Staggered PRI and Random Frequency Radar Waveform

Tel Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences Staggered PRI and Random Frequency Radar Waveform Submitted as part of the requirements towards an M.Sc. degree in Physics School of Physics and Astronomy, Tel Aviv University By: Yossi Magrisso The work has been carried out under the supervision of Prof. Nadav Levanon and Dr. Roy Beck-Barkay And with the assistance of Dr. Aharon Levi

1. Table of contents 1. Table of contents..........2 2. Abbreviations....4 3. Abstract..5 4. Introduction.......6 4.1. Definitions.. 9 4.2. Wide-Band Radar signals.............12 4.3. Generalized Ambiguity function.........16 4.4. Pulse Doppler train waveform Ambiguity Function........21 4.5. Linear Stepped-frequency waveform Ambiguity Function........26 4.6. Random Frequency Ambiguity Function.........28 4.7. Staggered PRI based waveform Ambiguity Function..........32 5. Staggered PRI and Random frequency Based Waveform...35 5.1. Description.......35 5.2. Optimization for minimum sidelobes, Normalized PSLR....39 5.3. Integration loss in the first range ambiguity zone using staggered PRI waveform. 41 5.4. Processing Staggered PRI waveform using perfect Reconstruction....43 5.5. Processing Staggered PRI with random frequency waveform. 47 5.5.1. Processing.. 47 5.5.2. Implementation complexity.. 49 5.6. Simulation results.55 5.6.1. Staggered PRI waveform........57 5.6.2. Single target (noise free and no clutter)........6 5.6.3. Single target in the presence of noise and clutter.......6 2

5.6.4. Two targets in the presence of noise.........62 5.7. Experimental results.65 5.8. Signal-to-Clutter Ratio considerations........ 7 5.9. Advantages and Disadvantages... 72 5.9.1. Advantages.... 72 5.9.2. Disadvantages.....74 6. Conclusions.....75 Appendix A Code Review.... 77 Bibliography... 82 3

2. Abbreviations Radar WF PRI PRF SNR SCR RF IF PSLR NPSLR RCS CPI FFT DFT NUFFT RDM RVM ECM ECCM RMS STD CW BW SF LPI SAR ISAR PDF CFAR RAdio Detection And Ranging Wave-Form Pulse Repetition Interval Pulse Repetition Frequency Signal-to-Noise Ratio Signal-to-Clutter Ratio Radio Frequency Intermediate Frequency Peak-to-Sidelobes Level Ratio Normalized PSLR Radar Cross Section Coherent Processing Interval Fast Fourier Transform Discrete Fourier Transform Non-Uniform Fast Fourier Transform Range-Doppler Map Range-Velocity Map Electronic Counter Measures Electronic Counter-Counter Measures Root Mean Square Standard Deviation Constant Wave Band Width Stepped-Frequency Low Probability of Intercept Synthetic Aperture Radar Inverse Synthetic-Aperture Radar Probability Distribution Function Constant False Alarm Rate 4

3. Abstract Radar systems are electromagnetic sensors that are in nowadays one the most important remote sensing tools in civilian and military uses. One of the main aspects of Radar systems design is the Radar waveform, which defines the modulation of the signal the Radar transmits. The Radar waveform directly affects the performance of all types of Radar systems, from detection capability and accuracy of surveillance Radars, to image quality and resolution of imaging Radars. Two of the main issues discussed in the literature of Radar waveform development are the sidelobes and recurrent lobes in the ambiguity function. High sidelobes and recurrent lobes in the ambiguity function usually lead to degradation and even limitations to Radar systems performance, so the main goal of the research in the area is to find waveforms having an ambiguity function that aspires the perfect "thumbtack" ambiguity function, which has no sidelobes and no recurrent lobes. In this work a new type of Radar waveform is proposed, one that utilizes a staggered Pulse Repetition Interval (PRI) and random frequency pulse train, and its performances are examined. The examination includes introduction of a new generalization of the ambiguity function one that can represent wide-band physical signals, detailed simulation results for different scenarios including analysis to evaluate the performance and implementation feasibility of the waveform, and experimental data analysis. Advantages and disadvantages of the waveform are presented, including implementation consequences in possible Radar application. 5

4. Introduction A Radar is an electromagnetic device aimed to sense objects from a distance by transmitting electromagnetic waves towards an object and measuring the reflections scattered from it. By precise time difference measurement between the time of transmission and time of arrival of the scattered echo, one can measure with high accuracy the range to the object, given that the speed of light in the medium is known. One of the main challenges in remote sensing using electromagnetic waves is that the radiation power attenuates proportionally to in each direction (marking the range from the transmitter to the object), giving a total attenuation proportional to between the transmitted power and the power of the reflected radiation by the object received at the location of the transmitter [1]. This drastic attenuation limits the maximal range in which a Radar can sense objects, since the reflected radiation always exists in an environment of noises added to it from different sources (thermal, solar, other man made transmitters, etc.), and from some range it will weaken enough to be masked by them. In order to increase this maximal detectable range, estimation methods are usually used to filter the noises as much as possible. The common method used is the Matched-Filter aimed to maximize the Signal-to-Noise Ratio (SNR) defined by: { } meaning the ratio between the power of the reflection and the RMS of the noise added to it (E{X} symbols the average of the random variable X). By maximizing the SNR we can reach optimization in the sense of maximum detection probability given a defined false detection probability (known also as the "Neyman Pearson lemma" [1,2,3]). The matched-filtering action is actually an optimally weighted integration of the power in time, and can be done in a coherent manner or in a non-coherent one. If the transmitter is capable of transmitting a coherent radiation and the measurement instrumentation is able to measure the phase of the returning reflections - than an optimal complex matched-filter can be applied, yielding a full complex integration of the wanted signal and averaging the noise to the minimum. Coherent 6

transmission and reception capabilities also enabled the measurement of the Doppler frequency shift of the reflected radiation, enabling the measurement of the radial velocity of the object relative to the transmitter. The Doppler effect on the Radar measurements will be further discussed in the following chapters 4.1.2 and 4.2. Since a coherent Radar can measure the range and velocity to an object, the following challenges are: 1. Measuring them at a good precision. 2. Providing the Radar with the ability to differentiate between different objects in the medium (placed at different ranges and/or moving at different velocities), by being able to distinguish between each object's reflection. In order for the Radar to have these capabilities, we aspire to maximize the Radar range and velocity resolutions. In order to achieve a higher range and velocity resolutions, one has to consider the waveform of the transmitted signal - meaning its amplitude and frequency modulation. High range resolution, meaning high time compression resolution, can only be achieved if the waveform modulation has a wide enough bandwidth. If the bandwidth of the modulated signal is much smaller than the carrier frequency of the signal, the signal is said to be a "narrow-band signal". However if the bandwidth is in the scope of the carrier frequency, the signal is said to be a "wide-band signal". We will further elaborate on wide-band Radar signals in the following chapter 4.2. Since the first Radar invention in the late 19 th century, different Radar applications have evolved and today include a wide span of implementations, amongst them: search and detection, targeting, triggering, weather sensing, navigation, mapping and imaging. Even though the different applications can have a very different purpose and function, they all share a common need for high SNR and high resolution. Unfortunately, there are some unwanted artifacts in the outputs of Radar systems utilizing matchedfilter in their signal processing the existence of sidelobes and recurrent-lobes (ambiguities) added to the main-lobe peak, that can lead to false detection or false reflectors in the Radar image [2]. The reason for the existence of the side-lobes and recurrent-lobes in the processed signal, usually has to do with the finite time-frame in which the data is collected and analyzed. The finite time frame is, in most cases, a natural limitation regarding the physics of the scenario the Radar has to operate in. For example, a search Radar meant to detect and locate a flying airplane, has only a few seconds or even less to do so before the airplane will fly out of its detectable region, meaning that the maximal time frame for the Radar's operation in this case can be only a few seconds at best. This whole time frame 7

