October 23, 2010 at 8:42 pm 2 comments

Radar is a remote-sensing system that is widely used for surveillance, tracking, and imaging applications, for both civilian and military needs. In this chapter, we focus attention on future possibilities of radar with particular emphasis on the notion of cognition. As an illustrative case study along the way, we consider the radar surveillance problem.

According to the Oxford English Dictionary, cognition is “knowing, perceiving, or conceiving as an act. . .“. Given three distinct capabilities,

  1. the inherent ability of radar to sense its environment on a continuous basis and thereby getting to perceive it,
  2. the ability of phased-array antennas to electronically scan the environment in a fast manner, and
  3. the ever-increasing power of computers to digitally process signals,

it is our conviction that it is not only feasible but also highly beneficial to build a cognitive radar system using today’s technology. Indeed, if ever there was a remote sensing system well suited for cognition, radar is it.

From the moment a surveillance radar system is switched on, the system becomes electromagnetically linked to its surrounding environment, in the sense that the environment has a strong and continuous influence on the radar returns (i.e. echoes). In so doing, the radar builds up its knowledge of the environment fromone scan to the next, andmakes decisions of interest on possible targets at unknown locations in the environment. The locations are not known before the radar is switched on, but they become determined by the radar receiver once the targets under surveillance are declared.

From signal-processing and control theory, we know that it is not necessary for the radar to keep the entire record of past data. Rather, by adopting a state-space model of the environment, and recursively updating the state vector representing an estimate of certain parameters pertaining to the environment, the need for storing the entire history of radar data on the environment is eliminated. The challenge is how to formulate the state-space model of the environment.

The requirement to update estimation of the environmental state is necessitated by the fact that the radar environment is nonstationary. Primary causes of nonstationarity include statistical variations in the weather, the presence of unknown targets at unknown locations, and the ever-present radar clutter, which refers to radar returns from unwanted objects. Recursive updating of a state is synonymous with adaptivity, which is the natural method for dealing with nonstationarity. In current designs of radar systems, however, adaptivity is usually confined to the receiver. For the radar to be cognitive, adaptivity has to be extended to the transmitter too, hence the need for a feedback channel from the receiver to the transmitter. Moreover, the radar has to learn from experience on how to deal with different targets, large and small, and at widely varying ranges, all in an effective and robust manner. We may therefore say that a cognitive radar implies adaptivity, but not the other way round.


The dictionary definition of cognition mentioned above also includes “conceiving,” which might be taken to mean the following statement:

“The formulation of a hypothesis, and then testing that hypothesis for the likelihood of its correctness”

This statement is in the spirit of the Bayesian approach to state estimation, with a probabilistic rating of alternatives. We are therefore emboldened to embrace the idea of Bayesian inference under the umbrella of cognitive radar.

This way of thinking leads us to the block diagram of Fig. 2.1, which depicts a picture of the cognitive radar signal-processing cycle. The cycle begins with the transmitter illuminating the environment. The radar returns produced by the environment are fed into two functional blocks: the radar-scene analyzer, and the Bayesian target-tracker. The tracker makes decisions on the possible presence of targets on a continuing time basis, in light of information on the environment provided to it by the radar-scene analyzer. The transmitter, in turn, illuminates the environment in light of the decisions made on possible targets, which initiates the next cycle of operation. The cycle is then repeated over and over again. Unlike a communication system, the feedback mechanism — a necessary requirement of a cognitive

system — is easy to implement as the radar transmitter and receiver are usually co-located. Note also that although the process of target detection is not explicitly shown in the cognitive cycle of Fig. 2.1, it is part and parcel of the Bayesian target-tracker, which performs “detection through tracking” as explained later.

Based on the picture depicted in Fig. 2.1, a cognitive radar distinguishes itself from an adaptive radar in three important respects:

  1. The radar continuously learns about the environment through experience gained from interactions of the receiver with the environment and, in a corresponding way, continually updates the receiver with relevant information on the environment.
  2. The transmitter adjusts its illumination of the environment in an intelligent manner, taking into account such practical matters as the size of the target and its range, and consequently, making adjustments to the transmitted signal in an effective and robust manner.
  3. The whole radar system constitutes a closed-loop dynamic system, encompassing the transmitter, the surrounding environment, the feedback channel, and the receiver. In other words,we have global feedback acting around the whole system.

It is well known that feedback is like a double-edged sword, in that it can become harmful if it is used improperly. Care must therefore be exercised in how the transmitter is designed in relation to the environment and receiver, so as to maintain a stable and reliable operation at all times.

