What Is a Band Reject Filter? Theory, Design, and When to Use a Notch

A band reject filter is an electronic circuit that allows all signal frequencies to pass through unchanged except for a specific target range (the stopband), which it heavily attenuates. In a real circuit, it changes the frequency spectrum by carving a deep "hole" or notch at a target frequency, dropping its amplitude by 20 dB to 60 dB while leaving adjacent frequencies intact. Beginners commonly confuse it with a band-pass filter (which does the exact opposite, keeping only the target band and rejecting everything else) or assume all band reject filters are "notch" filters. In reality, a notch filter is simply a band reject filter with a very narrow stopband and a high Q-factor (Quality factor).

The Core Mechanics: Carving Out the Stopband

Every filter is defined by its transfer function, which dictates how the output voltage relates to the input voltage across different frequencies. For a band reject filter, the transfer function features a "zero" at the center frequency ($f_c$). At this exact frequency, the circuit creates destructive interference or routes the signal to ground, resulting in maximum attenuation.

The width of the rejected band is defined by the Q-factor.

• Wide Band Reject (Low Q): Rejects a broad swath of frequencies (e.g., blocking everything between 300 Hz and 3 kHz). Often used in telecommunications to clear out a wide noisy band.
• Narrow Band Reject / Notch (High Q): Rejects a razor-thin slice of the spectrum (e.g., blocking exactly 59.5 Hz to 60.5 Hz). This is the most common variant on the electronics bench, used to eliminate specific interference like AC mains hum without degrading the surrounding audio or sensor data.

Worked Numeric Example: Killing 60 Hz Mains Hum

Let’s design a passive Twin-T RC notch filter to eliminate 60 Hz AC mains hum from a sensitive analog sensor signal. The Twin-T topology uses two T-shaped RC networks in parallel: one low-pass and one high-pass. When their outputs are summed, the phase shifts cancel out the target frequency.

First, we select a standard capacitor value to keep the resistor values in a practical range (avoiding massive electrolytics or tiny parasitic-prone picofarads). Let’s choose C = 100 nF (0.1 µF).

Solving for R:
$R = \frac{1}{2 \pi \times 60 \times 100 \times 10^{-9}}$
$R = \frac{1}{0.000037699}$
R ≈ 26,525 Ω (26.5 kΩ)

The standard Twin-T topology requires specific ratios: two resistors of value R, one shunt resistor of R/2, two capacitors of value C, and one shunt capacitor of 2C.

Where You Meet This in Practice

You will rarely see a band reject filter used for power delivery; they are almost exclusively signal-conditioning components. Here is where they earn their keep on the bench and in the field:

• Audio Engineering & Instrumentation: Removing 50 Hz (EU/UK) or 60 Hz (US) mains hum from microphone preamps, guitar pedals, and analog synthesizers without destroying the fundamental bass frequencies.
• Biomedical Sensors: ECG and EEG amplifiers use aggressive 50/60 Hz notch filters to reject the massive electromagnetic interference generated by hospital lighting and power grids, which would otherwise drown out the microvolt-level biological signals.
• Software Defined Radio (SDR): Rejecting a powerful local FM broadcast station that is overloading the front-end LNA (Low Noise Amplifier) and creating intermodulation distortion across the rest of the RF spectrum.
• Vibration Analysis: In industrial IoT, accelerometers mounted on motors use mechanical or digital band reject filters to ignore the known, constant rotational frequency of the motor shaft, allowing the DSP to focus only on the higher-frequency harmonics that indicate bearing wear.

Decision Path: Choosing Your Filter Topology

Selecting the right band reject filter depends entirely on your frequency range, signal type, and whether you are working in the analog or digital domain. Use this decision matrix to lock in your approach.

The Concrete Bench Pick

If you are building an analog sensor or audio interface on the bench and need to kill 50/60 Hz hum, do not rely on a purely passive Twin-T. The output impedance will interact with your next stage and ruin the notch depth.

Default Recommendation: Build an active Fliege or Twin-T notch filter using the Texas Instruments OPA1612 dual op-amp. The OPA1612 offers ultra-low noise (1.1 nV/√Hz) and extremely low THD, making it the definitive choice for precision audio and sensor filtering. Pair it with Susumu RG series 0.1% thin-film resistors and Kemet C0G/NP0 film capacitors. This combination guarantees a notch depth exceeding -50 dB at exactly 60.0 Hz, which you can verify on a bench spectrum analyzer. For further reading on active filter design topologies, refer to the Texas Instruments filter design guide and the Analog Devices Linear Circuit Design Handbook.

Common Pitfalls and Component Realities

When simulating a band reject filter in LTspice, the notch will look like a perfect, infinitely deep V-shape dropping to -120 dB. In the physical world, parasitic effects and component realities limit your performance.

• Capacitor Microphonics: Never use X7R or Y5V multilayer ceramic capacitors (MLCCs) in the signal path of an audio notch filter. These dielectrics are piezoelectric; they act as microphones and will inject physical vibration noise directly into your signal. Always use C0G/NP0 ceramics or polypropylene film capacitors.
• Op-Amp Slew Rate Limiting: If your signal contains high-frequency transients, an op-amp with a low slew rate will distort the signal before it even reaches the filter network. Ensure your op-amp’s gain-bandwidth product (GBWP) is at least 100x higher than the highest frequency present in your passband.
• Ground Loops: A 60 Hz notch filter will not save you if your ground loop is injecting 500 mV of hum into a 10 mV sensor signal. The filter will attenuate the hum, but the signal-to-noise ratio is already destroyed. Fix the grounding topology (star grounding, differential signaling) before relying on a notch filter as a band-aid.