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Zen in the Art of Archery. Radar Signals Book Title :Radar Signals A text and general reference on the design and analysis of radar signals As radar technology evolves to encompass a growing spectrum of applications in military, aerospace, automotive, and other sectors, innovations in digital signal processing have risen to meet the demand.
In style and methodology it follows the approach used by Levanon in his text Radar Principles see Chapter 1 references. These include a constant-frequency pulse, a linear-FM pulse with a weight window , and a coherent pulse train. Chapter 6 provides a very comprehensive presentation of phase-coded signals, starting from Barker codes binary and polyphase , minimum peak sidelobe codes, noiselike codes PRNs , chirplike codes Frank, Zadoff—Chu, P-codes , and codes suggested by Golomb, Ipatov, Huffman, and others.
The chapter contains an important section on bandlimiting schemes in which the rectangular code element is replaced by a Gaussian windowed sinc or by a quadriphase waveform. In Chapter 7 we expand on the most popular radar signal: a coherent train of linear-FM pulses.
We offer a detailed analysis of its delay—Doppler performance, including intra- and interpulse weighting matched or on-receive for range and Doppler sidelobe reduction. Diversity is widely used in radar signals, beginning with diversity of the pulse repetition interval Chapter 8 , which mitigates blind speeds. More elaborate diversity schemes are presented in Chapter 9, with emphasis on steppedfrequency pulses both unmodulated and linear FM.
The stepped-frequency signal is also used to demonstrate the important stretch-processing concept. The chapter ends with sections on complementary and orthogonal pulses. Continuous-wave radar signals have experienced a revival in both military and civilian applications and are the subject of Chapter Their analysis tool is the periodic ambiguity function, revisited in this chapter.
Both analog and digital coded continuous-wave signals are presented and analyzed. Chapter 11 is devoted to multicarrier radar signals. Multicarrier is well known in communications but is a relatively new signal concept in radar. It offers the designer more degrees of freedom and more dimensions through which to introduce diversity. However, it entails variable amplitudes. Mastering the use of these programs is well worth the effort.
Basic and advanced radar signals are presented and analyzed. The various chapters include many, if not most, of the radar signals described in the open literature. In the history of radar, signal ideas usually preceded implementation by many years, because of processing complexity and hardware limitations. Only when the necessary transmitter components e. Among them are such questions as: 1. Can the transmitter support complex waveforms? Is the transmitter suitable for pulsed- or continuous-wave transmission?
Are there unavoidable bandwidth limitations in the transmitter, receiver, or antenna? Is frequency shifting from pulse to pulse practical? In an attempt to extend its relevance, in this book we allow considerable freedom in the various characteristics of waveforms. Fairly early range has expanded to include direction to the target and radial velocity between the radar and the target. Presently, more information on the target can be sought, such as its shape, size, and trajectory. Determining the spatial direction to the target depends in a stationary radar on the antenna and its tracking system.
Range is associated with the delay of the signal received. Range rate is associated with the Doppler shift of the signal received. These relationships are discussed next. The factor 12 is due to the fact that the radar signal traverses the distance R twice round trip.
Equation 1. In the lower atmosphere, Cp is not a constant but changes with altitude; hence the radar signal propagates a slightly longer distance along a bent path.
Since the effect is minor and not very related to radar signals, it will be ignored. The Doppler shift is developed in Box 1A with the help of Fig. Rewriting 1A. Note that R1 in 1A. Figure 1. The signal was a pure sinusoid at a frequency f0. What happens when the signal contains modulation: that is, other frequencies?
In other words, can we talk about a single Doppler shift when the signal has considerable bandwidth? In wide or ultrawide bandwidth signals we have to go back to equation 1A. A popular rule of thumb says that if the signal bandwidth is less than one-tenth of the carrier frequency, the signal is considered a narrowband signal and it is reasonable to assume that the target motion causes only a Doppler shift of the carrier frequency according to 1. Otherwise, time scaling should be considered, which also affects the envelope of the signal.
