Radar has evolved from a very simple device for the detection of aircraft in the 1930's to the present almost wholly automatic systems in which the radar sensor is an integrated part of a computer complex with completely programmed functions, decision-making operations, and self-check features. The major impetus for the rapid development of radar was the urgent air-defense requirements of the Second World War; this stimulus led to many advances in radar technology in a relatively short period of time. Radar was raised to a level of sophistication comparable to older technologies in a period of less than ten years.

Waveforms normally encountered in radar are essentially time-limited and band-limited. (With proper interpretation, this characterization also applies to CW radar systems.) Those familiar with Fourier transform theory will realize, of course, that a waveform cannot be simultaneously time and band-limited. But in many practical cases this approximation permits a better understanding of radar performance and simplifies calculations. Hence we shall be interested in approximate mathematical representations based on this assumption.

The specific value of a noisy waveform at any instant of time is inherently unpredictable. Some values, however, tend to occur more often than others. The frequency with which each value occurs can be characterized by a probability density function. This concept and other related probabilistic concepts are presented in this chapter to provide a theoretical framework for subsequent discussions. Readers desiring a more detailed and rigorous exposition should consult the references recommended at the end of the chapter.

In radar and communication, signal reception is made difficult by randomly fluctuating waveforms that modulate or add to the desired signal. These waveforms, which change unpredictably as a function of time, are conveniently described statistically. They are observables of physical processes that are apparently controlled by a random mechanism; such functions of time are called random processes. There are many random processes in nature, such as the voltage generated across a resistor by thermally excited electrons, the spatial position of a particle undergoing Brownian motion, atmospheric pressure fluctuations, and so on. The mathematical representation of random processes introduced in this chapter is one of the tools required in the remainder of this book. Those wishing a broader and deeper understanding of this subject may consult the bibliography at the end of the chapter.

In the infancy of radar, signal-to-noise ratio was a popular measure of the effectiveness of a radar receiving system for combating noise. Increased signal power, with respect to average noise power, made it easier to distinguish a signal from background noise. Further, the parameters of a target echo, such as amplitude, (range) delay, and doppler frequency shift, could be estimated more accurately with increased signal-to-noise ratio. It was apparent that both parameter estimation and detection were dependent on signal-to-noise ratio - at least in a qualitative sense.

In this chapter the following topics are discussed: colored RC noise; rational kernel; integral equation; mixed kernel; infinite observation interval; and maximum signal-to-noise ratio.

In the a posteriori theory of reception, a posteriori probabilities and probability densities provide a complete description of the results of the receiving process. An ideal receiver, by definition, calculates and presents the a posteriori distribution of a desired quantity or quantities to an observer. Owing to the dependence of the a posteriori distribution on a priori information, which may not always be available, the concept of a sufficient receiver was introduced. A sufficient receiver calculates and presents to the observer a sufficient statistic; this statistic contains the essential information in the received waveform and permits the a posteriori distribution to be computed when the a priori statistics are known. When reception occurs in additive Gaussian noise, the a posteriori theory leads to sufficient receivers that employ matched filters. However, the theory has been found to be useful in many situations where the optimum receiver structure is not obvious.

The application of statistical decision theory to problems in communications and radar is being actively pursued. Despite the power of the method, certain limitations restrict its range of application. These limitations result from requirements on the system model that can never be completely satisfied in practice.

In this chapter, we consider the detection of signals in noise based on a single observation. It was shown that the structure of the optimum detector for the exactly known signal in white Gaussian noise is a matched filter followed by a sampling and decision circuit.

In Chapter 9 we investigated the structure and performance of an optimum receiver on the basis of a single observation. In this chapter we examine the optimum Bayes receiver structure and performance when many observations are available for detection. The radar waveform transmitted is assumed to be identical within each observation interval; a sequence of waveform echoes from a target is called a pulse train.

This chapter continues the study of detection based on multiple observations begun in Chapter 10. The same basic approach is now applied to the detection of fluctuating incoherent pulse trains. Fluctuating pulse trains occur often in practice. When a radar target consists of several relatively strong reflecting surfaces displaced from one another by the order of a wavelength, the amplitude and phase of the composite radar echo are sensitive to the spatial orientation of the target. If the target has relative motion with respect to the radar, such as translation, pitch, yaw, tumble, it presents a time varying radar cross section.

The basic quantities that determine radar range performance are related by the radar equation. With this equation, received radar echo power can be calculated from known characteristics of the transmitter, receiver, and antenna and with knowledge of target range, target cross section, and the propagation characteristics of the transmission medium. Other forms of the radar equation are useful for computing signal-to-noise ratio, which permits detection performance to be predicted from the Pd-Pfα curves of Chapters 9 through 11. (Signal-to-noise ratio also enters into the prediction of parameter estimation accuracy when the fundamental radar system parameters have been specified).

In this chapter, target detection is considered when data from several scans are available for processing. Cumulative detection probability denotes the probability that a target is detected at least once in M scans. We will compute the cumulative detection probability for targets that are essentially stationary with respect to the radar and for targets with appreciable motion relative to the radar.

In this chapter a postdetection integrator is considered that provides slightly poorer performance than the ideal integrator discussed in Chapters 10 and 11. In this integrator the received signal is quantized so that it can be processed by simple digital circuits. The integration technique is called binary integration, or double-threshold detection; both titles are descriptive of its operation.

Elementary search radars employ a continuous periodic pulse transmission with a rotating antenna. As the antenna scans across a target, the amplitudes of both transmitted and received pulses are modified by the antenna beam pattern; this converts the uniform periodic pulse train generated by the radar transmitter to an amplitude-modulated pulse train of finite duration. The pulse train duration is determined by the antenna scan rate and beamwidth; the number of pulses in the train is a function of both train duration and the radar pulse repetition frequency. Pulse-train amplitude modulation results in a variation of received pulse signal-to-noise ratio as a function of time. In previous chapters, optimum receiver processing and performance were investigated for uniform pulse trains; these results are extended to amplitude-modulated trains in this chapter.

In this chapter the number of observations required for a decision is a function of the test results obtained after each observation. Thus, the test length is a random variable; this is the principal distinguishing characteristic of sequential detection.

In this chapter Bayes decision rules are developed for optimizing the detection of multiple targets. We show that the strategy consists of an independent threshold test at each expected time of arrival for all targets in the radar field of view. This separable test strategy is shown to be optimum for the limiting cases of large and small signal-to-noise ratios; however, a different threshold level is required, in general, for large signal detection and for small signal detection. The separable test strategy is compared to the averaging solution, which is obtained from a Bayes strategy with cost assignments that are independent of parameters. The averaging approach is shown to be unsuitable for most radars when echo arrival times are unknown.

This paper discusses the conventional Doppler approximation for a narrowband representation of echoes from moving target.

In this appendix the false-alarm rate is evaluated for a sampled range-bin receiver and a receiver that employs a continuous threshold comparison, assuming input narrowband stationary Gaussian noise. In the latter type of receiver a false alarm occurs when envelope V(t) of the narrowband noise process crosses threshold V* from below (positive slope).