Noise radars employ random waveforms in their transmission as compared to traditional radars. Considered as enhanced Low Probability of Intercept (LPI) radars, they are resilient to interference and jamming and less vulnerable to adversarial exploitation than conventional radars. At its simplest, using a random waveform such as bandpass Gaussian noise as a probing signal provides limited radar performance. After a concise review of a particular noise radar architecture and related correlation processing, this paper justifies the rationale for having synthetic (tailored) noise waveforms and proposes the Combined Spectral Shaping and Peak-to-Average Power Reduction (COSPAR) algorithm, which can be utilized for synthesizing noise-like sequences with a Taylor-shaped spectrum under correlation sidelobe level constraints and assigned Peak-to-Average-Power-Ratio (PAPR). Additionally, the Spectral Kurtosis measure is proposed to evaluate the LPI property of waveforms, and experimental results from field trials are reported.
Keywords: noise radar; noise waveforms; waveform design; PAPR reduction; Low Probability of Intercept
Noise radars transmit random signals, as opposed to traditional radars having deterministic waveforms. This randomness provides advantages in many ways both for the civil and military fields. In civilian applications such as automotive radar or marine navigation, noise radars mitigate mutual interference [[
The use of random signals for detecting objects dates back to the late 19th and early 20th centuries. In 1897, Alexandr S. Popov used random pulses in his radio experiments and reported that ship detection was possible in a bistatic configuration [[
The pioneering work on the coherent reception of random signals was performed by R. Bourret in 1957 [[
Much of the early work included physical delay lines to match the delay of the reference signal to the object range and thus could only offer limited range. However, this technique has long been of interest due to the fact that intercept receivers observe only noise from the transmitting radar. Moreover, the nonrepetitive nature of transmissions would remove range and Doppler ambiguities on reception. However, due to the technical difficulties and lack of noise generating apparatus, the noise radar concept was not quite realizable by radar engineers at that time. In the late 20th century, with the technological advances in microwave and digital electronics, there was a major shift from analog to digital and it became possible to generate random signals and perform correlation processing fully digitally. Over the past three decades, there has been a considerable interest in noise radar technology and significant progress has been made by research communities [[
Random signals are realizations of a random process either generated physically by an analog source or digitally by a pseudo random noise generator. Once a realization of this random process is employed as the modulating code, the radar knows what has been transmitted and so can detect its presence in the received signal via matched filtering. The theory of matched filter tells us that noise radars can have similar performance merits in terms of range and Doppler resolution as conventional radars, i.e., the range resolution depends on signal bandwidth, and the Doppler resolution depends on coherent integration time. Another incentive for researchers has been its appropriateness for covert radar as stealth requirements can be met by low power and long duty-cycle noise waveforms.
Noise radars have many advantages compared to conventional radars [[
- (i) It is impractical for an adversary to intercept and exploit the transmitted signal due to the unpredictable and random nature of noise signals.
- (ii) The mitigation of mutual interference among radars sharing the same spectral band due to the (quasi) orthogonality of signals.
- (iii) The thumbtack shape of its Ambiguity Function allows high resolution both in range and Doppler space.
Despite these advantages, noise radar comes at a cost of computational burden as it requires a large bandwidth for processing wideband noise waveforms on return and fast digitizers to provide a sufficient sampling rate compared to traditional FMCW radars.
Noise radars can operate either in pulsed or continuous wave (CW) mode depending on the application. However, the use of CW emissions best matches the LPI properties of the noise signal because of the absence of edge modulations in the signal. No matter which emission regime is employed, the choice of transmit waveform has a significant effect on the radar performance. Using a random waveform such as a bandpass Gaussian noise (zero–mean and unit-variance) provides limited dynamic range (as shown later in Section 1.3) in radar sensitivity due to its sidelobe level and degrades radar detection performance. Therefore, noise radars require synthetic waveforms tailored at will to provide adequate resolution, dynamic range and detection performance.
