5.02 Butterworth Filter Approximation | 5.04 Active Filter Realization & Sallen-Key RC Networks
5.03 Chebyshev & Elliptic Filter Approximations
Core Idea
While Butterworth filters offer a perfectly flat magnitude response at the cost of a wide transition band, Chebyshev and Elliptic filters trade flatness in the passband and/or stopband to achieve dramatically narrower transition widths {sharper roll-offs} for the same filter order [5.2.6]. By distributing mathematical ripples across the operating bands, these approximations provide highly optimized hardware realizations [5.2.3, 5.2.4].
1. The Philosophical Shift: Ripple vs. Transition Width
When designing practical filters, we are constrained by a fundamental trade-off: passband flatness versus transition roll-off steepness [5.2.6].
- Butterworth: Maximally flat passband response, but requires a very high order to achieve steep attenuation [5.2.2].
- Chebyshev Type I: Permits controlled ripples in the passband to secure a much faster roll-off in the transition region and a monotonic stopband [5.2.3].
- Chebyshev Type II (Inverse Chebyshev): Maintains a perfectly flat (monotonic) passband, but introduces ripples in the stopband to maximize transition steepness [5.2.3].
- Elliptic (Cauer): Allows ripples in both the passband and stopband, yielding the absolute steepest transition band possible for any given filter order [5.2.4, 5.2.6].
2. Chebyshev Type I Approximation (Passband Ripple)
The magnitude-square response of an -th order Chebyshev Type I filter is mathematically defined by [5.2.3]:
Where:
- is the ripple factor {controls the depth of passband ripples} [5.2.3].
- is the passband edge frequency [5.2.1].
- is the -th order Chebyshev polynomial [5.2.3].
2.1 The Chebyshev Polynomials
The Chebyshev polynomials are defined piecewise as [5.2.3]:
\cos(N \cos^{-1}(x)), & |x| \le 1 \quad \text{(Passband)} \\ \cosh(N \cosh^{-1}(x)), & |x| > 1 \quad \text{(Stopband)} \end{cases}$$ These polynomials can be generated recursively using the recurrence relation [5.03]: $$T_N(x) = 2x T_{N-1}(x) - T_{N-2}(x) \quad \text{with} \quad T_0(x) = 1, \ T_1(x) = x$$ #### First Five Chebyshev Polynomials: * $T_0(x) = 1$ * $T_1(x) = x$ * $T_2(x) = 2x^2 - 1$ * $T_3(x) = 4x^3 - 3x$ * $T_4(x) = 8x^4 - 8x^2 + 1$ * $T_5(x) = 16x^5 - 20x^3 + 5x$ ### 2.2 Deriving Ripple Factor ($\epsilon$) and Order ($N$) 1. **Ripple Factor ($\epsilon$):** At the passband edge ($\Omega = \Omega_p$), $T_N(1) = 1$. The passband attenuation $\alpha_p$ (in dB) is [5.2.3]: $$\alpha_p = 10 \log_{10}(1 + \epsilon^2) \implies \epsilon = \sqrt{10^{0.1 \alpha_p} - 1}$$ 2. **Filter Order ($N$):** At the stopband edge ($\Omega = \Omega_s$), the attenuation must satisfy $\alpha_s$ [5.2.3]: $$\alpha_s = 10 \log_{10}\left(1 + \epsilon^2 \cosh^2\left(N \cosh^{-1}\left(\frac{\Omega_s}{\Omega_p}\right)\right)\right)$$ Solving for $N$ yields the critical design formula [5.2.3]: $$N \ge \frac{\cosh^{-1}\left(\frac{\sqrt{10^{0.1 \alpha_s} - 1}}{\epsilon}\right)}{\cosh^{-1}\left(\frac{\Omega_s}{\Omega_p}\right)}$$ Where $\cosh^{-1}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)$ is evaluated algebraically [5.2.3]. ### 2.3 Pole Locations on the s-Plane Semicircle Unlike Butterworth poles (which lie on a circle), Chebyshev Type I poles lie symmetrically on a **s-plane ellipse** [5.2.3]. The stable Left-Half s-Plane (LHP) poles are given by [5.2.3]: $$p_k = \sigma_k + j \omega_k, \quad k = 1, 2, \dots, N$$ Where [5.2.3]: $$\sigma_k = -\sinh(\beta) \sin\left(\frac{(2k-1)\pi}{2N}\right)$$ $$\omega_k = \cosh(\beta) \cos\left(\frac{(2k-1)\pi}{2N}\right)$$ $$\beta = \frac{1}{N} \sinh^{-1}\left(\frac{1}{\epsilon}\right) = \frac{1}{N} \ln\left(\frac{1}{\epsilon} + \sqrt{\frac{1}{\epsilon^2} + 1}\right)$$ ### 2.4 Normalized Chebyshev Type I Denominator Polynomials (Ripple = 1 dB) For $\alpha_p = 1\text{ dB} \implies \epsilon = 0.5088$, the normalized s-domain transfer functions are [5.2.3]: | $N$ | Denominator Polynomial $D(s)$ | Scaling Constant $H_0$ | | :--- | :--- | :--- | | **1** | $s + 1.9652$ | $1.9652$ | | **2** | $s^2 + 1.0977s + 1.1025$ | $0.98261$ | | **3** | $s^3 + 0.9883s^2 + 1.2384s + 0.4913$ | $0.49131$ | | **4** | $s^4 + 0.9528s^3 + 1.4539s^2 + 0.7426s + 0.2756$ | $0.24565$ | | **5** | $s^5 + 0.9368s^4 + 1.6888s^3 + 0.9744s^2 + 0.5805s + 0.1228$ | $0.12283$ | > [!important] The $H_0$ Scaling Rule > > To ensure correct passband scaling at DC ($s=0$), the numerator constant is calculated as [5.2.3]: > * For **odd $N$**: $H_0 = \prod |p_k| = D(0) \implies H_s(0) = 1$ > * For **even $N$**: $H_0 = \frac{\prod |p_k|}{\sqrt{1+\epsilon^2}} = \frac{D(0)}{\sqrt{1+\epsilon^2}} \implies H_s(0) = \frac{1}{\sqrt{1+\epsilon^2}}$ --- ## 3. Chebyshev Type II (Inverse Chebyshev) Approximation Chebyshev Type II filters keep a perfectly flat passband (like Butterworth) but introduce equal-amplitude ripples in the stopband to maintain sharp transition properties [5.2.3, 5.2.6]. The magnitude-square response is given by [5.2.3]: $$|H(j\Omega)|^2 = \frac{1}{1 + \epsilon^2 \frac{T_N^2(\Omega_s/\Omega_p)}{T_N^2(\Omega_s/\Omega)}} = \frac{T_N^2(\Omega_s/\Omega)}{T_N^2(\Omega_s/\Omega) + \delta_s^2 T_N^2(\Omega_s/\Omega_p)}$$ ### 3.1 Zeros and Poles Formulation 1. **Transmission Zeros ($z_k$):** Because the response decays to finite ripples instead of zero at high frequencies, the transfer function possesses zeros on the imaginary axis [5.2.3]: $$z_k = j \frac{\Omega_s}{\cos\left(\frac{(2k-1)\pi}{2N}\right)}, \quad k = 1, 2, \dots, N$$ *(Note: For odd $N$, the zero at $k = \frac{N+1}{2}$ lies at infinity).* 2. **Poles ($p_k$):** The poles of a Type II filter are the **reciprocals of a prototype Type I filter's poles** [5.2.3]: $$p_k = \frac{-\sigma_k \Omega_s}{\sigma_k^2 + \Omega_k^2} + j \frac{\omega_k \Omega_s}{\sigma_k^2 + \Omega_k^2}$$ Where $\sigma_k$ and $\omega_k$ are the prototype elliptical coordinates computed with stopband ripple parameter $\delta_s = \frac{1}{\sqrt{10^{0.1 \alpha_s} - 1}}$ [5.2.3]. --- ## 4. Elliptic (Cauer) Filter Approximation The **Elliptic Filter** is the most mathematically efficient analog filter approximation [5.2.4, 5.2.6]. By distributing equiripples across **both** the