Filters and Signal Processing

Every real electrical signal occupies a range of frequencies. Speech sits between roughly 300 Hz and 3.4 kHz; the audible bandwidth of music is 20 Hz to 20 kHz; AM radio broadcasts occupy 9 kHz channels around carriers between 540 kHz and 1.6 MHz; FM broadcasts occupy 200 kHz channels around carriers between 88 MHz and 108 MHz; Wi-Fi runs at 2.4 GHz and 5 GHz. Almost every electronic system needs to select a particular band of frequencies and reject the rest — to separate a wanted signal from interference, hum, hiss, or adjacent channels. The device that does this is a filter. The simplest filters consist of just a resistor and a capacitor connected in two arrangements: the low-pass filter, which passes signals below a cutoff frequency and attenuates higher frequencies; and the high-pass filter, which does the opposite. More elaborate filters combine these blocks with op-amps to give sharper roll-offs and to add band-pass and band-stop responses. This lesson develops the mathematics of first-order RC filters, derives the −3 dB cutoff frequency, introduces Bode plots in dB and log frequency, explains the universal −20 dB-per-decade roll-off, and surveys the active filters that extend these ideas to higher orders.

Spec mapping (AQA 7408 §3.13.3 — Analogue signal processing, option E, continued): This lesson covers passive RC low-pass and high-pass filters; the −3 dB cutoff frequency f_c = 1/(2πRC); the asymptotic gain and phase responses (Bode plots); the −20 dB/decade roll-off characteristic of a first-order filter; active filters using op-amps for sharper roll-offs and unity-gain pass-band response; band-pass and band-stop filter concepts; and the use of filters in audio, radio, and instrumentation signal chains. (Refer to the official AQA specification document for exact wording.)

Synoptic links: RC filters connect directly to the time-constant picture from Year 12 capacitor charging/discharging (§3.7.1) — the same RC product τ that sets the half-life of a discharging capacitor sets the cutoff frequency of a filter through f_c = 1/(2πRC). Active filters build on the op-amp topologies of Lesson 2 — placing a capacitor in the inverting feedback path produces an integrator (effectively a low-pass with unbounded DC gain) and the more elaborate Sallen–Key topology produces second-order active filters. Filters are also essential to the digital-signal lesson (Lesson 1) — every ADC needs an anti-aliasing low-pass filter and every DAC needs a reconstruction filter; the choice of filter order sets how close the sampling rate must sit above the Nyquist limit.

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The First-Order RC Low-Pass Filter

The canonical low-pass filter is a single resistor in series with a single capacitor, with the output taken across the capacitor:

V_in ──/\/\──┬─── V_out
       R     │
             C
             │
            GND

At low frequency the capacitor's impedance Z_C = 1/(jωC) is very large; almost no current flows; V_out ≈ V_in (the filter passes the signal unchanged). At high frequency Z_C → 0; the capacitor short-circuits the output to ground; V_out → 0 (the filter blocks the signal). The cross-over happens at a characteristic frequency at which |Z_C| = R.

Derivation of the Transfer Function

By the voltage-divider rule with complex impedances:

V_out / V_in = Z_C / (R + Z_C) = (1/jωC) / (R + 1/jωC) = 1 / (1 + jωRC)

The transfer function of the filter is:

H(ω) = 1 / (1 + jωRC)

Take the magnitude:

|H(ω)| = 1 / √(1 + (ωRC)²)

And the phase:

φ(ω) = −arctan(ωRC)

The Cutoff Frequency

The cutoff frequency is defined as the frequency at which |H| has fallen by a factor of √2 from its DC value — that is, the gain has dropped by 3 dB. This corresponds to ωRC = 1, or equivalently:

f_c = 1 / (2πRC)

At f = f_c:

• |H| = 1/√2 ≈ 0.707 (i.e. −3 dB)
• φ = −45° (the output lags the input by 45°)

For f ≪ f_c: |H| ≈ 1, φ ≈ 0° — the filter is "transparent". For f ≫ f_c: |H| ≈ 1/(ωRC), φ ≈ −90°. The gain falls in inverse proportion to frequency.

