LPI Radar Overview

Currently, many radar users are specifying Low Probability of Intercept (LPI) as an important tactical requirement [PAC09], [STO13]. The term LPI (whose meaning is not absolutely precise) [SCH06], [WIL06], is the property of a radar that, because of its low power, high duty cycle, ultra-low sidelobes, power management, wide bandwidth, frequency/phase modulation, and other design attributes, makes it difficult to be detected by means of intercept receivers, such as electronic support receivers, electronic intelligence receivers, and radar warning receivers [WSQ19]. The goal of the LPI radar is to detect targets at longer ranges than the intercept receiver can detect the LPI radar. It is important to note that defining a radar to be LPI necessitates defining the corresponding intercept receiver. That is, the success of an LPI radar is measured by how hard it is for the intercept receiver to detect and intercept the radar’s emissions.

One formal definition is as follows: A low probability of intercept (LPI) radar is defined as a radar that uses a special emitted waveform intended to prevent a non-cooperative intercept receiver from intercepting and detecting its emission [PAC09].

The LPI emitter has established itself as one of the premier tactical and strategic radars in the military spectrum. In addition to surveillance and navigation, the LPI emitter also operates in the time-critical domain for applications such as fire control and missile guidance [WIL06].

LPI Radar Characteristics

Some of the characteristics of the LPI radar are power management, ultra-low side lobes, and pulse compression.

Power management is the radar’s ability to control the power level so that it emits only the necessary power for detection of a target. An intercept receiver is used to seeing an increase in power as the radar approaches. If a power managed LPI radar decreases the power as it approaches the target, an intercept receiver may incorrectly assume that the radar is not approaching, thereby deciding that no response management is necessary, which could be a deadly decision [WIL06], [SHC19].

Ultra-low side lobes are used to prevent an intercept receiver from detecting radar emissions from the side lobes of the radar. Ultra-low side lobes are generally required to be -45dB or lower [SON22].

Pulse compression is a very important LPI radar characteristic. For frequency modulation LPI radars, the transmitted Continuous Wave (CW) signal is coded with a reference signal that spreads the transmitted energy in frequency, making it more difficult for an intercept receiver to detect and identify the LPI radar. The reference signal can be a linear frequency modulated signal or an Frequency Shift Keying (FSK) (frequency hopping). The most popular implementation has been the Frequency Modulated Continuous Wave (FMCW) [LIX23].

If the radar uses an FMCW waveform, the processing gain (apart from any noncoherent integration) is the sweep or modulation period $t_m$, multiplied by the sweep (input) bandwidth, $\mathrm{\Delta }F$ (see equation (1)). That is:

The LPI receiver compresses (correlates) the received signal from the target using the stored reference signal, for the purpose of performing target detection. The correlation receiver is a ‘matched receiver’ if the reference signal is exactly the same duration as the finite duration return signal [PAC09], [WIL06].

LPI Radar Waveforms

This section looks at the FMCW radar waveform.

FMCW is a signal that is frequently encountered in modern radar systems [VGS17]. The frequency modulation spreads the transmitted energy over a large modulation bandwidth $\mathrm{\Delta }F$, providing good range resolution that is critical for discriminating targets from clutter. The power spectrum of the FMCW signal is nearly rectangular over the modulation bandwidth, making non-cooperative interception very difficult. Since the transmit waveform is deterministic, the form of the return signals can be predicted. This gives it the added advantage of being resistant to interference (such as jamming), since any signal not matching this form can be suppressed [WIL06]. Consequently, it is difficult for an intercept receiver to detect the FMCW waveform and measure the parameters accurately enough to match the jammer waveform to the radar waveform [PAC09].

The most popular linear modulation utilized is the triangular modulated FMCW emitter, since it can measure the target’s range and Doppler [MIL18]. Triangular modulated FMCW is the waveforms that is utilized in this paper.