One other important comment is in order. In reality, cognition is a two-way process, one being inside-out and the other being outside-in. These two parts of the cognitive process are so referred to, depending on whether the source of information leading to cognition resides inside or outside the receiver, respectively, as explained in the following:

  1. The “inside-out” part of cognition is represented by prior knowledge on the environment; it is an integral part of the receiver, as shown in Fig. 2.1. The form of prior knowledge is naturally application-dependent. For example, it may take the form of a geographic map, a clutter map of the environment, an elevation model, or kinematics of noncooperative targets. The Bayesian target-tracker retrieves information from the prior-knowledge base and utilizes it for improved radar performance on a need-be basis. Prior knowledge may therefore be viewed as the long-term memory of the receiver.
  2. In contrast, the “outside-in” part of cognition may be viewed as short-term memory, which is developed by the receiver on the fly. It is initiated by the radar-scene analyzer in response to information-bearing signals gathered on the outside environment by the radar itself as well as other sensors working cooperatively with the radar.


The function of the radar-scene analyzer is to provide the receiver with information on the environment on a continuous basis. This information is of critical importance to the decisions made by the receiver on possible targets of interest. This function builds on two sources of information-bearing signals:

  1. radar returns, which are produced by the environment in response to the radar’s own transmitted signal.
  2. other relevant information on the environment (e.g. temperature, humidity, pressure, sea state), which is gathered on the fly by sensors other than the radar itself.

These two sources of inputs constitute the stimuli for the outside-in part of radar cognition.

In a surveillance scenario, radar performance is affected significantly by the unavoidable presence of interference. Typically, the interference is dominated by clutter (i.e. radar returns produced by undesired targets). Accordingly, to design a target tracker that embodies target detection, we need two kinds of information: one pertaining to the clutter acting alone, and the other pertaining to the target plus clutter.

In order to describe how these two pieces of information can be addressed in specific terms, consider the case of a coherent radar dwelling on a particular patch of the ocean surface. With the radar being coherent, the radar returns contain amplitude as well as Doppler information on that patch. Correspondingly, the baseband version of the radar returns will be complex-valued. Now, the dwelling process can be of a longterm nature, in which case the nonstationary character of the radar returns becomes quite noticeable. In situations of this kind, we may be forced to avoid modeling the actual Doppler spectrum (i.e. plot of average power versus frequency) of the radar returns, and do so by exploiting the following intuitively satisfying observations:

“The Doppler spectrum of clutter by itself is relatively smooth, whereas the spectral content of the radar echo from a target appears essentially as a line component”

However, when the target cross-section is small and the target-to-clutter power ratio is therefore low, we need to enhance the line component due to the target. This enhancement may be achieved by performing the following transformation:

“Divide the average power in each Doppler bin of the spectrum (pertaining to the range azimuth resolution cell of interest) by the mean of its neighboring bins, say k in number”

This transformation has the desired effect of accentuating the narrow peak of the line component due to the target and, at the same time, lowering the relatively wide peak of the clutter. Inspiration for the transformation, called a “peak filter,” is traced to the “grouped periodogram test” described by Priestly, which was itself inspired by earlier work by Tukey in 1949. The statistics of the peak filter output, in the absence of a target, may now be evaluated under three assumptions:

  1. None of the k neighboring Doppler bins in the power spectrum contains a target.
  2. Inside a spectral window encompassing (k + 1) Doppler bins, the continuous clutter power spectrum (that is always present) is approximately constant.
  3. All (k + 1) ordinates of the power spectrum are sampled independently.

Under these three assumptions, the individual ordinates of the actual power spectrum have ax2 distribution with two degrees of freedom. Correspondingly, the peak-filter output, which divides each spectrum ordinate by k others, has a hypergeometric distribution, specifically an F-distribution with (2, 2k) degrees of freedom. On this basis, the clutter statistics are described by the distribution F2, 2k(z), where z is a random variable (i.e. average clutter power measurement). It is noteworthy that in reference 5, a similar observation is made using stochastic differential equation theory.

Turning next to the target, which is typically unknown, modeling its statistics is unfortunately not straightforward. For ease of implementation, and due to a lack of detailed knowledge about the target, it may be prudent to assume that the target has the same distribution that governs the clutter, but with a difference. (This assumption may hold in the case of a small target moving on an ocean surface, in which case the underlying dynamics of the clutter and the target are closely coupled.) Accordingly, if the clutter distribution is described by F2,2k(z), the target distribution is taken to be 1/y F2,2k(z/y), where z is a power spectrum measurement and y is the target-to-clutter power ratio, the scalar parameter z is not to be confused with the vector z introduced later.

In addition to the target statistics, the receiver needs to have a model that accounts for the motion of the target. To this end, we may assume that the target has a Gaussian-distributed acceleration with variance s2, which characterizes the agility of the target. For a low standard deviation s, the target is seen by the radar when it is not accelerating. On the other hand, for a high s, the task of target detection may become difficult due to possible confusion of the target with small clutter peaks, hence the likelihood of the radar making a decision error. In summary, for an ocean environment under surveillance by a coherent radar, information on radar returns processed by the radar-scene analyzer for a particular range-azimuth cell may be modeled as follows:

  1. Clutter-statistics, described by the F-distribution F2,2k(z), where z is a power spectrum measurement and k is the number of neighboring Doppler bins over which the measurement is averaged.
  2. Target-plus-clutter statistics, described by the scaled F-distribution 1/y
    ,where g is the target-to-clutter power ratio.
  3. Target motion, described by a Gaussian-distributed acceleration with a variance σ2, which accounts for the target’s agility.