The objective of this article is to introduce the reader to a particular type of waveform which can improve the dynamic range while boosting LPI characteristics and ensuring orthogonality. The remainder of this article is organized as follows. In Section 1 (1.2–1.4), we introduce the reader to the particular continuous noise radar architecture that we are interested in and its related correlation processing implementation. In the ensuing parts, we address the dynamic range problem, correlation sidelobes and leakage effects, and Peak-to-Average-Power-Ratio (PAPR) considerations, and next, we provide a clear rationale for the usage of tailored noise waveforms. In Section 2, we present the steps of the novel Combined Spectral Shaping and Peak-to-Average Power Reduction (COSPAR) waveform generation method along with a cyclic PAPR reduction procedure. In Section 3, we provide a discussion on the range resolution and mainlobe broadening for COSPAR signals. Then, we show the mutual orthogonality of COSPAR signals in Section 4. Next, the Ambiguity Function and Doppler sensitivity of COSPAR signals are compared with those of a typical LFM signal in Section 5. After that, we propose the Spectral Kurtosis (SK) measure to assess the LPI characteristics of signals and compare the time–frequency distributions of COSPAR and LFM waveforms in Section 6. In Section 7, we report experimental results obtained from the field trials. Finally, the article ends with concluding remarks in Section 8.
Before we proceed to tailored noise waveform design, we find it helpful to introduce the reader to the general system architecture for noise radar technology. This will help the reader better understand the concept of noise radar technology and its associated signal processing.
Noise radar emits noise signals, which are generated either by analog noise sources (temperature-controlled resistors, Zener diodes, gas discharge tubes ...) or digital pseudo random noise generators. Since analog noise sources provide limited performance in terms of resolution and dynamic range, in this work we are solely interested in digital waveforms generated synthetically on computing devices such as computers, DSPs or FPGAs. Thus, we particularly refer to a typical noise radar architecture, shown in Figure 1, with collocated transmit and receive antennas. In this architecture, the waveform is generated or stored in a digital device and it is fed to the transmitter by the Digital-to-Analog Converter. It should be noted that analog filters and signal conditioning components are omitted in this simplified block diagram. Interested readers may refer to [[
In Figure 1, the complex baseband signal (noise waveform) is first generated and then up-sampled and digitally up-converted (complex-to-real conversion) to the radar intermediate frequency (IF) band. Next, the digital real IF band signal is driven into the Digital-to-Analog Converter (DAC) at DAC sampling rate
In this architecture, it is customary to generate the complex baseband signal at
The noise radar described here transmits and receives simultaneously and this leads to an unavoidable leakage into the receiver due to the coupling of collocated antennas. This leakage is very important in noise radar applications and as such it has an effect on the dynamic range, which will be discussed later in Section 1.2. Moreover, the noise radar utilizes a continuous wave (CW) emission, granting covertness with an absence of pulse edges (which can be easily detected by ESM receivers) in its emissions. The continuous emission leads to a continuous processing of return signals, necessitating fast and low latency signal processing in the receiver.
In noise radar, the fundamental way of calculating the cross-correlation is performed by matched filtering. This procedure is also called "pulse compression" as it compresses the signal energy into a much narrower correlation mainlobe than the duration of the signal. The matched filter is known as the optimal linear filter for maximizing the signal-to-noise ratio (SNR) when the received signal is contaminated by additive white noise [[
In signal processing, a matched filter is derived by correlating a template signal (delayed and possibly Doppler compensated) with an unknown signal to search for the existence of the template signal in the unknown signal. This is identical to convolving the received signal with a conjugated time-reversed version of the transmit signal. In this respect, we define the continuous time autocorrelation function (ACF)
(
In practice, considering the discrete time case, the waveform is both time and band limited. Therefore, from the waveform point of view, there could be two options for the correlation processing. One is the periodic correlation, which assumes the waveform is periodic and thus the correlation is calculated via cyclic convolution. The other one assumes the signal is aperiodic and therefore the correlation is calculated via linear convolution [[
For the ease of digital implementation, given the discrete sequence
(
where
A digital hardware implementation for a periodic correlator is shown as a block diagram in Figure 2. In the block diagram, all arithmetic operations including complex conjugation and complex multiplication are performed on digital hardware such as FPGA. The Hilbert Transform is implemented as an FIR filter structure and FFT/IFFTs are implemented by Radix-2 or Radix-4 decompositions for the calculation of DFT and IDFT, respectively. Such an example implementation is shown in [[
In Figure 2, we also illustrate the calculation of a periodic autocorrelation of a signal and show the power spectrum at each step during low IF sampling and Hilbert filtering. Here, we consider an exemplary analog IF band (200–300 MHz) signal is being injected to both ADCs where the sampling rate is
That being said, we note that all synthetic waveforms shall be generated in conjunction with its digital correlator, i.e., the samples of complex baseband signal should be generated at the sampling rate (e.g.,
Having shown the correlation processing implementation, we wish to discuss the dynamic range requirement for noise radars. There are two important factors affecting the receiver sensitivity, i.e., the choice of waveform and the system noise bandwidth.