The −20 dB-per-Decade Roll-Off

In the high-frequency limit |H| ≈ 1/(ωRC). Expressed in decibels:

|H|_dB = 20 log₁₀ |H| = −20 log₁₀(ωRC)

So when ω increases by a factor of 10 (one decade), |H|_dB falls by 20 dB. This is the signature first-order roll-off: gain falls at exactly −20 dB per decade above cutoff. A second-order filter (two cascaded poles) gives −40 dB/decade; a fourth-order filter −80 dB/decade. The order of the filter sets how steeply it rejects out-of-band signals.

Worked Example — Designing a Low-Pass

Question: Design a passive RC low-pass filter with a cutoff frequency of 1.0 kHz. Choose a sensible R and C.

Solution: f_c = 1/(2πRC) = 1000 Hz, so RC = 1/(2π × 1000) = 1.59 × 10⁻⁴ s. Choose R = 1.59 kΩ and C = 100 nF (RC = 1.59 × 10³ × 100 × 10⁻⁹ = 1.59 × 10⁻⁴ s ✓).

Equally valid: R = 16 kΩ, C = 10 nF; or R = 160 Ω, C = 1 μF. The choice within the constraint is governed by:

• R too small loads the source heavily.
• R too large is sensitive to bias currents and noise; the capacitor at the output also sees a high source impedance, slowing the response further.

A typical sensible choice is R ≈ 1–10 kΩ with C chosen to match.

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The First-Order RC High-Pass Filter

Swap R and C: take the output across the resistor instead of the capacitor.

V_in ──┤├──┬─── V_out
        C   │
            R
            │
           GND

At low frequency Z_C is huge; almost no current flows; V_out ≈ 0 (the filter blocks DC and low frequencies). At high frequency Z_C → 0; the capacitor passes the signal to the resistor; V_out → V_in (the filter passes the signal).

Transfer Function

By the voltage-divider rule:

V_out / V_in = R / (R + 1/(jωC)) = jωRC / (1 + jωRC)

H(ω) = jωRC / (1 + jωRC)

Magnitude: |H| = ωRC / √(1 + (ωRC)²)

Phase: φ = 90° − arctan(ωRC)

Cutoff Behaviour

Same cutoff frequency formula:

f_c = 1 / (2πRC)

At f = f_c: |H| = 1/√2 = −3 dB, φ = +45° (output leads input by 45° — high-pass filters introduce phase advance rather than phase lag).

For f ≪ f_c: |H| ≈ ωRC, φ ≈ +90° — the gain falls into the stop-band at +20 dB/decade as you approach DC. For f ≫ f_c: |H| ≈ 1, φ ≈ 0° — the filter is transparent.

Application — AC Coupling

The most common use of a high-pass filter is AC coupling: blocking the DC component of a signal while passing the AC. An audio amplifier between stages often has a coupling capacitor that removes the DC bias of the previous stage but lets the audio signal through. A typical 1 μF coupling capacitor into a 10 kΩ input impedance has f_c = 1/(2π × 10⁴ × 10⁻⁶) = 16 Hz — comfortably below the audio band.

Worked Example — AC Coupling

Question: An audio signal is coupled through a 0.10 μF capacitor into the 100 kΩ input impedance of the next stage. (a) Calculate the cutoff frequency. (b) Will the lowest audible frequency (20 Hz) be attenuated significantly?

Solution:

(a) f_c = 1/(2π × 10⁵ × 10⁻⁷) = 1/(2π × 10⁻²) = 15.9 Hz.

(b) At 20 Hz, ωRC = 20/15.9 = 1.26. |H| = 1.26/√(1 + 1.26²) = 1.26/1.61 = 0.78 — about 2 dB of attenuation. This is barely audible but a fastidious designer would drop f_c to 5 Hz by increasing C to 0.33 μF.

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Bode Plots

A Bode plot displays the frequency response of a filter (or any linear system) on log–log axes:

• Horizontal axis: log₁₀(frequency)
• Left vertical axis: gain in dB
• Right vertical axis: phase in degrees

The advantage of these axes is that asymptotic behaviour becomes straight lines that are trivial to sketch.

Bode Plot of a Low-Pass Filter

|H|_dB
   0  ━━━━━━━━━━┓
              ╲
              ╲ slope −20 dB/decade
              ╲
 −20            ╲
              ╲
 −40            ╲
              ╲
              └─────────────────→ log f
              f_c       10f_c     100f_c

Key features:

• The pass-band (f ≪ f_c) is a horizontal line at 0 dB.
• The cutoff frequency f_c sits at −3 dB (the actual response sags from the asymptotic 0 dB line by 3 dB at this point).
• The stop-band (f ≫ f_c) is a straight line falling at −20 dB/decade.
• The "corner" or "knee" of the asymptotes is at f_c.