has to also be divided into even smaller time frames in order to collect enough data needed to acquire the wanted range and velocity resolutions and unambiguous spans. An instrument used to analyze the properties of the Radar waveform, including resolution, sidelobes and recurrent-lobes is the Ambiguity Function - as defined in equation (25) in chapter 4.3. An ideal ambiguity function is a single spike, with no sidelobes or recurrent lobes, centered in the range- Doppler domain. It is referred to in the Radar literature as thumbtack ambiguity function. Its physical realization would yield superior target-resolution and clutter-rejection capabilities for the Radar. Finding Radar waveforms that can produce ambiguity functions having characteristics close to that of the ideal thumbtack ambiguity function is a major research subject in the field, and is also one of the goals of this work. Chapter 4.3 will be dedicated to the subject of the ambiguity function, also expanding it to wide-banded signals case. Another important feature that has to be taken into account in designing modern day military Radars and Radar waveforms, is the Radar's ability to deal with ECM (Electronic Counter Measures) meant to disrupt and confuse it, leading it to malfunction. Some ECM systems operate by detecting and studying the Radar's waveform, then transmitting matched counter signals that are received and analyzed in the Radar as a false target or even many false targets [4]. In order to prevent these ECM systems from disrupting the Radar's operation, some Radars are equipped with ECCM (Electronic Counter-Counter Measures) capabilities, using the Radar's waveform as major tool for that [5]. Two of the ways to provide the Radar waveform with ECCM capability are: 1. Helping it to avoid detection by Low Probability of Intercept (LPI) techniques. 2. By being unpredictable, and making it very hard for the ECM to synthesize disruptive signals. One of the ways proposed to do both these things is by using a random noise-like Radar waveform [6,7]. In addition to its ECCM capabilities, the noise-like waveforms also possess a good trait of uniformly distributed noise-like sidelobes in their ambiguity function, that can reduce the probability of false detections. The random characteristics of the waveform proposed in this work will enable the Radar using it to have similar ECCM and noise-like sidelobes properties. 8

4.1. Definitions Velocity In this work we will analyze the effects of a search Radar waveform on the ability to detect a moving target, and on the target's range and range-rate measurements. The range-rate of a target (also referred to as the 'radial velocity'), is directly related to its three dimensional Cartesian velocity relative to the position of the Radar system, in the following way: ( ) is the distance from the target to the Radar (where denotes the three dimensional Cartesian relative position of the target), and will simply be referred to as the "target's range". is the target's range-rate relative to the Radar, and is not necessarily identical to the target's relative velocity. A target moving toward the Radar will have a negative range-rate, whereas a target moving away from the Radar will have a positive one. For convenience, however, throughout this work the range-rate of the target will be denoted simply as the "target's velocity". Wherever the word "velocity" is mentioned, the interpretation should always be of "range-rate", regardless of the context. 9

Velocity [m/sec] Range and Velocity Profiles The Range-Doppler Map (RDM) or Range-Velocity Map (RVM) are, in many cases, the final output of a Radar signal processing flow. They represent the amplitude of the filtered signal as function of the range and velocity. A reflecting target placed at a certain range and moving in a certain velocity will yield a peak in a range-velocity cell in the map (see Figure 1a). Throughout this work, RDMs or RVMs will be presented as images. In some cases it is constructive to examine the range or velocity profiles of the map for some range (showing all the amplitude values for the different velocities in that range), or for some velocity (showing all the amplitude values for the different ranges in that velocity). These will be referred to as the "Velocity Profile" and as the "Range Profile", respectively. The range and velocity profiles are shown in Figure 1b. -1-5 -2-3 -4-5 -1-15 -2 [db] -6-25 5 1 15 2 25 3 35 4 45 5 Range [m] -3 Figure 1a An example of a Range-Velocity Map (RVM). This RVM is a simulation output of a target placed at 25 m range and moving at a velocity of -3 m/sec, in the presence of noise and clutter (in this case, many stationary strong reflectors places at different ranges). The dotted-red line marks the velocity "slice" of all ranges in velocity -3 m/sec (Range Profile), whereas the dashed green line marks the range "slice" of all velocities in range 25 m (Velocity Profile). 1

Velocity-Profile Amplitude [db] Range-Profile Amplitude [db] -1-2 -3-4 -5 1 2 3 4 5 6 Range [m] 2-2 -4-1 -2-3 -4 Velocity [m/sec] -5-6 -7 Figure 1b Range and velocity profiles of the marked lines in Figure 1a. The Range-Profile is the amplitude in the map for velocity -3 m/sec and for all the ranges, and the Velocity-Profile is the amplitude in the map for range 25 m and for all the velocities.

4.2. Wide-Band Radar signals A wide-band signal is a signal containing a frequency-span not much smaller than the carrier frequency itself. The use of wide-band signals in Radar applications can be attractive because they can produce very high range resolution outputs. Besides its obvious advantage in yielding high measurement precision and better scatterers separation, high range resolution enables adding to Radars systems some enhanced capabilities, such as object classification and target recognition [8,9]. However, the Doppler behavior of wide-band electromagnetic signals can be quite different than that of narrow-band ones, and this has to be taken into account. An electromagnetic signal transmitted from a moving object relative to a receiver, has a frequency shift known as a Doppler shift. If the moving object transmits the real part of the complex signal: The complex signal received at a stationary receiver will be: ( ) ( ( ) ) Where is the time delay caused by the wave propagation time. In free space, assuming the transmitter moves in a constant velocity (denoting the speed of light) and that the frequency is constant in time, the time delay is: thus: ( ) And the received signal will be: 2

( ( ) ) ( ( ) ) ( ) denoting The received signal has a Doppler frequency shift: where is the carrier wavelength, and is the attenuation of the signal as a result of the wave's propagation and the system losses. The approximation in equations (9-11) assume narrowband signal in which the frequency is the carrier frequency of the transmitted signal. In the case of Radar signals, reflected from an object moving at a constant velocity and received at the same stationary point of the transmitter, the two way delay is: The received signal will be: ( ( ) ) ( ( ) ) ( ) denoting the object's complex Radar Cross Section (RCS), and the two-way Doppler frequency shift. 3

The calculations presented above are true for a narrow band signal (relative to the carrier frequency) reflected from a slow moving object (relative to the velocity of light). However, if the signal is a wide-banded signal, or alternatively the object moves at a high velocity, the assumptions leading to equation (13) are no longer valid. In the general case, we have to recalculate the received signal as function of a more accurate time-dependent delay (equation (6)). Using a time symmetry property, assuming the object moves at a constant velocity and that the signal propagation delay is equal for both directions (to and from the target) [1,11], the delay fulfills the relation: ( ) ( ( )) ( ) It is interesting to notice that using only the symmetry property of the time delay, the wide-band time shift is consistent with the one predicted by the Special Relativity theory. The one-way relativistic time dilation transformation is given by: If is the time period of the waves emitted from the Radar, then the moving object sees waves hitting it with a period of. The object then reflects the waves having the same period, and these reflections are received back in the Radar with a secondary period shift of [12]: 4

Finally, a wide-band signal reflected by a moving object and received at the same spatial point as the transmitter's location, is given by [13,14,15]: ( ) ( ) ( ( ) ) where denotes: The frequency can be also a function of time, so the time dependency of the received frequency ( ) can be different than the time dependency of the transmitted frequency ( or simply ): ( ) In that case the received signal will be: ( ) ( ( ) ( ) ) 5

4.3. Generalized Ambiguity function The standard ambiguity function defined as [2]: Denoting the complex conjugate signal with a time shift, and the normalization factor: is a tool meant to help a Radar waveform developer. If we inspect the correlation function between the signal and the signal, we get: So we can see that the amplitude of ambiguity function (that part of the ambiguity function that mostly interest us) is actually equal to the amplitude of the correlation function between the signal and the signal. As we know, the signal is the matched filter of the signal. 6