In noise radar, when using standard bandpass Gaussian noise as a probing waveform, the correlation processing of noise signals of duration
(
These "random sidelobe modulations" cover the whole range swath and thus mask the visibility of weak targets with low radar cross section (RCS), i.e., not only nearby targets but also those at a great distance. This is also known as the "masking effect".
On the other hand, considering the unavoidable leakage signal due to antenna coupling, the correlation processing generates a target at zero distance, the range sidelobes of which roughly define the dynamic range lower bound of the noise radar. This leakage signal can be regarded as the largest signal (in terms of power) going into the receiver and therefore cross-correlation calculations are normalized with respect to the peak response of the leakage signal. Additionally, the receiver gain is adjusted carefully such that the leakage signal does not saturate the receiver.
Aside from the range sidelobes of the leakage signal, the cross-correlation of the reference signal with the internal noise of the receiver shall be taken into consideration. Assuming a receiver gain
(
where K is the Boltzmann constant,
(
where
(
Neglecting the quantization noise due to the fixed wordlength of ADCs, finally the noise floor after correlation processing is defined as:
(
Assuming a positive valued
(
As a comparison, the dynamic ranges are shown on one-sided correlation plots (up to lag index 4000) in Figure 3a,b for two cases where bandpass Gaussian noise and tailored noise signals having the same bandwidth and duration (
Considering a 100 mW (20 dBm) transmit power with an antenna isolation level of 90 dB gives a leakage power level of −70 dBm at the receiver antenna. Assuming a receiver bandwidth of 100 MHz and a system temperature of 273
For both cases, the cross-correlation of the reference signal and thermal noise would give us a mean noise floor of:
which is consistent with the histogram shown in Figure 3d. In the case of the Gaussian noise waveform, the dynamic range of the radar is limited by the range sidelobes of the transmit signal itself, which has a mean level of −42.14 dB, as shown in Figure 3c. On the other hand, when a tailored noise waveform is used such as a COSPAR waveform with −70 dB peak sidelobe level, then the range sidelobes are far lower than
Another important metric in noise radar is the Peak-to-Average Power Ratio (PAPR) of signals. The PAPR of radiated signal
(
where
Signals with constant amplitude envelope (termed as "unimodular") have a PAPR level of 1, which makes them ideally power efficient. Exemplary waveforms belonging to this group can be listed as LFM, Barker and Polyphase codes. Such signals can drive the power amplifiers near saturation, thereby allowing power maximization during transmission. Any deviation from the unitary amplitude level leads to an energy loss in the correlation mainlobe, i.e., lowering the peak response level. This can be regarded as
(
The
Aside from SNR loss, signals with non-constant envelope are vulnerable to nonlinear distortion effects of amplifiers such as clipping, which impairs the good correlation properties of radiated signals. Therefore, signals with large PAPR necessitate highly linear converters, and amplifiers with wide dynamic range thus bring additional costs for radar system designers.
Having shown the particular noise radar architecture and the fundamental periodic correlation processing scheme, with some important aspects related to noise radar technology, here we propose a noise waveform generation method for controlling sidelobes and an accompanying PAPR reduction technique for SNR maximization and linear amplifier operation. The noise waveform generation method described here is based on having a good spectral shape (e.g., a Taylor window function), giving an opportunity to control the sidelobes in its transformed domain.
