Using Truncated Data Sets in Spherical-Scanning Antenna Measurements

Abstract

We discuss the mitigation of truncation errors in spherical-scanning measurements by use of a constrained least-squares estimation method. The main emphasis is the spherical harmonic representation of probe transmitting and receiving functions; however, our method is applicable to near-field measurement of electrically small antennas for which full-sphere data are either unreliable or unavailable.

1. Introduction

The transmitting function of an electrically small probe tends to be very broad, so that full-sphere data are needed to compute the spherical-harmonic representation of the pattern by use of standard methods [1]. Unfortunately, backward-hemisphere data are often unavailable or at best unreliable due to support-structure blockage. Simply setting the backward-hemisphere data to zero leads to ringing effects that degrade the accuracy of the probe-pattern representation in the forward hemisphere. In this paper, we approximate the spherical-harmonic expansion of transmit/receive functions using only forward hemisphere data. The solution presented is a least-squares fit with energy constraints (to limit backward-hemisphere radiation) and proper weighting of measurements (to compensate for clustering near the poles). The energy constraint is specified using an estimated value of directivity. Results are given that compare pattern representations obtained using both full-sphere and half-sphere data.

Others have also considered truncation error mitigation in various near-field measurement contexts [2–5].

2. Theory

We wish to approximate a physical quantity 𝐛 by a spherical-harmonic expansion𝐛𝐛̂((𝜃𝜃,,𝜑𝜑))==𝑏^𝜃_𝑀(𝜃,𝜑)𝜽+𝑏^𝜑(𝜃,𝜑)𝝋≈𝐛̂(𝜃,𝜑),(1)_𝑁𝜇_=−𝑀_{𝜈=𝑛(𝜇)}𝑎^1𝜈𝜇𝑛(𝜇)=max1,||𝜇||,𝐗||𝑀_𝜈𝜇(𝜃||,≤𝜑𝑁)+.𝑎^2𝜈𝜇𝐘_𝜈𝜇(𝜃,𝜑),(2) Here, 𝐗_𝑛𝑚 and 𝐘_𝑛𝑚=𝑖𝐫̂×𝐗_𝑛𝑚 are vector spherical harmonics [6, chapter 16]. For example, 𝐛(𝜃,𝜑) might be the measured transmitting or receiving pattern of a probe, or it might be the “measurement vector” 𝐰(𝜃,𝜑) for a spherical near-field scanning measurement [1, 7–9].

When full-sphere data are available, the orthogonality of vector spherical harmonics can be used to determine the coefficients 𝑎^1𝜈𝜇 and 𝑎^2𝜈𝜇 from discrete measurements of 𝐛(𝜃,𝜑) gathered on a uniform grid in 𝜃 and 𝜑. When so determined, 𝐛̂(𝜃,𝜑) is optimal in the sense that^𝜋0sin𝜃𝑑𝜃^02𝜋||𝐛(𝜃,𝜑)−𝐛̂(𝜃,𝜑)||²𝑑𝜑(3) is minimal.

In this paper, we assume that data are available only in the forward hemisphere 𝜃≤𝜋/2 on the grid𝜃_𝑛=𝜋2𝑃^𝜑_𝑚𝑛,0≤𝑛≤𝑃^,=2𝜋𝑄𝑚,0≤𝑛<𝑄,(4) with 𝑄𝑃^>>2𝑁𝑀,.(5) Requirement (5) preserves the standard number of measurement points by halving the maximum sample interval in 𝜃. Ideally, 𝑃^ and 𝑄 are products of small primes to allow efficient application of the fast Fourier transform (FFT) algorithm; however, factorization may not be important when antennas are “electrically small.” Our goal is to minimize the discrepancyΔ=^𝑃′_𝑛=0𝑤_𝑛^02𝜋||𝐛𝜃_𝑛,𝜑−𝐛̂𝜃_𝑛,𝜑||²𝑑𝜑,(6) where the 𝑤_𝑛 are positive weights. Equation (6) may be viewed as an adaptation of (3). Because the sampling theorem has been satisfied, the 𝜑 integral can be readily evaluated analytically. The 𝜃 integral, however, has been discretized. (We generally set 𝑤_𝑛=sin𝜃_𝑛).

