Spectra and Efficiency of Binary Codes Without DC

In digital transmission of binary (+l, 1 ) signals it is desirable that the stream of pulses which constitutes the signal have no dc, that is, that the power spectrum go to zero at zero frequency. It is desuable that, for a given efficiency or entropy, the spectrum rise slowly with increasing frequency. We have obtained the spectrum for selected blocks with equal numbers of plus ones and minus ones. For a given efficiency, this is better than the spectrum obtained by Rice, using the Monte Carlo method, for block encoding using polarity pulses. An algorithm given by Schalwijk should allow simple encoding into selected blocks. punch a hole anywhere in the spectrum by basing the choice of polarity pulses on a Fourier analysis.) In unpublished work, Rice has obtained the power spectrum analytically for a random signal stream for a block length of 2, and he made Monte Carlo calculations for block lengths of 2 , 4 , 6 , 8 and 16. The Monte Carlo points lay about the correct curve for n = 2, but all the Monte Carlo spectra were ragged. We have made estimates by inspection of the fraction of the Nyquist band at which the power spectrum rose to 1/2 of the flat spectrum amplitude for a sequence of random digits; these will be tabulated later. Another scheme is block encoding using only selected blocks which contain equal numbers of positive and negative pulses. In this case the efficiency or transmission rate E , in bits per digit, is given by The spectra (hitherto unpublished) and the efficiencies of 1 n ! In digital transmission the power spectrum of$he pulse train (9)' ' two schemes of encoding which give pulse streams with no dc are compared, and it is shown that one is superior to the other. can be shaped through the introduction of redundancy. Usually in so doing the spectrum is made zero at some particular Here is the length. For large Of n 7 frequency and small near that frequency. Frequencies commonly chosen ar 0 and N / 2 (the Nyquist bandwidth). Practi1 cal reasons for doing this include [ 1 ] E = log, _____ (1) n E ; 1 1og2n + 1 log,(:)+ o&). (2) 2n 2n 1) ( f = 0) avoidance of interference with audio and other Fortunately, the power spectrum p ( w ) can be found as the low-frequency signals; ensemble average for blocks chosen randomly with equal 2) ( f = 0) signals will pass through transformers; probabilities. It is derived in the Appendix and is such that 3) (f= 0 or N/2).allows transmission of a carrier or timing