It must be re-emphasized, however, that this model is appropriate for the specific case of a target moving on an ocean surface. For other environmental scenarios, the radar designer is challenged to develop appropriate statistical models to describe the information content of radar returns on clutter and targets.


Previously, we mentioned that the Bayesian paradigm is a logical choice for coherent radar. We now describe a Bayesian strategy for the coherent radar detection of small targets in the presence of sea clutter. Unlike conventional tracking algorithms that perform intermediate detections (i.e. hard decisions) on the radar returns, the new algorithm processes the radar returns directly.
Specifically, the algorithm, referred to as a
direct tracking algorithm, consists of three basic steps:

  1. For a given search area, radar returns are collected over a certain period of time.
  2. For each range-azimuth resolution cell in the search space, the probability that the cell contains a target is computed.
  3. With the evolution of the target probability distribution resulting from the recursive computation of step 2 over time, target tracks are detected and corresponding hard decisions on possible targets are subsequently made.

In effect, the algorithm (formulated in probabilistic terms) may be viewed as a soft decision procedure on target detection.

To set the stage for the Bayesian framework, let there be a total of R range-azimuth resolution cells in the search space S, and let r ε
S denote a resolution cell in question. Let denote the event of a single target occurring in resolution cell r at discrete time t. Let the vector zt denote the frame that is made up of the spectral measurements for all R resolution cells at time t. The matrix

denotes the full set of all the available frames extending up to and including time t. Then, according to this notation, the vector zt denotes the current frame and the remaining matrix Zt1 denotes the combined set of all past frames. By the same token, Zt+1 denotes the combination of a future frame zt+1, the current frame zt, and all past frames Zt1.

Following the traditional approach to state estimation, we may now identify three different forms of the Bayesian target-tracker:

  1. one-step predictor, whose output is described by the conditional probability
  2. filter, whose output is described by the conditional probability
  3. smoother, whose output is described by the expanded conditional probability

Smoothing uses more information than both prediction and filtering, and may therefore be more accurate than both of them in a statistical sense. On the other hand, however, only prediction and filtering can be implemented in real time.

One-Step Tracking Prediction

Consider the joint event which describes a target occurring in resolution cell q at time t-1 and then moving into resolution cell r at time t. From probability theory, we may express the output of the tracking predictor at time t as

However, given the fact that the event has occurred at time t-1, it makes the
previous measurements matrix
Zt1 irrelevant. In other words, occurrence of the
event conveys exactly the same amount of information as the joint event
Accordingly, Equation 2.1 reduces to the simpler form

The conditional probability is the output of the tracking filter working
on resolution cell
q at time t-1. We also recognize as the set of
probabilities that event follows event . This set of probabilities is referred to
as the
transition matrix of the tracker, the formulation of which exploits the statistical model of target motion as supplied by the radar-scene analyzer. It is noteworthy that the less agile the target is, the smaller the jumps the target is expected to make in the search space S, thereby causing the transition matrix to be more sparse. In any event, given the tracking filter output at time t-1 and the transition matrix, we may use Equation 2.2 to compute the output of the corresponding tracking predictor at time t.

Tracking Filter

Consider next the issue of computing the output of the tracking filter at time t, which is defined by the posterior probability Applying Bayes’ theorem to this probability yields

where is the conditional probability density function of the current measurements matrix Zt given the occurrence of event and is the prior probability of that event. The probability density function p(Zt) in the denominator is the evidence, which acts merely as a normalizing function. As, by definition, we may rewrite Equation 2.3 by expanding the numerator, as shown by

Recognizing that the occurrence of event makes past measurements Zt1 irrelevant, we may simplify Equation 2.4 by setting Thus,

The first term in the numerator of Equation 2.5 is the probability density function of measurement zt given that there is a target in cell r at time t. The second term is computed by using the recursive formula

where, as before, is the input supplied to the receiver by the radar-scene analyzer, and is the one-step delayed version of hence the reference to Equation 2.6 as a recursive formula. The matrix of probabilities is the inverse transition matrix, which is defined by the probabilities that event preceded event . The term “inverse” is used here merely to imply the role reversal of these two events with respect to the transition matrix under Equation 2.2.

