Filter Poles and Zeros Explained

Filter poles and zeros determine how a filter changes signal amplitude and phase with frequency.

This article explains their meaning through a first-order RC low-pass filter, then shows how design engineers can use pole-zero information to assess cutoff frequency, rejection and practical component selection.

Key Takeaways

Start by distinguishing locations in the complex frequency plane from operating frequencies on a measured frequency-response plot.

• Poles are singularities of a transfer function; zeros are values of complex frequency at which its numerator becomes zero, after common factors have been cancelled.
• The ideal unloaded RC low-pass filter has one pole at s = −1/(RC), unity DC gain and no finite zero.
• Real-frequency response is evaluated at s = jω. A pole on the negative real axis does not imply infinite gain at a real operating frequency.
• The RC cutoff is ω_c = 1/(RC) in rad/s, or f_c = 1/(2πRC) in Hz. At cutoff, voltage magnitude is 1/√2 of its DC value, approximately −3 dB.
• All poles, zeros and gain factors contribute to the complete response. Source impedance, loading, losses and parasitic elements must be included when applying the ideal model.

What Filter Poles and Zeros Mean

A filter transfer function describes the relationship between input and output in a linear, time-invariant model with zero initial conditions. The explanation below develops the RC example used in the original Knowles Precision Devices article and distinguishes the mathematical model from the response measured at real frequencies.

Let X(s) be the input signal in the Laplace domain, Y(s) the output signal and H(s) their transfer function. Complex frequency is s = σ + jω, where σ is the real component, ω is angular frequency in rad/s and j is the imaginary unit, with j² = −1. For a voltage transfer function, H(s) is the output voltage divided by the input voltage. [1]

For a rational transfer function, poles are roots of the denominator that remain after common numerator and denominator factors have been cancelled. Zeros are roots of the remaining numerator. Near an uncancelled pole, the magnitude of the mathematical transfer function becomes unbounded; at a zero, it becomes zero. These are statements about the complex s-plane, not necessarily about frequencies encountered in a sinusoidal measurement. [1]

For the stable filters considered here, the steady-state sinusoidal frequency response is obtained by setting σ = 0 and evaluating H(jω). This means moving along the imaginary axis, rather than through every possible location in the s-plane. Ordinary frequency f in Hz and angular frequency are related by ω = 2πf. [1]

The First-Order RC Low-Pass Example

A series resistor and a shunt capacitor provide a direct way to connect a pole location with a familiar cutoff frequency. Figure 1 shows the original RC low-pass circuit, with the output taken across the capacitor.

Let R be resistance in Ω and C capacitance in F. Assume ideal components, a zero-impedance voltage source and an unloaded output. The capacitor impedance is 1/(sC), so the voltage-divider relationship gives Eq. (2). These assumptions matter: a finite source impedance or load changes the circuit response. [1, 2]

Here a is a positive pole-rate parameter, with units of s⁻¹. At s = 0, Eq. (2) gives H(0) = 1: the ideal unloaded circuit has unity DC gain. Its denominator becomes zero at s = −a = −1/(RC), so it has one pole on the negative real axis. Its constant numerator has no finite root, so the circuit has no finite zero. Its response approaches zero as frequency tends to infinity, which is also described as a zero at infinity. [1]

Keep the first-order gain normalisation explicit

The original general first-order example used Y(s) = X(s)/(s + a). This is a useful algebraic form, but it is not automatically the unity-gain RC voltage response. More generally, introduce a numerator constant K with units that make H(s) dimensionless. Eq. (3) then makes the DC gain explicit.

For the RC voltage transfer function, K = a and Eq. (3) becomes Eq. (2). The numerator changes the gain but not the pole location. In a normalised, dimensionless treatment the numerator can instead be written as unity, provided the frequency and gain scaling are defined consistently. [1]

Evaluate the response along the imaginary axis

Replacing s with jω in the general first-order expression gives Eq. (4). The output still equals the transfer function multiplied by the input; Y alone should not be used as another name for H.

In the interpretation of Figure 2, the pole is at s = −a and the frequency-response evaluation point is s = jω. Their separation is represented by the vector a + jω, whose length is √(a² + ω²). For fixed K, increasing that distance decreases the magnitude in Eq. (4). The pole itself is not on the imaginary-axis path when a is positive. [1]

For the unity-DC-gain RC circuit, Eq. (5) gives both magnitude and phase. The angle φ is the output phase relative to the input.

• At ω = 0, the evaluation point is as close to the pole as it can be while remaining on the imaginary axis. The RC magnitude is at its maximum value of 1.
• As ω tends to infinity, the distance from the pole increases without bound and the ideal RC magnitude tends to zero.
• At ω = a, the distance is a√2. The magnitude is therefore 1/√2 of its DC value and the phase is −45°.

Recover the RC cutoff frequency

The cutoff condition is ω = a, not jω = a. Substitution into Eq. (5) gives Eq. (6), including the distinction between angular frequency and frequency in Hz. The decibel value is calculated from a voltage-magnitude ratio relative to the DC gain.

