Band Stop Filters
A band stop filter is the complement of a band pass filter. Where the band pass filter passes a specific frequency range and rejects everything outside it, the band stop filter rejects a specific frequency range and passes everything outside it. It blocks a "notch" in the frequency spectrum — which is why the most common form of band stop filter is called a notch filter.
Band stop and notch filters are indispensable tools for interference rejection. When a single unwanted frequency is degrading a signal — mains hum, a switching-supply harmonic, a clock that leaks into an analog circuit, an acoustic feedback squeal — a notch filter can carve that frequency out of the spectrum with minimal effect on the wanted frequencies nearby. Every audio equalizer uses band stop filters to cut specific frequency ranges, and many digital audio processors include automatic notch filters that find and remove a feedback tone. Understanding the physics behind these functions helps you use them effectively.
Band stop (notch) filter frequency response. The passband is flat both below and above the notch frequency f0. At f0, the signal is deeply attenuated — ideally to zero. The notch width is defined by the −3 dB bandwidth, and its depth is determined by the Q factor of the notch circuit.
View LargerThe Band Stop Concept: The Complement of Bandpass
Every filter type has a complement — a filter whose response is the inverse. The complement of a low pass filter is a high pass filter (and vice versa). The complement of a band pass filter is a band stop filter.
Mathematically, if HBP(f) is the transfer function of a band pass filter, then the band stop filter has transfer function:
HBS(f) = 1 − HBP(f)
Where HBP(f) = 1 (passband), HBS(f) = 0 (stopband)
Where HBP(f) = 0 (stopband), HBS(f) = 1 (passband)
This complementary relationship can be implemented directly with a bridge circuit (like the twin-T below), or indirectly by combining a low pass filter (passing everything below the notch) with a separate high pass filter (passing everything above the notch) and summing their outputs. In the summed output, only the notch frequency is missing — it was attenuated by both filters in their respective stopbands.
The key parameters of a band stop filter are identical in form to those of a band pass filter:
- Notch frequency f0: The frequency of maximum attenuation (the center of the notch)
- Notch bandwidth: The frequency range between the −3 dB points around the notch
- Notch depth: The maximum attenuation at f0, measured in dB
- Q factor: f0 divided by notch bandwidth — high Q gives a narrow, deep notch
The LC Band Stop Filter
The simplest RF band stop filter uses the dual of what you learned for band pass filters. Recall that a series resonant LC circuit presents minimum impedance at its resonant frequency — so if placed in series in a signal path, it short-circuits the signal to ground at that frequency, creating a notch. Alternatively, a parallel resonant LC circuit placed in series in a signal path presents maximum impedance at resonance, blocking the signal at that frequency and creating a notch.
Both configurations create a notch, but their implementations differ:
Series-Connected Parallel Resonant Circuit
Place a parallel LC tank in series with the signal path. At resonance, the tank presents maximum impedance (Z = RD = Q²RS), greatly attenuating the signal. At all other frequencies, the tank impedance falls and the signal passes. The notch frequency is:
f0 = 1 / (2π√(LC))
Notch depth ≈ −20 log10(Rsource / (Rsource + RD))
For deep notch: RD >> Rsource, requiring high Q
Shunt-Connected Series Resonant Circuit
Place a series LC circuit from the signal line to ground (in parallel with the load). At resonance, the series LC presents minimum impedance to ground (Z = RS), shorting the signal to ground and creating a notch. At other frequencies, the series LC presents high impedance and does not load the circuit. This is the more common RF notch configuration:
Notch depth ≈ 20 log10(RS / (RS + Rsource/2))
For deep notch: RS << Rsource, requiring high Q
Notch frequency: f0 = 1 / (2π√(LC)) — same formula
In both configurations, the notch frequency is determined by L and C. Tunable notch filters use a variable capacitor to sweep f0 across a frequency range, so the notch can be placed exactly on the unwanted signal.
A switching power supply generates a harmonic at 10.0 MHz that leaks into a 50 Ω measurement line and swamps the signal you are trying to measure. Design a shunt-connected series LC notch to attenuate it. The system impedance is 50 Ω. Choose C = 100 pF.