The following two points are noteworthy:

  1. The recursive formula of Equation 2.6 is identical to the hidden Markov model (HMM) filter for a Markov chain with transition probabilities
  2. Given the posterior probability distribution of Equation 2.4, the conditional mean estimate (i.e. minimum mean-square estimate) of the event over the entire search space S can be computed as the summation

We may also compute the conditional probability density function in
another way by recasting the recursive formula of Equation 2.6 as follows:

Figure 2.2 Block diagram of the Bayesian direct filtering system. Notation: t, discretetime; xtjt, filtered state vector of probabilities of targets being present in the search

space at t given spectral measurements up to and including time t. The other data vectors in the diagram are similarly defined.

Then, substituting Equation 2.7 into Equation 2.5, we obtain the new formula for computing the posterior probability at the output of the tracking filter:

where the probability is a delayed version of the tracking predictor
output, and the probabilities
are elements of the transition matrix.

On the basis of Equations 2.2 and 2.8, we may now construct the block diagram of Fig. 2.2 for the Bayesian direct filtering system. The diagram is in the form of a
closed-loop feedback system that operates by propagating a state vector of probabilities from one iteration to the next. Most importantly, the right relationship must be established between the radar parameters and statistical characteristics of clutter and target-plus-clutter for the tracker to maintain a stable operation.

Tracking Smoother

An attractive feature of the Bayesian tracker, described herein, is the fact that it is straightforward to make its operation conditional on both past and future spectral measurements. The result of this expansion is a target tracking smoother, for which the output is expressed as

The factorization of terms in the numerator of Equation 2.9 assumes that the radar is treated as a first-order Markov model, in which case the conditional dependence of the distribution of past measurements Zt-1 on the future measurements zt+1 may be ignored; that is, we may set equal to

The additional factor in the numerator of Equation 2.9 is computed
by running the right-hand side of the recursive Equation 2.6 backwards in time.
Thus, whereas the target-tracking filter operates in the forward direction only, the
target-tracking smoother operates in the forward as well as backward directions.
Accordingly, decisions made on possible targets using the tracking smoother
contain more information than the corresponding tracking filter, and may therefore
be more reliable. However, this improvement in performance is gained at the
expense of two factors: increased computational complexity and [2] nonreal-
time operation. Simply put, for every gain in radar performance, there is a
corresponding price to be paid.

Experimental Results: Case Study of Small Target in Sea Clutter

In references 2 and 3, the performance of the Bayesian target detector was evaluated using real-life radar data under varying conditions. The data were collected by means of the McMaster IPIX radar, which is a highly configurable coherent multifunction X-band radar built specifically for research purposes. For a subset of the database collected at a site in Dartmouth (Nova Scotia), the radar was operated in the dwell mode with a 18 pencil beam and fixed radio frequency of 9.39 GHz.

The radar was mounted about 30 m above sea level, with the target of interest being located about 2.5 km offshore. The target was a sphere (1 m in diameter) made up of wire covered in foam. Radar range was sampled at 15 m intervals, obtained by using a 200 ns rectangular pulse. (The actual range resolution of the radar was 30 m.) The pulse-repetition frequency (PRF) was 2 kHz, but the pulse alternated between horizontal (H) and vertical (V) polarization, so that the effective single-polarization PRF was 1 kHz. For each pulse, both H and V polarizations were recorded simultaneously, resulting in a matrix of four possible transmit/ receive polarizations: HH, HV, VH, and VV. For each combination in the matrix, the amplitude and phase of the radar returns were stored in the form of in-phase (I) and quadrature (Q) components.

In reference 3, three data sets from the Dartmouth database were used to test the Bayesian target detector. The results pertaining to one of those data sets are reproduced in Fig. 2.3. Figure 2.3a displays the Doppler-time image of the raw radar data set, using a 64-sample sliding window. Figure 2.3b displays the resulting output of the Bayesian direct tracking smoother. Each pixel in the image represents the probability of a target being present in the corresponding resolution cell. The darker the pixel, the higher the probability of target occurrence. Note also that the dark traces included along the 500 Hz line indicate the points in time where the target was invisible to the radar or when the radar failed to detect the target.

Figure 2.3 (a) A 64-sample sliding window Doppler-time image of raw radar data set 3 from reference 3. (b) Output of the Bayesian tracking smoother, where each pixel represents the probability of a target in the corresponding resolution cell.

Figure 2.3 and several other results reported in reference 3 attest to the effectiveness of the Bayesian direct tracker. In particular, even for a data set with an average target-to-clutter power ratio as low as 27 dB, Fig. 2.3 clearly demonstrates the visibility of the target most of the time.

