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# Antenna arrays and nulls

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#### TomerIKO

##### Newbie level 3
Hi,

I was looking for a material about mitigating interferences and found out about Null-Steering methods using antenna arrays.

As it was written in the article, an array of N elements can produce N-1 nulls (N-1 direction where the gain would be minimized).

However I couldn't find out the theory of why you can only have N-1 nulls and not more.

Can anyone explain it to me or direct me to an article about the number of nulls you can produce?

Tomer

Hi Tomer,

With a single antenna the pattern is fixed.
With one additional antenna you can manage the amplitude and phase of the combination so you can make than in a particular direction (and only one you want) the total gain is zero.
With two additional antennas, you gain a degree of freedom and you can decide the positions of two nulls.
An so one.
There are as many degrees of freedom as additional antennas. They must be independent; for instance, two identical antennas at the same location in space count as only onr.
Regards

Z

Hey Z,

I understand what u said but I'm looking for the math proof that with 2 antennas there is only 1 direction I can make the total gain zero, and generally with N elements N-1 directions.

Hi Tomer,

Suppose we have two antennas with complex gains A0(θ,φ) and A1(θ,φ).
We form the combination

A(θ,φ) = A0(θ,φ)+g1*A1(θ,φ)

where g1 is the unknown gain. If you want a null in the direction (θ1,φ1), solving for g1:

g1 = -A0(θ1,φ1)/A1(θ1,φ1)

If the two antennas have the same complex pattern (e.g. they are identical and at the same location) nulling for (θ1,φ1) nulls at all directions. For this they must be "independent".
Extending this idea, if you have N independent elements you can have a linear system with N-1 conditions to satisfy and N-1 unknowns (the gains gi). If the matrix is invertible, there is a single solution.

Regards

Z

TomerIKO

Points: 2