Beamforming, or signal phasing with multiple antennas, means understanding array delay through the complete system. Here’s how it works.

Today, we will dig into the simple math you need to understand how diversity reception with two antennas actually works. Whether you are running a phased array, phase shifting two different antennas against each other, or doing noise cancelling with an ANC-4, it’s basically the same trigonometry involved.

Basically, when you sum two signals with the same phase, these signals add together. Alternately, when you sum two signals that are 180 out of phase, they cancel each other. And everything in between. There are two time delays involved. The first is a fixed time delay caused by antenna spacing, and the fact that a distant waveform arrives at each antenna at a different time. Shown above the wave front arrives at antenna 1 (right) earlier than antenna 2 (left) with the angle of arrival θ shown.

With two antennas spaced 30 meters apart, the time delay is maximized when the angle of arrival is in the plane of the array (end fire, or θ=0°.) On the other hand if the wave front is perpendicular to the plane of the array, time delay is minimized, or zero (broadside, or θ=90º.) Thus, with 30 meter spacing, time delay runs from 0 to 100 nanoseconds. Calculate this as 30 meters x cosθ /speed of light, or d cosθ / c.

While the time difference between the two antennas is independent of frequency, the phase difference of the two received signals is not. To figure this out, we need to calculate the spatial frequency of the signal, known as wave number. You will find this is simply as k = 2π/λ cycles per meter. So, put this all together and the phase difference created by your two antennas is k d cosθ .

As an example, say I am listening to 11.6 MHz on two antennas spaced 30 meters apart arriving at 15° to the plane of the antennas. The wavelength of an 11.6 MHz signal is 25.8 meters (11,600,000 ÷ speed of light). So, the phase difference is (2π /25.8) * 30 * cos(15°) = 7 radians or 400 degrees. Similarly, a 1.2 MHz MW signal arriving at the same angle would have a phase difference between antennas 1 and 2 of 42°.

## Array Delay Phase Shift Receiver

The second time delay in a diversity receiving system is the phase shift you can create between each of the channels connected to an antenna, shown in blue above. Typically, your coherent receiver will let you advance or delay a channel by ±180°. Traditional (physical) phased arrays accomplished this with different lengths of feedline to create time delays between each antenna element. Today, you can just do this digitally in your receiver.

So, to null or reduce the signal at the receiver, you just adjust the controls to obtain an 180 phase reversal between the two channels and sum them together.

Your final formula for the array phase factor ψ combines both delays as follows. Your receiver phase shift is α and your element spacing phase shift is k d cosθ, where k is wave number, d is distance between elements and θ is angle of arrival against the plane of the antennas.

ψ = α + k d cosθ

Two things to consider when you are playing around with this. First, remember whether you are working in radians or degrees. Second, some textbooks show θ as the angle perpendicular to the plane of the loop.