Armed with the knowledge from the previous sections, this section aims to present a complete and ordered view of the reception process from a Mills Cross type system from a signal process- ing point of view. The purpose of presenting this material in detail is threefold. Firstly, to justify the simplified approach taken during the simulations discussed in chapters 4 and 5. Secondly, to lay the foundations for the radiation pattern simulations in chapter 6 and ultimately for the GLEAM algorithm presented in chapter 10. Thirdly, to attempt to shed a bit of light on the mind- boggling phasing issues relating to Mills Cross arrays, particularly when using them to observe spatially distributed sources. With reference to figure 2.11, we can identify the following steps:
1. Any given antenna (element) will respond to a signal from any given direction as described by the antenna’s radiation pattern. This pattern describes the antenna’s response to incom- ing signals in the two polarisation components as described by two phasors (we adapt an x-y description throughout this thesis, although any set of orthonormal base vectors will work equally well).
CHAPTER 2. ANTENNAS 22
2. Just like the radiation pattern itself, any incoming signal can be described using two pha- sors for the two polarisation components. Note that we need to use the same coordinate system for the next step to be meaningful.
3. Ignoring an arbitrary phase offset (which is constant for all directions), the antenna will exhibit a signal DirectivityX×conj(Ex_incoming) +DirectivityY×conj(Ey_incoming)
at its terminals.
4. Beamforming networks, such as the additive beamformer described in section 2.3.3, will combine (phase-shifted=delayed and tapered) versions of these signals to effectively form a new radiation pattern, that of a fan beam in the Mills Cross case.
5. A cross-correlator such as used in a Mills Cross configuration will multiply the signals from two such beamforming networks to produce the narrow pencil beam made up from identical parts of the signals from the two fan beam inputs. This is the only non-linear processing stage in the Mills Cross reception process.
It is at this stage, that phase offsets introduced by the beamforming networks cause a phase offset in the resulting cross-correlated ‘power’ value for any given direction. As long as only point sources are examined, this is not an issue, as we can always use the absolute power value as a measurement of the power received from that point source. As soon as we examine a spatially distributed source, however, this effect needs special consideration, this will be discussed in further detail in chapter 10.
6. Throughout this description, we work with the basic assumption that signals from different directions are uncorrelated. The cross-correlator will therefore never produce an output for signals from two different directions, and the overall received power for any given pencil beam can simply be calculated as the sum of all signals (from all directions). This sum is the signal visible at the output terminals of the cross-correlator. Note that this is a complex weighted sum according to the phase offsets mentioned above.
7. According to Kraus [Kra88] (from [Sin50]) the voltage responseV of an antenna to a wave of arbitrary polarisation is given by
V =kcosMMa
2 (2.12)
dealing with (incoming) signals of random polarisation,MMawill vary randomly between
0◦ and 180◦, averaging at 90◦. On average, for random polarisation, equation 2.12 will therefore result in a constant factor<V >=kcos 45◦which we can safely ignore for all considerations that are only interested in relative signal levels and/or phase relations.
In our modelling of the Mills Cross, we use the basic arrangement of figure 2.11. The following discussion shows that, for any given direction of interest, even for randomly polarised incoming waves, the phase difference detected by the cross-correlator will be constant and not dependent on the actual state of polarisation of the incoming wave. Furthermore, the amplitude of the cross- correlator output signal will be constant for any given power influx with random polarisation.
The incoming signal is received by the antenna, described by the radiation patternβ2,x(θ,φ) andβ2,y(θ,φ). The array patterns for arms A and B simply scale those signals in amplitude and phase. In reality, the signals are combined before being passed through the array beamformer, but this is a linear operation and could therefore equally well be performed separately for the two polarisation planes.
Moving the ‘reception’ stage after the beamformer does not change the signal in any way either, as this is again a completely linear operation. This is in fact why the notion of radiation patterns is such a useful one, as we can now describe the response of an array made up of antenna element patterns and cross-correlator by one resulting ‘combined’ radiation pattern. Let us now consider phase and amplitude separately:
As we have just seen, the reception process will introducethe same phase offset into both branches (representing both arms of a Mills Cross type antenna array) in figure 2.11. The cross- correlator will therefore detect exactly the same phasedifferencebetween the two signals, in- dependent of the actual received waveform, dependent only on the different phase offsets intro- duced by the different beamforming networks of the two arms for any given direction of arrival. This phase offset and how it depends on direction of arrival is an inherent system property.
Theamplitudeof the received signals in branches 1 and 2 will vary depending on how well the antenna and incoming signals are matched. However, in the case of randomly polarised in- coming signals, equation 2.12 tells us that, on average, we will see both signals attenuated by the same constant value. On average, the observed signal amplitude of the cross-correlator output for any given direction will therefore only depend on the properties of the underlying (combined) radiation pattern and, moreover, will be proportional to the overall ‘combined’ radiation pattern
CHAPTER 2. ANTENNAS 24
Figure 2.11: Signal-processing view of reception of signals by a Mills Cross antenna array from any one direction(θ,φ)and for one particular pencil beam b. Arm A consists of aerials 1...m, arm B consists of aerials n...p. inix andiniy are the x and y polarisation components of
the incoming radio wave (assumed to be identical for all aerials),β1,i are phase shifts (delays)
due to the location of the aerial in question relative to the phase centre of the array. β2,x and
β2,y describe the element radiation pattern in the x and y polarisation planes (assumed to be
identical for all aerials),aiare the tapering factors,β3,iare the delays introduced by the additive
beamformer for one particular (fan) beam,outbis the output signal for this particular direction