The discrepancy Δ is minimized subject to the “energy” constraint𝐸=_𝜈𝜇||𝑎^1𝜈𝜇||²+||𝑎^2𝜈𝜇||²,(7) in order to limit the power radiated into directions where there are no measurements. When 𝐛(𝐫̂) is proportional to the transmitting pattern of the test antenna, we have𝐸=4𝜋||𝐛(𝐫̂)||²𝐷(𝐫̂),(8) so that 𝐸 may be chosen using an estimate of the directivity 𝐷 in some measurement direction 𝐫̂. (Directivity and gain are approximately equal when ohmic losses are small).

Avoiding details for the moment, we write 𝐛≈𝐛̂=𝐀𝐱,(9) where 𝐛 and 𝐛̂ are vectors of measurements and predicted values, 𝐱 is a vector of coefficients to be determined, and the matrix 𝐀 represents a known linear relationship. We choose 𝐱 to minimize Δ=(𝐛−𝐀𝐱)^∗𝜿(𝐛−𝐀𝐱) subject to the constraint 𝐸=‖𝐱‖². Here, 𝜿 is a diagonal matrix of positive weights, and an asterisk implies Hermitian transpose. It is easy to show that this 𝐱 is determined by the equations 𝐀^∗𝜿𝐀𝐱−𝐱𝜆𝐱=𝐀^∗𝜿𝐛,(10)_∗𝐱=𝐸(11) The significance of the Lagrange multiplier 𝜆 is seen in the relation||𝜆||=‖‖𝐀^∗𝜿𝐀𝐱−𝐀^∗𝜿𝐛‖‖√𝐸(12) that is, |𝜆| is a measure of the residual of the fit. In the absence of the energy constraint (7) of course, we can choose 𝐱 to be the solution of the normal equations 𝐀^∗𝜿𝐀𝐱=𝐀^∗𝜿𝐛.

Since 𝐀^∗𝜿𝐀 is Hermitian nonnegative definite, we have the singular-value decomposition (SVD)𝐀^∗𝜿𝐀=𝐒𝐃𝐒^∗,(13) where 𝐒 is unitary, 𝐷_𝑖𝑗=𝑑_𝑖𝛿_𝑖𝑗, and 𝑑_𝑖≥0. From (13), (10), and (11), we are led to𝐸=_𝑖||𝑓_𝑖||²𝑑_𝑖−𝜆²𝐟≡𝐒_∗,𝐀^∗𝜿𝐛.(14) The right side of (14) tends to 0 as |𝜆|→∞, and because of poles at the singular values 𝑑_𝑖, there may be numerous solutions for 𝜆. According to (12), the best choice corresponds to the unique solution for which 𝜆<𝑑₁, where 𝑑₁ is the smallest singular value. In general, 𝜆<0, since the constrained solution is expected to have lower energy than the unconstrained solution. Finally, given 𝜆, we may compute 𝐱,𝑥_𝑖=_𝑗𝑆_𝑖𝑗𝑓_𝑗𝑑_𝑗−𝜆.(15) Numerical methods used in this paper are discussed in greater detail in [10].

At this point, we have discussed the main ideas of this paper. What follows are some rather unpleasant details that we summarize dutifully and concisely. To begin with, (1) can be written as a Fourier series𝐛(𝜃,𝜑)=^𝑀_𝜇=_−𝑀𝑏^𝜃𝜇(𝜃)𝜽+𝑏^𝜑𝜇(𝜃)𝝋exp(𝑖𝜇𝜑),(16) where the coefficients may be computed from the data with a discrete Fourier transform𝑏^{𝜇𝜃,𝜑}(𝜃)=1𝑄^𝑄−1_𝑚=0𝑏^𝜃,𝜑𝜃,𝜑_𝑚exp−𝑖2𝜋𝑄𝑚𝜇.(17)

Similarly, for (2), 𝐛̂(𝜃,𝜑)=^𝑀_𝜇=_−𝑀𝑏̂^𝜃𝜇𝑏̂(𝜃)𝜽+𝑏^𝜑𝜇̂(𝜃)𝝋exp(𝑖𝜇𝜑),(18)_{𝜇𝜃,𝜑}(𝜃)=^𝑁_𝜈=_𝑛(𝜇)𝑎^1𝜈𝜇𝐗𝑋^{𝜃,𝜑𝜈𝜇}(𝜃,0)+𝑎^2𝜈𝜇𝑌^{𝜃,𝜑𝜈𝜇}(𝜃,0)_𝜈𝜇,𝐘=𝑋^𝜃𝜈𝜇𝜽+𝑋^𝜑𝜈𝜇𝝋,_𝜈𝜇𝑌=𝑌^𝜃𝜈𝜇𝜽+𝑌^𝜑𝜈𝜇𝝋,_𝜃𝜈𝜇=−𝑖𝑋^𝜑𝜈𝜇,𝑌^𝜑𝜈𝜇=𝑖𝑋^𝜃𝜈𝜇.(19)