Practical Implications of the Bayesian Target Tracker

To the best of the author’s knowledge, the Bayesian target tracker described in detail in references 2 and 3 and highlighted herein is the first to be studied regarding the feasibility of direct target-tracking without intermediate detections. The use of a Bayesian approach to direct tracking, combined with complete reliance on soft decisions (i.e. avoiding hard decisions through intermediate detections), has some important practical implications:

  1. Unlike hard decisions, the soft decisions made by the Bayesian target tracker preserve the information content of the radar returns; this approach follows the “principle of information preservation,” inferred from Shannon’s information theory.
  2. The Doppler-time image produced by the Bayesian direct target tracker makes it possible for the radar to see the motion of the target in a manner comparable to the human eye. Indeed, we conjecture that an experienced human operator could not do a better job of following the target than the Bayesian tracker, especially so when the tracker is operated in the smoothing mode; in this mode, the tracker exploits the combined benefit of forward and backward computations.
  3. The basic idea behind the Bayesian approach is to view the information contained in the radar image as a probability distribution that characterizes the likelihood of a particular resolution cell containing a target. The distribution, in the form of a posterior probability density function, is determined in part by the statistical structure of the radar scene (i.e. the outside world), and in part by the way in which echoes from the world are actually encoded by the radar itself. Accordingly, the Bayesian approach distinguishes itself from other approaches by invoking an explicit statistical structure of the world that, in reality, is a fundamental necessity.
  4. The book by Knill and Richards [8] presents a number of theoretical frameworks for studying visual perception that, in varying degrees, are all founded on Bayesian principles. In a way, this book lends further support to the Bayesian radar-target tracker, the theory of which is embodied in equations 2.1 through 2.9, depending on the mode of operation.

One last comment is in order. Using two different real-life radar data sets and computer-simulated data, a comparative evaluation of the Bayesian approach to target-detection-through-tracking has been made against a new detection strategy called the correlation anomaly receiver, which follows from the theory of stochastic differential equations. The results of this evaluation, reported in reference 9, show that the Bayesian receiver’s performance is superior to that of the correlation anomaly receiver.


As it stands, there is no optimization being performed on the posterior probability distribution computed by the Bayesian target tracker. This important matter, however, can be taken care of by making the transmitter, responsible for illuminating the environment, adaptive.
The practical issue with adaptive radar illumination (transmission) is how to observe past radar returns and extract useful information in order to decide or select the radar waveform for the next transmission in some optimal fashion.

In an implicit sense, the present spectral measurements at time t, denoted by zt and the past measurements denoted by Zt1, are all dependent on the transmitted signal. This dependence suggests that the whole radar system can be made adaptive by adjusting certain parameters in the transmitted signal in response to the probabilistic decisions made by the Bayesian tracker on the environment under surveillance. Note, however, that by doing so, the radar system assumes the form of a stochastic control system involving a state-space model governed by the posterior distribution of Equation 2.4; the optimal solution to such “partially observable” stochastic-control problems is NP hard. Fortunately, there are suboptimal procedures such as reinforcement learning that can yield acceptable .

There are many ways in which parameters of the transmitted signal can be adjusted. One practical way is to use burst waveforms, with each burst made up of a sequence of uniformly spaced, nonoverlapping subpulses of fixed duration. The pulse amplitudes are held constant for two reasons:

  • Unforeseen difficulties with dynamic range requirements are avoided.
  • The target-to-clutter power ratio may not be sensitive enough to pulseamplitude adjustments.

The logical strategy is then to adjust the phase of each transmitted radio-frequency (RF) pulse in accordance with feedback sent to the transmitter from the receiver. Here we have the choice of a phase response that varies with time according to a square law that results in linear frequency modulation (FM), or a cubic law that results in nonlinear FM. Both of these configurations are well known for their pulse compression characteristics, with the nonlinear FM being more effective than the linear FM.


From the introductory section, we recall that a cognitive radar system embodies three fundamental ingredients:

  • learning from the environment through experience,
  • adjustment of the transmitted signal in an intelligent manner, and
  • feedback from the receiver to the transmitter to make this adjustment possible.

All these three features are part and parcel of the echo-location system of a bat. Accordingly, there is much that we can learn from the echo-location system of a bat. Most echo-locating bats are blind.6 To see the world around it, the bat uses sonar, which is an active echo-location system. In addition to providing information about how far away a target (i.e. flying insect) is, the bat’s sonar conveys information about the relative velocity of the target, the size of various features of the target, and the azimuth and elevation of the target. The complex neural computations needed to extract all this information from the target echo occur within a brain the size of a plum. Indeed, an echo-locating bat can pursue and capture its target with a facility and success rate that would be the envy of a radar engineer. How then does the bat perform all these remarkable tasks? The answer to this fundamental question lies in the fact that soon after birth, the bat uses its innate hard-wired brain to build up rules of behavior through what we usually refer to as experience, hence the remarkable ability of the bat for echo-location.