In the following chapters we use the narrowband assumption. Another assumption made above and used henceforth is lack of radial acceleration: i. In practice, the expected target acceleration is usually limited below some predetermined value characterizing the targets in question.
The design of radar signals should take this value into consideration such that the target does not change its Doppler in an amount higher than Doppler resolution during the coherence processing period. They can be explained with the example of a train of unmodulated pulses. As we will learn later in the book, Doppler resolution of a signal is a function of the total duration of the signal. A common approach to extending the signal is to repeat it periodically.
A single pulse has poor Doppler resolution because the Doppler shift creates little change during the pulse duration. On the other hand, a train of pulses exhibits good Doppler resolution because of the changes due to Doppler between the pulses.
This change is primarily in the Doppler-induced initial phase of each pulse. To extract this Doppler-induced phase change it is necessary for the receiver to know the original initial phase of each pulse.
That is what we mean when we refer to the pulse train as a coherent pulse train. A simple example of a coherent pulse train is shown in Fig. The simple example in Fig. An example is radar utilizing a noncoherent transmitting device, where each transmitted pulse has a randomly generated initial phase. In such radar systems it is common to lock on the transmitted pulse phase using a dedicated circuit and to use this memorized phase as a reference for the pulse received.
Implementations where the phase value is known only one pulse backward are usually referred to as coherent on receive. Having associated range and velocity with two signal parameters, delay and Doppler, we can now discuss how well we can determine them and how the signal design can help.
We need to measure the frequency of a sinusoidal signal. If the signal-to-noise-plus-interference ratio SNIR is very high i. A counter counts the number of cycles within a given time span or measures the time interval between several zero crossings. A counter will produce an erroneous result when there is additive noise or when other sinusoidal signals are present: that is, when the signal-to-noise ratio SNR is low or when there are interferences from other signals. The lower the SNIR, the bigger the measurement error will be.
Below a threshold SNIR, the counter will fail completely. The radar scenario is almost always a low-SNIR scenario. In some applications the radar performance is noise limited, while in other applications the performance is interference limited. Still, in many applications the target may stand alone and provide a high SNIR value e.
Indeed, in these cases it is practical to perform a measurement e. However, the output before and after the peak are strongly affected by the waveform. If the output level remains high over an extended delay, the threshold will be crossed in many delay cells, resulting in uncertainty as to which is the true delay. The interference level itself is a function of the nature of the interference.
The mainlobe width and sidelobe level requirements are a function of the expected target separation and expected target RCS difference. Two targets can be near each other in range e. Here again it is important to achieve a narrow response in Doppler, so that, for example, the moving target could be distinguished from the stationary background.
The tool for that is the ambiguity function Woodward, ; Rihaczek, , which describes that two-dimensional response. A basic question when designing a radar signal is: What is a good ambiguity function AF , and can it be obtained? Even if it is the ideal AF, it cannot be produced completely.
We learn in Chapter 3 that the AF peak at the origin cannot exceed a value of 1 and that the volume underneath the ambiguity function squared is a constant. If the AF is lowered in one area of the delay—Doppler plane, it must rise somewhere else. Several AF shapes are presented in Figs. Only two quadrants of the AF positive Doppler are plotted. Two adjacent quadrants contain all the information because the AF is symmetrical with respect to the origin.
The four plots are presented in order to demonstrate different possible distributions of the AF volume over the delay—Doppler plane. The corresponding signals are discussed in more detail in later chapters of the book. The delay axis is normalized with respect to T. The same type of normalization is used in Figs. The 0. The AF demonstrates the expected resolutions in delay and Doppler. The ambiguity function is zero for delays higher than the pulse width; thus no interference is expected with targets having range separation higher than the pulse duration.
The time—bandwidth product used for the plot is 48 the number of chips in the phase code. The normalized Doppler axis extends to only one-sixth of the time—bandwidth product, but a low sidelobe pedestal extends, in normalized Doppler, as far as the time—bandwidth product. Signals with a large time—bandwidth product such as the one described here and the one used for Fig.
