Filters

Filters pass wanted frequencies and reject others. Learn low-pass, high-pass, band-pass and notch responses, passive RC/LC and active op-amp filters, filter order and roll-off, Butterworth, Chebyshev and Bessel approximations, Sallen–Key design, and anti-aliasing filters for ADCs.

Filters are everywhere: removing 50 Hz hum from ECG signals, anti-aliasing before an ADC, separating bass and treble in speakers, selecting a radio channel. Active filters use op-amps to achieve sharp responses without inductors.

An analogy

A filter is like a sieve for frequencies: the mesh size (cut-off) decides which frequencies pass and which are caught.

Filter types

Low-pass (keep lows), high-pass (keep highs), band-pass (keep a band), band-stop/notch (remove a band, e.g. 50/60 Hz hum). Passive RC filters are simple but load-sensitive; active filters add gain and buffering.

Order and approximations

Higher order = steeper roll-off. Butterworth: maximally flat passband. Chebyshev: steeper but with passband ripple. Bessel: best phase (pulse shape), gentle roll-off. Elliptic: steepest with ripple in both bands.

Sallen–Key and MFB

Second-order sections using one op-amp; cascade sections for higher orders (each with a specific Q from tables). Multiple-feedback (MFB) topologies suit band-pass and inverting designs.

Anti-aliasing and digital alternatives

Before an ADC sampling at f_s, attenuate everything above f_s/2 below 1 LSB. Oversampling ADCs relax the analog filter. Digital FIR/IIR filters (DSP module) complement analog ones.

Formulas

• Butterworth magnitude: |H| = 1/√(1 + (f/f_c)^(2n))
• Roll-off: −20n dB/decade (−6n dB/octave)
• Sallen–Key LP (equal R): f_c = 1/(2πR√(C₁C₂)), Q = ½√(C₁/C₂)
• Butterworth 2nd order: Q = 0.707 → C₁ = 2C₂

Worked example: Sallen–Key Butterworth

Given: f_c = 1 kHz, R = 10 kΩ, unity gain, Q = 0.707

Find: C₁ and C₂.

• C₂ = 1/(2πR f_c √2) = 11.25 nF
• C₁ = 2C₂ = 22.5 nF

Answer: C₁ ≈ 22 nF, C₂ ≈ 11 nF (use 22 nF and 10 nF + 1 nF)

Worked example: Attenuation at 2 f_c

Given: Butterworth n = 2 and n = 4

Find: Gain at 2 kHz.

• n = 2: 1/√(1 + 2⁴) = −12.3 dB
• n = 4: 1/√(1 + 2⁸) = −24.1 dB

Answer: −12.3 dB vs −24.1 dB

Common mistakes

• Wrong: Choosing a first-order filter for strict rejection. — Right: Compute required order from attenuation spec.
• Wrong: Ignoring op-amp GBW in active filters. — Right: Use GBW ≥ 100 × f_c × gain for accurate response.

Applications

• ADC anti-aliasing: Remove content above Nyquist.
• Audio crossovers: Split bands for woofers and tweeters.
• Biomedical: ECG band-pass and 50/60 Hz notch.

Key takeaways

• LP, HP, BP, notch.
• Roll-off −20n dB/decade.
• Butterworth flat, Chebyshev steep, Bessel phase.
• Sallen–Key: C₁ = 2C₂ for Q 0.707.

Going further

• Filter tables: Normalised pole locations for each order.
• Switched-capacitor filters: Clock-tunable integrated filters.
• Sensitivity analysis: Component tolerance effects on Q.