Bode Plot of a High-Pass Filter

The mirror image: a straight line rising at +20 dB/decade up to f_c, then a flat pass-band at 0 dB above f_c.

Worked Example — Reading a Bode Plot

Question: A low-pass filter has f_c = 1 kHz. (a) State the gain in dB at 10 kHz. (b) State the gain in dB at 100 kHz. (c) State the gain in dB at f = 0.1 kHz.

Solution:

(a) 10 kHz is one decade above f_c. The asymptotic stop-band slope is −20 dB/decade, so |H| = −20 dB (within rounding; the exact value is 20 log₁₀(1/√101) = −20.04 dB).

(b) 100 kHz is two decades above f_c, so |H| = −40 dB.

(c) 0.1 kHz is one decade below f_c — comfortably in the pass-band, where |H| ≈ 1 = 0 dB. (Exact: 20 log₁₀(1/√1.01) = −0.04 dB.)

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graph LR
    A["Input signal<br/>(broadband)"] --> B["Low-pass filter<br/>f_c = 1 kHz"]
    B --> C["Wanted signal<br/>(0 - 1 kHz)"]
    B --> D["Rejected:<br/>noise > 1 kHz"]

    style C fill:#27ae60,color:#fff
    style D fill:#e74c3c,color:#fff

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Active Filters Using Op-Amps

Passive RC filters have two limitations: (i) the maximum roll-off of a single RC pair is −20 dB/decade, which is gentle; (ii) the filter's input impedance and output impedance vary with frequency, making cascading awkward. Active filters use op-amps to provide buffering, gain, and sharper roll-offs.

Active Low-Pass — Inverting with Feedback Capacitor

Take the inverting amplifier from Lesson 2 and replace the feedback resistor with a parallel combination of R_f and a capacitor C_f:

V_out / V_in = −(R_f / R_in) × 1/(1 + jωR_f C_f)

This is an inverting amplifier whose gain rolls off as a first-order low-pass with f_c = 1/(2πR_f C_f). It buffers the input (high input impedance) and the output (low output impedance), can have gain greater than 1, and is easily cascadable.

Sallen–Key Second-Order Low-Pass

The Sallen–Key topology uses one op-amp configured as a non-inverting amplifier with two RC stages around it to give a second-order response (−40 dB/decade roll-off) from a single op-amp. AQA does not require derivation of the Sallen–Key transfer function but you should recognise the topology by name and know it as the standard second-order active filter.

Why Second-Order Matters

A first-order filter rolls off at −20 dB/decade. To get from a 20 kHz cutoff to a 100× attenuation (−40 dB) requires the frequency to be 100× higher — at 2 MHz. A second-order filter rolls off at −40 dB/decade and reaches −40 dB only one decade above cutoff (200 kHz instead of 2 MHz). For an anti-aliasing filter feeding an ADC sampling at 48 kHz, a second-order or higher filter is essentially mandatory to avoid wasting most of the sample rate as guard band.

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Band-Pass and Band-Stop Filters

A band-pass filter passes signals within a band f_low to f_high and rejects all others. It can be built by cascading a high-pass (cutoff at f_low) with a low-pass (cutoff at f_high). In radio receivers, every tuned channel is selected by a band-pass filter centred on the carrier frequency.

A band-stop filter (or notch filter) rejects signals within a narrow band and passes everything else. The classic application is rejection of 50 Hz mains hum from an audio or biomedical signal: a narrow notch at exactly 50 Hz removes the mains pickup without significantly affecting wanted content above 100 Hz or below 30 Hz.

Worked Example — Designing a Band-Pass

Question: An AM radio receiver needs to select a 9 kHz-wide audio channel centred at 1 MHz. Sketch the response of a band-pass filter that would do this, and state the two corner frequencies.

Solution: f_low = 1.000 − 0.0045 = 995.5 kHz; f_high = 1.000 + 0.0045 = 1004.5 kHz. The pass-band is the 9 kHz between these. The filter response is flat between f_low and f_high and rolls off above and below. A second-order band-pass is the textbook minimum; commercial AM receivers use ceramic filters of order 4 or higher to achieve sharper skirts.

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A Worked Example — Designing a Two-Stage Audio Filter Chain