LOW PROBABILITY OF INTERCEPT (LPI) RADAR SIGNAL

IDENTIFICATION TECHNIQUES

Contents:

Abstract 2

1. Introduction 3

2. Signal processing techniques 4

a. Filter Bank and Higher Order Statistics 4

b. Wigner Distribution 5

c. Quadrature mirror filter bank 7

d. Cyclo-stationary spectral analysis 8

3. LPI Signal data analysis 10

4. Results 19

a. Barker 5 19

b. Polyphase code 25

c. No modulation signal 30

d. Frank signal 32

5. Conclusions 35

6. References 36

1

Abstract:

Low Probability of Intercept (LPI) radar is a system that represents a confluence between Radar

and Electronic Support (ES) technology. The objective of LPI radar is clear, that is, to escape

detection by the ES receiver. Low probability of intercept (LPI) is that property of an emitter that

because of its low power, wide bandwidth, frequency variability, or other design attributes,

makes it difficult to be detected or identified by means of passive intercept devices such as radar

warning, electronic support and electronic intelligence receivers, In order to detect LPI radar

waveforms new signal processing techniques are required. Higher Order Spectral Analysis

algorithms are used to extract useful information from the input signal. This includes the use of

Bispectrum, Bicoherence and Trispectrum techniques. The images (2D plots) produced by the

above algorithms are unique for each LPI signal and serves as a signature.

2

Introduction:

Many users of radar today are specifying a Low Probability of Intercept (LPI) as an

important tactical requirement.

The term LPI is that property of a radar that because of its low power, wide bandwidth,

frequency variability, or other design attributes, makes it difficult to be detected by means of

passive intercept receiver devices such as electronic support (ES), radar warning receivers

(RWRs), or electronics intelligence (ELINT) receivers. It follows that the LPI radar attempts to

provide detection of targets at longer ranges than intercept receivers can accomplish detection of

the radar. The success of LPI radar is measured by how hard it is for the receiver to detect the

radar emission parameters.

The LPI requirement is in response to the increase in capability of modern intercept

receivers to detect and locate a radar emitter. In applications such as altimeters, tactical airborne

targeting, surveillance and navigation, the interception of the radar transmission can quickly lead

to electronic attack (or jamming). The LPI requirement is also in response to the pervasive threat

of being destroyed by precision guided munitions and Anti-Radiation Missiles (ARMs). The

denial of signal intercept protects these types of emitters from most known threats and is the

objective of having a low probability of intercept. Since LPI radars typically use wideband CW

signals that are difficult to intercept and/or identify, intercept receivers have a difficult time

using only power spectral analysis and must resort to more sophisticated signal processing

systems to extract the waveform parameters necessary to create the proper coherent jamming

response.

This paper compares four intercept receiver signal processing techniques to detect the

LPI radar waveform parameters. To test the four techniques, a variety of LPI CW waveforms

were generated with signal-to-noise ratios of 0 and -6 dB.

LPI waveforms generated include: FMCW, P1 through P4, Frank code, Costas hopping and

combined PSK/FSK. Signal processing techniques compared include (a) filter bank processing

with higher order statistics, (b) Wigner distribution, (c) Quadrature mirror filter banks and (d)

Cyclostationary processing.

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The need to detect transient signals arises in various applications, such as in communications,

underwater acoustics, and seismic surveillance. When the waveshape and the arrival time of the

signal is known, the optimal detector consists of a matched filter followed by a threshold circuit,

the threshold level of which is chosen to optimize a performance criterion. A number of

suboptimal methods for detection have appeared. Methods based on the short-time Fourier

transform, the Gabor transform, the wavelet transform, and nonlinear methods (such as the

Wigner—Ville distribution) perform well only for a given class of signals and high signal-to-

noise ratio (SNR) assumptions.

The matched filter method requires that both the waveshape and the arrival time be

known, but in practice this is not always possible. In some applications neither the waveshape

nor the arrival time of the transient signal is known; in such cases no matched filter can be

designed. An example of this would be a transient sound, such as a knocking sound or a hydro

acoustic sound pulse, that does not have a well-defined waveshape. Furthermore, the waveshape

from the same source may vary from event to event. It is difficult, therefore, to design a matched

filter for such transient sound signals.

