Geometric Mean: The Secret to Multiband RDF Balance

Designing a multiband phased receive array—especially compact systems like EchoTriad (3-element triangle) or QuadraTus (4-square)—comes down to one hard truth: your physical spacing is fixed, but your array’s electrical spacing changes with frequency. What matters for pattern shape and RDF is not meters, but d / λ.

That’s why a spacing that looks “perfect” on one band can behave quite differently on another. As frequency increases, d / λ increases, and the pattern may broaden, shift, or develop less useful null and side-lobe behavior. Optimizing spacing strictly for the lowest band often pushes the upper bands away from the array’s most useful operating region.

The Core Idea

Instead of choosing spacing based on a single band, select a design wavelength that sits in the middle of your intended band range on a logarithmic scale. That midpoint is the geometric mean.

λ = c / f

For two bands:
λ_gm = √(λ₁ × λ₂) = c / √(f₁ × f₂)

Then choose spacing:
d = k × λ_gm

Here, k is the spacing fraction dictated by your array geometry. Typical values used in practice are approximately k ≈ 0.139 for a compact 3-element equilateral receive array, and k ≈ 0.20 for a compact 4-square. The geometric-mean step does not replace your chosen k; it defines where to anchor it when multiple bands must be covered.

Why the Geometric Mean Works for Multiband Arrays

If spacing is set to d = k·λ_gm, the normalized spacing d / λ deviates from the target k by the same ratio at the low-band and high-band edges. This creates a balanced compromise in log space—the same way RF engineers think about bandwidth, frequency ratios, and decibels.

Example: EchoTriad for 160 m and 80 m

Choose representative frequencies that reflect actual operating interest:

• 160 m: 1.83 MHz → λ ≈ 163.8 m
• 80 m: 3.60 MHz → λ ≈ 83.3 m

λ_gm = √(163.8 × 83.3) ≈ 116.8 m

Applying the EchoTriad spacing fraction:

d = 0.139 × 116.8 ≈ 16.2 m

This spacing is not the absolute RDF peak for either band edge, but it typically lands in a stable “good-on-both” region—strong directivity on 160 m without pushing 80 m into an over-spaced regime.

Example: Extending the Coverage Range

The same method applies when covering a wider frequency span. For example, from 160 m to 40 m (1.83 MHz to 7.05 MHz):

λ_160 ≈ 163.8 m
λ_40  ≈ 42.5 m

λ_gm = √(163.8 × 42.5) ≈ 83.5 m
d    = 0.139 × 83.5 ≈ 11.6 m

This yields a compact physical layout that generally avoids severe over-spacing on the upper band while maintaining meaningful directivity on the lowest band.

Practical Notes

• Choose frequencies deliberately. Use CW or DX windows if those matter to you.
• Keep expectations realistic. Geometric-mean spacing is a disciplined compromise, not a guarantee of peak performance.
• Validate in context. Ground conditions, element type, height, coupling, and phasing accuracy all affect the optimal result.

Takeaway

When a single receive array must serve multiple bands, geometric-mean spacing is the cleanest way to select a design wavelength. Apply your array-specific spacing fraction to that λ_gm and you typically achieve more consistent RDF behavior across the intended band range.

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