A continuous-time extension of welch~s lower bound on total squared correlation

In this paper, Welch´s lower bound on the total squared correlation (TSC) for a set of vectors is extended to a set of band-limited waveforms. First, a continuous-time equivalent of total squared correlation (CTE-TSC) is derived by examining the definition of the TSC for a set of vectors in multiple access communication system. The multiple-access system is assumed to employ these vectors as spreading sequences and matched filters as receivers. Then, a lower bound on the CTE-TSC is derived using a frequency domain approach. It is shown that the lower bound on the CTE-TSC has the same form as the Welch´s lower bound except that the dimensionality of the vectors is replaced by the average degrees of freedom of the vectorized Fourier transform of the band-limited waveforms. Unlike the dimensionality of a vector, this average degrees of freedom of a band-limited signal used as a transmit waveform for a linear modulation is not restricted only to positive integers. Interestingly, it turns out that the set of waveforms that achieves the lower bound on the CTE-TSC is also the set of the optimum transmit waveforms for overloaded multiple access communication system employing linear minimum mean-squared error (LMMSE) receivers. This extends to continuous cases the discrete counterpart that the Welch bound equality sequences are optimal sequences for overloaded synchronous code-division multiple-access system employing LMMSE receivers