The maximum of CRLB of time delay estimation

Cramer-Rao lower bound (CRLB) is very significant to evaluate parameter´s estimation results. This paper proposes to study the maximum of CRLB of time delay estimation (TDE) under certain signal to noise ratio, when we only need to get a reasonable range of estimation variance to test the validity of an algorithm without knowing the exact formula of signal, or to design a waveform which is difficult to be located. First, we deduce the estimation variance of time delay using the principle of least squares. Then, through comparing to CRLB in frequency domain, we know that they are same to express the lower variance bound for unbiased TDE. Finally, with time-frequency uncertainty principle, we could know that Gaussian envelope is the waveform to maximize CRLB of TDE under certain signal to noise ratio. Simulations also illustrate that Gaussian envelope is close to the maximum of CRLB of time delay.