Optimized cross correlation waveforms for first order correlation loops

The phase-locked loop (PLL) architecture is studied extensively. It is demonstrated that PLLs are optimal structures in terms of minimizing steady-state mean square tracking error (MSE) at high signal-to-noise ratios (SNR). A closely related device, the delay-locked loop (DLL), is useful for estimating the time delay, or phase, of nonsinusoidal signals. Both PLLs and DLLs fall into the general class known as correlation-loop architectures. It is shown that the steady-state mean square tracking error of first-order correlation loops is minimized by varying the locally generated cross correlation function produced by the voltage controlled oscillator. This analysis assists in the design of modified first-order DLLs used for pseudonoise code synchronization in direct sequence spread-spectrum receivers. The optimization is performed for periodic, but otherwise arbitrary, transmitted signals, for systems operating at arbitrary SNR. The optimal reference waveform is produced by passing a replica of the transmitted signal through a linear time-invariant filter