter.
• $\xi$ (xi) is the selectivity factor, defined as the ratio of the stopband edge frequency to the passband edge frequency ($\xi ={\omega }_s/{\omega }_p$). This parameter determines the steepness of the transition band.

The defining characteristic of the elliptic rational function $R_n$ is that it is a ratio of polynomials. This is a crucial difference from the Chebyshev polynomial $T_n$. Because $R_n$ is a rational function, its transfer function will have both poles and zeros, just like a Chebyshev Type II filter. However, in the elliptic filter, these poles and zeros are placed in a highly specific way to create ripple in both the passband and stopband.

Let's examine the response visually.

As you saw in the reading, the response looks like this:

The magnitude response of a 4th-order elliptic filter. Note the equal-height ripples in the passband and the equal-depth nulls (zeros) in the stopband, which define the equiripple characteristic in both bands.

Let's analyze this behavior:

• Passband ($0\le \omega \le {\omega }_p$): The elliptic rational function $R_n$ oscillates in this region, causing the magnitude $|G_N(j\omega )|$ to ripple between a maximum of 1 and a minimum of $1/\sqrt{1+{\epsilon }^2}$. This is very similar to a Chebyshev Type I filter.
• Stopband ($\omega \ge {\omega }_s$): The transfer function has zeros on the $j\omega$-axis, which cause the deep nulls you see in the plot. Between these nulls, the response rises to local peaks. The magic of the Cauer approximation is that the poles and zeros are placed so that these stopband peaks are all of equal height, creating an equiripple stopband.

2. Deriving the Elliptic Filter Approximation

The design of an elliptic filter is mathematically intensive and relies on a family of special functions known as Jacobi elliptic functions (e.g., sn, cn) and elliptic integrals. Your background in advanced mathematics will be helpful in appreciating the elegance of this formulation.

The process, similar to other filter types, involves three main steps:

1. Determine the minimum filter order $N$ to meet specifications.
2. Calculate the locations of the transfer function's poles and zeros.
3. Construct the final transfer function $H(s)$.

Let's delve into the mathematics.

The Mathematical Machinery

The elliptic rational functions are constructed from Jacobi elliptic functions. While a full derivation is beyond the scope of this lesson, it's important to recognize their role as the fundamental building blocks.

Step 1: Determining the Filter Order $N$

Given the desired passband ripple ($A_p$), stopband attenuation ($A_s$), passband edge (${\omega }_p$), and stopband edge (${\omega }_s$), the first task is to find the minimum filter order $N$ that can satisfy these constraints. This calculation directly involves the complete elliptic integral $K(k)$.

As you've just read, the formula for the required order $N$ is:
$N\ge \frac{K(k)K(k_1^{\prime })}{K(k^{\prime })K(k_1)}$
where $k={\omega }_p/{\omega }_s$, $k_1$ is related to the ripple specifications, and $k^{\prime }$ denotes the complementary modulus $\sqrt{1-k^2}$. This formula elegantly captures the relationship between the filter's specifications and the required complexity ($N$).

Step 2: Finding the Poles and Zeros

With the order $N$ determined, the next step is to find the locations of the $N$ poles and the associated zeros.

• The zeros are located on the $j\omega$-axis, within the stopband.
• The poles are located in the left-half of the s-plane on an ellipse, similar to the Chebyshev filter.

Pole-zero plot for an 8th-order elliptic lowpass filter. The zeros lie on the imaginary axis, creating stopband nulls. The poles lie on an ellipse in the left-half plane, creating the passband ripple.

The precise locations are calculated using the Jacobi elliptic sine function, sn(u, k).

Step 3: Constructing the Transfer Function $H(s)$

Once the left-half-plane poles $\{p_l\}$ and the imaginary-axis zeros $\{z_m\}$ are found, the transfer function is simply constructed as a rational function:

$H(s)=K\frac{{\prod }_{m=1}^M(s-z_m)}{{\prod }_{l=1}^N(s-p_l)}$

Here, $M$ is the number of zeros (for a low-pass filter, $M<N$), and $K$ is a normalization constant chosen to set the gain at DC to the appropriate value (either 1 or $1/\sqrt{1+{\epsilon }^2}$, depending on whether $N$ is odd or even).

The Mathcad resource you've been using (LINK, section 4) continues from here to construct the final transfer function and plot its frequency response, which provides a satisfying visual confirmation that the intricate mathematics correctly produces the desired equiripple behavior in both bands.

3. The Optimality of Elliptic Filters

The reason elliptic filters are so important is their optimality. For a given order and set of ripple specifications, no other linear time-invariant filter can achieve a faster rate of attenuation between the passband and stopband.

Comparison of 5th-order Butterworth, Chebyshev, and Elliptic filters. The elliptic filter clearly has the sharpest transition between the passband and stopband, achieved by tolerating ripple in both bands.

This graph perfectly illustrates the trade-offs we've been discussing throughout this module. The Butterworth is smooth but slow. The Chebyshev is faster but has passband ripple. The Elliptic filter is the fastest of all, representing the limit of what is possible for a given filter order.

Conclusion

In this lesson, we have explored the most efficient of the classical analog filter approximations, the elliptic (Cauer) filter. You have seen how it leverages sophisticated mathematics to achieve performance that is, in a very real sense, optimal.

Key Takeaways:

• Elliptic filters achieve the sharpest possible transition for a given order by featuring equiripple behavior in both the passband and the stopband.
• The filter's response is defined by elliptic rational functions, which are constructed from Jacobi elliptic functions and elliptic integrals.
• The transfer function has both poles (arranged on an ellipse in the LHP) and zeros (on the $j\omega$-axis), which are meticulously placed to create the dual-equiripple characteristic.
• The design process involves using elliptic integrals to calculate the required filter order $N$ and then using Jacobi elliptic functions to find the pole and zero locations to construct $H(s)$.

Preview of the Next Lesson:
We have now studied the "big three" analog filter approximations: Butterworth, Chebyshev, and Elliptic. Each embodies a different design philosophy and offers a unique set of trade-offs. In our next lesson, we will step back and systematically analyze the trade-offs between these filter types in terms of order, transition bandwidth, and ripple, helping you understand how to select the best filter for a given engineering problem.