at time t 0 and position u03b6 0 there will consequently at any given instant be a difference in the phase angles of the signals emanating from the two antennas given by where subscript 1 refers to the 1 st antenna 10 and subscript 2 refers to the 2 nd antenna 12 it is convenient to substitute a new variable u for the function sin u03b8 so that the variable u has a range u2212 1 u2266 u u2266 1 all angles are measured modulo 2 u03c0 in the range u2212 u03c0 lt u03c6 u2266 u03c0 therefore if k gt 0 5 u03b4 u03c6 u12 can fall outside the measurable range and wraps back to a measured phase difference where p is in general an unknown positive or negative integer as a consequence the determination of u and eventually u03b8 from a measurement u03b4 u03c6 m12 is ambiguous for the sparse array considered here with k of the order of 5 u03b4 u03c6 m12 can wrap up to four times when where p is unknown for k 5 p takes on integer values in the range u2212 2 u2266 p u2266 2 for an unambiguous determination of the arrival angle u03b8 some means must be found to determine p in the radar system described here p is determined by repeating the phase difference measurement at a second frequency see m skolnik u201c resolution of angular ambiguities in radar array antennas with widely spaced elements and grating lobes u201d antennas and propagation ire transactions on vol 10 no 3 pp 351 352 may 1962 the author describes the use of measurements at two frequencies to identify and eliminate echoes from targets in grating lobes of a scanning radar with a sparse array antenna this in effect changes the electrical antenna spacing.