The bat uses its mouth (or nose) to broadcast echo-location sounds and uses its auditory system as the sonar receiver. The emitted sounds consist of burst waveforms whose characteristics are highly diverse, varying with both species and being situation-specific. The transmitted sound characteristics are summarized as follows:

  • Duration: 0.3–300 ms
  • Frequency: 12–200 kHz
  • Structure: Frequency-modulated (FM) component, or constant-frequency (CF) component followed by FM component

The constant-frequency component can be single or multiple harmonic. The FM component can be of a downward or upward kind, the FM sweep varying linearly or nonlinearly with time. The use of FM is intended to improve the echo-location system’s resolution capability for the bat. (It is noteworthy that an echo-location bat’s emitted sounds consist of burst waveforms just as the adaptive transmission strategy used in the DeLong–Hofstetter algorithm also consists of burst waveforms.)

Broadly speaking, the adaptive behavior of bats may be categorized as follows :

  • Velocity-dependent adaptation, which involves adjustment of the transmitted sound frequency; this form of adaptation is most salient in species of constant frequency–frequency modulation (CF-FM) bats. These CF-FM bats also appear to make adjustments in temporal patterning as they close in on their targets.
  • Range-dependent adaptation, which involves adjustment of the emitted-sound duration, bandwidth, and repetition rate; this second form of adaptation is most salient in bats using only frequency modulation (FM). These bats also appear to make adjustments in the transmitted sound bursts during target approach.

Echoes from targets (i.e. insects) are represented in the auditory system by neuronal activities that are sensitive to different combinations of acoustic inputs produced
in response to the transmitted sound bursts. In particular, three principal dimensions
of the bat’s auditory representation have been identified :

  1. Echo frequency, which is initially encoded in the auditory periphery cochlea by place in the cochlear;
  2. Echo amplitude, which is encoded by the neuronal responses under [1] and other neurons tuned to different dynamic ranges in the central nervous system;
  3. Echo delay, which is encoded through neuronal computations that produce target-range tuning responses.

There are two principal (neuronal) computations that are performed by the bat’s brain for image-forming purposes. One is the spectrum of the incoming echo, which is intended for the extraction of target shape, which is a particularly noteworthy point in light of the spectral processing performed by the radar scene analyzer. The other is delay in the received echo with respect to the transmitted sound bursts, which is intended for the extraction of target range. To carry out these computations, frequency-based information contained in the incoming echo spectrum is converted into estimates of the spatial (time) structure of the target.

In short, the echo-location system of a bat is very plastic, in that the parameters of the transmitted sound bursts can be changed considerably during the different phases of the target-pursuit sequence. We are therefore justified in viewing the echo-location system of a bat as physical proof (albeit in neurobiological terms) of cognitive radar.


Three important conclusions can be drawn from the presentations made in this article:

  1. Intelligence is a necessary requirement for the radar to be cognitive. A striking difference is discernible between the presentations we have made on adaptive radar illumination and echo-location in bats. Simply put, in signalprocessing terms, the echo-location systems of bats are far more plastic than the adaptive radar systems that are currently in use or being contemplated. This important point is best illustrated by the spectograms shown in Fig. 2.4, which were produced by four different bat species in their respective target (insect)-pursuit sequences.

Figure 2.4 Spectrograms of sonar signals produced by four different species of bats as they advance from the search to approach and finally to the terminal phase of insect pursuit. (Reproduced from reference 20, with permission of the University of Chicago Press.)


The significant characteristic that is immediately apparent from this figure is that the transmitted signal duration decreases and the burst repetition rate increases as the bat gets closer to its target. In doing this, the bat is using acquired knowledge of the distance from its target to adjust the parameters of its transmitted sound bursts. For a radar systemto be cognitive, therefore, it is a fundamental necessity for the radar transmitter to do the following:

Learn from continuing interactions with the environment and intelligently use the information extracted by the receiver on targets under surveillance, all of this being done on the fly during the different phases of the target-track sequence.

  1. Feedback from the receiver to the transmitter is a facilitator of intelligent signal processing. We say feedback is a facilitator of intelligence, because it
    is through feedback from the receiver to the transmitter that cognitive radar
    is enabled to learn from interactions with the environment.
  2. The preservation of information in radar returns is of crucial importance to receiver performance, which is realized by the Bayesian approach to target detection through tracking. The results presented in Section 2.4 on
    the Bayesian target-tracker emphasize the signal-processing power of the
    Bayesian approach. This approach is the only statistical approach in which a
    model of the received signal accounts for two factors contributing to the
    specification of information:
  • The statistical nature of interference (i.e. radar clutter and noise);
  • The explicit statistical structure of the radar environment (i.e. outside world), including targets.

In the past, the Bayesian approach has been criticized for requiring a model that includes a statistical structure of the radar environment. In response to such criticism, we merely have to emphasize that if we are to account for the physical realities that are responsible for the generation of radar returns, then the inclusion of a statistical structure of the radar environment is a fundamental requirement for preserving the information content of the received signal.

