Radar signal processing is a field of perpetual innovation, crucial for advancements in navigation, control systems, and communication. One key aspect of radar technology is the use of ambiguity functions (AF) to assess the time response of signals subjected to Doppler shifts and delays. While the theory of ambiguity functions is well-established, there remains a gap in the practical assessment of various signals using these functions—particularly with the application of Levanon-Mozeson's code.

There are more than one definition of the ambiguity function or AF, especially in the radar systems. The most precise definition is its representation of the time response of a matched filter tuned to a specified limited signal of energy once the signal reaches accompanied with a Doppler shift (ν) and a delay (τ) with respect to the minimal amounts (zeros) predictable at a matched filter [1]. Besides, the mathematical representation of the ambiguity function has the most famous (if not the only formula)

 equation as follows. ……... (1)

Where, is the complex baseband pulse.

Within every radar structure, it has been anticipated to obtain sufficient resolution in both Doppler range domains. Over decades, numerous approaches have been invented to reach such purpose. Possibly, the leading proposal to progress resolution of range was in using a pulse with extra-little period. Obviously, it has been achievable to reach sufficient resolute range domain. Such condition is by this pulse, nevertheless when such length of pulse is little, the Doppler resolution is reduced, beside the fact that dual-functional radar-communications (DFRC) system links with an OFDM receiver while, at the same time, assessing target parameters by using the backscattering signals [2]. However, some researchers have established pulses having extended period, nevertheless having such a construction, which is able to accomplish great resolution for both Doppler and range domains. Possibly, there are some codes that are more suitable in such characteristic than others [1, 3]. Many researchers have been working on many signals and codes accompanied with applications in radar systems and communications engineering. For instance, Malek G. M. Hussain suggested a generalized ambiguity function (GAF) for a MIMO-radar system that can be derived using a set of quasi-orthogonal step-frequency UWB-throb signals [4]. Moreover, similar works have been done with different applications [5 - 10].

This work is to show the description of several functions and codes that were subjected to the ambiguity function. E. Mozeson and N. Levanon primarily wrote the MATLAB code of the ambiguity function in a paper published in 2002 [11]. Such program permits entering numerous diverse signals and offers governor to a lot of plotting factors. The program tolerates over-sample of the signal through far better resolute signals than required in computing the postponed signals. Such idea makes it likely to calculate a weakened image (less delay - points of Doppler grid) of the function of ambiguity through little arithmetic work during sampling the signal by an adequately big rate of sampling [11].

This paper addresses the need for a systematic evaluation of different signals through the ambiguity function using Levanon-Mozeson's code. The novelty lies in the application of this code to a diverse set of signals, revealing unique interactions and providing insights into their suitability for various radar and communication systems applications. This research fills the gap by offering a comprehensive assessment that goes beyond theoretical analysis, providing empirical data and detailed visualizations of the AF's performance across different signals.

The novelty of this research lies in its empirical approach to evaluating diverse signals via the Levanon-Mozeson code. Unlike previous studies that may have narrowly focused on theoretical or limited signal types, this study broadens the scope, providing new data and interpretations that could influence radar signal processing practices.

Moreover, this study introduces a rigorous analysis of the zero-cut points within AF, which are crucial for radar signal clarity and integrity. By meticulously examining these points, the research offers new perspectives on optimizing signal design for enhanced radar performance.

In radar signal processing, the ambiguity function is a critical tool for analyzing how different signals respond to time delays and Doppler shifts. While the theoretical foundations of the AF are well-established, there is a somehow lack of empirical studies that systematically evaluate how various radar signals interact with the AF, in particular using tools like the Levanon-Mozeson code. This gap restricts the practical application of AF analysis in optimizing radar signal design and performance.

To address such problem, this research aims to assess a various set of radar signals by applying the Levanon-Mozeson code to compute and visualize their ambiguity functions. By analyzing parameters such as amplitude, phase, signal duration, and propagation delays, we seek to provide a comprehensive understanding of each signal's interaction with the AF. This study will offer valuable insights into selecting and designing radar signals that enhance system performance, thus bridging the gap between theoretical AF analysis and practical radar signal optimization. Moreover, this investigation leads to further future research of developments and optimizations.

