Output Pattern Analysis

In this section, the achieved patterns are analyzed over the whole output grid (i.e.

instants 𝑡𝑡_{𝑜𝑜𝑓𝑓𝑜𝑜}[𝑘𝑘] (85) for all 𝑘𝑘), to complete the resampling process and achieve the

final goal of the method.

First of all, the MSE-VBS method described in Section 4.4.2 (or equivalently the

MSE-SNR-VBS one of Section 4.4.4 with 𝛼𝛼 = 0, i.e., given all emphasis to the grid

regularity) is considered as a solution to the resampling problem, so as to assess the closest possible implementation of the patterns.

21 _{This behavior also explains what was visualized in Figure 32: the change from 0.0 to 0.25 kept the pattern}

shape and phase errors (and hence the MSE) fairly constant while causing a visible difference in the gain (and hence the SNR); the changes from 0.25 up to 0.75 had visually little effect in both regards; and the final change to 1.0 led to a high-gain pattern (the best SNR figure) which is however completely different from the goal and shows a correspondingly high phase error (hence the high MSE).

22 _{This can be explained by the fact that the optimum MSE solution incurs a relatively high SNR penalty by}

using all possible extended manifold elements, including fairly uncorrelated ones (corresponding to distant pulses). These do not contribute to a great improvement of the MSE, while degrading the SNR considerably. With moderately low values of α, very similar patterns are achieved with a lower weight magnitude, what can be interpreted as a better distribution of the activation energy, made possible by disregarding such extended manifold elements.

The optimal MSE weights where obtained for every output sample, using as commom

pattern 𝐺𝐺_{𝑐𝑐𝑜𝑜𝑚𝑚}(𝑓𝑓_D) = 𝐺𝐺_{𝑠𝑠𝑓𝑓𝑚𝑚}(𝑓𝑓_D) as design goal. The output grid has per PRI cycle 93

patterns, depicted in Figure 35. The abscissa values correspond again to Doppler

frequency, and 𝐵𝐵𝑤𝑤_{𝑠𝑠𝑜𝑜𝑜𝑜𝑐𝑐} and 𝑃𝑃𝑅𝑅𝐴𝐴_{𝑚𝑚𝑓𝑓𝑒𝑒𝑜𝑜𝑖𝑖} are marked by red and black dashed lines,

respectively, in both plots. Figure 35 (a) shows the pattern’s power gain whereas

Figure 35 (b) depicts the phase error with respect to the goal phase ramp (defined in

(98) for each outputs sample’s phase center 𝑡𝑡_{𝑜𝑜𝑓𝑓𝑜𝑜}[𝑘𝑘] of (85)).

Figure 35. Set of output patterns obtained from the optimal MSE method against Doppler frequency. (a) Magnitude of patterns. (b) Phase after removal of the sample-specific linear phase ramp (cf. (98) and (85)), highlighting residual phase errors with respect to ideal regular sampling.

The similarity between the amplitude of the patterns in Figure 35 and the sum pattern in Figure 31 is clear, indicating that the implementation is close to the desired patterns. Within the main beam, the patterns show stable magnitudes and very low residual phase errors with respect to the desired phase center positions, further indicating that successful regularization was achieved over the grid.

Given the advantages of using the MSE-SNR-VBS method (cf. Section 4.4.4), the

implementation of the grid using (117), (118) with 𝛼𝛼 = 0.6 is also considered, both

directly and with the addition of the iterative method explained in Section 4.4.5.

Section 5.4 Synthesis of Full Output Grid: Comparison between Methods 119

The normalized MSE 𝜉𝜉_{𝐹𝐹𝑀𝑀𝑀𝑀}[𝑘𝑘]/𝑝𝑝_𝐺𝐺 and the SNR scaling Φ_{𝑀𝑀𝐼𝐼𝑆𝑆}�𝒘𝒘[𝑘𝑘]� are shown over

the output patterns for the three aforementioned methods in Figure 36.