With substitution of (16) and (18), the discrepancy (6) becomesΔ=2𝜋^𝑃′_𝑛=0𝑤_{𝑛𝑀𝜇=}_−𝑀|||𝑏^𝜃𝜇𝜃_𝑛−𝑏̂^𝜃𝜇𝜃_𝑛|||²+||𝑏^𝜑𝜇𝜃_𝑛−𝑏^𝜑𝜇̂𝜃_𝑛||².(20) This may be rewritten in the formΔ=2𝜋^𝑀_𝜇=_−𝑀𝐛_𝜇−𝐀_𝜇𝐱_𝜇^∗𝜿_𝜇𝐛_𝜇−𝐀_𝝁𝐱_𝝁.(21) Here𝐱_𝜇=⎛⎜⎜⎝𝐱^1𝜇𝐱^2𝜇⎞⎟⎟⎠,𝐛_𝜇=⎛⎜⎜⎝𝐛^𝜃𝜇𝐛^𝜑𝜇𝐱_𝜇1,2⎞⎟⎟⎠,=⎛⎜⎜⎜⎜⎝𝑎𝑎⋮^{1,2𝑛(𝜇),𝜇}_1,2𝑁𝜇⎞⎟⎟⎟⎟⎠,𝐛_{𝜇𝜃,𝜑}=⎛⎜⎜⎜⎜⎝𝑏^{𝜇𝜃,𝜑}𝑏_{𝜇𝜃,𝜑}⋮𝜃₀𝐀_𝜇𝜃_𝑃^′⎞⎟⎟⎟⎟⎠,=⎛⎜⎜⎝𝜶^𝜇𝜷_𝜇𝑖𝜷_𝜇𝑖𝜶_𝜇𝜶_𝜇⎞⎟⎟⎠,=⎛⎜⎜⎜⎜⎝𝑋^{𝜃𝑛(𝜇),𝜇}𝑋⋮𝜃₀,0⋱⋯𝑋^𝜃𝑁𝜇⋮_{𝜃𝑛(𝜇),𝜇}𝜃₀,0𝜷𝜃_𝑃^′,0⋯𝑋^𝜃𝑁𝜇𝜃_𝑃^′,0_𝜇⎞⎟⎟⎟⎟⎠,=−𝑖⎛⎜⎜⎜⎜⎝𝑋^{𝜑𝑛(𝜇),𝜇}𝑋⋮𝜃₀,0⋱⋯𝑋^𝜑𝑁𝜇⋮_{𝜑𝑛(𝜇),𝜇}𝜃₀,0𝜿𝜃_𝑃^′,0⋯𝑋^𝜑𝑁𝜇𝜃_𝑃^′,0_𝜇⎞⎟⎟⎟⎟⎠,𝐰=⎛⎜⎜⎜⎜⎜⎜⎝𝑤₀=⎛⎜⎜⎝𝐰𝟎𝐰𝟎⎞⎟⎟⎠,0𝑤0₁⋯0⋮0⋱⋯⋱⋱0𝑤⋮0_𝑃′⎞⎟⎟⎟⎟⎟⎟⎠.(22) Finally, with the definitions𝐀=⎛⎜⎜⎜⎜⎜⎜⎝𝐀^−𝑀𝟎𝐀𝟎_−𝑀+1⋯𝟎⋮𝟎⋱⋯⋱⋱𝟎𝐀⋮𝟎_𝑀𝜿=⎛⎜⎜⎜⎜⎜⎜⎝𝜿_−𝑀⎞⎟⎟⎟⎟⎟⎟⎠,𝟎𝜿𝟎_−𝑀+1⋯𝟎⋮𝟎⋱⋯⋱⋱𝟎𝜿⋮𝟎_𝑀𝐱=⎛⎜⎜⎜⎜⎝𝐱_−𝑀⎞⎟⎟⎟⎟⎟⎟⎠,𝐱⋮_𝑀⎞⎟⎟⎟⎟⎠,𝐛=⎛⎜⎜⎜⎜⎝𝐛_−𝑀𝐛⋮_𝑀⎞⎟⎟⎟⎟⎠,(23) we have𝐛𝐸≈=𝐛̂‖=𝐱‖𝐀₂𝐱,Δ=(𝐀𝐱−𝐛)_∗,𝜿(𝐀𝐱−𝐛).(24) Thus, the measurement truncation problem has been reduced, explicitly, to the constrained least-square optimization problem discussed earlier. In particular, (10) and (11) can be written as𝐀^∗𝜇𝜿_𝜇𝐀_𝜇𝐱_𝜇−𝜆𝐸𝐱_𝜇==𝐀^∗𝜇_𝑀𝜿_𝜇𝐛_𝜇,−𝑀≤𝜇≤𝑀,(25)_𝜇=_−𝑀‖‖𝐱_𝜇‖‖².(26) Because of the block-diagonal structure, the overall SVD can be broken into smaller parts, one 𝜇 at a time, resulting in an algorithm with computational complexity 𝒪(𝑁⁴). Comparing unfavorably with the 𝒪(𝑁³) complexity for the full-sphere case, our method may be less useful for larger antennas. The order of 𝐀_𝜇 is 2(𝑃^+1)×2(𝑁−𝑛(𝜇)+1), so there will be more equations than unknowns in every block |𝜇|≤𝑀 as long as 𝑃^≥𝑁. Actually, there is always a unique solution to our constrained optimization problem; however, the quality of this solution is observed to degrade dramatically as 𝑃^ is decreased below the threshold 𝑃^=𝑁.

