Multifrequency OFDM SAR in Presence of Deception Jamming
© J. Schuerger and D. Garmatyuk. 2010
Received: 31 May 2009
Accepted: 12 January 2010
Published: 7 March 2010
Orthogonal frequency division multiplexing (OFDM) is considered in this paper from the perspective of usage in imaging radar scenarios with deception jamming. OFDM radar signals are inherently multifrequency waveforms, composed of a number of subbands which are orthogonal to each other. While being employed extensively in communications, OFDM has not found comparatively wide use in radar, and, particularly, in synthetic aperture radar (SAR) applications. In this paper, we aim to show the advantages of OFDM-coded radar signals with random subband composition when used in deception jamming scenarios. Two approaches to create a radar signal by the jammer are considered: instantaneous frequency (IF) estimator and digital-RF-memory- (DRFM-) based reproducer. In both cases, the jammer aims to create a copy of a valid target image via resending the radar signal at prescribed time intervals. Jammer signals are derived and used in SAR simulations with three types of signal models: OFDM, linear frequency modulated (LFM), and frequency-hopped (FH). Presented results include simulated peak side lobe (PSL) and peak cross-correlation values for random OFDM signals, as well as simulated SAR imagery with IF and DRFM jammers'-induced false targets.
Synthetic aperture radar (SAR) technology has been used since the 1960's for the purposes of imaging landscapes and seascapes—both for the civilian and military purposes . In the latter scenarios, the enemy may frequently use specific electronic countermeasures (ECM)  to introduce false imagery into the radar-collected data to prevent accurate battle scene assessment. Such methods of ECM are classified as deceptive . Deceptive ECM techniques—or deception jammers—operate by sensing incoming radar signals and reproducing them to the best of jammer's capabilities, then resending resultant pulses in a particular fashion, so as to hinder correct imaging of enemy targets. False Target Generation (FTG) is one such commonly used form of deception jamming. The replicated and delayed SAR waveform is transmitted at the next expected arrival of the radar signal and is seen as an actual target after image reconstruction. This type of FTG can be accomplished using a digital radio frequency memory (DRFM) repeat jammer . Another possible approach to FTG is to generate the replica waveforms by determining the instantaneous frequency (IF) of the incoming radar signal . The effectiveness of ECM can be degraded using electronic counter-countermeasures (ECCM) techniques [2, 4, 6, 7]. One of the most robust ECCM methods against deception jamming is pulse diversity of radar signals, for example, multi-tone phase modulation and slowly varying chirp rate of linear frequency modulated (LFM) chirps are explored in . Another method involves coding signals in such a fashion that a transmitted waveform at an arbitrary pulse repetition interval (PRI) will produce a low value of peak cross-correlation with the waveform at the previous PRI, thus severely limiting the effectiveness of deception jamming during the correlation process—one example of such an approach is random noise radar [5, 8–11]. Orthogonal frequency division multiplexing (OFDM) can also be employed in this fashion; it is currently being implemented in multiple commercial communications systems [12–14], however its applications to radar have been somewhat limited to this day [15–20]. Advances in sampling technology have made ultra-wideband (UWB) wave shaping a possibility for OFDM systems . In our paper we contrast multifrequency UWB SAR signals based on OFDM with several common types of wideband SAR waveforms with certain similarities to OFDM. LFM chirps [22, 23], while easily implemented and widely used for SAR processing due to the relatively simple and efficient range-Doppler algorithm, experience high susceptibility to jamming because of the linear nature of the IF. Frequency-hopping (FH) signals [24, 25], similarly to OFDM, change spectral composition at each PRI, thus limiting the effectiveness of DRFM jammers. However, because their IF is constant, if the hopping interval is known, this type of signal may also be affected by IF jamming. On the other hand, ultra-short Gaussian monopulses , while allowing for submeter resolution and exhibiting good multipath performance, are usually processed using a different technique, backprojection, which is computationally expensive and varies significantly from range-Doppler processing employed in our simulations, thus these UWB SAR signals are not considered in the paper.
This paper investigates the ECCM capabilities of an UWB OFDM signal and the benchmark LFM chirp and FH signals against an IF jammer and a DRFM repeat jammer. Section 2.1 describes the OFDM signal model, Section 2.2 explores peak side lobe (PSL) and peak cross-correlation performance of wideband OFDM radar pulses with random subband composition, and Section 2.3 describes signal characteristics for the benchmark waveforms. Sections 3.1 and 3.2 describe deception jammer models for an IF estimator and a DRFM repeat jammer, respectively; Section 3.3 discusses how target image reconstruction is performed. Section 4 presents the simulation results, while conclusions are offered in Section 5.
