Phase-comparison monopulse is a technique used in radio frequency (RF) applications such as radar and direction finding to accurately estimate the direction of arrival of a signal from the phase difference of the signal measured on two (or more) separated antennas or more typically from displaced phase centers of an array antenna. Phase-comparison monopulse differs from amplitude-comparison monopulse in that the former uses displaced phase centers with a common beam pointing direction, while the latter uses a common phase center and displaced beam pointing directions. In phase-comparison monopulse, typically an array is subdivided into sub-arrays, and then a "sum" and a "difference" or "del" channel are formed. For a linear array, these subarrays would each be half of the elements, divided in the middle. For a planar array, these sub-arrays would be the four quadrants of the array, each with 1/4 of the array's elements. In a linear array, the output of each sub-array is summed to form the "sum" channel, and the same outputs are subtracted to form the "del" channel. The monopulse ratio is formed by dividing the imaginary part of the del channel by the real part of the sum channel. This ratio gives an error signal that indicates to a high degree of accuracy the actual target angle as compared to the center of the beam. For a planar array, one sum channel is formed as the sum of the outputs of all four quadrants, but two del channels are formed, one for the elevation dimension and one for the orthogonal azimuth dimension. Two monopulse ratios are formed just as with a linear array, each one indicating the deviation angle in one dimension from the center of the beam. There are some common misconceptions about phase comparison monopulse. First, only one beam is formed. Monopulse processing is done entirely with the received signal in the array manifold and beam forming network. Speaking in terms of only one dimension for clarity, such as with a linear array, the signal is received by the array and summed into each of two subarrays with displaced phase centers. The sum channel is formed simply by adding these two subarray outputs, and the result is exactly the same as if the entire array was initially summed in one step. The del channel is formed simply by subtracting these same subarray outputs. Second, phase-comparison monopulse doesn't technically actually do a phase comparison, 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:
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