Modulation waveforms with continuous derivatives in conjunction with a receiver whose impulse response has continuous derivatives is shown to generate crosstalk interference in a correlation demodulator that decreases at least as the cube of the carrier spacing between waveforms. Constant envelope, continuous-frequency modulations called Continuous (Frequency) Shift Keying (CSK) are proposed which not only achieve this asymptotic crosstalk behavior but are better then Phase Shift Keying (PSK) and Minimum Shift Keying (MSK) for carrier spacings of at least 1.1 times and 2.0 times the data rate, respectively. CSK can be demodulated in Gaussian noise with no loss with respect to optimum hard-decision receivers of antipodal signals, i.e., with the same performance as PSK and MSK.

This paper discusses single-input, single-output binary sequential circuits composed of adders, multipliers and delay units. The principal results are: 1) Finite feedforward circuits are characterized by a response function (rf) which permits the determination of the output for an arbitrary input. 2) Corresponding to every rf is a transfer function (tf) which is obtained by a multidimensional Z-transform of the rf. This tf permits the determination of the response of a feedforward circuit by transform domain techniques. 3) Every tf may be synthesized as a physical circuit. 4) A canonic form is defined for every tf. With this canonic form is identified an implementation which contains a minimum number of delay units and adders. 5) Nonfeedforward circuits are called well defined if they meet certain physically meaningful restrictions. 6) Well-defined circuits may be classified as those which have inverses (class I) and those which do not. Membership in class I is shown to be a property of circuits which established one-one maps of inputs onto outputs. Circuits which do not belong to class I have neither left nor right inverses. 7) An intimate connection between inverses and feedback circuits is established. It is shown that any well-defined feedback circuit and a related circuit in I are mutually inverse. 8) It is shown that the necessary and sufficient condition for a feedback circuit to be well defined is that the circuit in the feedback loop contains delay.

By relating the average probability of error to the distortion measure of a source-sink pair, we prove a converse to the channel coding theorem. This converse lower-bounds the probability of error per source letter. It differs from the more familiar "weak" and "strong" converses which bound the probability of error of an entire message. The result is applicable to all stationary sources, all channels, and all block lengths. Lower-bounding the rate distortion function of the source-sink pair with which the channel is to be used reduces the new result to a lower bound on the achievable probability of error per source letter expressed in terms of the source entropy, alphabet size, and maximum achievable average mutual information on the channel. This latter result had been proved previously only for a memoryless channel operating with an independent letter source.

A scheme is described which sequentially encodes the output of a discrete letter source into the input symbols of a discrete input memoryless channel, with adjacent channel symbols mutually constrained over a length,n. The encoder permits desired channel input symbol probabilities to be approximated closely. Decoding at the receiver is accomplished with delaynby means of sequential tests on potential transmitted sequences with reject criteria set so that incorrect sequences are likely to be rejected at short lengths and the correct sequence is likely to be accepted. Averaged over a suitably defined ensemble of encoders, the decoding scheme has an average probability of error with an upper limit whose logarithm approaches-nE(R)for largen. E(R)is dependent only on the data rateR. For a channel symmetric at its output with equally likely inputs, the exponentE(R)is optimum for rates greater than a rate calledR_{crit}. For such symmetric channels, a computation cutoff rateR_{comp}is defined. ForR < R_{comp}, the average number of decoding computations does not grow exponentially withn, but algebraically.R_{comp}is also defined for an asymmetric channel. In this case too, the average number of decoding computations grows algebraically withnforR < R_{comp}.