By sampling zero crossings at the Nyquist rate (i.e., the predetection bandwidth) the phase of a narrow-band signal may be extracted directly with respect to an arbitrary, stable reference. Analysis, simulation, and experimental results are presented validating this all-digital technique for acquisition and coherent tracking of a Doppler-shifted carrier at a low carrier/noise ratio and a large percentage bandwidth. The key advantage of this scheme for microcomputer-based implementation is that phase comparison is accomplished by arithmetic subtraction, without the coordinaterotation transformation which is necessary for conventionalI,Qsampling. Also, the phase-sampling process is independent of the phase error, which exists only as a numerical representation in the digital tracking algorithm. The technique is applicable to Costas suppressed-carrier tracking and to BPSK bit synchronization and demodulation, by programming the appropriate digital algorithms.

Casting phase tracking into a binary decoding problem, as in a previous paper, a sequential decoding algorithm is applied to a third-order system. The goal is to reduce the "tracking threshold" below that of a phase-locked loop. Simulation results are obtained by microprogramming the algorithm for a HP-2100A minicomputer.

If a Costas loop is forced to have a relatively wide bandwidth, for example, to track phase variation due to platform motion, binary phase-shift keying (BPSK) demodulator performance is significantly degraded. However, if the data bits were known, the Costas loop would become equivalent to a phase-locked loop, which has approximately a 6-dB lower tracking threshold for the same loop bandwidth. A delayed-bit estimation algorithm with a recursive structure similar to the Viterbi algorithm is described. The memory in the tracking loop is exploited to correct preliminary bit decisions, as a means of realizing most of this theoretical threshold reduction. Experimental results for a practical digital implementation of the new algorithm for a 50-bit/s data rate and a 20-Hz loop noise bandwidth show a 4-dB improvement in demodulator threshold.

A Costas loop for tracking a BPSK signal does not acquire when the initial frequency error is comparable to the loop bandwidth. This paper describes methods for deriving control proportional to frequency error so as to improve the frequency acquisition capability by an AFC augmentation. A composite AFC/Costas loop is realized by combining the individual loops. Pull-in from a frequency error much greater than the Costas loop bandwidth is now feasible. The increase in phase error due to the AFC capability is evaluated by a linearized analysis. If the bandwidth of the AFC portion is sufficiently narrow, the degradation is negligible. Improved frequency acquisition can also be realized by a simple modification of the Costas loop to remove the low-pass filter on the quadrature phase detector channel, with negligible degradation to phase tracking.

Algorithms are described for estimating carrier phase by a maximum-likelihood procedure similar to the Viterbi algorithm for decoding a digital sequence. These algorithms avoid the constraint of physical realizability imposed by a phase-locked loop and allow a processing lag in estimation. An approximate analysis suggests that threshold extension can be achieved thereby. Computer simulations demonstrate the improved performance for demodulation of FM and for tracking of phase modulation due to dynamic maneuvers.

Convolutional encoding with biphase modulation is an attractive combination of modulation/coding for the Gaussian channel, but a means of tracking carrier phase must be provided. The performance of combined sequential decoding and suppressed carrier (Costas loop) tracking in the receiver is difficult to analyze but can be obtained by computer simulations. Results of such simulations are presented in the form of a Pareto fit to the computation distributions for various tracking loop bandwidths and demodulator quantization (soft decisions). These distributions are applied to predict probability of error performance for typical decoder parameters (buffer size, speed factor). The anticipated advantage of quantized demodulation is observed. However, narrow tracking bandwidths are required, with sufficient interleaving to overcome effects of correlated phase perturbations due to noise.

For a coherent channel transmitting binary digits, a worst distribution under an average power constraint can be defined. A solution for the very noisy channel is found to be Gaussian, and the minimax demodulator utilizes correlation detection to guarantee an exponential bound on probability of error.

A simplified analysis is presented to explain physically the change of signal-to-interference ratio which occurs in a band-pass limiter. The analysis utilizes the concept of sideband resolution into symmetric and anti-symmetric parts and considers only the asymptotic case where the signal-to-interference ratio is small in comparison with unity. Wide-band correlation-detection systems are discussed, as well as ordinary band-pass systems. The important conclusion is reached that the degradation is highly dependent on the statistics of the interference amplitude fluctuations. However, when the signal is weak compared to the interference, the maximum possible degradation is 6 db and occurs for constant-amplitude interference. Degradation with noise interference in a wide-band correlation-detection system has been obtained for arbitrary signal and noise bandwidths. It is found that the degradation ranges between 0.6 db and 1.0 db, the latter figure being for the case where the signal bandwidth is greater than approximately three times the noise bandwidth.

The sampling theorem enables a band-limited signal to be expressed in terms of a set of sample point values, which occur at the Nyquist rate. The sampling theorem has been generalized to include nonuniform sampling and the use of derivatives of the signal. In the present paper, a sampling theorem has been developed which utilizes information related to the zeros of the signal. The concept of complex zeros is introduced to show that the zeros occur at the Nyquist rate. This sampling theorem can be of use for enabling the transmission of binary signals, such as facsimile and infinitely-clipped speech, over a continuous band-limited channel. The result indicates the desirability of developing a completely general theory of sampling applicable to the various situations which may arise in practice.