If the following conditions are fulfilled: a. Matched filtering is used to detect the signal b. The target's velocity is small relative to the speed of light c. The signal is narrow-banded it can be argued that the ambiguity function represents well enough the matched-filter output of zerorange, zero-velocity object reflections with different range-velocity hypothesis filters. However, this is not necessarily true if one or more of the conditions above are not fulfilled. In the case one of them is not fulfilled, we have to find and work with a generalization of the ambiguity function. Several such functions have been proposed in the past [16,17], and here we develop a generalization corresponding to the wide-band signal reflections. As we've seen in chapter 4.2, the reflections received from a point target are (equation (24)). In this case the correct matched filtering output of a zero-range, zero-velocity object reflections using different range-velocity hypothesis filters, is: ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( ) Where is the envelope's complex conjugate. In order to simplify the last expression in equation (3) we assume the signal envelope changes relatively slowly in time, therefore we can use the approximation: 7

(Here is constant in time) ( ) ( ) In this approximation we define the Generalized Ambiguity function: ( ) that is, in fact, the correlation between and the function: ( ) The difference between the standard ambiguity functions and the Generalized ambiguity function in the case that one or more of the three conditions mentioned above are not complied, can be seen in the following example (in this case the signal has a wide-band, namely condition c. is not fulfilled). The waveform's parameters are: Waveform Stepped-frequency, 3 pulses per batch Number of pulses 1 Carrier frequency 4 khz Bandwidth* 5 khz Pulse Repetition Interval 325 μsec Pulse width 4 μsec Sampling rate 4 khz Velocity bin size 3, m/sec * The Bandwidth represents the whole frequency span of the different frequency steps. In this case the transmitted carrier frequencies (in khz) are [4, 425, 45, 4, 425, 45, 4, 425, 45,4], spanning over a total bandwidth of 5 khz. 8

f * CPI Base Band frequency [Hz] Amplitude (Real part) Figure 2a shows the pulse modulation and frequency as function of time, and Figures 2b,2c the waveform's standard ambiguity function and Generalized ambiguity function respectfully. The poor velocity resolution in this examples stems from the low carrier frequency (4 khz) 1.5 -.5-1 5 1 15 2 25 3 35 Time [ sec] 6 x 14 4 2 5 1 15 2 25 3 35 Time [ sec] Figure 2a A stepped-frequency waveform, 3 pulses in batch. -25-2 -15-1 -5 1.9.8.7.6 5 1 15 2 25 -.25 -.2 -.15 -.1 -.5.5.1.15.2.25 t / PRI.5.4.3.2.1 Figure 2b Standard Ambiguity function of a wide-band stepped-frequency modulated waveform. 9

Velocity [m/sec] x 1 6-3 -2-1 1.9.8.7.6 1 2.5.4.3.2.1 3 -.25 -.2 -.15 -.1 -.5.5.1.15.2.25 t / PRI Figure 2c - Generalized Ambiguity function of a wide-band stepped-frequency modulated waveform. This figure demonstrates the difference between the Generalized Ambiguity function and the standard ambiguity function shown in Figure 2b, regarding the behavior of both the sidelobes and the recurrent lobes. Simulation code for calculating the Generalized ambiguity function presented here is reviewed in Appendix A. Sibul and Titlebaum have discussed the wide-band ambiguity function volume properties [18], and showed that they can be quite different than those we know of the ordinary ambiguity function [2]. For example, they showed that the integrated volume of the wide-band ambiguity function can in fact be larger than 1, as opposed to the volume of the ordinary ambiguity function that is equal to 1. 21

4.4. Pulse Doppler train waveform Ambiguity Function A well-studied ambiguity function is of the Pulse-Doppler Waveform. The Pulse-Doppler waveform is a very common Radar waveform, used in a wide span of Radar applications. The way a pulsed Radar generally works is by transmitting a short time-span high energy pulses, and then shutting the transmission down and listening to the returning echos. A major advantage of the pulsed waveform over a Constant-Wave (CW) waveform, is that Radars using CW waveforms usually suffer from high transmission leaks to the receiver channel (producing unwanted "self-noise"), reducing the SNR of the received signals. Radars using pulsed waveforms usually suffer less from this problem since they usually do not transmit anything while listening. If several pulses are transmitted coherently one after another, than the waveform is actually a pulse-train waveform. A solution that can helps get some intuition as to how a pulse train ambiguity function may look like is the general pulses train periodic ambiguity function given by [2]: where denotes the PRI and denotes the ambiguity function of a single pulse. However, the periodic ambiguity function assumes matched filtering an infinite train of identical pulses with a finite pulses train, therefore equation (35) can serve only as a simple approximation to the case in which the matched filtering is between two finite pulse trains. In Figure 3a, the amplitude and base-band frequency (frequency shift relative to the RF frequency) of a single pulse waveform are drawn as function of time. In this example we see the transmission of the pulse lasts for 4 µsec and then shuts down. The pulsed signal does not have any frequency shift, hence the base-band frequency is a constant zero. The single pulse waveform Generalized ambiguity function is shown in Figure 3b. Time and frequency profiles of the Generalized ambiguity function at the zero time zero frequency point are shown in Figure 3c. 2

f * CPI Base Band frequency [Hz] Amplitude (Real part) 1.5 2 4 6 8 1 12 14 Time [ sec] 1.5 -.5-1 2 4 6 8 1 12 14 Time [ sec] Figure 3a An example of a single rectangular pulse waveform. The top figure shows the amplitude of the signal, and the bottom figure shows the base-band frequency shift relative to the RF frequency (in this case constant zero shift), as function of time 1-1.9.8-5.7.6.5.4 5.3.2 1.1 -.6 -.4 -.2.2.4.6 t / PRI Figure 3b The Generalized ambiguity function of the single pulse waveform. Time and frequency profiles of the function are shown in Figure 3c. 22

Frequency-Profile fot t = Time-Profile for f = 1.5 -.8 -.6 -.4 -.2.2.4.6.8 t / PRI 1.5-15 -1-5 5 1 15 f * CPI Figure 3c Time and frequency profiles of the Generalized ambiguity function of the single pulse waveform at the zero-time and zero-frequency point. If we look at the time profile for f = we will see the "correlation triangle" which is the product of a correlation of a rectangular time-window with itself. If we look at the frequency profile for t = we will see a Dirichlet periodic sinc pattern formed by a Discrete Fourier Transform of the same rectangular time-window. The next example shows the Generalized Ambiguity function of a 1 pulses train waveform with the parameters: Waveform Pulse - Doppler Number of pulses 1 Carrier frequency 5 GHz Bandwidth MHz Pulse Repetition Interval 2 μsec Pulse width 4 μsec Sampling rate 4 khz Velocity bin size 2 m/sec 23

Velocity [m/sec] Velocity [m/sec] Base Band frequency [Hz] Amplitude (Real part) In Figure 4a, the amplitude and base-band frequency of the 1 pulses train waveform are drawn as function of time. Here again the pulse train signal does not have any frequency shift, hence again the base-band frequency is a constant zero. The 1 pulses waveform Generalized ambiguity function is shown in Figure 4b. 1.5 2 4 6 8 1 12 14 16 18 2 Time [ sec] 1.5 -.5-1 2 4 6 8 1 12 14 16 18 2 Time [ sec] Figure 4a 1 Pulses train waveform with uniform PRI -2 1-2 1-15.9-15.9-1 -5.8.7.6-1 -5.8.7.6.5.5 5.4 5.4 1 15.3.2.1 1 15.3.2.1 2-8 -6-4 -2 2 4 6 8 t / PRI 2 -.4 -.3 -.2 -.1.1.2.3.4 t / PRI Figure 4b The Generalized Ambiguity function of the 1 pulses train Pulse-Doppler waveform. The figure on the right shows a zoom on the zero-time mainlobe. In this case in addition to the recurrent lobes in the velocity axis, recurrent lobes in the range axis also appear. 24

In this case not only do the sidelobes and recurrent lobes exist in the velocity axis, but now because of partial correlations between the train and the time shifted train time recurrent lobes also arise. Sidelobes and recurrent lobes could lead to false detections or distortions in a Radar image if not treated properly. 25

