Harmonic Rejection Filter Overview

Updated 25 January 2026

Harmonic rejection filters are engineered systems designed to suppress specific harmonic frequencies by introducing notches in the transfer function, ensuring improved signal integrity.
They are applied in power electronics, RF/microwave circuits, and signal processing through passive, active, and hybrid designs that demonstrate significant THD reductions.
Advanced techniques such as wavelet transforms and adaptive control algorithms enable real-time harmonic disturbance rejection under dynamic operation conditions.

A harmonic rejection filter is an engineered system or circuit that selectively suppresses or cancels specific harmonic components within a signal, typically to improve power quality, electromagnetic compatibility, or communication fidelity. Depending on the application domain—power electronics, RF/microwave circuits, or signal processing—varied methodologies and implementations are used, spanning from passive RLC arrangements to active filters employing control algorithms, wavelet transforms, or discrete-time adaptivity.

1. Principles and Definitions

A harmonic is a spectral component of a signal whose frequency is an integer multiple of a fundamental frequency. Harmonics are pervasive in systems involving nonlinear loads (industrial power grids), switching regulators, and RF front-ends utilizing frequency translation. Harmonic rejection filters (HRFs) are designed to provide:

Deep attenuation at targeted harmonic frequencies (e.g., 2nd, 3rd, 5th).
Minimal impact on desired (fundamental or signal) band.
High selectivity and robustness to system or environmental variations.

The core principle is to design a transfer function—for continuous- or discrete-time domains—such that the magnitude response exhibits notches (zeros) at undesired harmonics, thus shunting, cancelling, or reflecting these components while preserving desired signal content.

2. Harmonic Rejection in Power Systems

2.1 Passive Filters

Single-tuned (notch) and second-order high-pass filters are widely deployed for industrial harmonic mitigation. A single-tuned filter, a parallel R‑L‑C branch, is connected at the point of common coupling and sized such that its resonant impedance is minimized at a target harmonic $f_r$ :

$f_r = \frac{1}{2\pi\sqrt{LC}}$

High-order harmonics are addressed with second-order high-pass filters, consisting of a series capacitor and parallel R‑L, with corner frequency $f_c$ given by design constraints. Tuning, quality factor $Q$ , and proper detuning for system-impedance robustness are critical; see the design approach in (Memon et al., 2016).

2.2 Active and Hybrid Filters

Shunt active power filters (SAPF) inject compensating currents determined by instantaneous power theory or pq methods. The filter synthesizes a current $i^*(t)$ whose harmonic content is anti-phase with the measured load harmonics, tracking it with a closed-loop controller (often hysteresis-based). Quantitatively, simulated and experimental THD reductions of over 90% are reported—e.g., from 40% to 3% in (Adrian, 2010).

Hybrid active power filters combine passive filter branches (for dominant harmonics) with a voltage-source converter capable of dynamic compensation of residual harmonics and power factor correction, tracked via instantaneous power theory and implemented using a hysteresis controller. IEEE-519-compliant THD reductions (e.g., from 20.77% to 1.97%) are achieved, favorably balancing cost, efficiency, and adaptability (Memon et al., 2016).

2.3 Wavelet-based Harmonic Detection

In applications where harmonics are time-varying or nonstationary (as in accelerator magnet supplies), wavelet-based active power filters (APFs) decompose measured current signals using a discrete wavelet transform (DWT, e.g., Daubechies-5 with five levels). Harmonic components are isolated in frequency bands, then cancelled by injecting a compensating current. With $f_s=2048$ Hz and DWT computation on a DSP/FPGA, this approach delivers sub-2 ms response, THD suppression from 72% to <5%, and high robustness to load transients (Xiaoling et al., 2013).

3. Harmonic Rejection in RF and Microwave Circuits

3.1 N-Path Filters and Harmonic Responses

N-path filters, based on periodically switched capacitor banks controlled by multi-phase local oscillators (LO), operate as highly selective band-pass filters. However, standard rectangular (1/N duty) clocks create strong responses not only at the desired LO frequency $f_{\mathrm{LO}}$ but also at its odd harmonics (3 $f_{\mathrm{LO}}$ , 5 $f_{\mathrm{LO}}$ ). For open-loop N-path filters, attenuation at harmonics is limited (e.g., for $N=8$ , only 2–6 dB suppression at k=3 or 5) (Rayudu et al., 2021).

3.2 PWM-LO and Frequency-Translation Feedback

Recent advancements employ pulse-width modulated (PWM) LO waveforms that approximate sinusoidal mixing while suppressing higher harmonics. Frequency-translation feedback loops are used to sense harmonic downconversion products (e.g., at 3rd and 5th harmonics), upconvert them using a PWM-LO, and feed them back to the RF input with negative feedback—effectively notching the input impedance at those frequencies and achieving >50 dB rejection ratios (Rayudu et al., 2021, Rayudu et al., 2019). Precise multi-phase PWM generation (with sufficient resolution in delay-locked loops) is essential.

RF Harmonic Rejection Filter	Open-Loop HRR (dB, 3rd/5th)	Closed-Loop HRR (dB, 3rd/5th)	Key Method
N-path + HR combiner	3 / 6	–	Polyphase summing
N-path + Freq-Trans Feedback	3 / 6	50 / 60	PWM-LO, feedback, HR comb.
PWM-LO-driven N-path	75 / 90 (sim)	–	PWM spectral shaping

3.3 Harmonic-Rejecting Resonators and BPFs

Microstrip harmonic rejection is often achieved through tailored distributed-element band-pass filters. T-shaped open-circuit stubs, inserted in parallel-coupled line filters, impose deep transmission zeros at specific harmonics (e.g., 2 $f_0$ , 3 $f_0$ ) by setting stub lengths such that $\ell_{t,m} = v_p/(4 m f_0)$ . Measured suppression can surpass 40 dB at targeted harmonics without adverse impact on passband insertion loss or return loss (Ghamsari et al., 2021).