SIGNALS SUBJECTED TO THE AMBIGUITY FUNCTION

Roughly, any signal that can be transmitted and/or received through an antenna can be subjected to the ambiguity function. However, some signal codes exhibit more consistency with the throughput of the ambiguity function than others do. Here is a list, but is not limited to, of several signals that are considered useful in radar systems as well as other applications.

1. Constant-frequency pulse

    a. Costas frequency coding

    b. nonlinear frequency modulation

2. Linear frequency-modulated pulse

3. Coherent train of identical unmodulated pulses

4. Phase-coded pulse

    a. Barker codes

    b. Chirplike phase codes

    c. Asymptotically perfect codes

    d. Golomb’s codes with ideal periodic correlation

    e. Epatov code

    f. Huffman code

5. Coherent train of LFM pulses

6. Diverse PRI pulse trains

7. Coherent train of diverse pulses

8. Stepped-frequency train of LFM pulses

9. Stepped-frequency train of unmodulated pulses

10. Train of complementary pulses

11. Train of orthogonal pulses

     a. Orthogonal-Coded LFM-LFM pulse train

     b. Orthogonal-Coded LFM-NLFM pulse train

12. Frequency-modulated CW signals

13. Multicarrier phase-coded signals

Some signals were selected to be treated by the code representing the ambiguity function. However, Table (1) represents the signals subjected to the AF code and they are as following, and they possess the different parameters regarding Code of signal, Code of frequency, K Doppler [11]. 

Table 1. List of signals and corresponding parameters

Hence,

F*Mtb describes the scope of the designed axis of Doppler in regularized Doppler, which is Doppler times the whole period of the signal (Mtb).

K is the amount of lattice points in the axial Doppler of the scheme.

 is the propagation delay

1. Costas 7 signal

The first Costas code was defined in 1965 via John Costas. It is a sort of coding of time-frequency waveforms. Arrays of Costas were presented as a way to improve the act of sonar and radar systems. Moreover, such codes are a type of coding frequency waveforms and are burst of attached waveforms of uncoded pulse. Fig. 1 shows the Costas signal with seven bits. It can be noticed how the three-dimensional plot of the ambiguity function reacts to the code along with other relating information, such as the zero-cut Doppler depiction.

Fig. 1. Costas signal with AF

2. Pulse signal

In signal processing, any pulse is a fast, passing alteration in the signal amplitude from a base-line number to a lower or a higher number, tailed by a quick reappearance to the base-line number [12]. Fig. 2 shows how the ambiguity function works on such signal, which is fluctuating at this time.

Fig. 2. Effect of AF on pulse signal

3. Weighted LFM: We can define Linear frequency modulation or (LFM) as a means to decide two minor targets or goals, which are situated at extended period by right minor spacing flanked through them [13]. In order to gain a harmonic beamforming in antenna array, time – modulated amplitude – phase weighting is carried out [14]. Fig. 3 shows how the Weighted LFM reacts to the AF.

Fig. 3. Effect of AF on weighted LFM

4. Barker 13 (13-bit): Any Barker code is a sequence of numerals in the form of  of span  such that

Barker codes are utilized to compact pulsations of signals of radar, whose lengths extends up to 2, 3, 4, 5, 7, 11, and 13 [15]. It is one of the likelihoods of intra-pulse biphasic modulation to compress pulses of radar equipment to develop resolute range for comparatively extended broadcast pulsations [16]. Fig. 4 depicts the AF and its relating for such code.

Fig. 4. Effect of AF on Barker 13

5. Frank 16 signal (16-bit): The Frank Code is a poly-phase coding modulated formation and is used in compressing pulses. It can be in the form of sets of Frank sequences as a poly-phase order through a perfect intermittent auto-correlation purpose that together has optimal cross-correlation function [17]. Fig. 5 shows how the AF react to Frank 16 signal.

Fig. 5. Effect of AF on Frank 16 signal

6. P4 25 signal (25-bit): This is a class of waveform that is phase-coded and can be numerically treated and sampled at receiving, where such signal is derived from the waves of analogue chirp. In addition, it provides a type of frequency-based phase-coded functions [18]. Fig. 6 indicates the activity between the AF along with its relating and P4 25 signal.

