On the power spectral density of self-synchronizing scrambled sequences

We derive a closed-form expression for the power spectral density of amplitude/phase-shift keyed bit sequences randomized through self-synchronizing scrambling when the source sequence is a stationary sequence of statistically independent bits. In addition to the dependence on the symbol pulse shape, duration, and the signal space values with which symbols are represented, we show that the power spectral density is dependent only on the probability of logic ones in the source bit stream, the period of the impulse response of the scrambling shift register, and the number of logic ones in this period. Our results confirm that optimum randomization results with use of primitive scrambling polynomials and poorest randomization occurs with "two-tap" polynomials of the form x/sup D/+1.