Detection of LPI Radars: Intercept Receiver Overview

In this section we switch from the topic of LPI radars, to the topic of those devices that detect and characterize LPI radar signals – which are the intercept receivers.

The three main types of intercept receivers are: electronic support receivers, electronic intelligence receivers, and radar warning receivers.

Electronic intelligence is the result of observing the signals transmitted by radar systems in order to acquire information about their capabilities: it is the remote sensing of remote sensors. Through electronic intelligence, it is possible to obtain valuable information while at the same time remaining remote from the radar itself. Identification is performed by comparing the intercepted signal signature against the signatures contained within its threat library [CLA13]. So, the underlying basic function of electronic intelligence is to determine the capabilities of the radar, so that decisions can be made as to what threat it poses [GRA11]. Electronic intelligence receivers are the least time critical of the three intercept receivers. The output of the electronic intelligence receiver can be analyzed through off-line software-based analysis tools.

Radar warning receivers are designed to give nearly immediate warning if specific threat signals are received (e.g., illumination of an aircraft’s warning receiver by the target tracking radar of a threatening system). The warning receiver typically has poor sensitivity and feeds into a near-real-time processor that uses a few parameter measurements to identify a threat. Usually, rough direction (e.g., quadrant or octant) is determined for the threat and the operator has a crude display showing functional radar type, direction, and relative range (strong signals displayed as being nearer than weaker ones). This type of receiver does not provide the kind of output that is analyzed using the methods described later in this paper.

Electronic support receivers encompass all actions necessary to provide the information required for immediate decisions involving electronic warfare operations, threat avoidance, targeting, and homing [ASC16], [WIL06].

Intercept Receiver Signal Analysis Techniques

This section describes some of the classical time-frequency analysis techniques as well as the reassignment method employed in this paper.

Time-Frequency Analysis

Time-frequency signal analysis concerns the analysis and processing of signals with time-varying frequency content. Such signals are best represented by a time-frequency distribution, which is intended to show how the energy of the signal is distributed over the two-dimensional time-frequency plane [ZML16]. Processing of the signal may then exploit the features produced by the concentration of signal energy in two dimensions (time and frequency), instead of only one-dimension (time or frequency) [BOA15]. Since noise tends to spread out evenly over the time-frequency domain, while signals concentrate their energies within limited time intervals and frequency bands; the local SNR of a noisy signal can be improved simply by using time-frequency analysis [BOA15]. Also, the intercept receiver can increase its processing gain by implementing time-frequency signal analysis [GHA20].

Time-frequency distributions are useful for the visual interpretation of signal dynamics, through which an experienced operator can quickly detect a signal and also extract the signal parameters by analyzing the time-frequency distribution [BOA15].

Spectrogram

The spectrogram is defined as the magnitude squared of the STFT [FLA15], [STS17]. For non-stationary signals, the STFT is usually in the form of the spectrogram [FLA15], [STS17].

The STFT of a signal $x(u)$ is given in equation (2) as:

Where $h(t)$ is a short time analysis window localized around $t=0$ and $f=0$. Because multiplication by the relatively short window $h(u-t)$ effectively suppresses the signal outside a neighborhood around the analysis point $u=t$, the STFT is a ‘local’ spectrum of the signal $x(u)$ around $t$. Think of the window $h(t)$ as sliding along the signal $x(u)$ and for each shift $h(u-t)$ we compute the usual Fourier transform of the product function $x(u)h(u-t)$. The observation window allows localization of the spectrum in time but also smears the spectrum in frequency in accordance with the uncertainty principle, leading to a trade-off between time resolution and frequency resolution. In general, if the window is short, the time resolution is good, but the frequency resolution is poor, and if the window is long, the frequency resolution is good, but the time resolution is poor.

The STFT was the first tool devised for analyzing a signal in both time and frequency simultaneously. For analysis of human speech, the main method was, and still is, the STFT. In general, the STFT is still the most widely used method for studying non-stationary signals [COH95].