The time—bandwidth product is 8. The normalized Doppler axis extends as far as that time—bandwidth product. Similar AF shapes are obtained by using ordered frequency or phase coding e.
Instead of a uniform sidelobe pedestal, the response energy is essentially concentrated at the area of the diagonal ridge, causing lower sidelobes outside the ridge. The plotted delay axis is normalized with respect to the pulse repetition interval Tr and extends over more than two periods.
The AF volume is now spread at many recurrent lobes, almost as high as the mainlobe at the origin. This shape is referred to as a bed of nails. This type of signal achieves good resolution in both delay and Doppler but creates both range and Doppler ambiguities.
Table 1. The constant volume under the ambiguity function squared puzzled many researchers and yielded several unfounded variations of the ambiguity function, which are no more than TABLE 1. Like slums, ambiguity has a way of appearing in one place as fast as it is made to disappear in another. That it must be conserved is completely accepted but the thought remains that ambiguity might be segregated in some unwanted part of the time—frequency plane where it will cease to be a practical embarrassment.
The question of which ambiguity function should be preferred depends not only on the desired delay and Doppler resolution, or on the complexity of the required processor, but also on where the clutter or competing targets are located in the delay—Doppler plane i. The environment that the radar encounters may consist of a variety of clutter conditions, countermeasure interference such as chaff or deliberate electronic emissions , and interference from neighboring radars.
The environmental diagram details spectral, spatial, and amplitude characteristics of the radar environment and is used as the basis against which the ambiguity diagram is played in selecting a waveform design. Nathanson et al. An example of an environmental diagram of surveillance radar located at a coastal site is illustrated in Fig. Radial velocity Doppler is given on the ordinate and the target extent delay is indicated along the abscissa.
Diagrams such as Fig. This environmental diagram shows only the regions in which land or sea clutter, rainstorm, and high-altitude chaff can be expected. This additional information could be presented using a three-dimensional plot similar to an AF plot, taking into account the radar antenna pattern and direction of interest.
For example, looking in the direction of the sea, the land clutter is received only through the antenna sidelobes or even backlobes , whereas the sea clutter is in the mainlobe area. The basic environmental diagram gives a pictorial view of the clutter in range and velocity that the radar must contend with. By selecting the target trajectories expected within the R—V diagram and superimposing the ambiguity diagram of a particular waveform, it is possible to evaluate certain desirable characteristics inherent in the waveform.
In Fig. As the target follows a particular trajectory, the ambiguity diagram will move accordingly, and AF ambiguous peaks will enter and exit the chaff and rainstorm space. Aasen extended the concept of environmental diagrams to include other electromagnetic radiations from transmitters within the general locality of a radar site.
Interference signals will appear to the radar receiver as signals with particular velocity and range characteristics. For example, a stable CW signal within the receiver bandwidth would appear as a horizontal straight line in an environmental diagram.
If the CW signal were frequency modulated, the width of the line would increase according to the modulation bandwidth. A different and far more complicated environmental diagram is of airborne radar. In this case the ground clutter is amplitude, range, and Doppler modulated due to the platform velocity, altitude, and antenna direction. The Doppler spread of the SLC is determined by the platform velocity vector direction relative to the ground plane.
The altitude line return is concentrated in range close to the platform altitude. If we want narrow response in one dimension, we have to accept either poor response in the other dimension or additional ambiguous peaks.
We will learn that if we want the ambiguous AF peaks to be well spaced in delay, we have to accept them as closely spaced in Doppler and vice versa. If we want high range resolution, we should expect a penalty of wide spectral width.
If we want good resolution in Doppler, the penalty is long coherent signal duration. One of the trade-offs in radar signal design is between constant amplitude and AF sidelobes. On the other hand, sidelobe reductions in range or Doppler usually require amplitude variations weighting.
We will learn that such a mismatch carries a penalty in the output SNR. In Section 6. The resulting real amplitudes of a bit signal are plotted in Fig. The rectangular bit shape results in a constant-amplitude pulse dotted line , while the Gaussian-windowed sinc results in a variable-amplitude pulse solid line. The phase evolution is quite similar in both signals. The resulting AF and autocorrelation function are also very similar. What differ dramatically are the spectrums, plotted in Fig.