4.5. Linear Stepped-frequency waveform Ambiguity Function The linear stepped-frequency waveform is a pulse train, having each pulse in the train modulated by linearly increasing frequencies (see Figure 5a). Stepped frequency waveforms are generally used when a large bandwidth is required in order to achieve high range resolution, but it is impossible to increase the single pulse bandwidth using inter-pulse modulation. The linear stepped-frequency waveform and it's properties are discussed in detail by Levanon in [2,19]. Combining a linear SF with linearly increasing pulse intervals waveform was proposed in order to reduce ambiguity levels [2]. The next example shows the generalized ambiguity function of a waveform with the parameters: Waveform Stepped-frequency Number of pulses 1 Carrier frequency 5 GHz Bandwidth 5 khz Pulse Repetition Interval 2 μsec Pulse width 4 μsec Sampling rate 4 khz Velocity bin size 2 m/sec Looking at the generalized ambiguity function of this waveform (Figure 5b) and comparing it to the generalized ambiguity function of the single frequency pulse train, we can see some of the next features: a. Improvement in the time resolution due to the use of a wider bandwidth. b. Appearance of additional time/velocity recurrent lobes, closer to the mainlobe. c. Some reduction in the amplitude of the far time recurrent lobes, due to the miss correlation between pulses carrying different frequencies. d. Appearance of additional sidelobes in the time/velocity domain. 26

Velocity [m/sec] Velocity [m/sec] Base Band frequency [Hz] Amplitude (Real part) 1.5 -.5-1 2 4 6 8 1 12 14 16 18 2 Time [ sec] 6 x 14 4 2 2 4 6 8 1 12 14 16 18 2 Time [ sec] Figure 5a 1 pulses train waveform with a linear stepped-frequency modulation. The waveform consists of 2 batches of 5 pulses per batch, having a 1 khz frequency step between each two consecutive pulses in the batch. This creates a batch with a total bandwidth of 5 khz. -2 1-2 1-15.9-15.9-1.8.7-1.8.7-5.6-5.6.5.5 5.4 5.4 1 15.3.2.1 1 15.3.2.1 2-4 -3-2 -1 1 2 3 4 t / PRI 2 -.4 -.3 -.2 -.1.1.2.3.4 t / PRI Figure 5b The Generalized Ambiguity function of the 1 pulses train with stepped-frequency modulated waveform. The figure on the right shows a zoom on the zero-time mainlobe. When comparing it to the ambiguity function of a pulse-doppler train (shown in Figure 4b), we can see that the range resolution has improved due to the wider bandwidth, but at the price high and close recurrent lobes. 27

Base Band frequency [Hz] Amplitude (Real part) 4.6. Random Frequency Ambiguity Function In order to minimize the velocity and range sidelobes and ambiguities of the linear steppedfrequency waveform, several non-linear frequency series were proposed. Costas proposed a theoretical optimal series [19,21]. The use of random stepped-frequency series was also proposed [22,23]. The next example shows the generalized ambiguity function of a random stepped-frequency waveform with the parameters: Waveform Randomized Stepped-frequency Number of pulses 1 Carrier frequency 5 GHz Bandwidth 5 khz Pulse Repetition Interval 2 μsec Pulse width 4 μsec Sampling rate 4 khz Velocity bin size 2 m/sec As shown in Figure 6a, in this case a total bandwidth of 5 khz is also achieved but now by having each pulse in the train carrying a different frequency, and sorted in a random non-linear fashion. The waveform's general ambiguity function is shown in Figure 6b. 1.5 -.5-1 2 4 6 8 1 12 14 16 18 2 Time [ sec] 6 x 14 4 2 2 4 6 8 1 12 14 16 18 2 Time [ sec] Figure 6a Random frequency 1 pulses train waveform. 28

Velocity [m/sec] Velocity [m/sec] -2 1-2 1-15.9-15.9-1 -5.8.7.6-1 -5.8.7.6.5.5 5.4 5.4 1 15.3.2.1 1 15.3.2.1 2-4 -3-2 -1 1 2 3 4 t / PRI 2 -.4 -.3 -.2 -.1.1.2.3.4 t / PRI Figure 6b The Generalized Ambiguity function of the random frequency 1 pulses train waveform. The figure on the right shows a zoom on the zero-time mainlobe. Comparing the general ambiguity functions shown in Figure 6b and Figure 5b we can see that once we randomizing the frequency, significant decorrelation occurs and that the energy of the time recurrent lobes existing in the linear stepped-frequency waveform are now spread randomly all over the velocity axis, and suppressed in their amplitude. The reason that the decorrelation does not happen in the linear stepped-frequency case can be better explained by the following calculation: The total number of pulses in the Coherent Processing Interval (CPI) is: Marking the PRI as, and the frequency step as has a phase component of:, each pulse in the linear stepped-frequency batch 29

Correlating the whole batch with a reference signal (matched filter) will yield the integration output: Denoting as the index of the stepped-frequency batch in the CPI ( =,1,2,, ), and it's corresponding coherent phase. The meaning of equation (38) is full coherent integration of the pulses of the batch. Correlating the reference with a similar signal, only delayed by exactly one pulse interval, will yield the output: The result is that the partial correlation between the shifted pulses and the reference signal yields a high integration result, smaller than main-lobe only by the factor of The random SF waveform, however, does not have this partial correlation and therefore its' time recurrent lobes are low and spread all over the velocity axis. As can be seen in Figure 6b, because of the uniform PRF sampling and the low bandwidth - the velocity recurrent lobes in the zero-time vicinity remains high. If a gradually higher and higher bandwidth will be used, the velocity ambiguities will also gradually disappear as each reflected pulse in the train will have an increasingly different Doppler shifts and the train will not be integrated properly at non-zero velocities (see Figure 6c). 31

Velocity [m/sec] Velocity [m/sec] Velocity [m/sec] Velocity [m/sec] The random SF waveform was recently suggested for different applications, amongst them Synthetic Aperture Radar (SAR) [24,25] and Inverse SAR (ISAR) [26] imaging for improved point-spread function, and also for improved multiple target detection by using an iterative maximum-likelihood based algorithm [27]. -2-1 BW/f = 1e-5-5 x 1 5 BW/f =.3 1.9.8 1.7 2 -.4 -.2.2.4 t / PRI x 1 6 BW/f =.9 5 -.4 -.2.2.4 t / PRI x 1 6-4 BW/f =.33333.6.5.4-1 -2.3.2 1 2.1 -.4 -.2.2.4 t / PRI 4 -.4 -.2.2.4 t / PRI Figure 6c Different Generalized Ambiguity function of the random frequency 1 pulses train waveform, zooming on the zero-range main-lobe, for different bandwidth-to-transmission frequency ratios. In this case different ratios are achieved by reducing the carrier frequency while keeping the bandwidth constant. 3

4.7. Staggered PRI based Waveform Ambiguity Function In order to reduce the range and velocity ambiguities in the ambiguity function, a non-uniform, intra cycle staggered PRI was proposed [23]. When using this waveform, a spreading of the velocity ambiguities all over the volume of the general ambiguity function is caused by: 1. The non-uniform sampling of the Doppler phase that prevents the aliasing phenomenon. 2. The reduction of the range ambiguities is caused by the non-constant range wraparound of the far distance objects reflections preventing their proper integration. This waveform type was suggested to enable Radars suppress clutter [28,29] and interferences [3]. Lately the use of a staggered PRI waveform was also proposed for SAR applications in order increase the imaging coverage using high resolution, without the need for a long antenna to do so [31]. The next example shows the generalized ambiguity function of a waveform with the parameters: Waveform Pulse Doppler, Non-uniform PRI Number of pulses 1 Carrier frequency 5 GHz Bandwidth khz Pulse Repetition Interval 2 μsec + ΔT ΔT ~ U[ -T PRI /2, T PRI /2]* T PRI * 12 μsec Pulse width 4 μsec Sampling rate 4 khz Velocity bin size 2 m/sec *ΔT is a random time shift in each of the pulse intervals from the average PRI of 2 μsec, distributed uniformly between the values [ -T PRI /2, T PRI /2] (~U[a,b] symbols uniform distribution between the real values a and b). T PRI defines the time frame length around the average PRI of the waveform, in which each pulse interval is randomized. 32