Step 1 — Find L for f0 = 10 MHz:
L = 1 / (4π² × f0² × C) = 1 / (39.478 × 10¹⁴ × 10⁻¹⁰)
= 1 / (39.478 × 10⁴) = 1 / 394,784 = 2.53 µH
Step 2 — Reactance at resonance:
XL = 2π × 10×10⁶ × 2.53×10⁻⁶ = 158.9 Ω
Step 3 — Notch depth (assuming Q = 100 for a good air-core coil):
RS = XL / Q = 158.9 / 100 = 1.59 Ω
With a shunt series-resonant notch in 50 Ω system, notch depth ≈ 20 log10(2 × RS / Rsource) = 20 log10(3.18/50) = 20 log10(0.0636) = −23.9 dB
Result: Approximately 24 dB of notch depth. For a 10 MHz interference signal that is, say, 30 dB above your wanted signal, this reduces it to 30 − 24 = 6 dB above — a significant improvement but possibly not enough. Adding a higher-Q inductor (Q = 200) would halve RS and approximately double the notch depth to ~30 dB. Curing the interference at its source — better filtering on the power supply — is usually the better fix.
The Twin-T Notch Filter
The Twin-T is the most popular audio-frequency notch filter design. It gets its name from its circuit topology: two T-networks (each shaped like the letter T) connected in parallel but with opposite phase responses. At the notch frequency, the two paths cancel, producing a null in the output. At other frequencies, one path or the other dominates and passes the signal.
The Twin-T consists of three resistors and three capacitors arranged in a specific bridge configuration:
Component values: R, R, R/2 (resistors) and C, C, 2C (capacitors)
Notch frequency: f0 = 1 / (2πRC)
With ideal components, notch depth is infinite (complete cancellation at f0)
In practice, component tolerances limit notch depth to 40–60 dB with standard components, 60–80 dB with matched 1% components
The Twin-T is passive (no active components required) and produces a very deep notch at a fixed frequency determined by R and C. Its limitation is that the notch frequency cannot be varied without changing multiple components simultaneously — adjusting any one R or C shifts the notch and degrades the depth. This makes it suitable for fixed-frequency interference rejection (like 60 Hz hum or a known interference frequency) but not where the notch must be tuned continuously.
A tunable version (the active Twin-T or Wien-bridge notch) uses operational amplifiers and ganged variable components, but this is more of an audio electronics topic. For RF applications, LC notch circuits are generally preferred because they work at higher frequencies with simpler implementation.
Notch Depth and Q
The depth of a notch filter — how many dB of attenuation it provides at f0 — is determined primarily by the Q of the resonant element and the impedance matching to the circuit. This is a critical practical consideration, because a shallow notch (10–15 dB) may be insufficient to eliminate strong interference, while an extremely deep notch (>60 dB) requires high-Q components and careful construction.
| Notch Implementation | Typical Notch Depth | Tunable? | Frequency Range |
|---|---|---|---|
| Shunt series-LC (air core, Q = 100) | 20–30 dB | Yes (variable C) | High frequency (1–200 MHz) |
| Twin-T (1% resistors/capacitors) | 40–60 dB | No (fixed) | Audio (1 Hz–100 kHz) |
| Twin-T (matched 0.1% components) | 60–80 dB | No (fixed) | Audio (1 Hz–100 kHz) |
| Crystal notch | 60–80 dB | No (fixed frequency) | Around crystal frequency |
| DSP notch (digital audio processor) | 60–100 dB | Yes (fully tunable) | Audio IF (any baseband freq) |
In practice, component tolerances and stray reactances prevent achieving the theoretically infinite notch depth of a perfect Twin-T or perfectly matched LC circuit. A notch depth of 40 dB is considered excellent for a passive analog notch filter; 20–30 dB is typical for a simple LC notch in an RF circuit.
DSP Notch Filters in Digital Audio
Digital audio processors, hearing aids, conference systems and PA feedback suppressors include automatic notch filters that work on the audio after it has been digitized. These are fundamentally different from the analog LC or RC filters described above, though they accomplish the same goal.
The DSP notch operates by identifying the frequency of a continuous tone (a feedback squeal, a whistle, or power-line hum) in the audio spectrum and applying a digital band stop filter precisely at that frequency. Because the filter is implemented in software, it can:
- Achieve very deep notch depths (60–100 dB) regardless of component tolerances
- Be tuned to any audio frequency without changing hardware
- Track a drifting interferer by continuously updating the notch frequency
- Apply multiple simultaneous notches (feedback suppressors often place eight or more)
- Use adaptive algorithms that automatically detect and notch interference without user intervention
Understanding the analog notch filter principles in this lesson helps you understand what the DSP is doing mathematically — it is implementing the same transfer function that a Twin-T or LC notch would provide, but with perfect components that cannot be built in the analog domain.
