On the uncertainty relation for real signals

We investigate the form assumed by the uncertainty relation when the signal involved is real and the uncertainty in frequency is defined as the variance of |F + (ω)|2, where F + (ω) is the positive frequency spectrum of the signal. The corresponding uncertainty product is found to obey an inequality involving the value |F + (0)|2. If F + (ω) is obtained by suppressing the negative frequencies in a suitable Gaussian function, the uncertainty product as just defined can be made appreciably less than ½, the lower bound of the uncertainty product as conventionally defined. In this paper, uncertainty relations are understood to be inequalities involving certain measures of the time duration and frequency bandwidth of signals, and to have no implications about the simultaneous accuracy of time and frequency measurements other than what can be derived from these inequalities