Maximum Likelihood Energy Detection of Mary Orthogonal Signals

The literature on energy detection is extended by applying it to the processing of M'ary orthogonal communication signals of arbitrary time-bandwidth product. Guassian noise channel transition probabilities are derived for maximum likelihood energy detection, modified to the extent of including an erasure threshold. Relations and computational techniques are described for determining the symbol erasure and error probabilities for general signal alphabet sizes (M) and time-bandwidth products. When time of arrival is known exactly and Doppler is negligible, gible, it is determined that energy detection is inferior to noncoherent matched filter detection. For time-bandwidth products of the order of 100 and error probabilities around 10 3, a loss of about 5 dB occurs which is attributable to a lack of knowledge of the detail signal structure. However, for problems where time of arrival and/ or Doppler are unknown, energy detection will perform nearly as well as matched filter detection of, for example, spread spectrum signals, and is also simpler to implement. General energy detection performance curves are given in terms of required signal energy for specific error and erasure probabilities, as a function of M'ary and time-bandwidth product.     

Chirp Waveform Generation Using Digital Samples

This paper describes two methods of generating an analog frequency-modulated waveform by the use of a small number of digital samples of the ?chirp? waveform. The number of digital samples required is a function of the time-bandwidth product. For certain values of time-bandwidth product, this type of signal generation becomes extremely efficient. Several proofs are offered which show how to select ?optimum? values of time-bandwidth products. Two hardware implementations are suggested. One is based on the use of modulo arithmetic and a small stored memory table. The second method utilizes the inherent signal symmetries available if ?optimum? time-bandwidth products are selected. The symmetrical signal patterns are stored in recirculating reversible shift registers which can be read out at high speeds.     

Degradation of Probability of Error Due to IF Filtering

The probability of error P(e) is computed in a binary communication system with a single-pole IF filter. The effect of time-bandwidth product foT and IF frequency to band width ratio f1/Ifo on P(e) is shown. The sampling time is optimized and the effect of nonoptimal sampling time on P(e) is calculated.     

A Matched-Filter Pulse-Compression System Using a Nonlinear FM Waveform

The realization of a rectangular pulse-compression waveform having low time sidelobes and zero mismatch loss due to spectral weighting is discussed. The theoretical aspects of the design of such a waveform are presented with particular reference to frequency modulated, rectangular pulses. The design and performance of a matched-filter pulse-compression system having essentially zero mismatch loss are presented. The system discussed has a time-bandwidth product of 22 and time sidelobes suppressed at least 27 dB; the measured mismatch loss is 0.1 dB. The difficulty of achieving the required nonlinear time delay dispersion is overcome by synthesizing the dispersive network as a cascade of all-pass networks.     

Stability of Parameter Estimates for a Gaussian Process

Maximum-likelihood estimates for the levels of the mean value function and the covariance function of a Gaussian random process are investigated. The stability of these estimates is examined as the actual covariance function of the process deviates from the form assumed in the estimators. It is found that the time-bandwidth product for stationary processes represents an upper bound on the number of estimator terms that can be safely used when estimating with uncertainty about the process covariance function. This result is consistent with other interpretations of the time-bandwidth product and tempers the conclusion that, in principle, an infinite number of estimator terms can be used to obtain a perfect estimate of the covariance level. In practice, the estimate of the level can never be perfect, and the accuracy of the estimate depends on the observation interval. Finally, conditions are established to ensure asymptotic stability of the estimates and physical interpretations are presented.     

Likelihood-ratio detection of frequency-hopped signals

The optimum system for intercepting frequency-hopped signals uses a channelized receiver and a likelihood-ratio test (LRT). Previous results on the performance of the optimum system have been based on Gaussian assumptions, which are generally valid for transmissions having large time-bandwidth areas. Results, obtained by Monte Carlo simulation, for relatively small time-bandwidth transmissions are given here. The signal model is a simple one known as “pure frequency hopping.” Comparisons with the LRT show that energy detection loss increases when the time-bandwidth product of the transmission is increased by increasing the number of frequencies, even when the number of pulses is also increased. The loss decreases when only the number of pulses is increased. Over the parameter range observed, binary detection loss tends to increase with the number of pulses and decrease with the number of frequencies. Results are included for a moving-window version of the LRT. A parameter of the LRT is the signal-to-noise ratio (SNR). The effect of using a design value not equal to the true SNR is shown     

Linear FM Signal Formats for Beacon and Communication Systems

This paper examines the capabilities of the class of linear FM spread-spectrum signals within the context of potential communications systems usage in order to establish some performance criteria and bounds that permit comparison with other spread-spectrum formats. A systematic basis is provided for parameter selection for this class of signals by examining the interaction a mong the frequency-modulation indices, time-bandwidth product, and cross-talk criteria that determine the number of effective linear FM signals (or channels) that can be used within the constraints of a bounded time-frequency region. A general expression is derived relating N, the number of useful signals, R2, a cross-talk parameter, ToWo, the mean time-bandwidth product, and ?max and ?min, the maximum and minimum FM rates of the signal set. Canonic signal processor structures are described for ensembles of linear FM signals that have either constant duration or constant bandwidth. It is then shown that the signal modulation format can be modified in accordance with classical paired-echo theory to expand the utility of this class of signals in both synchronous and nonsynchronous operations to yield the equivalent of time-division and code multiplexing. Possible applications for this signal format are discussed.     