Figure 36. Normalized MSE (left column) and Φ_{𝑀𝑀𝐼𝐼𝑆𝑆}(right column) over output patterns/samples for different regularization methods. (a) MSE-VBS. (b) Non-iterative MSE-SNR-VBS. (c) Iterative MSE- SNR-VBS. For the last two methods, 𝛼𝛼 = 0.6 was adopted.

(a)

(c) (b)

The plots in Figure 36 (a) refer to the MSE-VBS method of Section 4.4.2. The ones in Figure 36 (b) were obtained with the MSE-SNR-VBS method of Section 4.4.4,

evaluated with 𝛼𝛼 = 0.6. Finally, the plots in Figure 36 (c) show the results for the

iterative method of Section 4.4.5, again using the cost function 𝜉𝜉_𝐽𝐽[𝑘𝑘] of (117) with

𝛼𝛼 = 0.6. For all cases, performance is worse for the samples near the gaps (indices 5 to 8, and 87 to 90) and around the edges of the sequence (indices 0 and 92), which is a consequence of the choice of using only the samples within a PRI cycle for resampling.

A comparison of the results in Figure 36 (a) and (b) highlights once again the compromise between the MSE and the SNR described in Section 4.4, embodied by the

design parameter 𝛼𝛼. Introducing the iterative procedure (compare Figure 36 (b) and

(c)) enhances, on average, both MSE and SNR, with a larger improvement for the

worst cases, as was the goal.

The ripple in Φ _{𝑀𝑀𝐼𝐼𝑆𝑆} over the samples is also reduced, indicating that a more uniform

performance was achieved. In all cases, the performance for the samples within the region of the blockage-induced gaps is clearly worse. This is expected and due to the larger phase center shift with respect to the input grid required to fill those gaps.

Further illustration of the iterative method is provided in Figure 37. In this case, the

convergence criterion is an average MSE step Δ𝑀𝑀𝑆𝑆𝐸𝐸 = 𝑀𝑀𝑆𝑆𝐸𝐸������_𝑖𝑖-𝑀𝑀𝑆𝑆𝐸𝐸������_𝑖𝑖−1 of less than

0.05 dB, which requires a total of 11 iterations. The magnitude of the common

patterns 𝐺𝐺_{𝑐𝑐𝑜𝑜𝑚𝑚}𝑖𝑖 (𝑓𝑓_D) is seen in Figure 37 (a) for different iteration numbers. The gradual

modification of the patterns consists mostly of a change in the first sidelobes and attenuation at the borders of the main beam. This is the result of two effects: first, the incorporation of the mean residual distortion as a part of the design goal and second,

the feedback of the SNR emphasis parameter 𝛼𝛼 into the design goal. Figure 37 (b)

shows how the MSE evolves with the iterations. Iteration 𝑖𝑖 = 0 is equivalent to the

non-iterative MSE-SNR-VBS method. The blue line shows 𝑀𝑀𝑆𝑆𝐸𝐸������_𝑖𝑖, which is monitored

Section 5.4 Synthesis of Full Output Grid: Comparison between Methods 121

which is also reduced, as intended, indicating a better approximation is achieved over

the grid. Figure 37 (c) shows the average SNR scaling 𝛷𝛷������_{𝑀𝑀𝐼𝐼𝑆𝑆}

𝑖𝑖, which is also seen to

improve, reflecting the feedback of the SNR emphasis parameter 𝛼𝛼.

Figure 37. Iterative MSE-SNR-VBS method with 𝛼𝛼 = 0.6. (a) Magnitude of common patterns 𝐺𝐺𝑐𝑐𝑜𝑜𝑚𝑚𝑖𝑖 (𝑓𝑓D) for different iterations 𝑖𝑖. (b) MSE (mean over grid 𝑀𝑀𝑆𝑆𝐸𝐸������𝑖𝑖 in blue and worst-case over grid in

red) as a function of iteration number. (c) Mean SNR Scaling 𝛷𝛷�������_{𝑀𝑀𝐼𝐼𝑆𝑆𝑖𝑖} as a function of iteration number.