1. Signal Processing Techniques

The signal processing techniques used to extract the LPI radar parameters are described

briefly below.

2.1 Filter Bank and Higher Order Statistics:

The filter bank and higher order statistic technique (shown in Figure 1 is based on the use

of a parallel array of filters and higher order statistics (cumulants). The objective of the filter

bank is to separate the input signal in small frequency bands, providing a complete time-

frequency description of the unknown signal. Then, each sub-band signal is treated individually

and is followed by a third-order estimator that helps suppress the noise. The parameters extracted

can then be used to create the proper jamming waveform to attack the radar [1].

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Figure 1

The main objective of this method is the detection of a transient signal s(n) from a single

recorded noisy signal x(n). We consider the signal plus noise model for the received signal, i.e.,

x(n) = s(n)+v(n)

The signal s(n) is a transient signal of unknown waveshape and arrival time, v(n) is an

additive zero-mean colored noise of unknown symmetric probability distribution.

We consider the following hypothesis test,

H0 : x(n) = v(n)

H1 : x(n) = s(n)+v(n)

The hypothesis H1 shows the presence of signal plus noise against the null hypothesis

H0, noise alone.

2.2 Wigner Distribution:

The Wigner distribution (WD) has been noted as one of the more useful time-frequency

analysis techniques for LPI waveform parameter extraction. The Wigner distribution W, ( t , w )

is defined as

where t is a time variable, w is a frequency variable, and * denotes the complex conjugate.

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The Wigner distribution is a two-dimensional function describing the frequency content

of a signal as a function of time, and possesses many interesting properties, such as:

1) It is always real

2) It possesses marginal distributions:

And

where X ( U ) is the Fourier transform of x( t ) ;

3) Its firstorder moments with respect to t and w give two important functions, the instantaneous

frequency and the group delay and

4) It does not suffer from interaction between time and frequency resolutions. It can be shown

that the windowed Wigner distribution can be represented as

where W ( t , w ) is called the pseudo-Wigner distribution, and W,(t, w ) and W,,.(t, w ) are the

Wigner distributions of the signal x ( t ) and the window w ( t ) , respectively.

The last equation indicates that W ( t , U ) is a convolution of two Wigner distributions in

the frequency domain. Therefore, only frequency resolution is affected by the windowing,

whereas the time resolution remains unchanged.

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The Wigner distribution (with cross-terms included) can reliably extract the waveform

parameters with only a moderate amount of processing needed to derive the kernel function.

2.3 Quadrature mirror filter bank

In many applications, a discrete-time signal x[n] is split into a number of subband signals

by means of an analysis filter bank. The subband signals are then processed. Finally, the

processed subband signals are combined by a synthesis filter bank resulting in an output

signal y[n].

If the subband signals are bandlimited to frequency ranges much smaller than that of the

original input signal x[n], they can be down-sampled before processing. Because of the lower

sampling rate, the processing of the down-sampled signals can be carried out more

efficiently.

After processing, these signals are then up-sampled before being combined by the

synthesis filter bank into a higher-rate signal. The combined structure is called a quadrature-

mirror filter (QMF) bank.

If the down-sampling and up-sampling factors are equal to or greater than the number of

bands of the filter bank, then the output y[n] can be made to retain some or all of the

characteristics of the input signal x[n] by choosing appropriately the filters in the structure. If

the up-sampling and down-sampling factors are equal to the number of bands, then the

structure is called a critically sampled filter bank. The most common application of this

scheme is in the efficient coding of a signal x[n].

Figure below shows the basic two-channel QMF bank-based subband codec (coder/decoder)

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Figure 2

The analysis filters H0(z)and H1(z) have typically a lowpass and highpass frequency

responses, respectively, with a cutoff at p/2 as shown below

Figure 3

2.4 Cyclo-stationary spectral analysis

Digitally modulated signals are cyclostationary. This means the probabilistic parameters

(mean value and correlation) vary in time with single or multiple periodicity. One property

that extends from this is that they have spectral correlation or the signal is correlated with

frequency-shifted versions of itself at certain frequency shifts.

