Cramer–Rao Lower Bound for Frequency Estimation of Sinusoidal Signals

Frequency estimation of sinusoidal signals, with wide-ranging applications, has been a fundamental topic in signal processing for some time. The Cramer–Rao lower bound (CRLB) is widely known as the threshold for the minimum variance when estimating the frequency of sinusoidal signals. Numerous previous studies simplified the numerical evaluation of CRLB, often assuming a linear relationship between the CRLB and the length of utilized data records. The actual values of CRLBs for the frequency estimation of sinusoidal signals are derived and calculated using the original formula of the CRLB and numerical testing methods. Experimental results indicate the following: 1) at a specific frequency, the value of CRLB is a range determined by the initial phase and recording length $N$ ; and 2) the waveforms used to obtain the maximum and minimum values of CRLB exhibit either even or odd symmetry at different frequencies. This attribute is used to obtain the value of the CRLB without significant computations. Given the widespread use of zero-initial-phase signals in engineering and research, we focused on this scenario, examining the CRLB values across various frequencies for potential future applications.