Velocity [m/sec] Base Band frequency [Hz] Amplitude (Real part) In Figure 7a the waveform is shown, having different random time delays between each two sequential pulses, and carrying the same frequency. Figure 7b shows the generalized ambiguity function of the waveform, and in it we can see a significant reduction of the both the time and velocity recurrent lobes. 1.5 5 1 15 2 25 Time [ sec] 1.5 -.5-1 5 1 15 2 25 Time [ sec] Figure 7a 1 pulses train with a staggered PRI waveform. -2-15 -1-5 5 1 15 1.9.8.7.6.5.4.3.2.1 2-4 -3-2 -1 1 2 3 4 t / PRI Figure 7b The Generalized Ambiguity function of the 1 pulses train with a staggered PRI waveform. 33

Velocity [m/sec] To emphasize the effects, Figure 7c shows the generalized ambiguity function of the same waveforms, but with a length of 4 pulses. In it we can see the spreading of the time and velocity recurrent lobes all over the volume, but that the time recurrent lobes are still relatively high in the zero velocity line due to some partial correlation in the time domain. -2-15 -1-5 5 1 15 1.9.8.7.6.5.4.3.2.1 2-4 -3-2 -1 1 2 3 4 t / PRI Figure 7c The Generalized Ambiguity function of the 4 pulses train having a staggered PRI waveform. One of the main challenges in implementation of the staggered PRI waveform is its signal processing. Li and Chen [32] propose to use the Non-Uniform Fast Fourier Transform (NUFFT) algorithm for the processing. In chapter 5.4 we propose a method of processing it using perfect reconstruction. 34

5. Staggered PRI and Random frequency Based Waveform 5.1. Description In order to gain from both worlds in terms of ambiguity rejection, we propose a waveform based on the combination of a staggered PRI, and the random frequency shift between sequential pulses in the train. The following example shows the Generalized Ambiguity function of a waveform with the parameters: Waveform Randomized Stepped frequency, staggered PRI Number of pulses 1 Carrier frequency 5 GHz Bandwidth 5 khz Pulse Repetition Interval 2 μsec + ΔT ΔT ~ U[ -T PRI /2, T PRI /2] * T PRI * 12 μsec Pulse width 4 μsec Sampling rate 4 khz Velocity bin size 2 m/sec *ΔT is a random time shift in each of the pulse intervals from the average PRI of 2 μsec, distributed uniformly between the values [ -T PRI /2, T PRI /2] (~U[a,b] symbols uniform distribution between the real values a and b). T PRI defines the time frame length around the average PRI of the waveform, in which each pulse interval is randomized. In Figure 8a the waveform is shown, having both different random time delays between each two sequential pulses, with each one carrying different frequency in a random order. Figure 8b shows 35

Velocity [m/sec] Velocity [m/sec] Base Band frequency [Hz] Amplitude (Real part) the generalized ambiguity function of the waveform, and in it we can see again a significant reduction of the both the time and velocity recurrent lobes. 1.5 -.5-1 2 4 6 8 1 12 14 16 18 2 Time [ sec] 6 x 14 4 2 2 4 6 8 1 12 14 16 18 2 Time [ sec] Figure 8a 1 pulses train with staggered PRI and random frequency waveform. -2 1-2 1-15.9-15.9-1.8.7-1.8.7-5.6-5.6.5.5 5.4 5.4 1 15.3.2.1 1 15.3.2.1 2-4 -3-2 -1 1 2 3 4 t / PRI 2 -.4 -.3 -.2 -.1.1.2.3.4 t / PRI Figure 8b The Generalized Ambiguity function of the 1 pulses train with staggered PRI and random frequency waveform. The figure on the right shows a zoom on the zero-time mainlobe. 36

Velocity [m/sec] Velocity [m/sec] -2 1-2 1-15.9-15.9-1 -5.8.7.6-1 -5.8.7.6.5.5 5.4 5.4 1 15.3.2.1 1 15.3.2.1 2-4 -3-2 -1 1 2 3 4 t / PRI 2 -.4 -.3 -.2 -.1.1.2.3.4 t / PRI Figure 8c The Generalized Ambiguity function of the 4 pulses train with staggered PRI and random frequency waveform. The figure on the right shows a zoom on the zero-time mainlobe. To emphasize the effects, Figure 8c shows the generalized ambiguity function of the same waveforms, but with 4 pulses train. In it we can see the spreading of the recurrent lobes all over the range and velocity axes. In this example it is also apparent that the far velocity recurrent lobes (in this example located at and ) are spread locally, but not entirely all over the velocity axis. The reason is that the PRI, although not uniform, is still localizes around 2 μsec for each pulse in the CPI. If we use a wider stagger in the PRI (by increasing T PRI ) the spreading will increase, but as a consequence we might also affect other parameters in the system (caused by a decrease in the minimal pulse interval and increase in the maximal pulse interval in the CPI). Figure 8d shows the effects of different staggers on the velocity recurrent lobes in the generalized ambiguity function. 37

Velocity [m/sec] Velocity [m/sec] Velocity [m/sec] Velocity [m/sec] -2 T PRI = sec -2 T PRI = 5 sec 1-1 -1.9.8 1 1.7 2 -.4 -.2.2.4 t / PRI T PRI = 1 sec -2 2 -.4 -.2.2.4 t / PRI T PRI = 2 sec -2.6.5.4-1 -1.3.2 1 1.1 2 -.4 -.2.2.4 t / PRI 2 -.4 -.2.2.4 t / PRI Figure 8d Four Generalized Ambiguity functions of the 4 pulses train of staggered PRI and random frequency waveform, with different PRI staggers (zooming on the zero-range main-lobe). In this case the mean pulse interval is 2 μsec. 38

5.2. Optimization for minimum sidelobes, Normalized PSLR In the case strong targets or clutter are present, the ambiguities or side lobes of the ambiguity function might unwantedly be detected as false targets. Therefore one of the main objectives in finding a good waveform is by reducing these unwanted artifacts down to the minimum. When using random series for the pulse intervals and frequencies, an obvious question rises: Are there optimal series in the sense of minimal ambiguities / side lobes in the ambiguity function? A useful quality criterion for the "goodness" of the side lobes level is the main-lobe to Peak Side- Lobe Ratio (PSLR), in other words the ratio between the amplitude of the main-lobe and the amplitude of highest side lobe. Here we will treat the recurrent lobes (or ambiguities) as unwanted sidelobes. Using a waveform that spreads the ambiguities all over the ambiguity function, the PSLR will be increased as we increase the length of the pulse train. The reason is that the main-lobe is the product of coherent integration of all the samples of the target reflections in the pulse train (generating an integration power proportional to, where is the number of pulses in the CPI), as opposed to the sidelobes that are also integrated, but in a non-coherent manner (generating an average integration power proportional to ). This produces a PSLR proportional to in power and in amplitude. Hence, a better criterion for the "goodness" of the waveform regardless of the length of the pulse train can be a Normalized PSLR (NPSLR), which we defined as: The higher the NPSLR, the better. Figure 9 shows histograms of 4, 1 and 2 pulses train with different PRI staggers ranges, T PRI. In it we can see that for large T PRI s the NPSLR depends only mildly on the pulse train length. 39

PDF PDF PDF Exercising an exhaustive search in order to find good series for PRIs and frequencies (located on the histogram's tail at high NPSLRs), can help us find the best NPSLR achievable as function of the PRI stagger range for different train lengths, as well as the good series itself. 1 4 4 pulses T PRI = 2 sec T PRI = 7 sec T PRI = 13 sec 1 2 T PRI = 32 sec 1.1.2.3.4.5.6.7.8 1 4 1 pulses 1 2 1.1.2.3.4.5.6.7.8 1 3 2 pulses 1 2 1 1 1.1.2.3.4.5.6.7.8 NPSLR Figure 9 NPSLR Histograms (proportional to the Probability Distribution functions, or PDFs) of different pulse-train lengths, of the staggered PRI (showing different PRI staggers) and random frequency waveform. As can be seen in Figure 9 the maximal NPSLR found in the exhaustive search was ~.7, and in most cases represents the maximal achievable suppression of the ambiguities. The maximal theoretical statistical value of the NPSLR is of course 1. 41