Band Stop Filters in Practice
Acoustic Feedback Suppression
When a microphone picks up its own sound from a loudspeaker, the loop rings at whichever frequency has the most gain, producing a howl. A narrow notch at that frequency removes the howl while leaving the rest of the sound almost untouched, allowing more volume before feedback. Sound engineers "ring out" a system by finding these frequencies and notching them on an equalizer.
60 Hz Hum Elimination
Power-line hum at 60 Hz (and its harmonics at 120, 180, 240 Hz) can couple into audio chains through ground loops, inadequate shielding, or power supply ripple. A Twin-T notch at 60 Hz provides high attenuation of the hum frequency while leaving voice and music largely unaffected. Twin-T notches for 60 Hz hum rejection are straightforward to build and can dramatically improve audio quality in a noisy environment — though curing the ground loop is always the better solution.
Removing Switching and Clock Interference
Interference from switching power supplies, LED drivers, solar inverters and digital clocks often appears at specific frequencies. A tunable LC notch filter inserted in a signal path can remove a specific interferer, and a fixed notch at a known switching frequency is sometimes built into the input of a sensitive measurement circuit.
Biomedical and Sensor Signals
ECG (heart) monitors and other biomedical instruments pick up mains interference through the body. A 50 or 60 Hz notch filter is a standard part of their input circuitry, removing the hum without distorting the much slower heart signal.
Experiment: Build a Twin-T Notch for 60 Hz Hum
You need: Three 27 kΩ resistors, one 13.5 kΩ resistor (or two 27 kΩ in parallel), two 100 nF capacitors, one 200 nF capacitor (or two 100 nF in parallel), breadboard, audio oscillator or smartphone tone generator app, oscilloscope or AC voltmeter.
Theory: f0 = 1/(2πRC) = 1/(2π × 27,000 × 100×10⁻⁹) = 58.9 Hz ≈ 60 Hz
Circuit: Top T: two 27 kΩ resistors in series (R−R), with 100 nF from center tap to ground. Bottom T: two 100 nF capacitors in series (C−C), with 13.5 kΩ from center tap to ground. Input connects to the left ends of both T networks; output connects from the right ends of both.
Test:
- Apply a 60 Hz tone at 1 V and measure output. Should be greatly reduced.
- Apply tones at 30 Hz, 120 Hz, 300 Hz, 1 kHz. Output should be near input level at these frequencies.
- Sweep slowly through 50–70 Hz to find the exact notch frequency — the deepest null is f0.
Expected result: The 60 Hz tone is attenuated by 30–50 dB depending on component tolerances. The Twin-T demonstrates how analog notch depth is limited by component matching — the closer the R and C values are to the ideal ratios, the deeper the notch.
Frequently Asked Questions
What is the difference between a band stop filter and a notch filter?
In common usage, the terms are interchangeable. Technically, "notch filter" emphasizes a very narrow rejection bandwidth relative to the center frequency (high Q) — it notches out a single frequency or very narrow band. "Band stop filter" is the more general term that includes both narrow notch filters and wider-bandwidth stop bands. In practice, "notch filter" is used most often because the application usually involves rejecting a single tone or narrow interference source, which requires a high-Q narrow notch.
If a notch filter eliminates a frequency, does it also eliminate sidebands near that frequency?
Yes — any signal component within the notch bandwidth is attenuated, not just the exact center frequency. The notch filter's 3 dB bandwidth defines the range of frequencies that are more than 3 dB attenuated. If the notch is too wide, it also attenuates desired signal content near the interference frequency. This is one reason that high-Q notch filters are preferred — they suppress the interferer more precisely without significantly affecting nearby desired frequencies. A DSP notch can be made very narrow (a few hertz) because it is not limited by component Q.
Why does a Twin-T have two capacitors of value C and one of 2C, rather than three identical capacitors?
The Twin-T achieves its null by creating two signal paths that cancel at f0. The top T carries the signal through two equal resistors and shunts a capacitor to ground at the center; the bottom T carries the signal through two equal capacitors and shunts a resistor to ground at the center. For perfect cancellation at f0, the component ratios must satisfy specific mathematical conditions. Working through the network analysis shows that the center shunt capacitor must be 2C (and the center shunt resistor R/2) to produce the exact 180-degree phase shift and equal amplitude required for null. Any other ratio produces imperfect cancellation and a shallower notch.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.