Consecuencias de la desesperanza

Since uniform arrays can be thought of as spatial sampling, often times temporal techniques that are effective in traditional signal processing algorithms can also be applied to array systems. Removing frequency ambiguities in the temporal domain was accomplished by utilizing a staggered PRT algorithm. A staggered spatial sampling technique will be described here that yields surprising results.

Grating lobes are ambiguities that cast doubt on the true angle of arrival for a phased array output. Another way to view the problem is to think of the main lobe as having a positive and negative steerable angular region in which it can discern targets at their true positions. This region is ±θgl/2. As the mainlobe

arrives at +θgl/2, the grating lobe originating at −θgl has now shifted to −θgl/2.

This is similar to the wrapping effect seen in radial velocity measurements for Doppler radars. In a manner similar to the technique presented in Section 3.1.4.1, the observable region of the radar can be increased by utilizing staggered spacing. An ambiguity waveform can be established in which the wrapping angles are

θ_gl1 = 0.5 sin−1λ/d1 (3.89)

θ_gl2 = 0.5 sin−1λ/d2 (3.90)

resulting in an effective increase in the viewing region, or shifting of the grating lobes, to θs

gl = mθgl2 = nθgl1, giving a staggered spatial ratio of κ = n/m.

With staggered array spacing, the shape of the radar beam is altered in a manner that is similar to the staggered PRT ambiguity function. Recall that in Figure 3.10, the ambiguity peaks along the Doppler axis were defocused and the amplitude was reduced. A similar effect can be seen in Figure 3.28. The green

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Staggered Spacing Algorithm λ=0.03 m, 36 elements d₁=1.8λ, d₂=2.7λ d=1.8λ (a) −10 −8 −6 −4 −2 0 2 4 6 8 10 −80 −70 −60 −50 −40 −30 −20 −10 0 Angle (deg) Power (dB)

Staggered Spacing Algorithm λ=0.03 m, 36 elements

d₁=1.8λ, d₂=2.7λ

d=1.8λ

(b)

Figure 3.28: Simulated beam pattern for a ULA (green) and a staggered spacing (blue) case. Much like the irregular spacing scenario explored earlier, the stag- gered array deforms the grating lobes allowing a dealiasing algorithm to increase the array field-of-view. A closer view of the main lobe and near-in sidelobes is given in (b).

line represents the ULA with d = 1.8λ where the blue line represents a staggered array with d1 = 1.8λ and d2 = 2.7λ, giving a staggered ratio of 2/3. Note the

presence of the grating lobes in the ULA case. The grating lobes split and reduce in amplitude after the staggered spacing is implemented. A closer view of the mainlobe and near sidelobes is given in (b). It is the splitting of the grating lobes that will ultimately be exploited for the dealiasing procedure.

To perform angular dealiasing, an ambiguity waveform must be generated. An example ambiguity waveform is presented in Figure 3.29. The same temporal rules

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True Angle (deg)

Expected Angle (deg)

Grating Lobe Aliasing Diagram d

1=1.8λ

d₂=2.7λ

∆θ

Figure 3.29: An illustration of the spatial ambiguity waveform derived from the staggered spacing algorithm. The blue line represents the ambiguity waveform for the elements with d1 = 1.8λ spacing, while the red line represents the elements

with d2 = 2.7λ. The resulting staggered ratio is κ = 2/3. The difference between

the two ambiguity waveforms is plotted as the green line and will help dealias the estimated DOAs.

is the data processing procedure. For temporal cases, the short pulse pairs are processed individually and an average ˆR(Ts) value is computed. It is this value

that is used in Equation (2.64). Likewise, the same is repeated for the long pulse pairs and the resulting velocity estimates are used with the dealiasing rules to produce a new velocity estimate.

In the spatial case, the processing is carried out in a similar fashion: pairs of elements are processed and an average value is used to determine an angle of arrival for both the short and long spacing. To test the theory, a simulation was created using modified code from Stoica and Moses (2005). A single source is placed in the far field with unity power. Noise was removed from the simulation by setting σ2

N = 0 and only one pulse is utilized. 36 equally spaced elements are

used as a baseline for comparison while the staggered setup utilizes 30 elements, approximately matching the aperture size of the ULA. As a result, there are 15 short space pairs and 14 long space pairs. Angular aliasing occurs at approxi- mately ±16.12◦ _{for the short pairs and at ±10.87}◦ _{for the long pairs.}

Results from the simulation with a target positioned at -3.2◦ _{are presented in}

Figure 3.30. The black line represents the return from the single target as seen from a ULA. Note that the ambiguity generated by the grating lobes is not a concern in this case because the target is within the vertical blue dashed lines, which denote the unaliased angular region. Once the target leaves this region, the grating lobes will produce false readings within the angular field-of-view. The blue and red lines represent the short and long element pairs, respectively. The angle of arrival is selected as the maximum power point for the average pattern that lies within the respective unaliased region, denoted by the vertical dashed lines. The maximum position is indicated with a star in the corresponding color. As can be seen, the maximum values match with the true position of the target, -3.2◦_{. The}

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Staggered Array Spacing

λ=0.03 m, Target at −3.2o

ULA Short Pairs Long Pairs

Figure 3.30: Simulated results from a staggered array. The black line represents the angular power distribution from a ULA for a target located at -3.2◦ _{and a}

power of unity. No noise was added to the signal for this simulation. The blue and red lines represent the power distribution from processing the short and long pairs of elements, respectively. The dashed vertical lines correspond to the angular ambiguity points and define the field-of-view for the element pairs. The ∗ points represent the maximum value within the pair field-of-view. The maximum points are the estimates used in conjunction with Figure 3.29 to produce the accurate dealiased DOA of -3.2◦_.

produce the angle of arrival estimate, ˆθ. In this example, the dealiasing rules indicate that the estimated angle is ˆθ = ˆθ1 since the target is not outside of either

unaliased boundary.

A second simulation tests the effectiveness of the algorithm with a target well outside the unaliased regions. The results of the test for a target placed at 30◦ are presented in Figure 3.31. The three lines show a maximum value at

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Staggered Array Spacing

λ=0.03 m, Target at 30o

ULA Short Pairs Long Pairs

Figure 3.31: Similar to Figure 3.30 except for a target located at 30◦_{. Note the}

short pair peak coincides with the ULA due to the spacing of 1.8λ. The offset peak found through the long pair processing facilitates the accurate DOA estimate of 30◦ _{via the chart in Figure 3.29.}

30◦_{, which is where the target is actually located. However, within the unaliased}

boundary, the ULA shows a target at approximately -3.2◦_{. The staggered array}

spacing algorithm will provide a method for resolving the ambiguity generated by the grating lobe. Since the short space pairs are separated by a distance that

lobe. However, the long space pairs produce a peak that is offset from the ULA grating lobe, at 7.5◦_{. The combination of the two peak values with the dealiasing}

rules produce an angle estimate of ˆθ = 30◦_{, the actual target angle value.}

While in theory the staggered array spacing technique is shown to work, there are concerns with some artifacts of the processing that must be addressed. Due to the pair processing, the beamwidth of the individual pairs is extremely large, on the order of 9.4◦ _{and 4}◦ _{for the short and long pairs, respectively. Resolving}

a single target in the absence of noise proves the methodology and theory, how- ever, in practice the technique may not be achievable. In the case of multiple or distributed targets, an average position of all scatterers would be produced as the peak value for each set of pairs. Thus, it is likely that a nearly flat angu- lar spectrum would be produced, causing the dealiasing algorithm to fail. More work in studying the effects of the staggered array spacing is required to identify potential applications and implementations.