It is well known by the Wiener–Khinchin theorem that the Power Spectral Density (PSD) of a signal is the Fourier transform of its Auto-Correlation Function (ACF). Thus, the PSD and ACF form a Fourier pair. Using this relation, if one can find a good spectral shape (square root of PSD) with desired ACF sidelobe properties, then adding an arbitrary phase spectrum to that spectral shape and taking the inverse Fourier transform leads to a single realization of a noise-like waveform. Using this technique, it is possible to generate infinitely many waveforms having the same PSD and ACF by randomizing the phase term for signal spectra at each realization. In this respect, we resort to the Taylor window function for the spectral shape synthesis and after each realization we apply our proposed PAPR reduction method. An illustration of the procedure is shown in Figure 5 and the steps of the COSPAR method are explained in the following subsections.
We begin the COSPAR procedure by constructing the spectral shape, which will provide us the desired correlation properties. For this purpose, we refer to window functions where designers can adjust the coefficients or window weights to make tradeoffs between mainlobe width, peak sidelobe level and sidelobe fall-off rate parameters.
Window functions are widely used in digital signal processing for controlling undesirable sidelobes or suppressing spectral leakage and Gibbs oscillations. Also called taper or apodization functions, researchers have developed various window functions throughout the years to achieve certain sidelobe properties. Thus, there exists an extensive literature on window functions and they are widely used in applications such as spectral analysis, beamforming and antenna design [[
In the COSPAR procedure, we resort to the Taylor window function in particular [[
- (i) Peak sidelobe level (PSL) can be specified according to desired correlation PSL dictated by the radar design.
- (ii) Near-in sidelobes can be specified for the desired range where constant sidelobe response is required.
- (iii) Tradeoff between mainlobe width and sidelobes does not have a substantial effect on the radar resolution. The mild broadening of the mainlobe is discussed in Section 3.
- (iv) The good roll-off factor and monotonic decrease in the sidelobes beyond the constant sidelobe zone allow the visibility of weak targets at far range.
In fact, Taylor windows are similar to the Chebyshev window, which is known to have the narrowest mainlobe for a given sidelobe level; however, the Chebyshev window comes at a cost such that flat equiripple sidelobes lead to edge discontinuity in the window function (in our case, the spectral shape), which is not desirable for LPI radar operation. Avoiding edge discontinuity, the Taylor window allows us to make tradeoffs between the sidelobe level and the mainlobe width.
Having discussed the good features of the Taylor window, we finally define a parameterized discrete Taylor window [[
- (i)
- (ii)
- (iii)
The
(
The cosine weights in Equation (
(
where
(
(
(
An exemplary discrete Taylor window with parameters
Having synthesized the spectral shape according to the desired level of correlation sidelobes in Step I, we have
(
where
An example will illustrate the COSPAR waveform generation procedure up to this step. Later in Section 2.2 we will show how to reduce the PAPR of the synthesized signal by optimizing phase samples of the signal spectrum.
Example 1
As a demonstration of the design procedure, let it be desired to create a −60 dB sidelobe level
Having specified the parameters, we synthesize our spectral shape using Equation (
Once we generate the Taylor-shaped magnitude spectrum, we append a random phase vector to the modulus of the spectrum and then take the IDFT, which leads to a single realization of the noise signal shown in Figure 7a. The Taylor-shaped power spectrum and the one-sided autocorrelation of the noise signal are shown in Figure 7b,c, respectively. We note that the signal has a PAPR = 9.13, which could be regarded considerably high for linear noise radar operation.
Post synthesis, the waveforms look like noise as they do not possess a deterministic structure neither in envelope nor in quadrature and in-phase components. Furthermore, randomizing the phase term of the modulus of the Taylor-shaped spectrum in Step II, let us generate an infinite number of distinct noise waveforms with different time–frequency distributions at each realization. However, in terms of power efficiency, the waveforms synthesized with this method are far lower than classical waveforms such as LFM or NLFM, which have constant modulus, i.e., unimodular envelope, over the extent of the signal. By experiment, signals generated via COSPAR without the PAPR reduction step tend to have a PAPR value around 8–11.