Some Properties and Effects of Reverberation in Acoustic Surveillance

Following the work of Van Trees,[1] the effect of wide-sense stationary clutter on signal detectability with a matched filter is determined. The improvement to be gained by a high time-bandwidth product in the transmitted waveform for the detection of low-velocity targets is clearly shown. The additional noise contributed by the clutter is reduced by a factor equal to the time-bandwidth product. This reducing effect occurs provided that the transmitted waveform is adjusted properly. The optimum transmitted waveform for detection of low-velocity targets turns out to be one whose energy density spectrum is flat over the bandwidth of interest. This derivation is made by a simple application of Schwarz's inequality rather than the application of the calculus of variations that was done by Manasse.[14] Computations were made of the loss encountered by a narrow-band single-frequency waveform and by a wide-band linear FM waveform, each used in a matched-filter detector. The contrast is especially marked for very low target speeds where the narrow-band waveform is very bad. Its loss drops off sharply with target speed while the loss of the wide-band waveform drops off very slowly in comparison. Beyond a certain small target speed, the narrow-band loss is negligible. However, with enough bandwidth, the wide-band waveform can be made to have acceptable loss at all target speeds.     

Adaptive array transient sidelobe levels and remedies

The transient sidelobe level of a sidelobe canceler (SLC) is a function of the external noise environment, the number of adaptive auxiliary antennas, the adaptive algorithm used, auxiliary antenna gain margins, and the number of samples used to calculate the adaptive weights. An analytical result for the adaptive sidelobe level is formulated for the case when the adaptive algorithm is the open-loop, sampled matrix inversion (SMI) algorithm. The result is independent of whether concurrent or nonconcurrent data processing is used in the SMI algorithm's implementation. It is shown that the transient sidelobe level is eigenvalue dependent and increases proportionally to the gain margin of the auxiliary antenna elements with respect to the quiescent main antenna sidelobe level. Techniques that reduce this transient sidelobe level are discussed, and it is theoretically shown that injection independent noise into the auxiliary channels significantly reduces the transient sidelobe level. It is demonstrated that using this same technique reduces the SMI noise power residue settling time     

The CLFM: A Method of Generating Linear Frequency-Coded Radar Pulses

A simple method of generating a coherent linear frequency-modulated pulse, suitable for use in a radar pulse-compression system, is described. The method, termed CLFM, employs a swept-frequency oscillator with closed-loop control of its phase. The phase-error voltages for the loop are obtained by sampling the waveform at regular intervals. The sampling technique itself is unconventional and avoids the need for large bandwidth circuitry. Results obtained with a demonstration model of the CLFM, generating a signal with a time-bandwidth product of 1000, are described.     

Time-Frequency HOP Signals Part I: Coding Based upon the Theory of Linear Congruences

Time-frequency hop codes are developed based upon the theory of linear congruences. These codes can be used for multiuser radar and asynchronous spread spectrum communications systems. A uniform upper bound is placed on the cross-correlation function between any two elements of the code set. The upper bound is minimized by choice of time-bandwidth product and is shown to diminish as 2/N, where N is the number of elements in the code set. The size and position of spurious peaks in the autocorrelation functions are discussed. The results are extended to narrowband ambiguity functions.     

Pulse Compression Results Using Metallic Reflective Array Lines

Radar pulse-compression results are presented for the first reflective-array compressor (RAC) dispersive delay lines (DDL) with both metallic reflecting arrays and phase-compensating films. The time-bandwidth product of the devices reported is approximately 400. Operation in a recirculation loop with a 37.5-dB Taylor weighting filter yielded 36-dB near-in range sidelobes. RMS phase errors less than 0.71 degrees across the band were achieved. Greater than 50-dB rejection of spurious response is achieved in the far-out range gate region. The potential for high-quality cost-effective fabrication of metallic RAC DDL for system applications is explored.     

Detection of Narrowband Signals Using Time-Domain Adaptive Filters

The detection performance of a conventional narrowband analyzer is compared with two adaptive processor mechanizations based on the Widrow least mean squares algorithm. Comparisons are based on both analysis and extensive digital simulation. With a narrowband signal in stationary, white background noise, the performance of the three systems is shown to be essentially the same. With nonstationary background noise, the performance of the conventional system degrades by an amount proportional to the processing time-bandwidth product. The adaptive systems appear to be less sensitive to the nonstationary background, resulting in a potential performance advantage relative to the conventional system.     