Fig. 6. Effect of AF on P4 25 signal

7. Complementary pair signal: In terms practical mathematical sciences, complementary sequences or (CS) are couples of sequentials and the beneficial merit in which their non-phasic non-periodic auto-correlation constants equal to nothing. Such signal represents a radar signal, which accomplishes the eventual range side-lobe discount null side-lobe [19]. In Fig. 7, we can see the AF and its relating with the Complementary pair signal.

Fig. 7. Effect of AF on Complementary pair signal

8. Pulse train signal: A pulse train is an episodic pulse waveform through a duration consisting of quadrangular pulses with period T. The cycle of duty of an intermittent pulse sequence is descripted by T/To [20]. Fig. 8 indicates the relationship between the AF and the pulse train signal.

Fig. 8 Effect of AF on Pulse train signal

9. Pulse train signal 2: This signal is very similar to the signal in point (8), but it differs in several parameters including delays; hence, in Fig. 9 we can see how the AF and relatings have changed.

Fig. 9. Effect of AF on Pulse train signal

10. Stepped-frequency pulse train: Stepped-frequency signal is a substitute method, which is able to reserve acceptable resolute range and signal to noise ratio at the incoming signal minus outsized immediate bandwidth. The stepped-frequency signal consists of a collection of chirping pulses at increased-up supported frequencies [21]. In Fig. 10, we can observe how the AF relates to the stepped-frequency pulse train signal. 

Fig. 10. Effect of AF on Stepped-frequency pulse train

The results from our analysis using the Levanon-Mozeson code provide suitable disclosures about the behavior of different radar signals when subjected to the ambiguity function. Precise striking is the performance of the Costas signal, which demonstrated a remarkably flat Doppler response across its entire range that indicates a high level of stability and minimal variance at zero-cut Doppler points. This contrasts sharply with pulse train signals, which exhibited significant decorrelation that suggested their reduced suitability in scenarios where signal integrity is critical. Such differential behaviors show the discrete interactions between various signal types and the ambiguity function, underscoring the critical role of signal selection in radar systems design. Such findings not only validate the theoretical foundations of the ambiguity function but also pave the way for optimized signal processing strategies in practical radar applications. By carefully selecting signals that align with the desired operational profiles, engineers can enhance both the accuracy and reliability of radar systems, particularly in complex environments where signal clarity and stability are principal.

Trade-offs in Radar Signal Design

One of the fundamental problems in radar signal design is achieving a balance between:

Range Resolution: The ability to distinguish between two targets separated by small distances.

Doppler Clarity: The ability to detect and track targets with high-speed variations.

Traditional approaches have relied on short-duration pulses for enhanced range resolution, but these compromise Doppler accuracy. Conversely, long-duration pulses improve Doppler clarity but reduce range resolution. The question arises: Can specific signals optimize both parameters simultaneously, and if so, how do they perform in practical scenarios?

This study addresses the problem by systematically evaluating the ambiguity function of diverse radar signals using the Levanon-Mozeson code. The novelty lies in applying a consistent computational background to assess signal performance across multiple parameters, including amplitude, phase, Doppler shifts, and propagation delays.

This investigation has effectively demonstrated how different radar signals interact with the ambiguity function when analyzed through the Levanon-Mozeson code. The analysis confirmed separate behaviors among the signals, especially that is highlighted by the Costas signal, which exhibited a nearly flat AF across the complete Doppler spectrum, demonstrating robustness with minimal variation that is particularly at zero-cut Doppler points. In contrast, pulse train signals showed high decorrelation and variability and indicating less stability and clarity in their AF profiles.

In a notable way, the Costas signal maintained consistent AF patterns with an amplitude peak at -60 Hz, and a propagation delay variability of 1.1 milliseconds across a 60-second interval, and it underscores its potential for reliable radar applications. On the other hand, signals like the Barker 13 and Frank 16, despite of having lower amplitude fluctuations, exhibited broader Doppler spreads and phase instabilities, where it challenges their effectiveness in scenarios demanding high signal integrity.