5.3. Integration loss in the first range ambiguity zone using staggered PRI waveform In many cases Radars work only in the first range-ambiguity zone, meaning they receive and process reflections of a transmitted pulse from targets located at close distances, and filter the reflections of the same pulse returning from farther objects - after the following pulse was transmitted. The filtering can be achieved by different methods such as transmitting in large frequency shift between two consecutive pulses, then applying a filter matched to the second pulse that rejects reflections of the first pulse returning from farther objects. In that case, the maximal range from which we receive reflections and integrate them coherently (and without loss) will be restricted by the pulse interval duration. Here we also assume, of course, that there is no reception while transmitting. The relation between the PRI and the maximal first ambiguity-zone range (in the case of uniform PRI waveform) is given by: The assumption is that any reflection returning from a farther distance than will be filtered. In the case of using a staggered PRI, different pulse intervals will dictate different maximum ranges, although the smallest pulse interval will not necessarily be the constraint to the maximal detectable range: 1. Integration level of reflections returning at delays smaller than the smallest pulse interval in the cycle batch will not be affected (therefore there will be no additional losses). 2. Some losses will be inflicted to reflections returning at times between the minimal and the maximal pulse intervals. 3. Reflections returning at later times than the maximal pulse interval in the series will not be detected, since filtering of signals returning from ambiguity zones is assumed. 4

Integration loss [db] In Figure 1 the integration loss is shown as function of the time delay of the Radar echo, for different PRI staggers. It shows that for a waveform with no stagger at all (T PRI = ), the loss will be zero for all times between the pulse width and the PRI, but will be infinite for time larger than the PRI. When using a stagger in the PRI, however, some loss will occur at times smaller than the mean PRI because of the narrower pulse intervals in the train, but because of the wider pulse intervals in the train the loss at times larger than the mean PRI will not be infinite. This means for example that although suffering from significant loss, a target can be detected in that area. 35 3 25 2 15 1 5 T PRI = 2 sec T PRI = 1 sec T PRI = 5 sec T PRI = sec.2.4.6.8 1 1.2 1.4 1.6 1.8 t / mean PRI Figure 1 Integration loss as function of the time delay (proportional to the target's range) for different PRI staggers. The average pulse interval used in this example is 2 μsec, and the average duty cycle is.2 (the pulse width is 4 μsec). The integration loss is an additional parameter that has to be taken into account when optimizing the waveform. Large PRI stagger range can increase the NPSLR - reducing probability for a false detection, but will also increase the integration loss - reducing the probability of detection. 42

5.4. Processing Staggered PRI waveform using perfect Reconstruction One of the main challenges in implementation of the staggered PRI waveform is its signal processing. Different methods were proposed, amongst them direct Discrete Fourier Transform (DFT) processing [33] and the more efficient Non Uniform FFT (NUFFT) [32]. The NUFFT algorithm is described in detail at [34]. In this chapter we propose and analyze another method using non-uniform to uniform sampling interpolation followed by FFT. Perfect reconstruction of a periodic signal band limited to, from non-uniformly spaced samples ( ), sampled at times, is given by [35]: ( ) where: ( ( ) ) ( ( ) ) { ( ( ) ) ( ( ) ) ( ( ) ) Non-uniform to uniform sampling interpolation is given by resampling of the perfect reconstruction: ( ) ( ( ) ) ( ( ) ) { ( ( ) ) ( ( ) ) ( ( ) ) 43

In matrix representation the interpolation transformation is: In our case ( ) represents the different pulses in the CPI. Once ( ) is resampled uniformly as, the Doppler frequency (and the velocity) of the target can be estimated by an FFT just like in the uniform PRI case. Because DFT is also a linear operator that is applied on the samples, we can combine the two matrix multiplications, and therefore reduce the amount of calculations: { } Notice in this analysis that we assume all the pulses in the CPI are transmitted at the same carrier frequency. A waveform that contains both non-uniformity in its PRI and frequency shifts between its pulses, as described in chapter 5.1, cannot be simply processed by non-uniform to uniform interpolation and DFT as described here. The reason is that the frequency shifts dictates a nonlinear phase shifts from pulse to pulse, that cannot be compensated by simple Doppler processing. Therefore - when also using random frequency we need to take a different approach. Figures 11a-c show simulation results for three different cases: a. A single target b. Three targets having different frequencies and amplitudes (amplitudes of 1 at 5Hz, 2 at 1 Hz, and.5 at 14 Hz). c. Same three targets in the presence of noise. The noise Standard Deviation (STD) is 2, meaning that the stronger target has an SNR of db before integration (the weakest target has an SNR of ~ -12 db before integration). In all cases good reconstruction and Doppler estimation is demonstrated. 44

Normalized Amplitude [db] Amplitude (Real part) Normalized Amplitude [db] Amplitude (Real part) 1 True signal Non-uniformly sampled signal Interpolated signal.5 -.5-1.2.4.6.8.1.12.14 t [sec] -1 FFT of True signal Spectral estimation of Non-uniformly sampled signal -2-3 -4-5 2 4 6 8 1 12 14 16 18 2 frequency [Hz] Figure 11a Doppler estimation of single target with non-uniform PRI waveform by interpolation to uniformly sampled signal and DFT. 6 True signal Non-uniformly sampled signal Interpolated signal 4 2-2 -4-6.2.4.6.8.1.12.14 t [sec] FFT of True signal Spectral estimation of Non-uniformly sampled signal -1-2 -3-4 2 4 6 8 1 12 14 16 18 2 frequency [Hz] Figure 11b Doppler estimation of 3 targets at frequencies 5 Hz, 1 Hz and 14 Hz. 45

Normalized Amplitude [db] Amplitude (Real part) 6 True signal Non-uniformly sampled signal Interpolated signal 4 2-2 -4-6.2.4.6.8.1.12.14 t [sec] FFT of True signal Spectral estimation of Non-uniformly sampled signal -1-2 -3-4 2 4 6 8 1 12 14 16 18 2 frequency [Hz] Figure 11c Doppler estimation of 3 targets at frequencies 5 Hz, 1 Hz and 14 Hz in the presence of noise. 46

5.5. Processing Staggered PRI with random frequency waveform 5.5.1. Processing We take a waveform composed of pulses. Between each 2 pulses, pulse and pulse, there is a time interval (PRI) of ( ). In this notation. Each pulse is modulated by a different random frequency. Given that the first pulse is transmitted at, the timeline between the different pulses is: We also mark the sample time relative to beginning of each pulse transmission as, where is the sample index. The global sampling time is therefore: frequency. is the number of samples in each pulse interval, and depends on the sampling As was shown in chapter 4.2 (equation (24)), the relative phase between the transmitted and received signal is given by: ( ) ( ) ( ) ( ) ( ) 47

The RF signal is usually down-converted to base-band in order to sample and process it digitally. The demodulated received signal is: { ( )} { } { } The received signal amplitude, ( ), is also a function of the time delay. Assuming all the transmitted pulses have the same amplitude, we can treat each pulse as independent and regardless of : ( ) ( ) ( ) and get: ( ) { } { } If we also use the "Stop and Hop" approximation, neglecting the phase dependency on the intra-pulse time,, meaning: we get: 48