The advantage in the analysis of LPI waveforms using cyclostationary modeling, is that

non-zero correlation is exhibited between certain frequency components when their

frequency separation is related to the periodicity of interest (e.g., the symbol rate or carrier

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frequency). The value of that frequency separation is referred to as the cycle frequency. The

spectral correlation properties of a signal are evident in a plot of the Spectral Correlation

Density (SCD) function.

Figure 4

There are two differing approaches to the treatment of cyclostationary processes. The

probabilistic approach is to view measurements as an instance of a stochastic process. As an

alternative, the deterministic approach is to view the measurements as a single time series,

from which a probability distribution can be defined as the fraction of time that events occurs

over the lifetime of the time series. In both approaches, the process or time series is said to be

cyclostationary if its associated probability distributions vary periodically with time.

However, in the deterministic time-series approach, there is an alternative but equivalent

definition: A time series that contains no additive finite-strength sine-wave components is

said to exhibit cyclostationarity if there exists some nonlinear transformation of the signal

that produces positive-strength additive sine wave components.

Cyclostationary spectral analysis is based on modeling the signal as a cyclostationary

process rather than a stationary process. A signal is cyclostationary of order n if and only if

one can find some nth order nonlinear transformation of the signal that will generate finite-

strangth additive sine wave components that result in spectral lines. Many useful

characteristics of LPI signals can be determined and are reflected in the cyclic

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autocorrelation function and the spectral correlation density, which form the basis for the

cyclic spectral analysis.

3. LPI Signal data analysis:

3.1 Graphical User Interface (GUI)

A GUI has been developed to analyze the LPI signals (at IF level of 160 MHz, Sampled

at 500 MHz) of 10uS duration (5000 samples) generated from the Signal Source / samples

generated through software.

The following signals are considered

Barker

Polyphase codes

Frank

Signal with no modulation

3.1.1 Bispectrum:

In the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear

interactions. The Fourier transform of the second-order cumulant, i.e., the autocorrelation

function, is the traditional power spectrum. The Fourier transform of C3(t1, t2) (third-order

cumulant-generating function) is called the bispectrum or bispectral density. Applying the

convolution theorem allows fast calculation of the bispectrum

B(f1,f2) = X * (f1 + f2).X(f1).X(f2).

They fall in the category of higher-order spectra, or polyspectra and provide supplementary

information to the power spectrum. The third order polyspectrum (bispectrum) is the easiest

to compute, and hence the most popular.

Second order statistics deals with the the mean value m and the variance. These are

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defined by the expectation value operation, where x is the result of a random process:

If x is a discrete time signal, the second order moment, also known as the autocorrelation

function (ACF), is defined as

Beside these moments higher order statistics provides higher order moments (m3,m4, ...)

and non linear combinations of higher order moments, known as cumulants (c1, c2, c3, ...).

3.1.2 Bicoherence

Bicoherence is a squared normalised version of the bispectrum. The bicoherence takes

values bounded between 0 and 1, which make it a convenient measure for quantifying the

extent of phase coupling in a signal. It is also known as bispectral coherency. The prefix bi-

in bispectrum and bicoherence refers not to two time series xt, yt but rather to two frequencies

of a single signal.

The difference with measuring coherence (coherence analysis is an extensively used

method to study the correlations in frequency domain, between two simultaneously measured

signals) is the need for both input and output measurements by estimating two auto-spectra

and one cross spectrum. On the other hand, bicoherence is an auto-quantity, i.e. it can be

computed from a single signal. The coherence function provides a quantification of

deviations from linearity in the system which lies between the input and output measurement

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sensors. The bicoherence measures the proportion of the signal energy at any bifrequency

that is quadratically phase coupled.

Second order statistics can’t handle non–linear processes very well. A familiar example is

the quadratic phase coupling (QPC) problem. If a signal x(n) has two harmonics at the

frequencies f1 and f2 and is filtered through a system with quadratic characteristics,

e.g. y(n) = [x(n)]2, y(n) has harmonics at the frequencies f1, f2, f1 + f2 and f1 −f2.

The two additional harmonics are called distortion harmonics, they are phase coupled to

the original harmonics. They can be detected using 3rd and 4th order cumulants.

