An Integral Inequality Bounding the Autocorrelation of a Pulse or Sequence at a Known Lag

This paper gives best bounds for the ratio ∫ b−t a f(x) f(x + t) dx/ ∫ b a f(x) dx for any square-summable real function f(x) on the interval (a, b]. Similarly, bounds are established for the autocorrelation of any pulse or finite-length sequence at any known lag, and the family of pulses and sequences attaining these bounds is identified. The form of this family is related to a half-cycle of a sinusoid. Stronger bounds are suggested for pulses known to be non-negative and unimodal or concave.