Beam shaping

Figure 1 shows the excitation of a linear array antenna with a half wavelength element spacing (i.e.,  ) and four elements of amplitude , , , .

Figure 1  Geometry of the problem

$QUESTION$. Find the position of the pattern nulls.

The nulls are at

$MESSAGEFALSE$<<This is not the correct answer

Clue: The array factor is >>

$RESULT$

$QUESTION$. Modify the elements voltage excitations so that the pattern has nulls at -45° and 30°.

Let  the amplitude of the i-th element voltage excitation

$MESSAGEFALSE$<<This is not the correct answer

Clue: Every array factor of a linear array with commensurable spacing between elements can be represented by a polynomial in the complex variable , where . An array of N elements is represented by a polynomial of degree N-1. It follows then from the fundamental theorem of algebra that this polynomial has exactly N-1 roots and can be factored into the form , where the  ’s are the zeros (roots) of the polynomials. It displays explicitly the zeros in the array factor, since the zeros are those  values which correspond to , and so on. Of course, zeros in the array factor give zeros in the radiation pattern provided they are within the visible range. Since only two nulls are required we can put the remaining null outside the visible range.>>

$RESULT$

$QUESTION$. Sketch the original and new array factors as a function of γ.

Original array factor

$RESULT$

$subquestion$

New array factor

$RESULT$

$END$

Score = $score$

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