The Butler Matrix and its Use for Beamforming and MIMO Testing

Figure 1. A photo of the Ranatec RI 3101 Butler matrix

The Butler matrix is a black box with input and output connectors. An example with eight input and eight output connectors is shown in Figure 1. The signal injected into one of the input connectors is distributed in equal amounts to all output connectors with very specific phase relations. Let’s have a look at the origin and recent renaissance and lift the lid to have a look at the inner workings of this device.

Original Invention

Originally designed in the early sixties by Jesse Butler and Ralph Lowe, the Butler matrix was an innovative distribution network for generating fixed beams in array antennas. It replaced earlier constructions since it needs much fewer phase shifters. This was a big savings and increased robustness of the large arrays, typically 64 by 64 elements, in the aircraft defense radars developed at the time. The work was performed at Sanders Associates in Nashua, New Hampshire, which specialized in complex circuit assemblies on printed wiring boards. In fact, the original article calls the arrangement Sanders beam-forming matrix, but it is now only known as the Butler matrix.

All Butler matrices have an equal number of input and output ports. This number is always a power of two, i.e. 2, 4, 8, 16 and so on. If fewer ports are needed, the unused ports must be terminated with matched loads. Otherwise, there will be unwanted reflections back into the circuitry that severely distorts the performance.

Starting with the smallest Butler matrix, which has two inputs and two outputs, it is identical to the 90° hybrid coupler. An injected signal into port 1 is split into two signals at ideally -3 dB amplitude and with quadrature phase. Injecting into port 2 yields the same but with a mirrored phase. Thus it can feed a two-element antenna array and generate one beam steered to the left and one beam steered to the right, depending on the choice of feed port. The amount of steering is also dependent on the physical separation of the elements according to

𝑑 sin 𝜃 = ∆𝜑𝜆 / 2𝜋

where d is the separation, θ is the steering angle from boresight, ∆ is the phase difference (in radians) and λ is the wavelength.

The second smallest Butler matrix has four inputs and four outputs. A signal injected into port 1 comes out as -6 dB in each output with a progressive phase shift of 45°. For inputs 2, 3 and 4, the phase shifts are 135°, 225° and 315°, respectively. For phase angles, 225° is the same as -135° and 315° is the same as -45°. Thus we can generate two beams to the left and two beams to the right from boresight.

For the Butler matrices of higher order, such as 8, 16 and 32, the signal is split up with successively lower amplitudes of -9 dB, -12 dB, -15 dB etc. and successively smaller progressive phase shifts of 22.5°, 11.25°, 5.625° and so on. The number of beams that are generated is always the same as the order of the Butler matrix and their directions get denser as the order increases.

In practice, there is always some additional loss from components and transmission lines within the box. This loss will tend to increase with higher-order matrices since they will have more components and longer lines. The phase shifts will also have deviations from the ideal values, the more so for larger operational frequency bandwidths.

Revived Interest

In recent decades, the Butler matrix has found an additional application. It is used as an emulator of multipath propagation when testing multiple-input/multiple-output (MIMO) devices. Instead of performing the tests over-the-air (OTA), the antenna ports of the transmitter and receiver are connected to the input and output ports of a Butler matrix. Then each output signal from the transmitter will be split into portions and enter each antenna input of the receiver. The receiver’s ability to separate the data streams can thus be evaluated. Additional attenuators and phase shifters can be added to the signal paths to check the robustness of the performance.

In the development of 5G NR mm-wave base station antennas, the original Butler matrix application has found a renewed interest. Since the frequency of operation is then roughly ten times that of sub-6 GHz mobile communication, the path loss is about a hundred times greater for the same coverage. (This is a consequence of using the same size for the user equipment antennas.) A way to mitigate this loss is to split up the base station coverage by generating beams. Then the power can be directed into the beams where the user equipment resides.

Measured Performance

The performance of a Butler matrix depends on its actual design. There is always a trade-off between operational bandwidth and signal variation. We will have a closer look at the performance of the Ranatec 8+8 port matrix. It was designed for the band 2.4-8.0 GHz with the main purpose of supporting MIMO testing of Bluetooth and WiFi devices for present and future standards.