The spectrogram (the squared modulus of the STFT) is given by equation (3) as:

The spectrogram is a real-valued and non-negative distribution. Since the window h of the STFT is assumed of unit energy, the spectrogram satisfies the global energy distribution property. Thus we can interpret the spectrogram as a measure of the energy of the signal contained in the time-frequency domain centered on the point (t, f) and whose shape is independent of this localization.

Here are some properties of the spectrogram: 1) time and frequency covariance - the spectrogram preserves time and frequency shifts, thus the spectrogram is an element of the class of quadratic time-frequency distributions that are covariant by translation in time and in frequency (i.e. Cohen’s class); 2) time-frequency resolution - the time-frequency resolution of the spectrogram is limited exactly as it is for the STFT; there is a trade-off between time resolution and frequency resolution. This poor resolution is the main drawback of this representation; 3) interference structure - as it is a quadratic (or bilinear) representation, the spectrogram of the sum of two signals is not the sum of the two spectrograms (quadratic superposition principle); there is a cross-spectrogram part and a real part. Thus, as for every quadratic distribution, the spectrogram presents interference terms; however, those interference terms are restricted to those regions of the time-frequency plane where the signals overlap. Thus if the signal components are sufficiently distant so that their spectrograms do not overlap significantly, then the interference term will nearly be identically zero [COH95].

The Reassignment Method

Bilinear time-frequency distributions offer a wide range of methods designed for the analysis of non-stationary signals. Nevertheless, a significant point of these methods is their readability , which means both a good concentration of the signal components along with few misleading interference terms. A lack of readability [ZHF22], which is a known deficiency in the classical time-frequency analysis techniques, must be overcome in order to obtain time-frequency distributions that can be both easily read by non-experts and easily included in a signal processing application [BOA15]. Inability to obtain readable time-frequency distributions can lead to inaccurate signal metrics extraction, which in turn can bring about an uninformed and therefore potentially unsafe intercept receiver environment.

Some efforts have been made in that direction, and in particular, a general methodology referred to as reassignment.

The original idea of reassignment was introduced in an attempt to improve the Spectrogram [MIJ18]. As with any other bilinear energy distribution, the Spectrogram, as mentioned before, is faced with an unavoidable trade-off between the reduction of misleading interference terms and a sharp localization of the signal components.

We can define the Spectrogram as a two-dimensional convolution of the Wigner-Ville Distribution (WVD) of the signal by the WVD of the analysis window, as in equation (4):

Therefore, the distribution reduces the interference terms of the signal’s WVD, but at the expense of time and frequency localization. However, a closer look at equation (4) shows that $W_h(t-s,f-\xi )$ delimits a time-frequency domain at the vicinity of the $(t,f)$ point, inside which a weighted average of the signal’s WVD values is performed. The key point of the reassignment principle is that these values have no reason to be symmetrically distributed around $(t,f)$, which is the geometrical center of this domain. Therefore, their average should not be assigned at this point, but rather at the center of gravity of this domain, which is much more representative of the local energy distribution of the signal. Reasoning with a mechanical analogy, the local energy distribution $W_h(t-s,f-\xi )W_x(s,\xi )$ (as a function of $s$ and $\xi$) can be considered as a mass distribution, and it is much more accurate to assign the total mass (i.e. the Spectrogram value) to the center of gravity of the domain rather than to its geometrical center. Another way to look at it is this: the total mass of an object is assigned to its geometrical center, an arbitrary point which except in the very specific case of a homogeneous distribution, has no reason to suit the actual distribution. A much more meaningful choice is to assign the total mass of an object, as well as the Spectrogram value, to the center of gravity of their respective distribution [BOA15], [FAC18].