2. Signal Modeling and Characteristics
2.1. OFDM Signal Construction
The analog baseband OFDM radar signal is given as
where is the th data symbol, is the number of subcarriers, and is the signal duration. The signal is simply the sum of individual RF pulses known as subcarriers. Each subcarrier has a unique center frequency where is the frequency separation between each subcarrier. The individual spectra located at multiples of are known as subbands. Each th subcarrier has a corresponding subband which is described by a sinc function centered on . The subbands will overlap, but because of their orthogonality the peak of one subband will coincide with zeros for all other subbands. If we then sample we can obtain the following baseband discrete time signal expression:
2.2. Peak Side Lobe and Cross-Correlation Performance of Random OFDM Signals
Radar ambiguity function (AF) is an important tool for understanding performance of a waveform. Conventionally, narrowband form of AF is used and closed-form integral solution for analog cases is obtained before plotting the function . Such an approach, for example, was taken in  to plot AF of OFDM signal consisting of 8 subbands spread over 5 MHz bandwidth. However, UWB OFDM signal should be treated differently due to its wide bandwidth. The error resultant from using narrowband approximation for computation of wideband signal's AF is derived and discussed in , which uses conventional integral format of AF definition. In  a similar approach is used to derive and optimize a narrowband AF of OFDM radar signals with up to 7 orthogonal subbands. In this work we derive the discrete form of UWB OFDM radar signal AF, as information extraction in OFDM system is performed on a digital baseband waveform—that is, all analysis below is based upon the down-converted receiver signal vectors sampled at the prescribed rate.
Normalized point target return for any type of wideband radar signal can be written as(3)
where is the transmit signal, is roundtrip time delay. When target or radar platform (or both) is in motion, the roundtrip time delay is a function of target range where is point target's initial range and is the radial velocity, which is assumed to be constant during radar observation time; from ,
where is the velocity of light. Next, we need to convert (3) from general sample format into discrete time format; to translate sample indices into discrete time values we use , where is sampling interval, which is an inverse of D/A converter's sampling rate. This produces the expression for sampled transmit signal:
Then, substituting (5) into (3) into it, we express sampled OFDM received signal as function of time and target's radial velocity, as shown in (6):
Radar range profile reconstruction is performed via matched filtering. Instead of integral format, cross-correlation of sample-based data is a summation:
Calculating absolute value of (7) and squaring it we recognize the result as a discrete AF of UWB OFDM radar signal:
It is seen that with subband fill ratios above 50% and total number of subbands 128 or higher, OFDM signals exhibit better PSL performance than the benchmark same-bandwidth LFM pulse (more on the benchmark pulse construction is in Section 2.3 below). It is also evident from the plot that if PSL is desired to be 20 dB, a minimum number of subbands has to be 128 and a subband fill ratio greater than 80%. In Figure 4(b) "inverse" scenario refers to the simplest method for cross-correlation minimization, which is to generate two OFDM pulses with the following spectral domain properties,
Unfortunately, this method of signal generation is not optimal in practical applications, especially in jamming scenarios. Having only two unique signals dramatically increases the radar's susceptibility to deception jamming. Therefore, to penalize the jammer, it is essential that a radar system be capable of employing pulse diversity, while maintaining reasonable cross-correlation values. The ultimate pulse diversity can be achieved if the radar signal contains randomness, or, in the extreme case, bandlimited random noise can be used as a radar signal [5, 9–11]. Thus, if random OFDM subband distribution is assumed, it ensures transmission of a unique pulse at every PRI. As a comparison to other frequency-modulated schemes, in  it is noted, for example, that simultaneous minimization of AF everywhere except the point of origin (thus minimizing PSL) and XAF for every time-frequency point is challenging for frequency coded signals—signals with ideal AF (such as Costas arrays) will have poor CAF characteristics and vice versa—and the codes proposed by the authors of  exhibit maximum XAF peak of approximately 6 dB compared to the AF peak value. For simulated UWB OFDM signal this level is reached at approximately 30% of subband fill ratio and it improves with higher fill ratios. If the second pulse is constructed as described in (9) we can obtain even better performance, reaching 20 dB when the first pulse has nearly 90% subband fill ratio and, consequently, the second pulse has approximately 10% subband fill ratio. This, however, is a trade-off, as lowering the subband fill ratio for the second pulse to 10% will significantly degrade range resolution and ECCM characteristics of the radar, as discussed above.
2.3. Benchmark Signals Construction
An LFM chirp can be expressed as
where is the constant envelope used to equalize the energy, is the center frequency, is the modulation rate, and is the duration of the chirp. Basic analog LFM transmitter implementation initially requires a pulsed sinusoid waveform at to be amplified and passed through an up/down-chirp filter. Using a passive surface acoustic wave (SAW) chirp filter as in  greatly reduces hardware complexity and decreases power needed in the transmitter design. The chirped pulse is passed through a power amplifier (PA) and transmitted through the antenna. The constant envelope (CE) waveform of the LFM chirp gives it high tolerances against nonlinearities requiring less stringent constraints in the PA.
