Elliptical Filters — Maximum Stopband Performance

The elliptical filter — also called the Cauer filter after German mathematician Wilhelm Cauer — achieves the sharpest possible rolloff of any realizable filter for a given number of components. Like the Chebyshev, it accepts ripple in the passband. But it goes further: it also accepts a fixed minimum attenuation floor in the stopband rather than continuously increasing attenuation. In exchange for this bounded stopband performance, the elliptical filter can achieve its minimum stopband attenuation at a frequency much closer to the passband edge than either the Butterworth or Chebyshev — often within 1.1 to 1.3 times the passband edge frequency.

This property makes the elliptical filter invaluable for demanding ham radio applications: IF roofing filters that must cut off sharply just outside a 2.7 kHz SSB passband, band-pass filters that must have extremely high near-channel rejection, and any situation where you need maximum selectivity with minimum circuit complexity. The elliptical filter achieves this through transmission zeros — frequencies at which the filter response drops to exactly zero (theoretically infinite attenuation) before the stopband floor.

Transmission Zeros: The Key to Sharp Rolloff

A transmission zero is a frequency at which the filter's transfer function equals exactly zero — the signal is completely blocked. In an LC ladder filter, a transmission zero arises when a resonant circuit in the ladder is tuned to resonate at that frequency.

In a Butterworth or Chebyshev filter, all the resonances are at infinity — the ladder's shunt capacitors resonate infinitely far up in frequency and the series inductors present infinite impedance only at infinity. The rolloff is therefore gradual and asymptotic. The elliptical filter adds resonant circuits tuned to finite frequencies in the stopband, producing deep notches that dramatically increase attenuation at specific frequencies — and these notches are placed just above the passband edge, pulling the effective rolloff down to a much narrower transition band.

In a low-pass elliptical filter, these resonators appear as shunt tank circuits (a capacitor in series with an inductor, forming a parallel LC resonator to ground). At resonance, this tank presents a short circuit to ground, sending the signal current to ground at that frequency. In the schematic, you see the familiar series inductors and shunt capacitors of the Chebyshev ladder, but some shunt capacitors are replaced by shunt LC tanks. Each tank adds one transmission zero.

Elliptical (Cauer) low-pass filter response. The passband shows equiripple behavior (like Chebyshev), and the stopband shows equiripple between the transmission zero notches. The sharp transition band results from the finite-frequency transmission zeros placed just above f_c.

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The number of transmission zeros equals the number of resonators in the filter, which equals n/2 for even-order filters and (n−1)/2 for odd-order filters, where n is the total filter order. A 5th-order elliptical filter has two transmission zeros, producing two deep notches in the stopband. A 7th-order filter has three notches.

The Shape Factor

The shape factor is the single most important specification for comparing the selectivity of different filter designs. It is defined as the ratio of the −60 dB bandwidth to the −3 dB bandwidth (or sometimes −60 dB to −6 dB, depending on the convention used):

Typical shape factors for 5th-order LC filters:

For an SSB receiver with a 2.7 kHz IF filter, a 5th-order elliptical filter might achieve 60 dB rejection at only 3.5 kHz from the filter center — only 0.8 kHz outside the passband edge. The equivalent Butterworth would need to be 11 kHz away from the edge for the same rejection. The elliptical filter simply provides far greater adjacent-channel rejection for the same component count.

Elliptical Filter Topology

An elliptical low-pass filter looks like a Chebyshev ladder with some of the shunt capacitors replaced by series LC resonators (an inductor and capacitor in series, placed in the shunt position). In the schematic:

Each shunt resonator [Lₐ + Cₐ in series, shunt to ground] forms a series-resonant circuit that shorts to ground at its resonant frequency f₀ = 1/(2π√(LC)). This is the transmission zero frequency. By tuning each resonator to a specific frequency just above the passband edge, you create the notches that sharpen the rolloff.

5th-order elliptical low-pass filter schematic (top) vs. the equivalent 5th-order Chebyshev (bottom). The elliptical replaces the two simple shunt capacitors with series LC resonators. The resonators create transmission zeros — deep notches — just above the passband edge.

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The complexity increase is real: each transmission zero adds one inductor to the design. A 5th-order Chebyshev uses 3 inductors and 2 capacitors; the 5th-order elliptical uses 3 series inductors plus 2 resonator inductors plus 2 resonator capacitors plus the shunt capacitors — a total of 5 inductors and 4 capacitors, almost double the component count. However, you may achieve the same selectivity with a 3rd-order elliptical as a 7th-order Chebyshev, overall saving components.