This is exactly how the reassignment method proceeds: it moves each value of the Spectrogram computed at any point $(t,f)$ to another point $(\hat{t},\hat{f})$ which is the center of gravity of the signal energy distribution around $(t,f)$ (see equation (5) and (6)) [MIB16]:

and thus leads to a reassigned Spectrogram (equation (7)), whose value at any point $(t^{\prime },f^{\prime })$ is the sum of all the Spectrogram values reassigned to this point:

One of the most interesting properties of this new distribution is that it also uses the phase information of the STFT, and not only its squared modulus as in the Spectrogram. It uses this information from the phase spectrum to sharpen the amplitude estimates in time and in frequency. This can be seen from the following expressions of the reassignment operators:

where ${\mathrm{\Phi }}_x(t,f;h)$ is the phase of the STFT of $x$ : ${\mathrm{\Phi }}_x(t,f;h)=\arg (F_x(t,f;h))$. However, these expressions (equation (8) and (9)) do not lead to an efficient implementation, and have to be replaced by equation (10) (local group delay) and (11) (local instantaneous frequency):

where $T_h(t)=t\times h(t)$ and $D_h(t)=\frac{dh}{dt}(t)$. This leads to an efficient implementation for the Reassigned Spectrogram without explicitly computing the partial derivatives of phase. The Reassigned Spectrogram may thus be computed by using 3 STFTs, each having a different window (the window function h; the same window with a weighted time ramp t*h; and the derivative of the window function h with respect to time (dh/dt)). Reassigned Spectrograms are therefore very easy to implement, and do not require a drastic increase in computational complexity.

One of the most important properties of the reassignment method is that the application of the reassignment process to any distribution of Cohen’s class theoretically yields perfectly localized distributions for chirp signals, frequency tones, and impulses, since the WVD does so also. This is one of the reasons that the reassignment method is a good choice as a signal process analysis tool for analyzing LPI radar waveforms such as triangular modulated FMCW waveforms (which can be viewed as back-to-back chirps) and FSK waveforms (which can be viewed as frequency tones).

The reassignment method provides readability improvement. The components are much better localized and more concentrated. In order to rectify the classical time-frequency analysis deficiency of poor time-frequency localization, there needs to be a method that produces more concentrated distributions, which the reassignment method does. This squeezing quality of the reassignment method lead to improved readability - which leads to more accurate metrics extracted – which in turn, creates a more informed and safer intercept receiver environment.

The reassignment principle for the Spectrogram allows for a straight-forward extension of its use to other distributions as well [BOA15], [FAC18]. If we consider the general expression of a distribution of the Cohen’s class as a two-dimensional convolution of the WVD, as in equation (12):

replacing the particular smoothing kernel $W_h(u,\xi )$ by an arbitrary kernel $\mathrm{\Pi }(s,\xi )$ simply defines the reassignment of any member of Cohen’s class (equation (13) through (15)):

The resulting reassigned distributions efficiently combine a reduction of the interference terms provided by a well-adapted smoothing kernel and an increased concentration of the signal components achieved by the reassignment. In addition, the reassignment operators $\hat{t}(x;t,f)$ and $\hat{f}(x;t,f)$ are almost as easy to compute as for the Spectrogram [YUG19].

Again, for Cohen’s class, it can be shown that these modified distributions are also theoretically perfectly localized for chirps and impulses.

Once again, the smoothing and squeezing qualities of the reassignment method lead to improved readability, which in turn, leads to more accurate metrics extraction, which may create a more informed and safer intercept receiver environment.

The reassignment method utilized in this paper is the Reassigned Spectrogram.

Methodology

The methodologies detailed in this section describe the processes involved in obtaining and comparing metrics between the utilization of the Spectrogram and of the Reassigned Spectrogram time-frequency analysis techniques for the detection and characterization of low probability of intercept triangular modulated FMCW radar signals.

The tools used for this testing were: Matrix Laboratory (MATLAB) (version 8.3), Signal Processing Toolbox (version 6.21), Wavelet Toolbox (version 4.7), Image Processing Toolbox (version 7.2), and Time-Frequency Toolbox (version 1.0) (http://tftb.nongnu.org/).