This is conventionally called the −3 dB cutoff. For this single-pole circuit, the negative-real pole coordinate and the positive angular cutoff frequency have equal magnitudes. That simple correspondence should not be extended automatically to higher-order filters, whose response depends on all poles and zeros. [1]

Using Pole and Zero Information to Enhance Your Filter Designs

The single-pole example provides useful intuition, but a practical filter must be assessed as a complete transfer function. Pole and zero locations influence selectivity, phase and transient behaviour together, rather than acting as independent pass and reject controls. [1]

Interpret distance rules as individual contributions

For an individual denominator factor, approaching its pole increases that factor’s contribution to transfer-function magnitude. For an individual numerator factor, approaching its zero decreases that contribution. The total magnitude combines all numerator and denominator factors with the gain constant, so the complete response need not vary monotonically with distance from one selected pole or zero. [1]

A zero exactly on the imaginary axis can create an ideal transmission null at its corresponding real frequency, provided an uncancelled denominator factor does not coincide with it. Zeros away from that axis can shape rejection without forcing the real-frequency response to zero. Practical losses, tolerances and coupling can limit the depth of a measured notch. [1]

A simple zero example: the RC high-pass filter

To demonstrate a finite zero, exchange the positions of R and C in the simple voltage divider and take the output across R. Under the same ideal-source and unloaded-output assumptions, the resulting first-order high-pass response is Eq. (7). [1]

Eq. (7) has a zero at s = 0 and a pole at s = −a. Consequently, its DC output is zero and its magnitude approaches unity as frequency increases in the ideal model. The zero explains the DC rejection, while the denominator establishes the first-order transition. Compared with the low-pass example, the pole is unchanged but the numerator changes the filter type. [1]

Pole Q, stability and scope limits

For stable causal continuous-time rational filters, uncancelled transfer-function poles lie in the left half of the s-plane. Complex-conjugate poles close to the imaginary axis can produce sharper resonant features and more persistent ringing. Pole Q describes this behaviour for a second-order pole pair; it should not be confused with the Q of an individual capacitor or inductor. [1]

The RC example is a small-signal voltage-filter model. It does not establish the suitability of a mains EMI filter or the stability of a switch-mode power converter. In RF and signal applications, preserve the intended source and load conditions. In power applications, assess current-dependent component behaviour and the interaction with the connected converter or load separately.

Design and Selection Checklist

Use the ideal pole-zero model to establish the intended response, then verify that the actual components and terminations preserve it. Loading is especially important when passive sections are connected directly: the overall response is not generally the product of the isolated, unloaded section responses. [2]

• Define the signal path: specify whether the transfer quantity is voltage, current or a transmission parameter, and record the intended source and load impedances.
• Check the whole response: compare passband loss, cutoff or bandwidth, stopband rejection, phase and transient requirements rather than relying on a single pole location.
• For discrete RC filters, compare resistance and capacitance tolerance, temperature behaviour, voltage rating and resistor dissipation. Include changes in effective capacitance under the intended operating conditions where relevant.
• For LC or packaged filters, compare losses, impedance or frequency-response data, current and voltage ratings, parasitic behaviour and the manufacturer’s stated test terminations.
• Include the source, load and adjacent stages in simulation. Do not multiply isolated passive-section responses unless the loading assumptions are justified.
• Compare catalog and custom options against the same electrical, mechanical and environmental requirements. For a custom filter, agree the test conditions and acceptance limits before approving the design.
• Check required qualification and safety separately from frequency response. A satisfactory pole-zero model does not establish suitability for connection to mains or compliance with an equipment safety requirement.
• Measure the assembled circuit with its intended terminations. Include the measurement instrument and probe loading when comparing the result with the model.

Conclusion

Poles and zeros connect the algebra of a transfer function with frequency-response and transient behaviour. The RC low-pass example has a pole at s = −1/(RC), while its real-frequency cutoff occurs at ω = 1/(RC). A high-pass version retains that pole but introduces a zero at the origin.

Use these relationships to interpret and design the complete response, not to place poles and zeros independently of gain, stability or loading. The final design must be checked with realistic component behaviour, source and load impedances, and measurements appropriate to the application.

FAQ

These answers summarise the distinctions most relevant when interpreting filter transfer functions and selecting a practical implementation.

How to Check an RC Filter Using Its Pole

Use this procedure for the ideal first-order low-pass circuit, then extend the model to include the actual circuit and measurement conditions.

Further Reading

These related articles connect the pole-zero explanation with circuit fundamentals, quality factor and practical RF or EMI filter implementation.

• Basic Filter Circuits Explained: develops the voltage-divider and impedance relationships behind RC and LC filters.
• Filter Q Factor Explained: distinguishes component, bandpass and pole Q.
• RF Inductors and Filters: connects filter topology with RF component behaviour and implementation.
• Insertion Loss and Performance in EMI Filtering: explains why terminations, current and physical construction affect practical rejection.

References

The numbered references support the transfer-function relationships and loading limitations discussed above.

1. Analog Devices, A Filter Primer, Analog Devices technical article, 2002.
2. Analog Devices, Activity: Cascaded RC Low Pass Filters – ADALM2000, Analog Devices Wiki, approved revision, 2022.

Latest technical review and revision: 5 November 2026. Definitions, gain normalisation and cutoff notation corrected; practical design guidance, FAQ and verification procedure added.