Chapter III: Proximal-Field Radiation Sensors
3.5 PFRS Data Processing
3.5.2 Statistical Methods
54
Figure 3.44: Simulated gain patterns of the 2×2 array when∆ψ = 180◦for the two cases of broadside and off-axis radiation.
Essentially, PCA represents a data set in a way that best demonstrates the variance in it. If we imagine a large dataset ofNvariables as points in anN-dimensional space, i.e., one variable per axis, PCA provides a projection of the data with fewer dimen- sions and the most information. This is performed by reducing the dimensionality of the data set to its first few principal components through the transformation.
In order to better understand how we can use PCA in PFRS data processing context, we need to first define the problem in a consistent way to PCA. We know that many different parameters can potentially affect the performance of a radiator. These parameters include the phases and amplitudes of the signals at the driving ports (which in turn depend on various biasing signals that control the drive circuitry), dimensions of the substrate, relative placement of the electromagnetic structure on the substrate, etc. Any changes in one of these parameters or any subset of them directly affects the far-field radiation properties of the system. If, hypothetically, for every single such parameter, there existed a function of PFRS read-outs which could provide a specific mapping between its output and that particular parameter when applied to the sensors outputs, such a function could have been used in determination of the state of that parameter and readjustment of that, if possible, to improve system’s performance. This is what we wish to use PCA for.
First, we try to identify as many quantifiable parameters as we can, which could affect the performance. The nominal operation of the radiator happens when all the identified parameters take their nominal values, and as they vary the radiation performance deviates from the nominal operation. Also, let’s assume there are N PFRS outputs available on the integrated radiator. This means each set of values for the parameters results in a set of NPFRS read-outs. One way to interpret the read- outs is to assume an N-dimensional space whose N orthogonal axes correspond to the N PFRS outputs. Each set of N read-outs corresponds to a point in this
56 space. Any combination of these parameters results in a different point in the N-dimensional space.
The parameters that affect the performance do not necessarily change one at a time.
Often it is a combination of variations of a number of them, which results in a deviation of performance. However, through simulations and calibration measure- ments, we can investigate the effect of each parameter on the read-outs, assuming no other parameters have changed compared to their nominal values. A sweep of the parameter of interest (e.g. a biasing voltage, a phase control parameter, a physical length in the structure) over a range of possible values that it might take around the nominal value during the radiator’s operation, provides a large set of data points in the N-dimensional space. However, not all sensor outputs are fully uncorrelated, which means that significant redundancy exists in the data set. This is why we can use PCA to analyze the data. PCA maps the data into its essential components in the multi-dimensional space to eliminate redundancy and reduce the dimensions of data set. In fact, for small variations around the nominal value which allow assumption of linearity, since only variation of one parameter is used to generate this data set we expect that the data points mostly vary in one dimension, defined by a linear combination of N PFRS read-outs, i.e. the first principal component calculated by running PCA on the data set. We can do the same for all the identified parameters, which could potentially introduce error to the performance, and use the resulting functions as a tool to monitor the performance and adjust it back to the optimum setting.
As a demonstration, again consider the same 2×2 array of four-port transmitting antennas with nine slot-ring PFRS antennas, where all the transmitting antennas are driven by quadrature signals to radiate circular polarization. If the transmitting antennas are not operating in phase, the relative phase errors, θA, θB, θC, and θD would exist between the antennas A, B,C, and Dand an arbitrary phase reference, respectively. For simplicity, let’s assume these four parameters are the only param- eters that affect the radiation performance, and they can be adjusted through phase control units in the drive circuit to allow error correction. For proper operation we must have θA = θB = θC = θD = 0◦, and any deviation of each of them from the nominal zero value results in deviation of the radiation performance. We can use PCA and simulation to define the first principal component for each of these parameters. The resulting coefficients of this linear combination are then used to define a sensitivity function which shows the largest sensitivity to that particular
Figure 3.46: Magnitude of the principal components associated withθD calculated using (a) both phases and amplitudes and (b) only amplitudes of the PFRS read-outs.
Figure 3.47: Application of an exemplary phase error correction algorithm to a random combination of relative phase errors θA, θB, θC, and θD, using the first principal component for each parameter.
These sensitivity functions can be used in a basic algorithm to calibrate the drive cir- cuitry such that the phase errors caused by process variation, temperature variation, unexpected electromagnetic coupling to the adjacent objects, etc., are corrected.
Figure 3.47 shows an exemplary application of such an algorithm using the first principal components calculated based on both amplitudes and phases of the sen- sors’ read-outs to an arbitrary initial condition for θA, θB, θC, and θD. In each
58 iteration, the algorithm tries to reduce the calculated value for all four sensitivity functions associated with the phase errors and adjust the control knobs until they are all minimized. We can see that although the initial values forθi’s are very far from each other, the algorithm is still able to bring them all close to each other within a few iterations, which is a required condition for maximum broadside.
It should be noted that use of PCA or any other statistical analysis tool is not limited to this specific example. PCA can be used to characterize system’s sensitivity to a combination of multiple parameters. A variety of other methods based on different fundamentals can also be used.