Most importantly, referring to the closed-loop feedback system of Fig. 2.1, encompassing the radar transmitter, the propagation medium, the radar receiver and the feedback channel back to the transmitter, there is much tat we can learn from the way in which this system operates. In this context, when we speak of the cognitive signal-processing cycle, the iterative processing of radar signals proceeds on a cycle-by-cycle basis inside the feedback loop, with each cycle corresponding to a frame of radar pulses produced by the transmitter. Moreover, within each cycle, there is processing being performed on a pulseby- pulse basis. We thus have two forms of iterative processing that are performed side by side:

  • Processing on a pulse-by-pulse, which is performed in the receiver
  • Processing on a cycle-by-cycle basis, which is performed in the transmitter


Throughout this article, we have emphasized that learning is a basic ingredient of cognitive radar. In a generic sense, the learning process can take two different forms: off-line and on-line.

Through off-line learning, knowledge is acquired about the environment and then embedded in the receiver. In the radar context, an established way of accomplishing

this acquisition is to collect real-life data by conducting ground-truthed experiments on the environment under varying conditions. Then, by performing statistical analysis on the radar data and formulating models on clutter and targets, the acquisition of knowledge of the environment is. In any event, the off-line learning takes place through the intervention of the experimenter.

Among the many different on-line learning procedures, reinforcement learning stands out as a procedure well suited for cognitive radar. In the modern approach
to reinforcement learning, also referred to as approximate programming, Bellman’s
dynamic programming (rooted in control theory) provides the theoretical foundation
of the procedure. Bellman’s dynamic programming suffers from the “curse of dimensionality,”
which limits its practical utility. The use of approximate dynamic programming
(ADP) overcomes this limitation.


A discussion of cognitive radar would be incomplete without some applications where it has the potential to make a difference. In what follows, we address two applications of cognitive radar, one dealing with multi-function radars that are expensive, and the other dealing with noncoherent radars that are inexpensive.

Multifunction Radars

Multifunction Radars Thanks to continuing advances and improvements on two fronts, namely phased-array antennas and computers, multifunction radars are fast becoming, if not already, the norm in building sophisticated radar systems. For example, the radar may have to deal with a “fading target” due to

he presence of multipath produced by close proximity of the target (e.g. a seaskimming missile) to the sea surface in a hostile marine environment. One way of mitigating the fading problem is to increase the dwell time in order to track the target with adequate accuracy. In such an environment, we may identify two problems that require serious attention:

1. Agility, which mandates the use of phased-array antennas oriented to provide360° coverage (e.g. four arrays at 90° with respect to each other).

2. Fast response, which is attained by using powerful computers that enable the radar to adapt its transmission waveforms so as to detect, track, and paint the target rapidly enough for the engagement to occupy a range of no more than 30 s to a couple of minutes.

Typically, while attending to the fading target, the radar is also required to handle other threatening targets. The radar is therefore faced with a new problem—resource
management. Neurodynamic programming provides a “principled” approach for a solution (suboptimal, but perhaps adequate) to the resource management problem.

Noncoherent Radar Network

For an entirely different application that could benefit from the use of cognitive radar, consider the international border security problem. To be more specific, consider the Great Lakes St. Lawrence Seaway. There are two challenging problems with this large open border between the United States of America and Canada:

  • the protection of assets and populations of people from terrorism and
  • the prevention of illegal crossings across the border.

A cost-effective, all-weather, and all-day solution to both of these challenges is a cognitive noncoherent radar network. The network would be made up of inexpensive commercial off-the-shelf marine radars, which are distributed across the border. The only discriminant available for surveillance with such simple radars is amplitude, which severely limits the capability of the radar to detect noncooperative targets with small radar cross-section in the presence of lake clutter. To mitigate this serious problem, Weber et al. [26] depart from conventional radar signal processing by purposely setting low detection thresholds. Naturally, the false-alarm rates are raised to levels higher than a conventional processor, but, most importantly, the small noncooperative targets are now detectable. Then, through the use of a sophisticated tracking algorithm, the real targets are extracted and the false-alarm rates are reduced to an acceptable level.

Given a network of such noncoherent radars, which also incorporates a central base station, the real-target tracks computed by the component radars are transmitted by a communication channel (wireline or wireless) to that station. Consequently, we have yet another new problem, namely, multisensor fusion. Given the limited computing resources at the base station, the challenge here is how to design a cognitive radar network that produces a map in real-time for the entire Great Lakes St. Lawrence Seaway, which identifies the tracks of all noncooperative targets operating therein and does so in the most reliable manner possible.

In both of the applications addressed herein, another extremely challenging issue is that of knowing how to define a metric by means of which it can be said that the task in question has been accomplished? Stated in another way, what is the essence of the description of the environmental scene that is under surveillance? The traditional radar specifications, based on the probability of detection and the problem of false alarm (which are never measured anyway in a real-time setting) are unsuitable. Rather, we need a new metric that addresses specifically what the end-user needs to see. The formulation of this metric is further exasperated when the application at hand involves several tasks and the tasks have to be prioritized. Here again, a cognitive metric that learns over time may well provide an answer, as is often the case with humans (private communication, Prof. C. J. Baker, 17 January 2004).