Effects of Reduced SAR Azimuth Processing

Several approaches are evaluated for reducing the large data handling and processing loads associated with synthetic aperture radar (SAR) imaging systems. The effects of data abbreviation on SAR images were studied theoretically as well as experimentally with SEASAT-A data. Image degradation was measured in terms of azimuth resolution and signal-to-clutter ratio. It was found that the degradation of image quality to data rate reduction is graceful. With high time-bandwidth product signals, little or no azimuth resolution degradation was noticed with several data bandwidth reduction techniques that achieve data reduction factors of up to 8. The signal-to-clutter ratio was found to decrease slowly with increasing reduction in data rate.     

Reception of pulsed signals by detuned receiver: addenda andcorrections

The parameters of a pulse perturbed by passage through a filter were examined in a variety of cases in a previous paper by the author (ibid., vol.AES-4, p.486-9, May 1968). One case not covered is that of a rectangular pulse passed through a detuned Gaussian filter; that case is presented, as is the case of a swept Gaussian pulse passed through such a filter, which may be dispersive. Corrections are included for two tabular entries in the original paper concerning a Gaussian pulse fed to an ideal bandpass filter when the time-bandwidth product and detuning are large     

Recent advances in analog signal processing

Analog signal processors which perform convolution, correlation, matched filtering, and Fourier transformation are considered. Specifically, the authors focus on one- and two-dimensional signal processors using charge-coupled-device, surface-acoustic-wave, acoustooptic, and superconductive components. The device performances are compared in terms of parameters such as signal bandwidth, time-bandwidth product, and dynamic range. These analog components are compared with existing and planned digital processor designs. Potential applications are highlighted in wideband spread-spectrum packet-radio communication, radar range-Doppler imaging, wideband compressive receivers, and low-bit-rate image-coding systems     

A sampling-based approach to wideband interference cancellation

Classical adaptive array schemes which use only complex spatial weights are inherently narrowband and consequently perform poorly when attempting to suppress wideband interference. The common solution to this problem is the use of tapped delay line filters in each spatial channel to facilitate space-time adaptive processing (STAP). The higher performance provided by the STAP architecture comes at the cost of a considerable increase in complexity. This paper presents a simpler technique based on programmable time adjustable sampling (TAS) that provides a limited number of wideband degrees of freedom. Two TAS methods are introduced: TAS-sidelobe canceler (TAS-SLC) is based on the sidelobe canceler, while TAS-minimum variance beamformer (TAS-MVB) is derived from the minimum variance beamformer. TAS is implemented by adjusting the sampling instant at selected array channels. TAS-SLC consists of controlling the sampling in the main channel of the sidelobe canceler With TAS-MVB array complex weights are substituted with TAS time delays. The performance of TAS methods with wideband interference is compared to the conventional sidelobe canceler and minimum variance beamformers. It is shown that TAS-SLC provides better performance than the sidelobe canceler, while TAS-MVB outperforms the minimum variance beamformer     

Phase-Coded Interrupted CW Signals and Processors

High resolution radars require signals with large time-bandwidth product such as CW signal and coherent pulse train (CPT). We discuss a phase-coded interrupted CW (ICW) signal which is the combination of CW signal and CPT. Phase codes used here are with perfect periodic autocorrelation. The periodic ambiguity function of ICW signals is studied including single-carrier signal and multi-carrier signal. It is interesting that the gate function has different effects on two signals and contributes to a multi-carrier ICW signal which yields nearly perfect autocorrelation. Meanwhile we also suggest an efficient receiver approach to ICW signals, which can reduce the computational burden of the processor and utilize the good properties of P3 and P4 codes.     

Time-Frequency HOP Signals Part II: Coding Based upon Quadratic Congruences

High-efficiency multicomponent signals for maximization of signalto-noise ratio are investigated. Maximization of signal-to-noise ratio in colored noise requires control of volume distribution of the signal ambiguity function and transmission of unity efficiency signals. Signal efficiency is defined as the ratio of average power to the peak power. It is concluded that the signals must be frequency hop pulse trains. Quadratic congruences are chosen to place the components in time-frequency space. The number-theoretic properties of these signals provide bounds on the position and amplitude of the various peaks of the signal ambiguity function. The tradeoffs are shown between volume removal, number of component signals, and the time-bandwidth product.     

Performance Evaluation of Energy Detectors

The prediction of energy detector performance requires a complicated calculation or a tedious manipulation of nomograms. For a large time-bandwidth product WT, however, it is commonplace to use the formula (E/No) = d?WT to anticipate the required average input energy-to-noise spectral density ratio for a wanted signal detectability parameter d and thus avoid the computational difficulty. This paper proposes a modified formula (E/No) = ?d?WT that is applicable for all range of WT, where ? is the modification factor derived on an empirical basis. The Van Trees measure of the signal detectability parameter of the energy detector also is derived analytically and compared to the modified equation.