Bicoherence is defined as

Multi channel signals appear when two or more sensors yield the same signals which

only differ in their time delay. With second order statistics we can’t detect this delay,

whereas higher order auto– and cross–cumulants can achieve that.

3.1.3 Tri spectrum:

The trispectrum is a statistic used to search for nonlinear interactions. The Fourier

transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional

power spectrum. The Fourier transform of C4 (t1, t2, t3) (fourth-order cumulant-generating

function) is called the trispectrum or trispectral density.

The trispectrum T(f1,f2,f3) falls into the category of higher-order spectra, or polyspectra,

and provides supplementary information to the power spectrum. The trispectrum is a three

dimensional construct.

The trispectrum has been used to investigate the domains of applicability of maximum

kurtosis phase estimation used in the deconvolution of seismic data to find layer structure.

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The trispectrum is the non-zero stationary support for the four-dimensional non-stationary

trispectrum.

An .m file (sig_pulse.m) is run to create this GUI as shown in fig.5, the functions of the

buttons are explained here. In this GUI the functions of Bispectrum, Bicoherence and

Trispectrum are modified such that they can handle a complete pulse.

Figure 5

a). Load

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This button allows us to load and plot the signal samples saved in a “*.dat” format in the

GUI. A file can be selected from the specified path as shown in figure 6. The plot can be

saved in the path of user’s choice from file menu.

Figure 6

b). Add Noise (dB)

A text box is provided under this to enter the Gaussian noise in dB. The noise level in dB

is added relative to the power level of the signal, i.e., the power level of the signal is

measured and then specified dB of noise is added.

This button allows us to add Gaussian noise that is entered and plots the resultant

waveform before and after adding noise. The first plot displays the actual signal. The second

sub-plot shows the signal with noise added after measuring the power level of the signal. The

data related this plot is used for the further computation by HOSA. The HOSA algorithms

use 256 point FFT

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c). Bispectrum, Bispec_noise

These two buttons computes and plots the bispectrum of the signal without noise and

with the noise specified respectively. The flow chart for Bispectrum computation is shown

figure.7.

Figure 7.

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START

SET DEFAULT PARAMETERS

fft length,nfft=256,

length of Rao gabor window=5,

CREATE 2D WINDOW

(Rao Gabor)

STOP

COMPUTE BISPECTRUM BY ACCUMULATING

TRIPLE PRODUCTS H(f1).H(f2).H*(f1+f2)

PERFORM FREQUENCY DOMAIN

SMOOTHING

DRAW CONTOUR PLOT OF

MAGNITUDE BISPECTRUM

d). Bicoherence, Bicoher_noise

These two buttons computes and plots the bicoherence of the signal without noise and with

noise respectively. The flow chart for Bicohorence computation is shown figure.8.

Figure 8

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SET DEFAULT PARAMETERS

fft length,nfft=256,

Hanning window=5,

samples for segment=8,overlap parameters=50

samples for segment=8,overlap parameters=50

samples for segment=8,overlap parameters=50

COMPUTATION ON INPUT DATA VECTOR/MATRIX

CREATE 2D WINDOW

(Hanning)

STOP

COMPUTE BICOHERENCE BY ACCUMULATING

TRIPLEPRODUCTS H(f1).H(f2).H*(f1+f2)

PERFORM FREQUENCY DOMAIN SMOOTHING

DRAW CONTOUR PLOT OF MAGNITUDE

BICOHERENCE

START

e). Trispectrum and Trispect_noise

These two buttons computes and plots 2D slice of the Trispectrum of the signal without

noise and with noise respectively. The flow chart for Trispectrum computation is shown

figure.9.

Figure 9

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START

SET DEFAULT PARAMETERS

FFT length, NFFT=512,

Fixed frequency=0

STOP

COMPUTE TRISPECTRUM BY

TRIPLE PRODUCTS H(f1).H(f2).H(f3).H*(f1+f2+f3)

DRAW CONTOUR PLOT OF MAGNITUDE

TRISPECTRUM

f). Bispec_corr_test, Bicoher_corr_test, Trispect_corr_test

These buttons computes the cross correlation coefficient between the HOSA matrices

(plots) of reference data signals and the HOSA matrix of the test signal to be identified for a

chosen HOSA technique (Bispectrum or Bicohorence or Trispectrum). Initially three

matrices were developed by applying HOSA techniques to reference data i.e., for all 9

signals as mentioned above. Then while testing we have to apply one of the HOSA technique

to the signal, then the extracted features are correlated with reference data and the type of

modulation is declared in a dialog box. It can be done in two ways, with and without noise.