passband and the stopband, it yields the sharpest possible roll-off for a given order $N$ [5.2.4]. The magnitude-square response is defined as [5.2.4]: $$|H(j\Omega)|^2 = \frac{1}{1 + \epsilon^2 U_N^2\left(\frac{\Omega}{\Omega_p}\right)}$$ Where $U_N(x)$ is the **Jacobian elliptic function** of order $N$ [5.2.4]. ### 4.1 Elliptic Parameter Design Rules The sharp transition boundary is characterized by the **selectivity factor ($k$)** [5.2.4]: $$k = \frac{\Omega_p}{\Omega_s} \quad \text{(where } 0 < k < 1\text{)}$$ To estimate the required order $N$, we apply the following Jacobian coordinate algorithm [5.2.4]: 1. Compute the complementary selectivity parameter $k'$ [5.2.4]: $$k' = \sqrt{1 - k^2}$$ 2. Evaluate the initial coordinate parameter $\rho_0$ [5.2.4]: $$\rho_0 = \frac{1 - \sqrt{k'}}{2(1 + \sqrt{k'})}$$ 3. Calculate the modular constant $\rho$ [5.2.4]: $$\rho = \rho_0 + 2\rho_0^5 + 15\rho_0^9 + 150\rho_0^{13}$$ 4. Solve for the minimum required integer order $N$ [5.2.4]: $$N \ge \frac{\log_{10}\left(16 \frac{10^{0.1 \alpha_s} - 1}{10^{0.1 \alpha_p} - 1}\right)}{\log_{10}(1/\rho)}$$ --- ## 5. High-Yield Comparative Analysis To help you secure maximum marks on qualitative questions, here is the official ECE 2107 comparison of the five primary analog filter families [5.2.6, 5.2.7]: ### 5.1 The 5-Way Filter Property Comparison Matrix | Parameter | Butterworth | Chebyshev I | Chebyshev II | Elliptic | Bessel | | :--- | :--- | :--- | :--- | :--- | :--- | | **Passband Shape** | Monotonic (Flat) | Equiripple | Monotonic (Flat) | Equiripple | Monotonic | | **Stopband Shape** | Monotonic | Monotonic | Equiripple | Equiripple | Monotonic | | **Transition Width** | Extremely Wide | Medium-Narrow | Medium-Narrow | Steepest (Ultra-Narrow) | Extremely Wide | | **Phase Linearity** | Moderately Linear | Non-linear | Moderately Linear | Highly Non-linear | **Perfectly Linear** | | **Transient Response** | Moderate overshoot | Poor (Ringing) | Moderate overshoot | Severely degraded ringing | **Zero Overshoot** | ### 5.2 Order Requirements for Identical Specifications [Table 5.5] * **Target Specifications:** $f_p = 450\text{ Hz}$, $f_s = 550\text{ Hz}$, $\alpha_p = 1\text{ dB}$, $\alpha_s = 35\text{ dB}$ [5.2.6]: ``` [ORDER STEEPNESS COMPARISON] Butterworth (N = 24) ======================== Chebyshev I (N = 9) ========= Chebyshev II (N = 9) ========= Elliptic (N = 5) ===== ``` --- ## 6. Comprehensive Worked Examples (The Exam Killers) ### 6.1 Elliptic Parameter Design [PYQ 2025 - 13 Marks] **Question:** Design an Elliptic Low-Pass Filter with $f_p = 1\text{ kHz}$, $f_s = 1.5\text{ kHz}$, $\alpha_p = 1\text{ dB}$, and $\alpha_s = 40\text{ dB}$. Determine: 1. The filter order $N$. 2. The ripple factor $\epsilon$. 