Overall, the use of code has provided a valuable three-dimensional perspective on signal performance, and it enables the analysis of signal parameters like amplitude, phase, and delay in a comprehensive way to understand and guide future developments in radar system design.

The authors declare that they have no conflict of interest.

No funding sources.

The study was approved by the University of Samarra.

1. Levanon et al.; "Radar Signals" (2004), pp. 1-2.

2. Keskin et al.; "Limited Feedforward Waveform Design for OFDM Dual-Functional Radar-Communications" 69 (2021), pp. 1-2, DOI: https://doi.org/10.1109/TSP.2021.3074179.

3. Jansing et al.; "Introduction to Synthetic Aperture Radar: Concepts and Practice" (1st ed., 2004), pp. 1-2.

4. Hussain et al.; "Ambiguity Function of MIMO Radar Based on Quasi-Orthogonal Ultrawideband-Throb Signals" 58.7 (2020), pp. 1-2, DOI: https://doi.org/10.1109/TGRS.2020.2962776.

5. Ming-jiu et al.; "Non-linear Stepped Frequency Waveform Design Method Based on Range Ambiguity Function" (2017), pp. 1-2.

6. Zhu and Zhao; "Wideband Joint Range-Velocity Acceleration Ambiguity Function of Stepped Frequency Signal" (2012), pp. 1-2.

7. Fan et al.; "Wireless Hand Gesture Recognition Based on Continuous-Wave Doppler Radar Sensors" 64.11 (2016), pp. 1-2, DOI: https://doi.org/10.1109/TMTT.2016.2619044.

8. Jimi and Matsunami; "A Novel Spectrum Hole Compensation Using Khatri-Rao Product Array Processing on Random Stepped FM Radar" (2017), pp. 1-2.

9. Zhao et al.; "Performance Analysis of Joint Time Delay and Doppler-Stretch Estimation with Random Stepped-Frequency Signals" (2016), pp. 1-2.

10. Si and Li; "An Optimization Design Scheme for Multiple PRT and Multi-Frequency Coherent Doppler Velometry" (2016), pp. 1-2.

11. Levanon and Mozeson; "MATLAB Code for Plotting Ambiguity Functions" 38.3 (2002), pp. 1064-1068, DOI: https://doi.org/10.1109/TAES.2002.1035091.

12. Molina and González; "Pulse Voltammetry in Physical Electrochemistry and Electroanalysis-Theory and Applications" (2015), pp. 1-2.

13. Patel et al.; "Linear Frequency Modulation Waveform Synthesis" (2012), pp. 1-2.

14. Li et al.; "Harmonic Beamforming in Antenna Array With Time-Modulated Amplitude-Phase Weighting Technique" 67.10 (2019), pp. 1-2, DOI: https://doi.org/10.1109/TAP.2019.2923286.

15. Barker; "Group Synchronizing of Binary Digital Sequences" in Communication Theory (1953), pp. 273-287.

16. Borwein and Mossinghoff; "Wieferich Pairs and Barker Sequences, II" 17.1 (2014), pp. 24-32, DOI: https://doi.org/10.1112/S1461157014000133

17. Fan et al.; "Crosscorrelations of Frank Sequences and Chu Sequences" 30.6 (1994), pp. 477-478, DOI: https://doi.org/10.1049/el:19940308.

18. Qazi and Fam; "Doppler Detection Capable Good Polyphase Code Sets Based on Piecewise Linear FM" (2014), pp. 1-2.

19. Levanon et al.; "Complementary Pair Radar Waveforms—Evaluating and Mitigating Some Drawbacks" 32.3 (2017), pp. 40-50, DOI: https://doi.org/10.1109/MAES.2017.160271.

20. Grami; "Introduction to Digital Communications: Signals, Systems, and Spectral Analysis" (2016), pp. 1-2.

21. Wang et al.; "A Novel Scheme for Detection and Estimation of Unresolved Targets With Stepped-Frequency Waveform" 7 (2019), pp. 1-2, DOI: https://doi.org/10.1109/ACCESS.2019.2901832.