Now, the question is "how to reduce PAPR without affecting the PSD and ACF envelope of the signal?" Many PAPR reduction algorithms exist in the literature for wireless communication, especially for OFDM signals [[
In our method, we first initialize the vector with the sequence generated at Step II. Next, we apply a time domain constraint (i.e., clipping the overshooting sample) and then project it to the frequency domain whenever clipping occurs on a sample by sample basis. In the frequency domain, we apply the spectral constraint (i.e., force it to have the magnitude spectrum obtained in Step I while keeping the modified phase spectrum) and then we project it back to the time domain. The projection is performed via FFT/IFFT operations and this alternation is repeated for each overshooting sample in turn in the entire vector. This iteration continues until a desired stop criterion (e.g., PAPR value or PAPR improvement) is reached at the end of each pass over the entire vector. It should be noted that this cyclic iteration proceeds by one sample-by-one sample for each overshooting sample in order to minimize the regrowth of overshoot and to accelerate the convergence. However, the stop criterion is checked only once at each time the end of the sequence is reached. It is observed that the proposed procedure reduces PAPR at each pass over the entire frame and it converges to a desired PAR level. Consequently, nearly unitary (
Example 2
We initialize the PAPR reduction algorithm with
The final sequence shown in Figure 10 has the same power spectrum as of the initial sequence (see Figure 7a) and thus its ACF (see Figure 7b) has been preserved. It should be noted that when the waveform is up-sampled for the generation of an IF signal, the PAPR may increase by 2–3 dB [[
The complex baseband signal
(
where
Example 3
Using the COSPAR waveform synthesized in Step III (Example 2), we consider up-converting the baseband signal to the IF band (200 MHz–300 MHz) given the DAC sampling rate of 1200 MSPS. Thus, the oversampling factor is
In general, the radar range resolution
(
and the mainlobe broadening factor
(
where
Each COSPAR synthesis leads to a distinct realization of a noise signal and thereby makes it possible to generate a set of mutually orthogonal noise waveforms. When transmitted successively, these orthogonal waveforms resolve range ambiguities in the radar while boosting the LPI property. Moreover, it permits a plurality of radars working in the same area and sharing the same spectrum. In Figure 13, the cross-correlation of two distinct COSPAR signals with the same energy and
The Ambiguity Function (AF) of a signal is a representation of the signal's matched filter response against various time delays and Doppler shifts and it can be used for the evaluation of signal resolution in Doppler and range space and for assessing waveforms' Doppler tolerance [[
(
where
A comparison of COSPAR and typical LFM (chirp) signal AFs is shown in Figure 14. In this respect, the LFM waveform is Doppler tolerant such that the matched filter still outputs a peak when the return signal incurs a Doppler frequency shift despite some offset in the target's apparent range. This is called "range-Doppler coupling", and to identify the true range, most often an up and down chirp pair is used to average the detection offset from opposite slopes of each chirp AF [[
However, in a continuous wave noise radar, the waveform with appropriate integration time can still be Doppler tolerant for practical applications. For example, in the X-band (e.g., 9.3–9.4 GHz) marine radar case, the typical vessel velocity ranges between 0 and 30 m/s (0–60 kts), leading to a Doppler frequency shift
As for the temporal analysis, the noise-like properties of COSPAR signals are discussed in previous sections. From the spectral point of view, the COSPAR signal attains a deterministic spectral shape (e.g., Figure 7b) prescribed by the Taylor window function during its synthesis, which could be considered as a weakness in terms of Electronic Protection at first glance. However, unless the exact duration and bandwidth of the signal are known a priori, estimating the spectrum of the continuous emission COSPAR signal from the arbitrary capture of its transmissions is no easier than any other conventional LPI waveforms. Owing to the Taylor window function, there are no spectral discontinuities and its favorable smooth roll-off at band edges makes it a good candidate for LPI operation. Moreover, its strength lies in its time–frequency representation discussed herein.
Modern ESM systems are capable of generating time–frequency analysis maps apart from traditional spectral analysis methods [[
- Short Time Fourier Transform (STFT) [[
31 ]]. - Wigner–Ville Distribution (WVD) [[
32 ]]. - Choi–Williams Distribution (CWD) [[
33 ]].