The bottom subplot presents the spectrum of the constant-amplitude pulse. Such a long-tailed spectrum may violate spectral emission regulations, can cause interference to neighboring radars, and may be too wide for the next radio-frequency RF stage—the antenna. Examples of other radar signals with variable real amplitude are the Huffmancoded signal, discussed in Section 6. The few examples above suggest that removing the constant-amplitude restriction provides radar signals with an additional degree of freedom, which can be used to improve performance.
In it and the remaining chapters it is assumed that the reader has some radar background, from courses, general radar texts, or experience. In the following chapters we provide the basic mathematical tools required for analyzing and comparing different radar signals. Also discussed are useful building blocks and concepts e.
Chapter 11 is an example of utilizing many of the building blocks to design more complex waveforms. As already stated, the work or art of designing radar waveforms is based mostly on experience and expertise obtained through successive designs. This experience is gained by manipulating signal parameters while using special building blocks with desirable mathematical properties.
Finally, it is worth mentioning that the book does not cover at least three subjects related to radar waveforms. Selection of the radar center frequency or band and polarization are both well covered in other textbooks. Noise radars, or truly random waveforms, are not covered. Assume that the target is initially at zero delay i.
The signal transmitted 2. The signal received using the time compression equation 1. The signal received using Doppler shift calculated at the center frequency f0. Interpret the results. EMC, no. Cook, C. DiFranco, J. Levanon, N. Nathanson, F. Reilly, and M. Rihaczek, A. AES-7, no. Woodward, P.
The highest SNR happens at a particular instant, which is a design parameter. Most practical radar signals can be considered as narrow bandpass signals. We therefore begin this chapter by describing narrow bandpass signals with the help of their complex envelope. A narrow bandpass signal can be written in several forms.
Readers are encouraged to prove to themselves that gc t and gs t produced in Fig. Box 2A demonstrates an important case that we will meet in a later chapter. In most radar applications s t is a narrow bandpass signal. Furthermore, in radar the receiver usually knows what the transmitted signal was, and what actual carrier frequency was used in the modulation process. The complex envelope u t is also a baseband signal bounded by W.
The relative spectrums are depicted in Fig. The probability of detection is related to the SNR rather than to the exact waveform of the signal received. Hence we are more interested in maximizing the SNR than in preserving the shape of the signal. Consider the block diagram in Fig. Equation 2. Taking the inverse Fourier transform of 2.
A few examples are shown in Fig. From 2. The complete output signal is shown in Fig. Using the representation of s t as given in 2. Once uo t is obtained, so t is given through 2. Modern radar processing utilizes I and Q detection Fig. As discussed in Chapter 1, for narrow bandpass signals it is practical to treat the Doppler effect as a change in the carrier frequency.
Without exact knowledge of the Doppler shift, the radar receiver cannot modify its matched receiver to the new carrier frequency exactly, and mismatch occurs. The AF was introduced by Woodward and is the main tool in several important radar textbooks Cook and Bernfeld, ; Rihaczek, There are also differences with regard to the function as is, or its magnitude or its square or its magnitude squared. Representation of the AF of various signals is more often done through graphic plots than through analytic expressions.
In the plots there is an emphasis on sidelobes relative to the mainlobe. Check the agreement with equation 2. Do the two approaches yield the same result? Is s2 t a narrow bandpass signal? North, D.
Rihaczek A. Sinsky, A. AES, no. The ambiguity function is a major tool for studying and analyzing radar signals. It will serve us extensively in the following chapters, where different signals are described. This chapter presents important properties of the ambiguity function and proves several of them.
Proof of the four properties is provided in the next section. The first two properties assume that the energy E of u t is normalized to unity. Properties 1 and 2 imply that if we attempt to squeeze the ambiguity function to a narrow peak at the origin, that peak cannot exceed a value of 1, and the volume squeezed out of that peak must reappear somewhere else. More restrictions on volume dispersion will be discussed later. The next two properties apply to all signals, normalized or not.