As a final illustration of the method’s characteristics, the actual illumination of the reflector’s surface induced by the beamforming weights is plotted in Figure 38. The

MSE-SNR-VBS weights with 𝛼𝛼 = 0.6 (cf. Figure 36 (b)) are chosen as an example.

In each sub-figure ((a)-(d)), the abscissa values correspond to azimuth position, and two sub-plots are shown.

(a) (b)

(a)

Section 5.4 Synthesis of Full Output Grid: Comparison between Methods 123

Figure 38. Illustration of reflector illumination for the MSE-SNR method with 𝛼𝛼 = 0.6. (a) Output sample of index 𝑘𝑘 = 4: minor phase center shift is achieved with little contribution from other pulses. (b) Neighbor sample 𝑘𝑘 = 5: notice gradual migration of the illumination. (c) 𝑘𝑘 = 74: mainly two neighboring pulses are used to form the output pattern. (d) 𝑘𝑘 = 86: in this sample occurring during the second Tx-event induced gap, several pulses are combined, degrading the SNR.

(c)

In each case, the upper sub-plot shows the illumination (normalized power

distribution) at the reflector’s surface, as seen from above, for each of the 𝑁𝑁_{𝑒𝑒𝑓𝑓𝑓𝑓} = 31

non-blocked pulses. The color coding indicates the resulting power levels, represented separately for each pulse. The separation in the ordinate axis has no physical meaning, and is only for visualization purposes. The reason is that, as it can be seen from the upper sub-plots, the displacement of the reflector between neighboring pulses is considerably smaller than its size, leading to considerable overlap between the surfaces, hampering visualization. The vertical red dashed line indicates the desired output sample phase center position, which changes from sub-figure to sub-figure. The azimuth sampling as in Figure 32 is represented below for reference: pulse positions are highlighted by arrows, the individual physical channels’ phase centers by

blue circles and the desired output phase center by the red crosses. The 𝑁𝑁_𝑐𝑐ℎ = 3

physical channels are combined at each time to generate the corresponding

illumination, meaning the weight vector of dimension 𝑁𝑁_{𝑚𝑚𝑎𝑎𝑚𝑚𝑖𝑖𝑓𝑓𝑜𝑜𝑒𝑒𝑓𝑓} = 93 is split into 31

sub-sets, so that it can be understood from which pulse position the main contribution to the output patterns arise. It should be noted that the final pattern is a sum of the contribution of all pulses. The surfaces’ position in azimuth matches the sequence’s

𝑡𝑡𝑜𝑜𝑚𝑚[𝑖𝑖] (defined in (84)), and is highlighted by a cyan cross and the corresponding pulse

number.

The Figure 38 (a) show the illumination for output sample 𝑘𝑘 = 4, whereas Figure 38

(b) corresponds to its neighbor, output sample 𝑘𝑘 = 5. In both cases, the major

contribution to the illumination is from a single pulse, the second. The comparison between them shows the migration of the primary beam as the desired phase center changes, and illustrates the physical principle behind the method, described in Section

4.3. Figure 38 (c) corresponds to output sample 𝑘𝑘 = 74 and shows a different

scenario, in which the contributions to the illumination are evenly split between neighboring pulses, which advocates for the use of the extended manifold. As described in Section 4.4.2, performance is improved (with respect to the pulse- independent steering) by taking advantage of the fact that the signal is oversampled

Section 5.4 Synthesis of Full Output Grid: Comparison between Methods 125

and thus information from different pulses can be exploited. Finally, Figure 38 (d),

corresponding to output sample 𝑘𝑘 = 84 shows the worst-case performance, during the

second Tx-induced gap. The larger phase center shift means that the correlation between the output pattern and the inputs is smaller, and several pulses are employed to implement the pattern. As visible in Figure 36 (b), this worsens the pattern implementation (meaning higher MSE) and furthermore degrades the SNR performance, though this effect is to a considerable extent countered by the choice of the MSE-SNR method.