but rather simply divides the del channel by the sum channel to arrive at a ratio wherein the angle information is encoded. The following mathematical derivation should make it clear why this is so. Mathematics Sum Pattern We can define the beam pattern (array factor) of a uniform linear array (ULA) with N elements, as: , where is the array manifold vector and is a vector of complex weights representing amplitude and phase adjustments applied to each antenna element. The manifold vector, , fully encapsulates all of the spatial properties of the array. is the distance between elements of the array, and is the angle of arrival of an incident plane wave, defined from end-fire, i.e., is a signal from array broadside. It is common to perform a variable substitution to -space, where , and therefore we have: and we can more easily see that is simply the phase shift between adjacent elements. The term simply references the absolute phase to the physical center of the array. Notice that this result is the same if we instead first sum each half of the array, then add those results together. The weight vector is a combination of a steering vector that steers the beam in a steered direction, , using phase adjustments and an amplitude taper that is often applied to reduce sidelobes. Thus, , and , where . We can clearly see now that the beam pattern, in -space, is the spatial equivalent of the discrete time Fourier transform (DTFT) of the array amplitude tapering vector times a linear phase term. The advantage of -space is that the beam shape is identical no matter where it is steered, and is only a function of the deviation of the desired target phase from the actual target phase. Let us now assume an un-tapered, normalized array with . The beam pattern can be easily shown to be the familiar aliased sinc (asinc) function: This pattern is also known, for monopulse purposes, as the "sum" pattern, as it was obtained by summing all of the elements together. Going forward we will suppress the subscript and instead use only with the understanding that it represents the deviation of the steered target phase and the actual target phase. Difference Pattern Let us now develop the monopulse "difference" or "del" pattern by dividing the array into two equal halves called subarrays. We could have just as