Effects of Phase Errors on Resolution

The mathematical problem consists of determining the spread of the Fourier transform of function when the function is modified by a multiplicative factor exp jα(t), where α is a stationary random process. Let F(ω) be the Fourier transform of f(t) and Fm(ω) be the transform of f(t) exp jα(t). For example, f may be the illumination function of a linear antenna and α accounts for imperfect phasing of the antenna. The major results consist of simple formulas for the rms tilting (or shifting) of the pattern |Fm|2 and the rms radius of gyration (or beamwidth) of the pattern. These positional errors and resolution degradations are formulated in terms of the pattern in the absence of phase errors and the power density spectrum of α'. The problem of calculating the best obtainable resolution, i.e., minimizing the mean-square resolution over all possible illumination functions, requires numerical solution; however, it is shown that it is always possible to obtain a rms resolution better than the smaller of rms α' and √rms α'. The actual numerical solution is compared to this simple approximation for the case of sinusoidal phase errors. The general results have a broad scope of applications, and here the spreading of the ambiguity function in time and frequency in the presence of time phase errors and dispersion (frequency phase errors) is described with particular attention to linear FM pulses. Finally, some observations are made about quadratic phase errors, signal-to-noise performance, and mean-square point-target response.