Compact dual-band resonators (DBR), realized via microstrip coupled lines and DC-block capacitors, can be configured to present open circuits at both 2nd and 3rd harmonics, enabling harmonic recycling in rectification circuits and simplifying the filter block compared to cascaded solutions. This yields measured improvements of up to 18.4 dB (2nd harmonic) and 7.6 dB (3rd) with efficiency gains (Wu et al., 4 Jan 2026).

4. Adaptive and Control-Based Disturbance Rejection

In dynamic or uncertain environments, adaptive harmonic rejection filters are required. Estimation-based controllers (finite-time or adaptive harmonic steady-state, AHSS) operate by extracting the frequency, phase, and amplitude of unknown sinusoidal disturbances and synthesizing cancellation signals in real time.

AHSS algorithms (for SISO/MIMO LTI plants with known harmonic frequencies) update a complex weight $u_k$ to minimize output at the disturbance frequency, adaptively estimating the plant’s gain at that frequency (Kamaldar et al., 2016). Stability is enforced by Lyapunov-based analysis, and convergence yields >60 dB notch depth at the target frequency, with global asymptotic convergence except for certain pathological initializations.
Passivity-based finite-time adaptive controllers can identify and null arbitrary-frequency input disturbances in finite time, relying on regression-based parametrization and switching to a disturbance-matched internal model upon parameter convergence (Dobriborsci et al., 2020).

5. System-Level Harmonic Filter Planning

For multibus transmission systems with stochastic load models and resonance, the optimal placement and parameterization of harmonic rejection filters is a large-scale, risk-constrained optimization problem. Methodologies incorporating stochastic aggregate harmonic modeling, percentile-constrained THD indices (as per IEEE 519), and analytical impedance formulae (e.g., for C-type filters) are used. Restricted hierarchical direct search solves for filter locations and sizing, validated by Monte Carlo and modal analysis, with documented reductions in system-wide voltage THD and probabilistic risk boundedness (Akbari et al., 2019).

6. Design Guidelines and Performance Metrics

Optimal filter performance requires:

Precise tuning (capacitance, inductance, stub/electrical lengths) anchored in target harmonic frequencies.
Accurate models for control bandwidth, real-time computation (e.g., DWT latency, FPGA throughput), and feedback stability margins (particularly in active and adaptive schemes).
Robustness with respect to environmental and operational drifts, e.g., grid impedance, load variation, component tolerances, or LO jitter (in RF).
Quantitative performance evaluation: THD before/after filtering, suppression in dB at harmonics, system efficiency, response time, and compliance with power quality standards (e.g., IEEE 519).

Harmonic Order	Filter Type	L (H)	C (μF)	R (Ω)	Pre-Filter THDi (%)	Post-Filter THDi (%)
5th	Single-tuned	0.0365	11.09	0.54	20.8	4.3
...	...	...	...	...	...	...

7. Limitations and Extensions

Passive approaches are narrowband, sensitive to detuning, and susceptible to interaction with network impedance.
Active and hybrid topologies increase complexity, cost, and require advanced controllers for stability.
RF/microwave HRFs (N-path, PWM-LO, distributed stubs, DBRs) are bandwidth- and implementation-limited by device/fabrication tolerances and clocking constraints.
Adaptive solutions depend on accurate disturbance regression and require careful parameter selection to guarantee fast convergence and robustness to non-idealities (noise, model drift).
Multi-harmonic, band-adaptive, and self-tuning filters—using digital or programmable analog means—are active research directions.

References

Guo Xiaoling & Cheng Jian, "Application of Wavelet-based Active Power Filter in Accelerator Magnet Power Supply" (Xiaoling et al., 2013)
S. Irshad et al., "Design of Three-Phase Hybrid Active Power Filter for Compensating the Harmonic Currents of Three-Phase System" (Memon et al., 2016)
Kamaldar & Hoagg, "Multivariable Adaptive Harmonic Steady-State Control for Rejection of Sinusoidal Disturbances Acting on an Unknown System" (Kamaldar et al., 2016)
A. Nourian et al., "An N-Path Filter with Multiphase PWM Clocks for Harmonic Response Suppression" (Rayudu et al., 2019)
X. Chen et al., "Harmonic-Recycling Rectification Based on Novel Compact Dual-Band Resonator" (Wu et al., 4 Jan 2026)
F. Maestri et al., "Open-stub based spurious harmonic suppression method for Microstrip Coupled Line Filter" (Ghamsari et al., 2021)
S. Bhattacharya et al., "Harmonics Mitigation of Industrial Power System Using Passive Filters" (Memon et al., 2016)
A. Pashkevich et al., "Output adaptive control for linear systems under parametric uncertainties with finite-time matching input harmonic disturbance rejection" (Dobriborsci et al., 2020)
R. Rayudu et al., "A N-Path Receiver With Harmonic Response Suppression" (Rayudu et al., 2021)
M. Santamaria et al., "An Analytical Strategy for Passive Harmonic Filter Placement in Transmission Systems with Stochastic Aggregate Load Models Considering Resonant Conditions" (Akbari et al., 2019)
V. Gligor, "Design and Simulation of a Shunt Active Filter in Application for Control of Harmonic Levels" (Adrian, 2010)