IIR Filter Design: Butterworth, Chebyshev, Elliptic, and State-Variable Methods

Key Concepts and Design Methods

1. IIR Filters: Basics and Advantages

IIR (Infinite Impulse Response) filters use feedback (recursive part) and feedforward (non-recursive part) to achieve sharp frequency responses with fewer coefficients than FIR filters. Their transfer function is: $$H(z)=\frac{{\sum }_{k=0}^M{b}_k{z}^{-k}}{1-{\sum }_{k=1}^N{a}_k{z}^{-k}}$$ Advantages:

• Lower computational complexity (fewer taps).
• Steeper roll-off with fewer coefficients.
• Better suited for narrowband applications.

Disadvantages:

• Non-linear phase response (unless carefully designed).
• Potential instability if poles are outside the unit circle.

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2. Butterworth Filters: Maximally Flat Response

Butterworth filters have a monotonic frequency response (no ripples in passband or stopband). Their magnitude squared is: $$|H(j\omega ){|}^2=\frac{1}{1+(\omega /{\omega }_c{)}^{2N}}$$ where $N$ = filter order, ${\omega }_c$ = cutoff frequency.

Design Steps:

1. Specify passband (${\omega }_p$) and stopband (${\omega }_s$) frequencies and attenuation ($A_p,A_s$).
2. Calculate normalized cutoff frequency (${\Omega }_c$): $${\Omega }_c=\tan \left(\frac{{\omega }_c}{2}\right)$$
3. Determine filter order ($N$) using: $$N\ge \frac{\log \left(\frac{10^{A_p/10}-1}{10^{A_s/10}-1}\right)}{2\log ({\Omega }_s)}$$
4. Find pole locations using: $$p_k={\Omega }_c{e}^{j(\pi /2+(2k+1)\pi /(2N))},k=0,1,\dots ,N-1$$
5. Convert analog prototype to digital using bilinear transform: $$s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$

Worked Example: Butterworth Lowpass Filter

Given:

• Passband edge (${\omega }_p=1$ kHz), stopband edge (${\omega }_s=2$ kHz).
• $A_p=3$ dB, $A_s=20$ dB.
• Sampling frequency $F_s=10$ kHz.

Steps:

1. Normalize frequencies: $${\Omega }_p=\tan \left(\frac{\pi \cdot 1}{10}\right)=0.3249,{\Omega }_s=\tan \left(\frac{\pi \cdot 2}{10}\right)=0.6559$$
2. Calculate $N$: $$N\ge \frac{\log \left(\frac{10^{0.3}-1}{10^2-1}\right)}{2\log (0.6559)}\approx 3.5⟹N=4$$
3. Pole locations (for $N=4$): $$p_k=0.6889e^{j(\pi /2+(2k+1)\pi /8)},k=0,1,2,3$$
4. Bilinear transform to get digital coefficients.

Visual: Pole-Zero Plot

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3. Chebyshev Filters: Equiripple Response

Chebyshev filters have ripples in the passband (Type I) or stopband (Type II). Their magnitude response is: $$|H(j\omega ){|}^2=\frac{1}{1+{ϵ}^2{C}_N^2(\omega /{\omega }_c)}$$ where $C_N(x)$ is the Chebyshev polynomial of order $N$, and $ϵ$ controls ripple magnitude.

Design Steps (Type I):

1. Calculate $ϵ$: $$ϵ=\sqrt{10^{A_p/10}-1}$$
2. Find normalized cutoff frequency (${\Omega }_c$): $${\Omega }_c=\frac{1}{\sinh \left(\frac{1}{N}{\sinh }^{-1}(1/ϵ)\right)}$$
3. Pole locations: $$p_k=-{\Omega }_c\sinh \left(\frac{\pi }{N}\right)\sin \left(\frac{(2k+1)\pi }{2N}\right)+j{\Omega }_c\cosh \left(\frac{\pi }{N}\right)\cos \left(\frac{(2k+1)\pi }{2N}\right)$$

Comparison: Butterworth vs. Chebyshev

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4. Elliptic Filters: Optimized Transition Band

Elliptic filters have ripples in both passband and stopband, achieving the steepest roll-off for a given order. Their magnitude response is: $$|H(j\omega ){|}^2=\frac{1}{1+{ϵ}^2{U}_N^2(\omega /{\omega }_c)}$$ where $U_N(x)$ is the elliptic rational function.

