Question

The array factor of an N element linear uniform array is \(\newline\) (\(\Psi = \beta d \cos\theta + \alpha\), \(\beta = \frac{2\pi}{\lambda}\), d = spacing between elements and \(\alpha\) = inter element phase shift)

• A. \( \frac{\cos(n\Psi/2)}{\sin(\Psi/2)} \)
• B. \( \frac{\sin(N\Psi/2)}{\sin(\Psi/2)} \)
• C. \( \frac{\sin(N\Psi)}{\cos(\Psi)} \)
• D. \( \frac{\sin(\Psi/N)}{\sin(\Psi)} \)

Hint:

The array factor for an N-element uniform linear array is the sum of a geometric progression of phasors.
The magnitude is often expressed as \( |\sin(N\Psi/2) / \sin(\Psi/2)| \), where \(\Psi = \beta d \cos\theta + \alpha\) is the total phase difference between adjacent elements in direction \(\theta\).
This function is also known as the Dirichlet sinc function or aliased sinc function in some contexts.

Answer 1

The Correct Option is B

For an N-element linear uniform array with element spacing 'd', inter-element phase shift '\(\alpha\)', and wave number \(\beta = 2\pi/\lambda\), the phase difference between signals from adjacent elements in a direction \(\theta\) (measured from the array axis) is given by:

\( \Psi = \beta d \cos\theta + \alpha \). 

The array factor (AF) is the sum of the contributions from each element, considering these phase differences. If the element excitations are uniform (amplitude 1), the array factor can be expressed as the sum of a geometric series:

Array Factor: \( AF = \sum_{k=0}^{N-1} e^{jk\Psi} = 1 + e^{j\Psi} + e^{j2\Psi} + \dots + e^{j(N-1)\Psi} \)

This is a geometric series with first term \(a=1\), common ratio \(r=e^{j\Psi}\), and \(N\) terms. The sum of the series is:

Sum: \( AF = \frac{a(1-r^N)}{1-r} = \frac{1(1 - e^{jN\Psi})}{1 - e^{j\Psi}} \)

To get the magnitude form often presented:

Array Factor Magnitude: \( AF = \frac{e^{jN\Psi/2}(e^{-jN\Psi/2} - e^{jN\Psi/2})}{e^{j\Psi/2}(e^{-j\Psi/2} - e^{j\Psi})} = e^{j(N-1)\Psi/2} \frac{-2j\sin(N\Psi/2)}{-2j\sin(\Psi/2)} \)

After simplification:

Array Factor: \( AF = e^{j(N-1)\Psi/2} \frac{\sin(N\Psi/2)}{\sin(\Psi/2)} \)

The magnitude of the array factor is:

Magnitude: \( |AF| = \left| \frac{\sin(N\Psi/2)}{\sin(\Psi/2)} \right| \)

Often, the phase term \(e^{j(N-1)\Psi/2}\) is ignored when considering just the array factor as a pattern shape, or if the array is centered for symmetry. The expression \( \frac{\sin(N\Psi/2)}{\sin(\Psi/2)} \) represents the normalized magnitude pattern (or the unnormalized form if the phase factor is considered part of the overall complex AF).

Option (b) gives the correct normalized magnitude pattern:

\( \boxed{\frac{\sin(N\Psi/2)}{\sin(\Psi/2)}} \)