3. Experimental and Simulated Results

To investigate the utility of our least-squares technique for reducing the truncation error that results from zero filling in the rear hemisphere, we examined a number of typical probes for which full-sphere far-field patterns are available. In each case, we computed the far-field pattern using the spherical mode expansion obtained (a) from the standard algorithm with full-sphere data, (b) from the standard algorithm with zero-fill in the backward-hemisphere, and finally, (c) from the constrained least-squares fit to forward-hemisphere measurements described previously. Ideally, the far-field pattern calculated from the spherical-mode expansion should agree with the original pattern. For the full-sphere modal calculation, this is of course the case if the sampling theorem is satisfied. We show E-plane results only, since truncation effects are more important because the patterns tend to be broader. For all cases, the sampling is 2° in 𝜃 and 5° in 𝜙.

3.1. High-Gain Symmetric Probe

In this case, the input data were simulated using a far-field pattern calculated from a specified set of (𝑚=±1) modal coefficients. For this antenna, we used 𝑁=10, 𝑀=5. The directivity of 15.44dB was used in (8) to determine the energy constraint. Figures 1 and 2 indicate excellent agreement between the full-sphere and least-squares techniques in the forward hemisphere.

Figure 1 

Simulated probe: deviations from the full-sphere pattern (errors) are shown for the zero-fill (long dash) and least-squares (short dash) techniques. The full-sphere pattern is shown as a solid line.

Figure 2 

Simulated probe: comparison of the full-sphere E-plane far-field patterns obtained using zero-fill (long dash) and least-squares (short dash) techniques. The full-sphere pattern is shown as a solid line. The least-squares and full-sphere patterns are indistinguishable at this scale.

In this example, the constrained least-squares technique also performs well in the backward-hemisphere if the correct energy is specified. This is not necessarily true in the presence of noise. As a test, we added random errors with an RMS value of 0.1% relative to the noise-free RMS pattern level. An unconstrained fit resulted in an on-axis directivity of −32dB. In other words, the back hemisphere overwhelmed the forward hemisphere. On the other hand, a constrained fit forced the back hemisphere pattern to remain at a reasonable level even though the details were incorrect. Although an unconstrained fit may be somewhat better in the forward directions, a constrained fit is undoubtedly more reasonable given our knowledge of the directivity of the test antenna.

In probe-corrected spherical near-field measurements, for example, the complete probe pattern is required. We often argue that the forward hemisphere pattern is most important, but all bets are off when the back hemisphere pattern dwarfs the forward hemisphere pattern.

