Analysis of Uniform Phased Arrays

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In this page, we'll derive a general equation for the array factor or antenna array response for an N element uniformly spaced linear antenna array. The weights will be simple phased weights; when the antenna array is steered towards direction , the weights are given by:

phased weights

Assuming that element n is at location given by:

uniformly spaced antenna array

This implies that the inter-element spacing is constant and equal to d. Our goal now is to determine the response of the array when steered towards , when the weights are chosen using the equation above.

Using the definition of the array factor, we can write:

definition of array factor (AF)

The above can be simplified by recalling the definition of the wave vector:

phase delay as a function of frequency (wavenumber) and position

Substituting the above equation into the array factor equation,

array factor

In the above equation, G is a "dummy variable" that is simply given by:

dummy variable used for antenna theory

Recall the following sum formula, which will make our work simple:

mathematical sum formula

The array factor can be rewritten using the above identity as:

simplified array factor using sum formula

Really, we only care about the magnitude of the array factor. Hence, we can factor out terms from the numerator and denominator that will simplify the results when we take the magnitude:

magnitude of the array factor

Taking the magnitude of the above equation, the multiplying complex exponentials (which always have a magnitude equal to one) go away. In addition, using the following general formula for the sin() function:

formula for sine function sin(x)

The magnitude of the array factor reduces to:

reduced form of array factor

In the next section, we'll look at understanding this equation (which explains grating lobes), and extend the results to two-dimensional (planar) antenna arrays.

Next: Further Analysis of Uniform Arrays

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