Figure 10

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START

DATA INPUTS – BISPECTRUM/ BICOHERENCE /TRISPECTRUM RESULTS

The ref and actual : X[N] AND Y[N]

COMPUTATION OF MEAN OF X[N] AND Y[N]

STOP

COMPUTE CORRELATION SERIES WITH PARAMETERS maxdelay

COMPUTE CORRELATION COEFFICIENT

The correlation coefficient for the unknown signal is computed with respect to the

reference signal and displayed in MATLAB command window. The signal with maximum

coefficient value indicates the type of unknown signal. The flow chart for the computation of

correlation coefficient is shown in figure 10.

4. Results

Barker 5

Figure 11. Barker 5 signal (4 pulses)

19

Figure 12. Barker 5 – Bispectrum

Figure 13. Barker - 5 Bicoherence20

Figure 14. Barker-5 Trispectrum

Figure 15. Barker 5 (3dB Noise)21

Figure 16. Barker-5 Bispectrum with 3 dB Noise

Figure 17. Barker-5 Bicohorence with 3 dB Noise

22

Fig.18. Barker-5 Trispectrum with 3 dB Noise

Fig.19. Barker 5 ( -3 dB Noise)

23

Fig.20. Barker-5 Bispectrum with -3 dB Noise

Fig 21. Barker-5 Bicohorence with -3 dB Noise

24

Fig.22. Barker-5 Trispectrum with -3 dB Noise

P1 Signal( Poly phase code):

Fig.23. P1 Signal (4 pulses)

25

Fig.24. P1 Bispectrum

Fig.25. P1 Bicohorence

26

Fig.26. P1 Trispectrum

Fig.27. P1 with 3 dB Noise

27

Fig.28. P1 Bispectrum with 3 dB Noise

Fig.29. P1 Bicohorence with 3 dB Noise

28

Fig.30. P1 Trispectrum with 3 dB Noise

Fig.31. P1 Bispectrum with -3 dB Noise

29

Fig.32. P1 Trispectrum with -3 dB Noise

No Modulation Signal:

Fig.33. No-mod Signal (4 pulses)

30

Fig.34. No-mod Signal Bispectrum

Fig.35. No-mod Signal Bicohorence

31

Fig.36. No-mod Signal Trispectrum

Frank Signal:

Fig.37. Frank Signal (4 pulses)

32

Fig.38. Frank Signal Bispectrum

Fig.39. Frank Signal Bicohorence

33

Fig.40. Frank Signal Trispectrum

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Conclusions:

Higher order spectral analysis of the signals clearly distinguishes one signal from the

other.

Trispectrum computation is more complex than the others.

Bispectrum, bicoherence and tripectrum will distinguish the signal even in presence of

measured Gaussian noises of order upto -6db.while classifying signals effected by

Gaussian noise we have to go for all the three methods.

This system would be very efficient in detecting and classifying LPI radar signals.

The plots obtained in the above results can be used as signatures in identifying and

classifying signals at particular noise levels mentioned.

Current and future research in the field of higher-order spectra includes nonstationary and

cyclostationary signal analysis with HOS-based time-frequency representations, HOS-

based wavelet representations as well as utilization of higher-order statistics for the

analysis of chaotic signals.

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References:

[1] F. Sattar “On Detection using Filter Banks and Higher Order Statistics”, IEEE Transactions on Aerospace and

Electronic Systems, volume 36, No. 4, October 2000.

[2] A Wigner Spectral Analyzer for Nonstationary Signals - IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 38, NO. 5, OCTOBER 1989

[3] HOSA – Higher Order Spectra Analysis - G¨unther Hannesschl¨ager, Nov 24, 2004

[4] Monitoring Bicoherence - Steve Penn (HWS) & Vijay Chickarmane (LSU)

[5] Wikipedia - http://en.wikipedia.org/

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