The remaining two can be deduced from the symmetry property. Our AF plots will usually contain only quadrants 1 and 2 i. The meaning of the shear will be demonstrated following the proof of property 4. This important property is the basis for an important pulse compression technique. Most of the proofs follow Papoulis We explain the shearing caused by the LFM effect with the help of Fig. We use 3. Using 3. The shearing property of linear FM, which we just studied, reduces improves the delay resolution, as pointed out in Fig.
This implies that for positive LFM slope signal a positive error in estimating target range the target is assumed to be farther than it really is will translate to lower closing velocity negative Doppler. Consider first the cut along the delay axis. On the other hand, the ACF equals the inverse Fourier transform of the power spectral density.
Adding linear frequency modulation broadens the power spectrum, hence narrows the range window, as shown in Fig. The second interesting cut is along the Doppler frequency axis. In other words, this cut is indifferent to any phase or frequency modulation in u t ; it is a function only of the amplitude.
Thus, close target separability in one parameter is gained at the expense of spreading volume over a large interval of the other parameter. However, there are several types of signals that are periodically continuous. Two prominent examples are the periodic continuouswave CW radar signal and a coherent train of identical pulses.
An example of a typical coherent pulse train is shown in Fig. As shown in Fig. We must first explain what the word coherent implies.
When we introduced the complex envelope of a finite-duration signal e. Now, when we consider a train of separated pulses, we need to extend the assumption.
We need to assume that the carrier frequency remains the same for all the pulses and that we also know the initial phase of each pulse. A simple variation is to consider the pulses as an interrupted CW signal. Other than those phase shifts, 3. We now return to Fig. In Box 3A some other versions are described and compared to the definition used here.
The definition in 3. This definition is not unique. Alternative definitions can be adopted Freedman and Levanon, For all three definitions the multiperiod ambiguity function relation to the single-period ambiguity function defined in 3. Still these definitions are not equivalent, as described below.
Since the signal is periodic with period Tr , the PAF is also periodic in the delay axis direction. The period is Tr for the definitions given in 3. The period of the PAF defined in 3A. The other definitions of the PAF, given in 3. As in the nonperiodic ambiguity function, there is a constraint on the volume under ambiguity function magnitude. For all three definitions of the periodic ambiguity function described here the volume within a strip of width Tr on the delay axis is equal to 1.
For a larger filter length this volume is reduced according to the period-to-filter length ratio N. The unlimited reduction of the volume as N increases is the result of the unlimited improvement of the Doppler resolution. It can be shown Freedman and Levanon, ; Getz and Levanon, that a very simple and important relationship exists between equations 3.
The multiplying function is a function of the Doppler shift only. Figure 3. Because the function plotted in Fig. Modern pulse radars generally use pulse compression waveforms i. These kinds of waveforms are applied to obtain high pulse energy with no increase in peak power and large pulse bandwidth and, consequently, high range resolution without sacrificing maximum range, which is related to the pulse energy.
In the following chapters we acquire this knowledge. In Chapter 4 we develop the AF of several basic radar signals. The specific examples will contribute further understanding of the ambiguity function, its properties, and its significance to radar. The ambiguity function of some of the radar signals to be discussed can be derived analytically. However, many signals are too complicated, and only numerical calculation of their AF is feasible. The most practical means for displaying the numerical result is a three-dimensional plot.
The code makes it possible to input many different signals and provides control over many plot parameters. The program allows oversampling of the signal with much finer resolution than needed for calculating the delayed signal. This makes it possible to compute a diluted picture fewer delay—Doppler grid points of the ambiguity function with low computational effort while sampling the signal with a sufficiently large sampling rate.
Because of the symmetry of the AF with respect to the origin, the code plots only two of the four quadrants. This provides an opportunity to display the zeroDoppler cut of the AF, which is the magnitude of the autocorrelation function.