( ) { } { } ( ) { } Matched filtering of the signal is applying the filter on the received signal ( ). The product is the Range-Velocity Map (RVM, as opposed to the traditional Range-Doppler Map, RDM): ( ) { } { } ( ) The inner sum in the last expression of equation (6) represents a temporal short-time matched filter. The outer sum represents the "generalized" Doppler processing (now actually Range-Velocity processing), that carries out the coherent integration. Using digital signal-processing, the range and velocity hypotheses can be quantized in order to receive a finite data size. In addition, the grids of and do not have to be uniform, and can vary in different manners as needed by the application. Implementation of the waveform's signal processing in MATLAB code is reviewed in Appendix A. 5.5.2. Implementation complexity One of the most important aspects in the implementation of a Radar waveform is its signal processing computational complexity and memory usage. We will now compare between the processing complexity of an ordinary modulated pulse-doppler waveform, and that of the staggered PRI random frequency waveform. 49

The ideal signal processing in terms of maximal output SNR, is matching the received signal with all the possible range-velocity hypotheses: Where is the expected received signal transformed by the propagation from objects at different ranges and velocities. As was shown in chapter 4.3 (equation (3)), the correct matchedfilter response should be: ( ) ( ) ( ) Processing complexity of a basic pulse-doppler waveform If we use a simple modulated waveform, keeping all of the limiting conditions presented at chapter 4.3 fulfilled, then we can reduce the reference signal to a simpler form: ( ) ( ) Usually, in order to reduce the dependency of the signal on the high carrier frequency the signal is coherently down-converted to base-band, so the reference signal will actually have to be: ( ) The matched-filter output then has the form: ( ) 51

If the signal is sampled uniformly and the waveform is a pulse train with a uniform PRI, the matched-filter becomes a sum instead of an integral: ( ) Here is the uniform PRI, is the number of pulses transmitted in the cycle, and is the number of samples sampled at each pulse interval reception window. Here again marks the sampling time relative to beginning of each pulse transmission, and is the sample index. Two more assumptions are usually made: a. All the pulses in the train are identical to each other. Using this assumption the reference signal's amplitude modulation, ( ), does not depend on the pulse number. The sum then becomes: ( ) b. The phase change within a transmitted pulse reflection due to the Doppler shift is negligible (again, this is the "Stop and Hop" approximation) : In this case we can assume the additional phase term is constant: 5

and it will not have a major effect on interest us, only the relative phases). Finally we have: (taking into account that the absolute phase does not ( ) The signal processing computations of equation (7) includes the following: 1. Performing times (for each pulse interval) a matched filter on a sample set at a size of at least samples. We will mark the final output size of this process as - the number of range bins. The reason the number of range bins ( ) might be different than the number of sample at each pulse interval ( ) is that we might want to use a finer sampling by interpolating the data, or alternatively might not need all of the samples. 2. Performing FFTs at the size of - the number of velocity bins. Here might be different than for similar reasons as in the case of and. The complexity will be in the order of: However, if we are only interested in a subset of Doppler frequencies and not in all of them, digital filtering followed by decimation can be used, reducing the total necessary FFT size. If digital filtering is applied, its complexity has to be also taken in account and it will add FIR calculations. Assuming decimation factor of (in this case the number of velocity bins will be ), the computational complexity will be in order of: ( ) 52

In terms of space in both cases, sampled data before performing the FFT (and filtering). memory will be needed in order to remember all the For example, assuming that and and that no decimation is applied, the computational complexity will roughly be ( ), and memory units will be needed. Processing complexity of a staggered PRI random frequency waveform As shown in equation (6), the staggered PRI random frequency waveform's signal processing is given by: { } ( ) The signal processing computations includes the following: 1. Performing times (for each pulse interval) a matched filter on a sample set at a size of at least samples. We again mark the final output size of the process as - the number of range bins. 2. Performing DFTs of size, marking and the number of range bins and number of velocity bins (respectfully) that are inspected. Here and can be any number, because the ranges and velocities that are inspected can be arbitrary. In this case the complexity will be in the order of: In terms of space, of memory will be needed in order to remember all the sampled data before performing the DFT, plus all of the required DFT coefficients. 53

Using the previews example, assuming that and, and that no decimation is applied, the computational complexity will roughly be, and memory units will be needed. If we compare between the computational complexities of the two types of processing in this example, it seem that the second is much more complex in both terms of number of calculations and memory usage (about ~1 fold more calculations and ~1 fold more memory units). However, this is true only if and. In reality we might not need to do all of the calculations for the entire range and velocity spans, so it is possible to reduce the amount of calculations and memory usage in a controlled way by reducing, or both. 54

5.6. Simulation results A simulation was written in order to evaluate the performance of the staggered PRI random frequency waveform and its signal processing. The simulation includes the transmitted signals with the wanted frequencies and pulse intervals, propagation of the waves in free space reflected from moving objects, clutter and in the presence of noise, and the reception process including RF to IF conversion, filtering and sampling, signal processing and analysis. Figure 12a shows a blockscheme of the simulation. Figure 12b demonstrate the simulation output for a single target the Range Velocity Map (RVM). Figure 12c shows the range and velocity profiles of the target in the RVM. Targets Clutter Noise Simulation parameters Signal generation Physical medium IF Reciever and sampling Convertion to Base-Band Filtering and Decimation Signal processing Display and analysis Figure 12a A scheme of the simulation's modules and data-flow 55

Velocity-Profile Amplitude [db] Range-Profile Amplitude [db] Velocity [m/sec] -1-5 -2-3 -4-5 -1-15 -2 [db] -6-25 5 1 15 2 25 3 35 4 45 5 Range [m] -3 Figure 12b Range-Velocity Map by simulation of single target at 25 m range and -31 m/sec radial velocity. The waveform includes integration of 1 pulses. -1-2 -3-4 -5 18 2 22 24 26 28 3 Range [m] -2-4 -6-1 -2-3 -4 Velocity [m/sec] -5-6 -7 Figure 12c Range and velocity profiles of the target in the Range-Velocity Map shown in Figure 12b. The meaning of the range and velocity profiles is explained in chapter 4.1. 56

Velocity-Profile Amplitude [db] Range-Profile Amplitude [db] Velocity [m/sec] 5.6.1. Staggered PRI waveform We first demonstrate the cycle coherent integration using the method described in chapter 5.4 - nonuniform to uniform interpolation of staggered PRI sampled waveform (Figures 13a, 13b). Unfortunately the method cannot be applied in the case of using frequency hopping, so we show the results using only staggered PRI with a constant frequency modulated waveform. In this case of course we lose range resolution, and the effective range resolution is determined only by the pulse width. We compare the results to the general processing (equation 6) described in chapter 5.5 (Figures 13c, 13d). -2-5 -4-1 -6-8 -15 [db] -1-2 -12-25 -14-3 -16 5 1 15 2 25 3 35 4 Range [m] Figure 13a RVM of single target using staggered PRI with single frequency waveform, created by non-uniform to uniform interpolation and DFT. The waveform includes a 1 pulses. -35-1 -2-3 -4-5 16 18 2 22 24 26 28 3 32 Range [m] -2-4 -6-8 -2-4 -6-8 -1 Velocity [m/sec] -12-14 -16-18 Figure 13b Range and velocity profiles of the target in Figure 13a. 57

Velocity-Profile Amplitude [db] Range-Profile Amplitude [db] Velocity [m/sec] -5-5 -1-15 [db] -1-2 -25-15 5 1 15 2 25 3 35 4 45 5 Range [m] Figure 13c RVM of single target using staggered PRI with single frequency waveform (no frequency hopping), created by the general signal processing. The waveform includes a 1 pulses. -3-35 -1-2 -3-4 -5 18 2 22 24 26 28 3 32 Range [m] -2-4 -6-2 -4-6 -8-1 Velocity [m/sec] -12-14 -16 Figure 13d Range and velocity profiles of the target in Figure 13c. It appears that processing the signals using non-uniform to uniform interpolation may yield better results, but in fact it seems to be very sensitive to the exact PRI series, and demands greater computational resources than the general processing method. An example for a simulated PRI series for which the calculation did not yield the correct result is shown in Figure 13e. 58

