Information rates by oversampling the sign of a bandlimited process

A sequence of binary (±1) random variables is generated by sampling a hard-limited version of a bandlimited process at n times the Nyquist rate. The information rate ℐ carried by these binary samples is investigated. It is shown by constructing a specific nonstationary, bounded, bandlimited process (the real zeros of which are independent and identically distributed, isolated, and lying in different Nyquist intervals) that ℐ=log₂(n+1) bits per Nyquist interval is achievable. A more complicated construction in which L distinct zeros are placed in L consecutive Nyquist intervals yields achievable rates that approach (for L→∞) ℐ arbitrarily closely, where ℐ=log₂n + (n-1)log₂[n/(n-1)], n⩾2 (and ℐ=1 for n=1 and L=1). By exploiting the constraints imposed on the autocorrelation function of a stationary sign (bilevel) process with a given average transition rate, the latter expression is shown also to yield an upper bound on the achievable values of ℐ. The logarithmic behavior with n (n≫1) is due to the high correlation between the oversampled binary samples, and it is established that this trend is also achievable with stationary sign processes. This model may be used to gain insight into the effect of finite resolution on the information (in Shannon´s sense) conveyed by the sign of a bandlimited process, and also to assess the limiting performance of certain oversampling-based communication systems