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