Time–frequency plots provide a better insight into the signal than traditional methods. Thus, to evaluate and compare the time–frequency distribution of traditional LFM and COSPAR waveforms, the STFT and Wigner–Ville Distribution of these signals are shown in Figure 16a–d. Our first observation is that the energy of the COSPAR signal is well spread over the whole time–frequency plane, which makes signal analysis by the electronic reconnaissance system impractical. Therefore, it is believed that extracting signal features from the COSPAR signal analysis is a challenging task for ESM receivers. Then the question is "How can we say that the signal appears as noise and is suitable for LPI operation?"
In order to quantify the LPI properties, we refer to the Spectral Kurtosis (SK) measure in this work. Spectral Kurtosis is a statistical tool that can reveal non-stationary and non-Gaussian distribution in the frequency domain for each frequency component. A small value of Spectral Kurtosis for a frequency bin indicates stationary Gaussian noise is present, whereas a high positive value pinpoints the existence of a transient or structured signal at the corresponding frequency [[
The Spectral Kurtosis of a signal
(
where
(
where 〈
As shown in Figure 17, the distribution of frequency bins over time for the COSPAR signal tends to have a normal distribution, giving a Spectral Kurtosis value of nearly zero, indicating such a noise-like distribution (normal distribution) for all frequencies. On the other hand, the LFM signal possesses a high value of Spectral Kurtosis over the entire frequency band, which means the signal has a structured time–frequency support. Considering an ESM receiver, the COSPAR signal can be buried well under the ESM receiver's internal thermal noise, making it difficult for the ESM system to distinguish due to the signal's noise-like behavior.
In January 2021, marine trials were conducted in the shore-based test site of Turkish Naval Research Center Command (TNRCC) using the noise radar demonstrator developed by TNRCC. The demonstrator system conforms to the architecture shown in Figure 1. The demonstrator was set to work in X-band during trials and raw data from marine targets of opportunity were collected for offline analysis. A radar scan view of the scene generated using COSPAR recordings shows various ship targets at sea and land structures around the test site up to 8 km in Figure 18.
During these trials, the COSPAR signal (having a bandwidth
In this paper, the fundamental noise radar architecture and its correlation processing have been reviewed along with important performance aspects related to waveform choice for noise radars, and a novel noise waveform synthesis method (called "Combined Spectral Shaping and Peak-to-Average Power Reduction, COSPAR") including a cyclic PAPR reduction algorithm has been presented. Its main advantages are its low computational complexity and its simplicity for hardware implementation. The method allows parametric spectral shape synthesis such that designers can specify the Taylor window properties for spectral shape synthesis and adjust the corresponding ACF near-in sidelobes and peak sidelobe level accordingly.
The COSPAR waveforms are a special case within the general class of noise radar waveforms that while bearing a deterministic power spectral shape, every realization generates a different time domain signal, each having a distinct time–frequency distribution that makes them amenable for enhanced LPI operation. In this respect, these waveforms are non-repetitive and thus cannot be easily predicted or intercepted by ESM receivers.
COSPAR waveforms enjoy many valuable attributes, some of which can be summarized as follows:
- (i) COSPAR waveforms are noise-like. Although they possess a deterministic spectral shape, it has been shown that the signal energy is well distributed over their time–frequency plane. Signals which have structured time–frequency support or signals which concentrate energy within a certain band are easily intercepted at a great distance by the ESM receiver. In this respect, COSPAR waveforms possess good LPI attributes.
- (ii) Signal energy does not vary radically from pulse to pulse due to its pre-specified and fixed power spectrum. Therefore, target SNR fluctuation (due to inconstant transmit signal energy from pulse-to-pulse) is eliminated.
- (iii) Matched filter outputs are not affected by random range sidelobe modulation due to its pre-defined autocorrelation shape, and standard pulse compression radar signal processing techniques can be applied.
- (iv) Its Ambiguity Function tends to be thumbtack shape, which resolves range and Doppler ambiguities.
Moreover, the possibility to shape the underlying autocorrelation function by the parametric spectral shape synthesis (i.e., Taylor window) makes this type of waveform potentially useful for many sensing applications such as automotive radar, marine navigation radar, surveillance radar, SAR imaging and MIMO radar. Consequently, COSPAR signals represent a form of waveform diversity for noise radars offering degrees of freedom to control sidelobe level and crest factor that can be readily explored in many sensing measurements.