3.2. NIST Circularly Cylindrical Waveguide Probe

This cylindrical waveguide probe was designed and built for spherical near-field measurements at 3.3 GHz. The far-field pattern was measured over the entire sphere, although support-structure blockage limits the value of backward-hemisphere information. In this case, we used 𝑁=17, 𝑀=17. A directivity of 7.87dB was calculated using the full-sphere data. Deviations (Figure 3) between the full-sphere and zero-fill methods will lead to significant errors when the zero-fill results are used in spherical scanning measurements. Within the range ±75^∘, the maximum error in the zero-fill result is about 0.6dB, while the least-squares technique has a maximum error of 0.08dB. Figure 4 shows how the full-sphere, zero-fill, and least-squares results compare in the backward-hemisphere. In this case, the truncation level is as high as −12dB, which causes a greater discrepancy between the zero-fill and full-sphere patterns than observed in the previous example.

Figure 3 

Cylindrical waveguide probe: deviations from the full-sphere pattern (errors) are shown for the zero-fill (long dash) and least-squares (short dash) techniques. The full-sphere pattern is shown as a solid line.

Figure 4 

Cylindrical waveguide probe: comparison of the full-sphere E-plane far-field patterns obtained using the zero-fill (long dash) and the least-squares (short dash) techniques. The full-sphere pattern is shown as a solid line.

3.3. Rectangular Waveguide Probe

This probe is a section of WR-284 rectangular waveguide. Far-field patterns were obtained over the entire sphere at 3.3 GHz. In this case, we used 𝑁=17, 𝑀=17. Strictly speaking, rectangular waveguide probes do not have the correct symmetry for spherical near-field scanning applications; however, they often perform satisfactorily when the scan radius is more than about an antenna diameter. Thus, we include results for this type of probe. A directivity of 7.12dB, calculated from the full-sphere modal expansion, was used to determine the energy constraint. Measured gain was 6.7dB. The results are similar to those for the cylindrical waveguide case. As evident in Figure 5, within the angular range ±75^∘, the zero-fill error is greatest at about 40°, where it is 0.66dB. The error is only about 0.11dB for the least-squares technique at this angle. In the forward hemisphere, the zero-fill technique exhibits large errors, while the least-squares technique errors remain relatively small. Figure 6 provides a full-sphere comparison of the three techniques.

Figure 5 

Rectangular waveguide probe: deviations from the full-sphere pattern (errors) are shown for the zero-fill (long dash) and least-squares (short dash) techniques. The full-sphere pattern is shown as a solid line.

Figure 6 

Rectangular waveguide probe: comparison of the full-sphere E-plane far-field patterns obtained using the zero-fill (long dash) and the least-squares (short dash) techniques. The full-sphere pattern is shown as a solid line.

3.4. Choice of 𝐸

Several tests indicate that the quality of the least-squares fit in the forward hemisphere is relatively insensitive to the choice of 𝐸. In a typical case with 0.9𝐸₀≤𝐸≤1.1𝐸₀, the errors observed in the results were generally much less than 0.15dB within 20° of the on-axis direction. (Here, 𝐸₀ is the correct value). When gain is used to approximate directivity, experience indicates that the error in 𝐸 usually will be less than 10%.

4. Summary and Future Work

We have demonstrated a constrained least-squares technique for calculating the spherical mode coefficients of a small antenna from forward-hemisphere far-field data. This technique can significantly reduce truncation errors that arise when the standard algorithm is used with zero-fill in the backward-hemisphere. While the least-squares algorithm is not especially efficient, computational times are still acceptable for small antennas.

Our technique can also be used to process near-field data when reliable full-sphere measurements are not available. The method may be adapted to serve when measurements are made in the range 0≤𝜃≤𝜃₀, where 𝜃₀ is not restricted to 90^∘. We are considering an iterative algorithm to find both 𝐱 and 𝜆 simultaneously. For well-conditioned systems, this could effectively reduce the computational complexity to 𝑂(𝑁³). By beginning with a spherical-wave expansion and projecting onto a scanning surface, our method should be useful for mitigation of truncation error in planar and cylindrical scanning measurements, as well.

Acknowledgments

The authors thank Allen Newell, consultant, for bringing this problem to our attention. They also thank Brad Alpert of NIST for insightful discussions concerning the numerical methods employed in this work.