Entropy of Some Binary Sources

A sequence of independent identically distributed binary random variables in which the probabilities of the binary symbols 0 and 1 are not equal to 1/2 is applied to the homogeneous shift register with $k$, $k\leq 1$, memory locations (a $k$-register). The sequence of binary symbols from the $k$-register output is delivered to the input of a memoryless binary symmetric channel in which noise is independent of the $k$-register input sequence. We show that for $k=\overline{1,3}$ the entropy [R. Gallager, Information Theory and Reliable Communication, Wiley, New York (1968)] of the output sequence from the channel may increase with the increase of $k$.