On the Invariance of Cross-Correlation Peak Positions Under Monotonic Signal Transformations, with Application to Fast Time Difference Estimation

We present a theorem concerning the invariance of cross-correlation peak positions, which provides a foundation for a new method for time difference estimation that is potentially faster than the conventional fast Fourier transform (FFT) approach for real/complex sequences. This theoretical result shows that the peak position of the cross-correlation function between two shifted discrete-time signals remains unchanged under arbitrary monotonic transformations of the input signals. By exploiting this property, we design an efficient estimation algorithm based on the cross-correlation function between signals quantized into low-bit integers. The proposed method requires only integer arithmetic instead of real-valued operations, and further computational efficiency can be achieved through number-theoretic algorithms. Numerical experiments demonstrate that the proposed method achieves a shorter processing time than conventional FFT-based approaches.