Graph: Figure 1 Noise radar block diagram.
Graph: Figure 2 Periodic correlation processing in spectral domain. (
Graph: Figure 3 Illustration of dynamic range in noise radar. (a) Dynamic range is limited by waveform sidelobes. (b) Dynamic range is limited by noise floor. (c) Histogram of Gaussian noise range sidelobes in Figure 3a. (d) Histogram of correlation noise floor in Figure 3a.
Graph: Figure 4 SNR loss versus PAPR.
DIAGRAM: Figure 5 Block diagram of COSPAR waveform synthesis method.
Graph: Figure 6 Taylor window and its power spectral density.
Graph: Figure 7 (a) COSPAR signal. (b) Normalized power spectral density of COSPAR signal. (c) Normalized one-sided autocorrelation of COSPAR signal.
Graph: Figure 8 Gerchberg–Saxton phase retrieval algorithm.
Graph: Figure 9 PAPR evolution over iterations.
Graph: Figure 10 The PAPR optimized COSPAR signal (PAPR = 1.5). The power spectrum and ACF of the signal omitted here are the same as in Figure 7b,c.
Graph: Figure 11 Power spectrum of IF band COSPAR signal (200–300 MHz).
Graph: Figure 12 Mainlobe broadening factor for various SLL and n¯ values.
Graph: Figure 13 Normalized cross-correlation and autocorrelation of two COSPAR realizations.
Graph: Figure 14 (a) Ambiguity Function of COSPAR signal. (b) Ambiguity Function of LFM signal.
Graph: Figure 15 (a) Normalized correlation functions under varying Doppler velocities for 0 m/s, 5 m/s, 10 m/s, 20 m/s, 30 m/s. (b) Correlation sidelobes adjacent to mainlobe and SNR loss due to Doppler mismatch.
Graph: Figure 16 (a) STFT of COSPAR signal. (b) STFT of LFM signal. (c) Pseudo Wigner–Ville Distribution of COSPAR signal. (d) Pseudo Wigner–Ville Distribution of LFM signal.
Graph: Figure 17 (a) Spectral Kurtosis of COSPAR signal. (b) Spectral Kurtosis of LFM signal.
Graph: Figure 18 A surveillance scan view obtained using COSPAR signal. Ship targets are encircled in white. The large ship target marked with dashed white circle is used for range profile analysis.
Graph: Figure 19 Comparison of range profiles of the ship target for Gaussian noise and COSPAR waveform. COSPAR provides more room for better target SNR by having lower sidelobes than Gaussian noise.
Conceptualization, K.S.; methodology, K.S.; software, K.S.; validation, K.S., G.G., and G.P.; formal analysis, G.G. and G.P.; investigation, G.G. and G.P.; writing—original draft preparation, K.S.; writing—review and editing, K.S., G.G. and G.P. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Not Applicable.
Not Applicable.
Not Applicable.
The authors declare no conflict of interest.
The following abbreviations are used in this manuscript:
ACF Auto Correlation Function CA Cyclic Algorithm COSPAR Combined Spectral Shape and Peak-to-Average-Power Ratio Reduction CW Continuous Wave CWD Choi–Williams Distribution DSP Digital Signal Processor DFT/IDFT Discrete Fourier Transform/Inverse DFT EA Electronic Attack EP Electronic Protection ESM Electronic Support Measures FFT/IFFT Fast Fourier Transform/Inverse FFT FMCW Frequency Modulated Continuous Wave FPGA Field Processing Gate Array LFM Linear Frequency Modulation MF Matched Filter NLFM Nonlinear Frequency Modulation PAPR Peak-to-Average Power Ratio PSD Power Spectral Density PSL Peak Sidelobe Level PR Phase Retrieval RCS Radar Cross Section SNR Signal-to-Noise Ratio STFT Short Time Fourier Transform WVD Wigner–Ville Distribution
We thank the Italian University Consortium for Telecommunications—CNIT for its contribution to the publication of this work.
By Kubilay Savci; Gaspare Galati and Gabriele Pavan
Reported by Author; Author; Author