Response Comparison: All Four Classical Families

It is worth putting all four filter families side by side to understand when each is appropriate. This comparison assumes a 5th-order filter in each case with 50 Ω source and load:

Ham Radio Applications of Elliptical Filters

Receiver IF Roofing Filters

The most common ham radio application for elliptical filters is the IF roofing filter in a superhet receiver. A roofing filter is a wide-band bandpass filter placed immediately after the first mixer, before the first IF amplifier. Its job is to reject strong signals outside the desired band before they can saturate the IF amplifier chain. The roofing filter does not need to pass a narrow SSB or CW passband — that is done by a narrow crystal filter downstream — but it must have steep enough skirts to reject adjacent-band transmitters that might otherwise overload the first IF amplifier.

An elliptical bandpass roofing filter can achieve 60–80 dB rejection at frequencies just a few kilohertz outside a 30–500 kHz IF passband. This prevents the "reciprocal mixing" distortion that occurs when a strong nearby signal mixes with the receiver's VFO noise sidebands to produce noise within the desired passband.

Transmitter Band-Pass Filters

In a modern software-defined transmitter, the audio from the DSP chip is upconverted to RF. An elliptical band-pass filter on the transmitter output can simultaneously provide harmonic suppression (low-pass function) and adjacent-band rejection (high-pass function) with fewer total components than separate LPF and HPF stages. The narrow transition band means the filter can pass the entire 40m band (7.0–7.3 MHz) while providing strong attenuation at 6.9 MHz (below the band) and 7.4 MHz (above the band) to prevent interference from the transmit chain to other bands.

Preselector Bandpass Filters

A receiver preselector is a switched bank of band-pass filters placed before the first mixer to reduce out-of-band signals. For HF receivers covering 1.8–30 MHz in multiple bands, a bank of 6–8 elliptical bandpass filters can provide 60 dB or more of adjacent-band rejection, greatly reducing the risk of intermodulation from strong broadcast stations. Many high-performance HF transceivers (Icom IC-7610, Elecraft K3, FLEX-6000 series) include switchable preselector filters based on elliptical or near-elliptical LC designs.

Practical Limits and Tradeoffs

The elliptical filter's outstanding selectivity comes with real practical costs:

Component sensitivity: Elliptical filter performance degrades more rapidly with component tolerance than Butterworth or Chebyshev filters. A 5% capacitor tolerance might shift a Butterworth cutoff by 2%, but it can shift an elliptical filter's transmission zero by 5–10%, dramatically degrading the notch depth. For the best results, use 1% or 2% tolerance capacitors for the resonator elements, and measure and trim inductors to within 1% of design values. This makes elliptical filter construction more demanding than Butterworth or Chebyshev.

Stopband floor: Unlike Butterworth and Chebyshev filters whose attenuation increases monotonically above the stopband, the elliptical filter's response comes back up between the transmission zero notches and settles to a floor. If the signal you are trying to reject falls between two notches (in the "ripple" of the stopband), you may see much less attenuation than expected. Verify that the frequencies you need to reject fall within the notches or the stopband floor, not in the raised regions between notches.

Phase distortion: The elliptical filter has the worst phase response of all classical filter families. Near the transmission zeros, the phase changes extremely rapidly with frequency. This produces large group delay variation, which distorts pulse waveforms and digital signals. For ham radio voice and CW operation, this is rarely a concern. For digital modes (PSK31, FT8, etc.), filter the audio gently at audio frequencies before applying transmit equalization — avoid elliptical filters for this purpose.

Design complexity: Elliptical filter prototype values are not easily tabulated for all combinations of ripple, stopband floor, and filter order because there are three independent specifications (passband ripple, stopband attenuation level, and the frequency at which the stopband floor is first reached) rather than two. Filter design software or specialized tables (such as those in Zverev's "Handbook of Filter Synthesis") are recommended rather than hand calculation.

Using Design Software for Elliptical Filters

Because of the three-parameter specification and complex mathematics, elliptical filter design is almost always done using software. Several free tools are available:

Iowa Hills RF Filter Design is a free Windows application that computes Butterworth, Chebyshev, and elliptical LC filter component values for arbitrary specifications. Input the passband edge frequency, passband ripple, stopband edge frequency, and stopband attenuation; the software outputs normalized prototype values and scaled component values for any impedance level.

AADE Filter Design (Another Amateur Design Environment) is another free Windows tool popular among ham radio operators. It also outputs a SPICE simulation netlist so you can verify the response before building.

RF Tools online calculators and Analog Devices Filter Design Tool are web-based options that handle LC filter synthesis including elliptical types.

When using any filter design tool for an elliptical filter, always verify the transmission zero frequencies using SPICE simulation before ordering components. Small variations in the tool's optimization algorithm can place the zeros slightly differently than expected, which changes which component values are most critical.