All testing was accomplished on a laptop computer (HP EliteBook; Processor – AMD Ryzen 7 PRO 7730U with Radeon Graphics 2.00 GHz. Installed RAM – 64.0 GB; System Type – 64-bit operating system, x64-based processor).

Testing was performed for a triangular modulated FMCW waveform (parameters: sampling frequency=6KHz; carrier frequency=1.5KHz; modulation bandwidth=2400Hz; modulation period=.015sec). The waveform parameters were chosen for academic validation of signal processing techniques. Due to computer processing resources they were not meant to represent real-world values. The number of samples for each test was chosen to be 512, which seemed to be the optimum size for the laptop computer. Testing was performed at three different Signal-to-Noise (SNR) levels: 10dB, 0dB, and the lowest SNR at which the signal could be detected. The noise added was white Gaussian noise, which best reflects the thermal noise present in the IF section of an intercept receiver . Kaiser windowing was used, when windowing was applicable. 125 runs were performed for each test, for statistical purposes. The time-frequency analysis techniques used for these tasks were the Scalogram and the Reassigned Scalogram.

After each particular run of each test, metrics were extracted from the time-frequency representation. The different metrics extracted were as follows:

1) Percent detection: Percent of time the signal was detected - signal was declared a detection if any portion of each of the signal components (4 chirp components for triangular modulated FMCW) exceeded a set detection threshold (a certain percentage of the maximum intensity of the time-frequency representation).

Detection threshold percentages were determined based on visual detections of low SNR signals (lowest SNR at which the signal could be visually detected in the time-frequency representation) (see Figure [fig:1]).

(Note – using the methodology from Figure [fig:1], the detection threshold percentages were determined to be 60% for the Spectrogram, and 50% for the Reassigned Spectrogram).

For percent detection determination, these detection threshold values were included in the time-frequency plot algorithms so that the detection thresholds could be applied automatically during the plotting process. From the percent detection plot, the signal was declared a detection if any portion of each of the signal components was visible (see Figure [fig:2]).

2) Lowest detectable SNR: The lowest SNR level at which at least a portion of each of the signal components exceeded the set detection threshold listed in the percent detection section above.

For lowest detectable SNR determination, the detection threshold value was included in the time-frequency plot algorithms so that the detection threshold could be applied automatically during the plotting process. From the lowest detectable SNR plot, the signal was declared a detection if any portion of each of the signal components was visible. The lowest SNR level for which the signal was declared a detection is the lowest detectable SNR (see Figure [fig:3]).

3) Carrier frequency: The frequency corresponding to the maximum intensity of the time-frequency representation (see Figure [fig:4]).

4) Modulation bandwidth (modBW): Distance from highest frequency value of signal (at a manual measurement threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in Y-direction (frequency).

The manual measurement threshold of 20% maximum intensity was determined based on manual measurement of the modulation bandwidth of the signal in the time-frequency representation (based on 50 test runs for each time-frequency analysis technique, for each waveform). During each manual measurement, the max intensity of the high and low measuring points was recorded. The average of the max intensity values for these test runs, for each of the time-frequency analysis techniques, for each waveform, was 20%. This was adopted as the manual measurement threshold value and is representative of what is obtained when performing manual measurements. This manual measurement threshold of 20% maximum intensity was also adapted for determining the modulation period and the time-frequency localization (both are described below).

For modulation bandwidth determination, the manual measurement threshold of 20% maximum intensity was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the modulation bandwidth plot, the modulation bandwidth was manually measured (see Figure 5):

5) Modulation period (modPer): Distance from highest frequency value of signal (at a manual measurement threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in X-direction (time).

For modulation period determination, the manual measurement threshold of 20% maximum intensity was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the modulation period plot, the modulation period was manually measured:

6) Time-frequency localization: Measure of the thickness of a signal component (at the manual measurement threshold of 20% maximum intensity on each side of the component) – converted to % of entire X-Axis, and % of entire Y-Axis.