The program also produces a second figure with subplots of three characteristics of the signal: amplitude, phase, and frequency. Comment on the choice of r: Since the signal is described by a vector, with a well-defined length number of elements, referred to as M , it is often necessary to increase the number of samples repeats during each of these elements bits , in order to meet the Nyquist criterion.
This is the function of r. Note: When a continuation ellipsis. Getz, B. Mozeson, E. Papoulis, A. The ambiguity function will be the main tool in this study. For these three signals we will be able to develop closed-form expressions of the ambiguity functions AF , but their three-dimensional plots will be obtained numerically using the MATLAB code given in Appendix 3A.
The first two quadrants of the ambiguity function are plotted in Fig. A contour plot of the AF, covering all four quadrants, appears in Fig. Figure 4. The delay response reaches zero at the pulse width T. The zero-delay cut is less obvious from Fig. A numerical example will be useful here. With a pulse compression ratio of , the range resolution will be reduced to an acceptable 7.
This velocity encompasses all possible terrestrial velocities, reaching typical satellite velocities. The conclusion is that a single uncompressed pulse cannot usually provide sufficient range resolution or velocity Doppler resolution. Satisfactory range resolution will be reached using pulse compression and acceptable velocity resolution by using a coherent pulse train.
Examples of both are discussed. Resolution was practically defined by the first null, but beyond the first null the response may build up again see Fig.
The sidelobes of a return from a strong target can mask a small target. The first and highest Doppler sidelobe in Fig. Amplitude weighting techniques are usually used to reduce such high sidelobes. One more function of interest is the spectrum of the complex envelope of the signal. The voltage spectral density is the Fourier transform of u t. Because in the constant-frequency pulse u t is a real constant, the Fourier transforms of u t and u t 2 exhibit the same result, and Fig.
Here again we see poor performance. The signal occupies its bandwidth inefficiently, with relatively high spectral sidelobes. We saw poor performances of a constant-frequency rectangular pulse in many aspects: poor range resolution, poor Doppler resolution and high Doppler sidelobes, and inefficient spectral use. Our next basic signal, a linear-FM pulse, improves on some of those weaknesses.
The basic idea is to sweep the frequency band B linearly during the pulse duration T Fig. The ambiguity function AF is obtained by applying property 4 to the AF of an unmodulated pulse. The phase and frequency of the complex envelope are shown in Fig. Returning to Fig. The issue of spectral efficiency is demonstrated in Fig. The horizontal scale is frequency normalized with respect to the pulse width. The vertical scale is the spectral density in decibels. To obtain similar range resolution, the unmodulated pulse width was one-tenth of the LFM pulse width.
The absolute horizontal scales are therefore identical in both plots. Comparing the two plots clearly shows more efficient spectrum use in the LFM case.
Note that the spectrums plotted in Fig. Due to symmetry, it suffices to plot positive frequencies only. The improved delay resolution of LFM does come with a penalty, delay—Doppler coupling. It is expressed by the diagonal ridge seen in the threedimensional plot of the AF Fig. A contour plot Fig. From 4. In many applications the resulting range error is acceptable. The delay error of the shifted peak response is accompanied by a small decrease in the height of the peak, as evident in Fig.
This behavior is responsible for the Doppler tolerance property attributed to LFM. The importance of Doppler tolerance is discussed in Section 4. However, relatively strong sidelobes remain in the autocorrelation function ACF , as seen, for example, in Fig.
The ACF is related to the power spectral density of the signal through the Fourier transform. ACF sidelobes can be reduced by shaping the spectrum. The shaping should alter the spectrum from its nearly rectangular window shape to one of the wellknown Harris, ; Nuttall, weight windows e.
Spectral reshaping of LFM can be done using two different basic approaches: amplitude weighting or frequency weighting. Amplitude weighting is discussed here; frequency weighting is covered in Section 5. Spectral shaping through amplitude weighting makes use of the linear relation between instantaneous frequency and time along the pulse. If the amplitude of the signal at that instance is higher, the power spectral density of the corresponding frequency is also higher.
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