Schelkunoff Polynomial Method

Weighting Methods
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Instead of steering an antenna array (in which case we want to receive or transmit primarily in a particularly direction), suppose instead we want to ensure that a minimum of energy goes in particular directions. The weights of an antenna array can be selected such that the radiation pattern has nulls (0 energy transmitted or received) in particular directions. In this manner, undesirable directions of interference, jamming signals, or noise can be reduced or completely eliminated.

It turns out that this isn't real hard to do, either. In general, an N element array can place N-1 independent nulls in its radiation pattern. This just requires a little math to work through, and will be illustrated via an example. Let's assume we have an N element linear array with uniform spacing equal to a half-wavelength and lying on the z-axis.

To start, the array factor of a uniformly spaced linear array with half-wavelength spacing can be rewritten using a variable substitution as:

array factor (AF) for an antenna array of uniformly spaced antennas

The above equation is simply a polynomial in the (complex) variable z. Recall that a polynomial of order N has N zeros (which may be complex). The polynomial for the AF above is of order N-1 zeros. If the zeros are numbered starting from zero, the zeros will be 0, 1, ..., N-2. The AF is rewritten then as:

rewriting the array factor for an antenna array in terms of desired zeros or nulls

We've introduced variables, zeros or null locations and have gotten rid of the weights. Hence, we can choose the zeros selected zeroed or null directions in the antenna array's radiation pattern to be whatever we want, and then figure out what the weights should be to give us the same pattern.

To make the example concrete, let N=3. Suppose we want the array's radiation pattern to have zeros at 45 and 120 degrees. We can simply use the equation for z above, and substitute these values for the angle. We then obtain the corresponding zeros, zeros. The z values corresponding to the zeros are:

z values corresponding to null directions for radiation pattern of antenna arrays

For simplicity, we'll let weights selected for an antenna array. The AF then becomes:

array factor written as a function of the nulls or zeros

This AF must equal the original AF, so:

AF written out with the nulls or zeroed out directions

The weights then can be easily found to be:

weights for antenna array with nulls in specified directions or angles

We already know what the desired null directions are, so we automatically have the weights. Using these weights to plot the magnitude of the array factor, we obtain the result in Figure 1.

magnitude of array factor for array designed with nulls in particular directions

Figure 1. Magnitude of array factor.

Observe that the radiation pattern has zeros at 45 and 120 degrees, exactly as we specified. This method can be used for whatever directions you want; however if N-1 nulls are selected for an N element array, the designer no longer has control over where the maximum of the radiation pattern is.

This method can easily be performed on linear arrays with many more elements. This method can be The Schelkunoff Polynomial Method easily extends to planar and multi-dimensional arrays. The simplicity of placing nulls in the radiation pattern adds a powerful advantage for using arrays in practice.

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