Geometric-Mean Spacing for Multiband Receive Arrays

When one phased receive array must cover several bands, I use the geometric mean to choose where to start—not where to stop. It balances the band edges in frequency ratio. The useful pattern still comes from the complete installed array.

Physical Spacing Becomes Electrical Spacing

Let the physical element spacing be d, frequency be f, wavelength be λ and propagation speed be c. In free space:

The metres do not move, but d/λ rises in direct proportion to frequency. That changes the spatial phase between elements. It can move lobes and nulls, change beamwidth and alter which complex weights produce a useful forward response.

That frequency scaling is only the geometry. Real channels also include the installed element response, ground, mutual coupling, feedline transformation, amplifier or transformer response and the combiner's amplitude and phase. The array must be designed around all of them.

The Geometric Mean Balances Frequency Ratio

For lower and upper design frequencies f_L and f_H, define the geometric-centre frequency and its wavelength as:

If a trial layout uses d = kλ_g, then its normalized spacing is:

At the two band edges, the factors around k are reciprocal:

That is the useful part of the method. Neither edge is favoured in logarithmic frequency ratio. It does not say which k is best, and it does not say that equal ratio error produces equal RDF error. Array response is not generally linear or symmetric around that centre.

A Low-Band Example Shows Both the Value and the Limit

Suppose the design points are 1.83 MHz and 3.60 MHz. Their geometric centre is about 2.57 MHz, giving λ_g of about 116.8 m when c is approximated as 299.8 Mm/s.

A trial spacing selected as kλ_g becomes about 0.713k wavelengths at 1.83 MHz and 1.403k wavelengths at 3.60 MHz. Those factors are reciprocal, so the trial is log-balanced. They are not a performance prediction.

Stretch the same idea from 1.83 MHz to 7.05 MHz and the edge factors become about 0.510k and 1.963k. The physical spacing then spans almost a four-to-one range in normalized electrical spacing. A geometric centre still organizes the first trial, but the wider frequency ratio makes full-band optimization and validation much more important.

For three or more discrete bands, the geometric mean of all band centres is only one possible weighting. Equal importance per band, equal importance per hertz and priority for a specific operating window are different design problems. State the objective before choosing the centre.

Define RDF Before Optimizing It

For this article, receiving directivity factor uses the array's power response G(Ω) over solid angle Ω and a declared forward direction Ω₀. The average response is:

The forward RDF is then:

If the declared forward direction is the pattern maximum, this is peak directivity expressed in decibels. Using normalized power pattern P_n(Ω) = G(Ω)/G_max, the beam solid angle is Ω_A = ∫_4πP_n(Ω)dΩ and peak RDF becomes 10 log₁₀(4π/Ω_A).

The definition needs a three-dimensional pattern, a frequency, a polarization convention, a steering state and a reference plane. An azimuth cut alone cannot supply the sphere average. Front-to-back compares two directions; null depth describes one cancellation relative to a declared reference. Neither is RDF.

The Installed Array Is More Than an Array Factor

A compact expression for the response is:

The vector w contains the complex channel weights. The embedded-response vector a_emb contains what each installed element contributes with the other elements present and terminated in their operating states. It includes polarization, coupling, ground, support structures, feedlines and the channel response to the chosen reference plane.

Multiplying an isolated-element pattern by an ideal array factor can be a useful first sketch. In a compact array it can also hide exactly what matters. Mutual coupling changes element impedance and embedded pattern; the ground changes amplitude and phase by arrival angle; unequal feedlines and electronics change the intended weights. A triangle, square or line therefore has no universal spacing fraction that guarantees a particular RDF across bands.

Full-wave modelling can include conductors, ground, loads, networks and incident fields. Measurement must then test the actual installation. Neither a geometric mean nor an ideal polar plot substitutes for that chain.

Deep Nulls Magnify Small Channel Errors

Consider two nominally equal contributions intended to cancel. If one has small fractional amplitude error ε and small phase error δ in radians, the residual relative to one contribution is approximately:

This first-order relation explains why a main lobe can look reasonable while a deep null collapses. Cable loss, phase slope, group delay, component tolerance, temperature, termination impedance and coupling all matter more near cancellation. Fixed phase networks and physical delay lines also evolve differently with frequency, so every state must be characterized across its intended band.

Null direction matters as much as null depth. A deep notch at one azimuth and elevation can miss an interferer arriving by another path, or remove a wanted signal when propagation changes. Band-edge testing must therefore include null angle, width and stability—not just the lowest number on one trace.

RDF Is Not the Site's SNR Guarantee

RDF is a pattern metric. It is especially useful for comparing how strongly a pattern favours one direction over a uniform angular field. A real HF noise environment is rarely uniform. Atmospheric noise, local emitters, reradiation and receiver noise can have different directions and correlations.

If R_n is the channel-noise covariance matrix, the combined output noise power is:

The weights that maximize modelled RDF need not maximize wanted-signal SNR at one site. Correlated external noise, independent receiver noise, element sensitivity and amplifier headroom all change the answer. That is why I keep RDF, front-to-back, null depth and installed A/B/A SNR measurements in separate columns.

Validate the Centre, the Edges and the Space Between

A trustworthy multiband design record should include:

• Declared design points: the exact frequencies, operating windows and priority assigned to each.
• Installed geometry: element coordinates, height, orientation, feedline routes, support structures, ground model and nearby conductors.
• Embedded responses: element impedance, current and pattern with the remaining ports in their real operating terminations.
• Channel calibration: amplitude, phase and group delay at declared reference planes for every steering state.
• Three-dimensional patterns: forward direction, peak direction, beam solid angle, RDF, F/B, sidelobes and null position, width and depth.
• Error sweeps: sensitivity to amplitude and phase tolerance, temperature, cable or component swaps and termination changes.
• Noise behaviour: element sensitivity, receiver-noise contribution, channel covariance, overload margin and wanted-signal SNR.
• Band coverage: centre, band edges and enough intermediate frequencies to expose non-monotonic coupling or network behaviour.

IEEE measurement practice treats the radiation pattern as a fundamental antenna property and requires a suitable measurement arrangement. NIST's phased-array work shows how phase drift can destroy coherent beamforming until calibration restores it. Those systems operate far above HF, but the transferable lesson is simple: a coherent result belongs to the calibrated system and state in which it was established.

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