Binary Communication with Large Bandwidth to IF Frequency Ratios

The effect of I F filter bandwidth to I F frequency ratio on the probability of error in a binary bandpass communication system with nonreturn-to-zero (NRZ) and split-phase (SP) signals is investigated. ed. Explicit results are given for the case when the bandpas I F filter is transformed from a single-pole low-pass filter.     

An Analysis of the Performance of the Oscillating Limiter Driven by FM Signals Corrupted by Noise

The behavior of the oscillating limiter (OL) driven by FM signals is surveyed, and its performance with signal corrupted by noise is investigated. For high values of the carrier-to-noise ratio (CNR), if the frequency deviation of the signal is small in comparison with the locking range of the OL, it is calculated, and experimentally verified, that a system OL discriminator is equivalent to a system bandpass limiter discriminator followed by a linear network whose frequency response has been specified. When the frequency deviation is not so small, the baseband noise power increases with it; a formula is given that allows the calculation of this power when the signal is such that the circuit operates in quasistationary fashion. For low values of the CNR, a mathematical analysis presents unsurmountable difficulties. However, heuristic argumentation leads to an interpretation of the operation of the OL in the threshold region, which is substantiated by an experimental investigation. The results of this paper enable a comparative evaluation of a system OL discriminator and a system bandpass limiter discriminator, to which the former reduces when the feedback path in the OL is open.     

The Performance of Analog Correlators for Several Integrator Functions

A measure of analog correlator performance is considered and interpreted reted as an output signl-to-noise ratio. This quantity is shown to depend on the fourth-order expectation of the input signals and the characteristics of the integrating system. Based on the output signal-to-noise ratio, figures of merit are established for correlators utilizing an ideal low-pass filter, an RC filter, and a finite time integrator. These figures of merit indicate to what degree the correlators reject noise components. For jointly Gaussian inputs, the variation of the output signal-to-noise ratio with the input signal-to-noise ratio is shown to be independent of the integrating system. Finally, a graphical comparison indicates the characteristics of the three systems for several different parameters. This comparison shows that the finite time integrator is superior to the other two systems considered.     

Probability Density Function at the Output of the Delay-Lock Discriminator

The first-order probability density at the output of the correlation network of the delay-lock discriminator is determined using the methods developed by Arthur, Kac, and Siegert. The analysis assumes that the input signal is stationary and Gaussian and that the discriminator is locked on. No restrictions are made on the bandpass filter characteristics in the channels.     

Logarithmic Detection with Postdetection Filtering

An analysis is presented of the performance of a logarithmic detector or followed by a low-pass filter, under the assumption of a large ratio of IF bandwidth to postdetection bandwidth. This allows for the probability of detection and similar quantities to be computed in a straightforward manner. The mean and variance of the filter output are found by first deriving the autocorrelation function of the logarithmic detector output and then obtaining the power spectral ral density. A useful engineering approximation is made for the variance which is valid for all signal-to-noise ratios.     

The Effect of Hard Limiting an Angle-Modulated Signal Plus Noise

The effect of hard limiting an angle-modulated signal plus narrow-band Gaussian noise is analyzed. Several examples are considered?sinusoidal angle modulation, Gaussian angle modulation, and biphase angle modulation. The general conclusion is that when a zonal band-pass filter is used, which rejects dc and second harmonics, an angle-modulated signal plus Gaussian noise provides the same output signal-to-noise ratio as shown by Davenport for a CW signal plus Gaussian noise. However, when a narrow bandpass filter is used, which has a bandwidth approximately equal to the input angle-modulated signal, an angle-modulated signal plus Gaussian noise has a better output signal-to-noise ratio than a CW signal plus Gaussian noise.     

Performance Degradation of MTI Circuits Due to Amplitude Limiting

The loss in output signal-to-noise ratio (SNR) due to amplitude limiting is obtained for a radar circuit consisting of a bandpass limiter, coherent demodulator, matched filter, and moving-target-indicator (MTI) filter. The circuit is used in scanning MTI radars. The tandem connection of the limiter and coherent demodulator is a model for the saturation of the intermediate-frequency (IF) demodulator of an MTI radar. Results on special functions are used to obtain simple formulas for the loss in output SNR relative to a linear IF demodulator when the input SNR is less than -15 dB and the number of hits per 3-dB beamwidth exceeds 15.     

The Response of Hard-Limiting Bandpass Limiters to PM Signals

The signal-to-noise ratio of the output of a hard-limiting bandpass limiter to a PM signal is calculated by using the probability density function of the random phase variable. The signal-to-noise ratio transfer characteristics are plotted for comparison.     

