^{1}

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.

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 [

Others have also considered truncation error mitigation in various near-field measurement contexts [

We wish to approximate a physical quantity

When full-sphere data are available, the orthogonality of vector spherical harmonics can be used to determine the coefficients

In this paper, we assume that data are available only in the forward hemisphere

The discrepancy

Avoiding details for the moment, we write

Since

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, (

Similarly, for (

With substitution of (

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

In this case, the input data were simulated using a far-field pattern calculated from a specified set of (

Simulated probe: deviations from the

Simulated probe: comparison of the full-sphere E-plane far-field patterns obtained using

In this example, the

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.

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

Cylindrical waveguide probe: deviations from the

Cylindrical waveguide probe: comparison of the full-sphere E-plane far-field patterns obtained using the

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

Rectangular waveguide probe: deviations from the

Rectangular waveguide probe: comparison of the full-sphere E-plane far-field patterns obtained using the

Several tests indicate that the quality of the

We have demonstrated a constrained

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

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.