To cover such a wide frequency range, the requirements on return loss, insertion loss, amplitude flatness and phase deviation need to be adjusted accordingly.

Figure 2 shows the measured amplitude flatness for an 8+8 port Butler matrix. The specification is that it should be above -13 dB and with less than ±3 dB deviation from nominal within the operational frequency band. The requirement mask is shown in the Figure as grey areas showing the forbidden regions.

A reference signal is input into port 1R and the eight output signals are plotted as curves with different colours. All of them fulfill the specifications. A lossless split into eight equal amplitude signals would mean that they are all at -9 dB. In excess of that, we see that we have on average something like 1.5 dB insertion loss.

Figure 2. The magnitude of all eight output signals compared to the input signal in port 1R for the Ranatec 8+8 port Butler matrix

The phase of all eight outputs is shown in Figure 3. It varies rapidly over the band since there is a path length and therefore a time delay between the input port and each output port. This is, however, not important as long as the phase difference between the ports is close to the wanted values for each frequency within the operational band.

For the example shown, the progressive phase shift between the output ports should be 22.5°. It is hard to see in Figure 3 if this is fulfilled, so we need to postprocess the measured data a bit.

First, we compute the relative output phases to a chosen reference. The choice here is the output signal in port 1. The relative phases of ports 2-8 will then be fairly flat and close to their nominal values of 22.5°, 45°, 67.2°, 90°, 112.5°, 135° and 157.5°.

Next, we subtract the nominal phase value from the curve of each output port. Then they will all be close to 0° and the curves show the deviation from the nominal phase. The result is shown in Figure 4. We can now clearly see that the deviation from the nominal phase is indeed very low, below ±10° in the complete 2.4-8.0 GHz frequency band for all of the signals.

Figure 3. The phase of all eight output signals compared to the input signal in port 1R for the Ranatec 8+8 port Butler matrix.

Figure 4. The phase deviation from the ideal values of all eight output signals for an input signal in port 1R for the Ranatec 8+8 port Butler matrix

Inner Workings

A Butler matrix consists of a network of dividers/combiners and phase shifters. The network uses binary combination and branching in levels so that it can couple part of each input to each output. The level depth of the network is 1 for a 2+2 port matrix, 2 for a 4+4 port matrix, 3 for an 8+8 port matrix and so on. The scheme is very similar to the FFT computation algorithm.

Unfortunately, the network paths must cross (except for the 2+2 port). The crossings are either built-in multilayer networks or back-to-back hybrids. Putting two 90° hybrids in series (ideally) retains the input signals, but shifted left-to-right.

In a multilayer design, the layer changes must be carefully designed to preserve phase and impedance control of the transmission lines. Back-to-back hybrids, on the other hand, take up more room and are limited in bandwidth.

The number of hybrids needed is 1 for a 2+2 matrix, 2+2=4 for a 4+4 matrix, 4+4+4=12 for an 8+8 matrix and in general N²⁻¹ for a 2^N+2^N port matrix. In other words, each level of the Butler matrix has the same number of inputs and outputs as the black box itself. It is best to use 90° hybrids for the dividers/combiners because they give one of the needed phase shifts for free. There are several ways to design them with broadband behavior, for example as branch-line couplers, coupled lines or waveguide couplers.

The 2+2 port Butler matrix uses no phase shifters except for the 90° hybrid. The 4+4 matrix uses two 45° phase shifters in addition to the four hybrids. They are positioned in between the hybrid levels. The 8+8 matrix uses 22.5° and 67.5° between the first and second level and 45° phase shifters between the second and the third level. Higher-order matrices use successively finer phase shifts.

Practical Construction

Parts of a physical realization of a Butler matrix are a box with a lid, connectors and etched laminates that contain transmission lines, couplers and phase shifters. The couplers should be broadband 90° hybrids and the phase shifters of the Shiffmann type. They are built up by a coupled line delay section in parallel with a fixed-length transmission line in order to achieve the broad bandwidth. Care must be taken to get the proper length of all transmission lines.