Signal-to-Noise Ratios in Self-Biased-Pwer-Law Amplifiers with Saturation

The relation between the output signal-to-noise ratio (SNR) and the input SNR is presented for the bandpass self-biased third-law amplifier with saturation, when the input is composed of sinusoidal carrier and zero-mean stationary narrowband Gaussian noise. It is found that significant improvement in the output SNR at low input SNR's can be achieved by the self-biased third-law amplifier with saturation operated in class A. The results obtained are also verified experimentally.     

Maximizing the Usable Bandwidth of MTI Signal Processors

A technique is presented for maximizing the percentage of usable Doppler bandwidth throughout which a radar return can be detected while maintaining an acceptable clutter suppression. The technique employs the weighted Chebyshev approximation to the design of a transversal high-pass digital filter which has an optimal passband ripple for a given number of filter weights and associated integration gain consistent with the required increase in signal-to-noise ratio needed for acceptable probabilities of detection and false alarm. Conventional approaches to the design of a movingtarget arget indictor (MTI) filter which maximizes the improvement factor by clutter suppression typically improve the signal-to-background noise ratio over less than 50 percent of the range between dc and the pulse-repetition frequency fT. This technique can increase the usable bandwidth to 80 percent or more of fT. Two examples are included which utilize parameter values from the Army Missile Command's experimental radar and demonstrate the interactive influence of such filter parameters as the number of weights, passband ripple and bandedge, and stopband attenuation and cutoff.     

Detection of Known Signals in Nonstationary Noise

The basic design of a nonlinear, time-invariant filter is postulated for detecting signal pulses of known shape imbedded in nonstationary noise. The noise is a sample function of a Gaussian random process whose statistics are approximately constant during the length of a signal pulse. The parameters of the filter are optimized to maximize the output signal-to-noise ratio (SNR). The resulting nonlinear filter has the interesting property of approximating the performance of an adaptive filter in that it weights each frequency band of each input pulse by a factor that depends on the instantaneous noise power spectrum present at that time. The SNR at the output of the nonlinear filter is compared to that at the output of a matched filter. The relative performance of the nonlinear system is good when the signal pulses have large time-bandwidth products and the instantaneous noise power spectrum is colored in the signal pass band.     

Phase-Shift-Keyed Signal Detection with Noisy Reference Signals

This paper derives and graphically illustrates the performance characteristics of Phase-Shift-Keyed communication systems where the receiver's phase reference is noisy and derived from the observed waveform by means of a narrow-band tracking filter (a phase-locked loop). In particular, two phase measurement methods are considered. One method requires the transmission of an auxiliary carrier (in practice, this signal is usually referred to as the sync subcarrier). This carrier is tracked at the receiver by means of a phase-locked loop, and the output of this loop is used as a reference signal for performing a coherent detection. The second method is self-synchronizing in that the reference signal is derived from the modulated data signal by means of a squaring-loop. The statistics (and their properties) of the differenced-correlator outputs are derived and graphically illustrated as a function of the signal-to-noise ratio existing in the tracking filter's loop bandwidth and the signal-to-noise ratio in the data channel. Conclusions of these results as well as design trends are presented.     

Interference Effects of Pseudo-Random Frequency-Hopping Signals

When a pseudo-random frequency-hopping signal is intercepted by a conventional receiver operating within the same frequency band, the interfering signal has the form of a pulse-amplitude modulated signal. Each pulse amplitude is dependent upon the hopping frequency and the selectivity characteristic of the victim receiver. The probability density function for the interfering pulse amplitude prior to demodulation is determined when the probability density function for the hopping frequency is uniform and the victim-receiver characteristic is 1) ideal flat bandpass, 2) single tuned, and 3) Gaussian shaped. It is shown that the average interfering pulse amplitude and interference power decrease as the frequency-hopping bandwidth increases with respect to the victim-receiver bandwidth. Fast Fourier transform computer techniques are used to obtain the probability density function of the interference amplitude in a Gaussian receiver when several (from 2 to 10) pseudo-random frequency-hopping systems are simultaneously using the same frequency band. The probability that the interference exceeds a prescribed threshold value is computed from the derived probability density functions. This probability may be used in signal-to-interference ratio calculations, to describe the capture effect, or to compute the expected number of clicks produced in an FM discriminator.     