Simon Haykin

[1] S. Haykin. Cognitive radar: A way of the future. IEEE Signal Processing Magazine 2006; 23:30–41.

[2] R. Bakker, T. Kirubarajan, B. Currie, S. Haykin. Adaptive radar detection: A Bayesian approach. In: EPSRC and IEE Workshop on Nonlinear and Non-Gaussian Signal Processing, Peebles, Scotland, 8–12 July 2002.

[3] R. Bakker, G. Lopez, S. Haykin. Bayesian approach to the direct filtering of radar targets in clutter. ASL Report 02-02. Adaptive Systems Laboratory, McMaster University, Canada, August 2002.

[4] M.B. Priestly. Spectral Analysis and Time Series. Academic Press; 1981.

[5] T.R. Field, R.J.A. Tough. Stochastic dynamics of the scattering amplitude generating K-distributed noise. J. Math. Phys. 2003;44:5212–5223.

[6] M.G.S. Bruno, J.M.F. Moura. Multiframe detector/tracker: Optimal performance. IEEE Trans. Aerospace and Electronic Systems 2001;37:925–944.

[7] A. Viterbi. Wireless digital communication: a view based on three lessons learned. IEEE Communications Magazine 1991;29(9):33–36.

[8] D.C. Knill, W. Richards, editors. Perception as Bayesian Inference. Cambridge University Press; 1996.

[9] B. Currie, S. Haykin. Bayesian detector evaluation and comparison. Final Report prepared for Defence Research Development-Ottawa (DRDC) under contract W7714- 020683/001/SV, April 2004.

[10] D.T. Gjessing. Target Adaptive Matched Illuminate Radar: Principles and Applications. IEE Electromagnetic Waves Series 22. Peter Peregrinus Ltd: London; 1986.

[11] M.R. Bell, S. Monroq. Diversity waveform signal processing for delay-Doppler measurement and imaging. Digital Signal Processing (Special issue on defense applications) 2002; 12:329–346.

[12] P.Weber, S. Haykin. Non-linear FM pulse compression radar systems. CRL Report #120. Communications Research Laboratory, McMaster University, Canada, 1983.

[13] D.F. DeLong, E.M. Hofstetter. On the design of optimum radar waveforms for clutter rejection. IEEE Trans. Information Theory 1967;IT-13:454–463.

[14] D.F. DeLong, E.M. Hofstetter. The design of clutter-resistant radar waveforms with limited dynamic range. IEEE Trans. Information Theory 1969; IT-15:376–385.

[15] D.P. Bertsekas. Nonlinear Programming. Athenas Scientific, 1995.

[16] S. Haykin, T. Kirubarajan, B. Currie. Adaptive radar for improved small target detection in a maritime environment. ASL Report 03-01, 2003. Simulation setup for optimum energy-constrained and amplitude-constrained waveform design. ASL Report 03-02, 2003. Literature search on adaptive radar transmit waveforms. ASL Report 02-01, 2002. Adaptive Systems Laboratory, McMaster University, Canada. (An edited version of these reports is reproduced in Chapter 6 of reference 2.)

[17] J.A. Simons. The resolution of target range by echo-locating bats. Journal of the Acoustical Society of America 1973;54:157–173.

[18] N. Suga. Cortical computation maps for auditory imaging. Neural Networks 1990; 3:3–21.

[19] J.A. Thomas, C.F. Moss, M. Vater. Echolocation in Bats and Dolphins. Chicago: The University of Chicago Press; 2004.

[20] G. Schuller, C.F. Moss. Vocal control and acoustically guided behavior in bats. In: J.A. Thomas et al., editors. Echolocation in Bats and Dolphins. Chicago: University of Chicago Press; 2004. p. 3–16.

[21] S. Haykin, editor. Adaptive Radar Signal Processing. Wiley; 2007.

[22] S. Haykin. Neural Networks: A Comprehensive Foundation. 2nd ed. Prentice-Hall; 1999.

[23] J. Si, A.G. Barto, W.B. Powell, D. Wunsch II. Handbook of Learning and Approximate Dynamic Programming. Wiley; 2004.

[24] V. Krishnamurthy, R.J. Evans. Beam scheduling for electronically scanned array tracking systems using multi-arm bandits. IEEE Trans. Signal Processing 2001;49:2893–2908.

[25] S.L.C. Miranda, C.J. Baker, K.D. Woodbridge, H.D. Griffiths. Multifunction radar resource management. Chapter 10, this volume.

[26] P. Weber, A. Premji, T.J. Nohara, C. Krasnor. Low-cost radar surveillance of inland waterways for homeland security applications. 2004 IEEE International Radar
Conference, Philadelphia, PA, 26–29 April 2004.

Entry filed under: Electrical Engineering.

Common Linear Models Used in Model Predictive Control

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