3. The normalized s-domain transfer function $H_N(s)$. #### Step-by-Step Solution: 1. **Calculate Selectivity Factors:** $$\Omega_p = 2\pi(1000) = 2000\pi \text{ rad/s}, \quad \Omega_s = 2\pi(1500) = 3000\pi \text{ rad/s}$$ $$k = \frac{\Omega_p}{\Omega_s} = \frac{2000\pi}{3000\pi} = \frac{2}{3} \approx 0.6667$$ $$k' = \sqrt{1 - k^2} = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \frac{\sqrt{5}}{3} \approx 0.745356$$ 2. **Calculate Jacobian Coordinates:** $$\rho_0 = \frac{1 - \sqrt{0.745356}}{2(1 + \sqrt{0.745356})} = \frac{1 - 0.86334}{2(1 + 0.86334)} = \frac{0.13666}{3.72668} \approx 0.03667$$ Since $\rho_0$ is small, the higher order terms are negligible: $$\rho \approx \rho_0 = 0.03667$$ 3. **Evaluate Filter Order $N$:** $$N \ge \frac{\log_{10}\left(16 \frac{10^{0.1(40)} - 1}{10^{0.1(1)} - 1}\right)}{\log_{10}(1/\rho)} = \frac{\log_{10}\left(16 \frac{9999}{0.258925}\right)}{\log_{10}(27.269)}$$ $$N \ge \frac{\log_{10}(617875.2)}{1.43567} = \frac{5.7909}{1.43567} \approx 4.03$$ Rounding up to the next integer yields: $$\mathbf{N = 5}$$ 4. **Calculate Ripple Factor $\epsilon$:** $$\epsilon = \sqrt{10^{0.1 \alpha_p} - 1} = \sqrt{10^{0.1(1)} - 1} = \sqrt{0.258925} \approx \mathbf{0.5088}$$ 5. **Formulate Normalized Transfer Function:** From the normalized Elliptic reference table for $\alpha_p = 1\text{ dB}, \alpha_s = 40\text{ dB}$, and $N = 5$ [5.2.4]: $$H_N(s) = \frac{0.0470 s^4 + 0.2201 s^2 + 0.2299}{s^5 + 0.9234 s^4 + 1.8471 s^3 + 1.1292 s^2 + 0.7881 s + 0.2299}$$ --- ## 7. Common Mistakes That Cost Marks > [!danger] **The Even-Order DC Scaling Trap** > > In Chebyshev Type I designs, when the order $N$ is **even**, the magnitude response at $s=0$ does not start at $1.0$. It starts at $\frac{1}{\sqrt{1+\epsilon^2}}$ (due to even symmetry ending on a ripple trough) [5.2.3]. Failing to apply this scaling factor to $H_0$ will result in a **loss of 3–5 marks** on your transfer function. > [!warning] **The Inverse Chebyshev Ripple Location Mix-Up** > > Do not mix up the ripple regions of Type I and Type II Chebyshev filters. Type I has ripples in the passband and is monotonic in the stopband [5.2.3]. Type II has ripples in the stopband and is monotonic in the passband [5.2.3]. Examiners love testing this distinction via 4-mark short questions. --- ## 8. PYQ Bank — Verbatim Questions & Answer Plans ### 8.1 PYQ 2025 Question 5c [13 Marks] **Question:** Design an Elliptic Low-Pass Filter with $f_p = 1\text{ kHz}$, $f_s = 1.5\text{ kHz}$, $\alpha_p = 1\text{ dB}$, and $\alpha_s = 40\text{ dB}$. Determine: (i) order, n; (ii) ripple factor, $\epsilon$; (iii) Transfer function. * **Answer Plan:** Follow the step-by-step math layout shown in **Section 6.1**. Write out the selectivity variables, use the logarithm formula for order approximation, define the ripple factor, and write the final normalized transfer function to secure all 13 marks. --- ## 9. Self-Check Before Moving On - [ ] Can you recursively generate Chebyshev polynomials up to $T_5(x)$? [5.03] - [ ] Do you know how to calculate the stable s-plane poles for a Type I Chebyshev filter? [5.2.3] - [ ] Do you understand why Chebyshev Type II filters have zeros on the $j\Omega$ axis? [5.2.3] - [ ] Can you evaluate the Jacobian coordinate algorithm ($\rho_0, \rho$) to calculate Elliptic orders? [5.2.4] --- *Source: (k.Deergha Rao) signals and systems.pdf (Chapter 5), Signals, Systems, and Networks MOC Roadmap (Chapter 5).*