The Effect of Integrator Pole Position on the Performance of an Adaptive Array

The LMS adaptive array requires an integrator in each weight feedback control loop. In practice the integrator is often replaced by a low-pass filter, i.e., by a filter with a single pole at s = - ? (where s is complex frequency). The effect of this pole position on array performance is examined. It is shown that to obtain optimal performance from the array, ? must be less than k?2, where k is the loop gain and ?2 is the thermal noise power per element. When at exceeds k?2, the output signal-to-inter ference-plus-noise ratio from the array is degraded for intermediate values of interference power.     

Some Matched Filter Configurations for Infrared Systems

An analysis of the output of three alternative matched filter configurations in an infrared scanning system model is presented. The sensor is corrupted by thermal noise, generation-recombination noise, photon noise, and modulation noise, the latter providing an extreme discoloration in the signal passband. Expressions for the signal voltage density spectrum, signal pulse shape, noise power spectrum, and average noise power at the matched filter output are derived where the integral evaluations attendant to these derivations do not appear elsewhere in the literature. The paper also provides graphical displays of the signal-to-noise power ratio at the filter output versus various system parameters, noise power spectrum out of the matched filter versus ?, and the signal pulse shape out of the filter versus time. Also included are discussions of practically realizable approximations to the matched filters and curve fitting techniques for the signal pulse shape function.     

Hard Limiting and Sequential Detection Applied to Loran-C

The combination of an antenna, a 100 kHz bandpass filter, a hard limiter, and a sequential detector can supply highly accurate Loran-C data to a digital processor, even under low signal-to-noise-ratio conditions. For such a simple, low-cost receiver, calculations are given for the accuracy of the envelope and phase tracking of the Loran-C signal as a function of the signal-to-noise (Gaussian and atmospheric) ratio, averaging time, and radian speed of the observer with respect to the transmitter. Mentioned are the quasi-noise censoring effects of the hard limiter. Besides the Loran-C application, the hard limiter-sequential detector system can in general be applied for low-cost, synchronous signal detection under poor signal-to-noise ratio.     

A mean level adaptive detector using nonconcurrent data

Convergence results for a mean level adaptive detector (MLAD) are presented. The MLAD consists of an adaptive matched filter (for spatially correlated inputs) followed by a mean level detector (MLD). The optimal weights of the adaptive matched filter are estimated from one batch of data and applied to a statistically independent batch of nonconcurrent data. The threshold of the MLD is determined from the resultant data. Thereafter a candidate cell is compared against this threshold. Probabilities of false alarm and detection are derived as a function of the threshold factor, the order of the matched filter, the number of independent samples per channel used to calculate the adaptive matched filter weights, the number of samples used to set the MLD threshold, and the output signal-to-noise power ratio of the optimal matched filter. A number of performance curves are shown and discussed     

Noncoherent Detection of Split-Phase FSK by Multiple Predetection Filtering

When the cumulative drift in the center frequency of a binary split-phase FSK signal exceeds the peak deviation of the signal, a conventional noncoherent receiver (i.e., one provided with only two IF filters) may be unable to achieve the probability of error per bit which the designer desires. This limitation may be overcome if the receiver is provided with a bank of more than two contiguous filters (each followed by an envelope detector) tospan the total IF band the instantaneous IF signal might occupy. It is shown that the probability of error per bit for such a receiver is a function of 1) the ratio F of peak frequency deviation to peak frequency drift, 2) the number M of IF filter/detectors, and 3) the signal-to-noise ratio ? in the output of the filter containing the signal. It is further shown thatfor a given value of F an increase in M reduces the amount of transmitter power the communication system designer must provide to yield a given probability of error per bit.     

Phase Analysis in Multiple-Sensor Receivers with High Signal-to-Noise Ratio

The ideal phase detector characteristic is analyzed for estimating the phase difference between two stochastic input signals. This can essentially be described as a correlation process formed by multiplying the two input signals and extracting the phase. High signal-to-noise ratio conditions are assumed to linearize the system with respect to the noise. The effects of the nonlinearity on the signal are handled in terms of a series expansion and by using low-pass filtering on the receiver output. The mean square error of the system is calculated for some typical parameters.     

Output of a Balanced Frequency Discriminator Driven by a Sine Wave and Gaussian Noise

The output of a realizable balanced frequency discriminator is calculated for an input consisting of a sine wave plus Gaussian noise. Explicit autocorrelation and power spectra are found for one practical embodiment of the discriminator for various input carrier-to-noise ratios, with the carrier tuned to center frequency and also off-center. The formulas also permit the calculation of the output for the case of a spectrum of noise slowly swept through the discriminator. Although qualitatively similar to results previously obtained with an ideal discriminator, substantial differences are also found. Measurements are made that closely verified the theoretical results. No limiting is assumed.