For time-frequency localization determination, the manual measurement threshold of 20% maximum intensity was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the time-frequency localization plot, the time-frequency localization was manually measured:

The data from all 125 runs for each test was used to produce the actual, error, and percent error for each of these metrics listed above.

The metrics from the Spectrogram were then compared to the metrics from the Reassigned Spectrogram. By and large, the Reassigned Spectrogram outperformed the Spectrogram, as will be shown in the results section.

Results

Table [tab:1] presents the overall test metrics for the two time-frequency analysis techniques used in this testing (Spectrogram versus Reassigned Spectrogram).

Overall test metrics (average percent error: carrier frequency, modulation bandwidth, modulation period; average: percent detection, lowest detectable snr, time-frequency localization (as a percent of x axis and y axis) for the two time-frequency analysis techniques (Spectrogram versus Reassigned Spectrogram).

From Table [tab:1], the Reassigned Spectrogram outperformed the Spectrogram in every metrics category.

(As noted above – using the methodology from Figure [fig:1], the detection threshold percentages were determined to be 60% for the Spectrogram, and 50% for the Reassigned Spectrogram. These values were automatically set for the plots in Figure [fig:9] below).

Figure [fig:9] shows comparative plots of the Spectrogram (left) vs. the Reassigned Spectrogram (right) (triangular modulated FMCW signal) at SNRs of 10dB (top), 0dB (middle), -3dB (bottom).

Comparative plots of the triangular modulated FMCW low probability of intercept radar signals (Spectrogram (left-hand side) vs. the Reassigned Spectrogram (right-hand side)). The SNR for the top row is 10, for the middle row is 0, and for the bottom row is $-3dB$. In general, the Reassigned Spectrogram signals appear more localized (‘thinner’) than do the Spectrogram signals. In addition, the Reassigned Spectrogram signals appear more readable than the Spectrogram signals at every SNR level.

Discussion

This section will elaborate on the results from the previous section.

From Table [tab:1], the Reassigned Spectrogram outperformed the Spectrogram in every category. For the Spectrogram, the poorer signal localization (‘thicker’ signal), when compared with the Reassigned Spectrogram’s ‘squeezing’ quality (see Figure [fig:9]), can be accounted for by the Reassigned Spectrogram outperforming the Spectrogram in the areas of average percent error of: modulation bandwidth (2.58% to 6.19%), modulation period (0.43% to 0.52%), carrier frequency (3.17% to 10.37%), as well as average time-frequency localization (x and y-direction (as a percent of x axis and y axis)) (x: 1.37% to 2.86%; y: 1.08% to 3.06%). Note that average percent detection and lowest detectable SNR are both based on visual detections in the time-frequency representation. Figure [fig:9] shows that the signals in the Reassigned Spectrogram plots are more readable than those in the Spectrogram plots, which accounts for the Reassigned Spectrogram’s better average percent detection (74.22% to 66.82%) and better average lowest detectable SNR (-3.34dB to -3.06dB). The Spectrogram might be used in a scenario where a ‘quick and dirty’ check is needed to see if a signal is present, without the need for overly accurate extraction of its parameters. The Reassigned Spectrogram might be used in a scenario where accurate parameters are needed, in a low SNR environment, requiring a very quick time frame.

Conclusions

Digital intercept receivers, whose main job is to detect and extract parameters from low probability of intercept radar signals, have moved away from Fourier-based analysis and towards classical time-frequency analysis techniques, such as the Spectrogram, and Reassigned Spectrogram, for the purpose of analyzing low probability of intercept radar signals. Based on the research performed for this paper it was shown that the Reassigned Spectrogram by-and-large outperformed the Spectrogram for analyzing these low probability of intercept radar signals - for reasons brought out in the discussion section above. More accurate characterization metrics could well translate into saved equipment and lives.

Future plans include analysis of additional low probability of intercept radar waveforms, using additional time-frequency analysis and reassignment method techniques.