Design Steps:

1. Calculate $k$ (modulus): $$k=\frac{{\Omega }_s-{\Omega }_p}{{\Omega }_s+{\Omega }_p}$$
2. Find $N$ using elliptic integral tables or numerical methods.
3. Pole locations are computed using Jacobi elliptic functions.

Visual: Frequency Response Comparison

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5. Bilinear Transform: Analog-to-Digital Conversion

The bilinear transform maps the left-half $s$-plane to the unit circle in $z$-plane: $$s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$ Pre-warping is required to correct frequency distortion: $$\Omega =\tan \left(\frac{\omega T}{2}\right)$$

Worked Example: Pre-warping

Given:

• Analog cutoff frequency $f_c=1$ kHz.
• Sampling frequency $F_s=10$ kHz.

Steps:

1. Calculate pre-warped frequency: $$\Omega =\tan \left(\frac{2\pi \cdot 1000}{10000}\right)=0.6421$$
2. Design analog filter at $\Omega =0.6421$ rad/sample.
3. Apply bilinear transform to get digital coefficients.

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6. State-Variable Implementation

State-variable filters use integrators and summation to implement IIR filters. The block diagram for a second-order section is:

Advantages:

• Structured and stable implementation.
• Easy to tune coefficients.

Disadvantages:

• Requires more multipliers than direct form.

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7. Stability and Causality

• Stability: All poles must lie inside the unit circle ($|z|<1$).
• Causality: Filters must be non-anticipative (no future samples in output).
• Check stability: Ensure no poles are outside the unit circle in the $z$-plane.

Visual: Stability Region

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In the Real World

1. WhatsApp Voice Calls (Meta)

   • Uses IIR filters (Butterworth/Chebyshev) for echo cancellation and noise reduction in voice signals. The filters suppress background noise while preserving speech clarity by attenuating frequencies outside the human voice range (300 Hz–3.4 kHz).

2. Nepal Telecommunications Corporation (NTC) 4G/5G Base Stations

   • Deploy elliptic IIR filters in OFDM modulators/demodulators to separate adjacent frequency channels (e.g., 15 kHz spacing in LTE). The steep roll-off of elliptic filters minimizes inter-channel interference, improving data throughput.

3. Khalti Payment App (Nepal)

   • Uses digital filters (IIR) in audio-based OTP verification (e.g., "play a tone to confirm"). A bandpass IIR filter isolates the tone frequency (e.g., 1 kHz) from ambient noise, then applies Goertzel algorithm (a DFT approximation using IIR feedback) to detect the tone’s presence.

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Exam Tip

1. Memorize key formulas:

   • Butterworth order calculation: $N\ge \frac{\log (\text{stopband error}/\text{passband error})}{2\log ({\Omega }_s)}$.
   • Bilinear transform: $s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$.
   • Pre-warping: $\Omega =\tan (\omega T/2)$.

2. Pole-zero plots are critical:

   • Sketch the analog prototype poles (Butterworth: circle; Chebyshev: ellipse; Elliptic: clustered).
   • Show digital transformation after bilinear transform (poles move to unit circle).

3. Compare filter types:

   • Butterworth: Smooth response, higher order needed.
   • Chebyshev: Faster roll-off, passband ripples.
   • Elliptic: Best roll-off, ripples in both bands.

4. Worked examples:

   • Always show steps for order calculation, pole placement, and bilinear transform.
   • For state-variable filters, draw the block diagram and label coefficients.

5. Stability check:

   • Questions may ask to verify stability by checking pole locations. Use the ** Jury stability criterion** if poles are not obvious.

6. Real-world mapping:

   • Relate audio equalizers (e.g., in Spotify) to Butterworth filters for smooth tone control.
   • Link 5G channel separation to elliptic filters in exam applications.

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