Velocity-Profile Amplitude [db] Range-Profile Amplitude [db] -1-2 -3-4 -5 18 2 22 24 26 28 3 32 Range [m] -2-4 -6-2 -4-6 -8-1 Velocity [m/sec] -12-14 -16-18 Figure 13e A different result for a similar case as was shown in Figure 13d for a different PRI series, using non-uniform to uniform interpolation and DFT. The waveform includes a 1 pulses. For this PRI series the calculation did not yield the correct result. The failure to reconstruct the signal properly in some of the cases tested in the full simulation is due to the sensitivity of the reconstruction solution to the bandwidth requirement, presented at chapter 5.4. In reality, it is difficult to constrain signals to a strictly confined bandwidth with no leaks none so ever to higher frequencies. The perfect reconstruction method seems to be very sensitive to such frequency leaks, leading in some cases to a wrong signal reconstruction and making it a non-practical solution. 59

Velocity [m/sec] Velocity [m/sec] Velocity [m/sec] Velocity [m/sec] 5.6.2. Single target (noise free and no clutter) In this scenario we use the simulation to simulate a single target at different ranges and velocities. The range resolution of the simulated Radar is 32 m, and the velocity resolution is 1 m/sec. As shown in Figure 14, the target signal is coherently integrated regardless of the target range and velocity. Range: 18 m, Velocity: -1e-8 m/sec Range: 25 m, Velocity: -3 m/sec -2-2 -5-4 -4-6 -6-1 2 4 Range [m] Range: 7 m, Velocity: -2 m/sec 2 4 Range [m] Range: 43 m, Velocity: -5 m/sec -15 [db] -2-2 -2-4 -4-25 -6-6 2 4 Range [m] 2 4 Range [m] -3 Figure 14 Range-Velocity Maps of single target at different ranges and velocities. 5.6.3. Single target in the presence of noise and clutter In this scenario we simulate a single target in the presence of thermal noise and clutter. The clutter is simulated by many strong point reflectors at zero velocity and at different ranges. In Figures 15a, 15b simulation results are shown with the target standing out on the background of the clutter and noise after the signal processing. 61

Velocity-Profile Amplitude [db] Range-Profile Amplitude [db] Velocity [m/sec] -1-5 -2-3 -4-5 -1-15 -2 [db] -6-25 5 1 15 2 25 3 35 4 45 5 Range [m] -3 Figure 15a Simulated Range-Velocity Map of single target at 25 m range and -3 m/sec radial velocity, in the presence of thermal noise and clutter. -1-2 -3-4 -5 1 2 3 4 5 6 Range [m] 2-2 -4-1 -2-3 -4 Velocity [m/sec] -5-6 -7 Figure 15b Range and velocity profiles of the target in the Range-Velocity Map. Even though the target is approximately 11 db weaker than the clutter located in its range, after integration it is approximately 18 db stronger than the clutter's velocity sidelobes. 6

Velocity [m/sec] 5.6.4. Two targets in the presence of noise In this scenario we simulate two targets with the same RCS at: 1. Different ranges but same velocity (Figure 16a) 2. Different velocities but same range (Figure 16c) in the presence of thermal noise. As shown in their range and velocity profiles in Figure 16b and Figure 16d, the two targets are still separable although they are close to each other (relative to the resolution) in both dimensions. -1-5 -2-3 -4-5 -1-15 -2 [db] -6-25 5 1 15 2 25 3 35 4 45 5 Range [m] -3 Figure 16a Simulated Range-Velocity Map of two close targets at 25 m and 28 m ranges, and each at -3 m/sec radial velocity, in the presence of thermal noise. 62

Velocity [m/sec] Range-Profile Amplitude [db] -5-1 -15-2 -25-3 -35-4 -45-5 1 2 3 4 5 6 Range [m] Figure 16b Range profile of the targets in the Range-Velocity Map shown in Figure 16a. -1-5 -2-3 -4-5 -1-15 -2 [db] -6-25 5 1 15 2 25 3 35 4 45 5 Range [m] -3 Figure 16c Simulated Range-Velocity Map of two close targets at 25 m range, and at -3 m/sec and -32.5 m/sec radial velocity, in the presence of thermal noise. 63

Velocity-Profile Amplitude [db] -5-1 -15-2 -25-3 -35-4 -45-5 -7-6 -5-4 -3-2 -1 Velocity [m/sec] Figure 16d Velocity profile of the targets in the Range-Velocity Map shown in Figure 16c. 64

Frequency [MHz] 5.7. Experimental results In order to prove the implementation feasibility of a staggered PRI random frequency waveform, experimental data including real linear stepped-frequency Radar raw samples were manipulated to produce an effective random waveform, and a proper signal processing was applied to produce the required Range-Velocity maps. The original data processed was of a waveform similar to the one shown schematically in Figure 17a, made of a train of some 35 pulses. Processing the data using equation (6) produces the RVM shown in Figure 17b, in it we can see a real target located at 115 m range and moving at -34 m/sec velocity, and also strong static clutter (at zero velocity) located at all ranges. The range profiles of the target and clutter are shown in Figure 17c, and the velocity profile of the target is shown in Figure 17d. 6 5 4 3 2 1 5 1 15 2 25 PRI index Figure 17a A full stepped-frequency waveform data (scheme) 65

Range-Profile Amplitude of Clutter and Target [db] Velocity [m/sec] -1-1 -2-2 -3-3 -4-5 [db] -4-6 -7-5 -8 5 1 15 2 25 3 Range [m] -9 Figure 17b Range-Velocity Map created by processing of a full steppedfrequency train. In this example an target appears at 115 m range and at -34 m/sec velocity, in the presence of strong clutter (located at the zero velocity). The number of integrated pulses is ~35. -1 Target Clutter -2-3 -4-5 -6-7 -8-9 5 1 15 2 25 3 35 Range [m] Figure 17c Range profiles of the target (-34 m/sec velocity) and the clutter (zero velocity) in the Range-Velocity Map, created by processing of a full stepped-frequency train. 66

Velocity-Profile Amplitude of Target [db] -1-2 -3-4 -5-6 -7-8 -9-1 -2-3 Velocity [m/sec] -4-5 -6 Figure 17d Velocity profile of the target (at 115 m range) in the Range-Velocity map, created by processing of a full stepped-frequency train. An effective random waveform was created from the original data by selecting a random series of pulses from an entire linear stepped-frequency cycle. Because the selection of the PRI series is random, the pulse interval and the frequency difference between each two consequent pulses in the series are also random (although quantized by the basic original PRI and frequency step). The diluted data contains about 15 pulses of the original 35. This method is illustrated in Figure 18a. The RVM produced by processing the diluted data is shown in Figure 18b. In it we can still see the clutter at the zero velocity, but now additional velocity sidelobes of the strong clutter also appear. In fact, the clutter sidelobes level is so high (about -35 db under the clutter level, as expected see Figure 18c), that they reach the target's level and mask it. The range profiles of the target and clutter are shown in Figure 18d. Looking carefully at the range profile of the target we can still see it rising a little above the sidelobes level at 115 m range. If we were to use a larger pulse-train, or if the target was stronger enough, then the target's profile would have been more prominent on the background of the clutter sidelobes. We can also see that both the range profile of the clutter and its velocity resolution did not change significantly. This gives good indication that the random waveform is feasible for implementation. It is important to mention here that pulse cancelling technique, meant to reduce the clutter level, cannot be applied in this case due to the different frequencies of the consecutive pulses and also due to the different time interval between the pulses (creating changing phase differences between them). 67

Velocity [m/sec] Frequency [MHz] Frequency [MHz] 6 Before dilution 4 2 5 1 15 2 25 6 After dilution 4 2 5 1 15 2 25 PRI index Figure 18a An effective staggered PRI and "random" (disordered) frequency waveform achieved by random dilution of a steppedfrequency train data. -1-1 -2-2 -3-3 -4-5 [db] -4-6 -7-5 -8 5 1 15 2 25 3 Range [m] -9 Figure 18b Range-Velocity Map created by processing of a randomly diluted stepped-frequency train data. In this example the clutter's velocity sidelobes are so high, that they mask the target and it seems to disappear. 68