Page 206 - Fundamentals of Radar Signal Processing

(3.27)

Defining s = sinθ and α = D/λ, Eq. (3.27) can be rewritten as

(3.28)

which is a sinc-squared function. It follows immediately that its Fourier
transform is a triangle function in the normalized variable (x/λ), where x is the
spatial dimension of the antenna aperture (Bracewell, 1999). This function is
illustrated in Fig. 3.17.

FIGURE 3.17 Fourier transform of the two-way antenna voltage pattern for an
ideal rectangular antenna aperture with uniform illumination.

Because the Fourier transform of the antenna pattern has a width of 2α, the
Nyquist sampling interval in s must be

(3.29)

Recall that s = sinθ. To convert T into a sampling interval in θ, consider the
s
differential ds = cosθ dθ, so that dθ = ds/cosθ. Thus, a small interval T in s
s
corresponds approximately to an interval T = T /cosθ in θ. The minimum value
θ
s
for T occurs when θ = 0 so that T = T . Thus, the sampling interval in angle
θ
s
θ
becomes (using α = D/λ for the second step)

(3.30)

This is the Nyquist sampling interval in angle for a rectangular aperture of size
D with uniform illumination.
As a final step, this result can be expressed in terms of 3-dB beamwidths.