An FH radar signal is given as
The graphs exhibit the differences between the three types of signals from the perspective of an intercepting entity. It is easy to see that the LFM signal's time-frequency behavior not only can be exactly inferred, but it can also be predicted. The tolerances in choosing appropriate time window lengths and shapes and sampling frequency are very wide and it is intuitive that the clarity of the analysis will remain the same for a number of chirps—not just LFM, but also nonlinear FM chirps, such as quadratic, logarithmic, and so forth, Thus, qualitatively, FM chirp will require the least time for the intercepting jammer to analyze and reproduce the signal. The second signal—FH pulse—is admittedly more difficult to reproduce and predict, as its time-frequency representation does not follow any mathematical function. However, knowing the hop interval and starting point of the pulse we can choose the time window so that it coincides with the hop interval, providing for the graph shown in Figure 6. Locating -axis maxima within each time window then clearly shows that we can, indeed, recover time-frequency portrait of an FH signal—white dots overlaid on top of the spectrogram graph represent both the locations of maxima and the original values of hop frequencies in the FH signal. Of course, there are limitations to this perfect-case scenario: the interceptor is required to know the hop interval duration—which must be constant within a pulse—and oversampling of the received signal is required to ensure quality capture of the signal within each hop interval (in our case we collect 40 samples per hop interval of 8 ns). The third signal—OFDM pulse—is evidently the hardest to intercept and predict. In fact, its uniqueness is such that no amount of oversampling and no size of a fractional sample window will allow the interceptor to resolve the time-frequency characteristics of an OFDM signal precisely. This is due to the fact that the signal is inherently multifrequency.
3. Deception Jammer Model Implementation
3.1. Instantaneous Frequency (IF) Estimator
However, examples given in [29, 30] show that (13) may not be a suitable definition in all cases, particularly the case of multi-component signals. It was stated in  that (13) will give physically meaningful results only if the spectrum of the signal is symmetric about a center frequency. The UWB OFDM signal, LFM chirp, and FH signal all exhibit this characteristic and, therefore, (13) is sufficient for determining the IF of the waveforms.
which is the discrete derivative calculated by using the current and previous instantaneous phases along with the time sampling interval . The discrete waveform is delayed to give the signal a false range offset and stored in memory. At the next PRI the discrete waveform is sent through a D/A converter and transmitted in the final form,
The IF expressions for the LFM chirp, OFDM pulse and FH signal were derived in  and are given below:
where = is the instantaneous frequency deviation of the LFM chirp, and = . Comparing (16) to (17) we see that the IF generated by the jammer precisely matches the theoretically defined IF for the LFM chirp, whereas (16) and (18) will not match. It is important to note that the frequency hopping interval determines at any given time and is crucial in determining the IF of the FH signal—we assume that is known to the jammer.
3.2. DRFM-Based Architecture
where t d is the time delay of the signal.
DRFM jammer simulation will consist of copying and delaying the complete radar signal by a certain time period to introduce the false range offset. It is assumed that the jammer is capable of producing an exact replication of the intercepted radar signal.
3.3. Target Image Construction
Both range and cross-range profile reconstructions are achieved via matched filtering, but the domain cross-correlation is performed in is slow-time—as opposed to range reconstruction, performed in fast-time (terminology and approach are per ). Following derivation of spherical phase-modulated (PM) signal within limited synthesized aperture  representing return signal phase characteristic along the cross-range coordinate as shown in Figure 9, we express the single-frequency return signal in slow-time domain as
The dependence of (21) on radar position demands that the jammer have the ability to accurately predict the PRI to ensure pulse-to-pulse phase coherence between transmitted jammer signals—otherwise, the jammer and radar reference phase function will be poorly correlated and the false target will not appear in the resulting reconstructed image. The jammer echoed signal must have the form
where the is the reflectivity coefficient of the false target and , are the range and cross-range of the false target, respectively, as designated by the jammer. It is also evident from (22) that radar transmit position must be known to the jammer. Thus, the generated jammer signals for the OFDM, LFM chirp, and FH waveform are given, as in ,
4. False Image Simulation Results
A multifrequency UWB OFDM signal model was developed and compared with two common radar imaging transmit signals, the LFM chirp and FH signal. A discrete AF of an UWB OFDM signal with random subband distribution was used to obtain PSL and cross-correlation performance characteristics in radar imaging scenarios. It was established that for a 500 MHz-bandwidth OFDM signal the minimum number of subbands and subband fill ratio required to exceed the performance of a same-bandwidth LFM chirp were 128 and 80%, respectively. Random spectral composition of such a signal ensures strong ECCM capabilities in presence of deception jamming, at the resultant PSL and peak cross-correlation values of approximately 22 dB and 12 dB, respectively. IF and DRFM jammers were modeled and used to introduce false targets into the imaging area of the radar system at SJR 0 dB. The two jammer models tested pulse diversity of each transmit signal, specifically frequency diversity, and the ability of each signal to suppress jammer waveform effects during image reconstruction. The appearance of false targets in both jamming scenarios for the common LFM chirp clearly demonstrates the lack of ECCM capabilities. The frequency agility of both the OFDM and FH signals proved useful in the DRFM repeat jammer scenario. Although the jammer could produce replicas of the signals, the orthogonality of the waveforms from adjacent transmission intervals amounted to weak correlation between jammer and radar signals. If the IF jammer knows the hop interval of a FH signal, false target image introduction will result—however, due to the spectral structure of the OFDM signal, no false target images are found in OFDM SAR image simulations. These qualities make UWB OFDM waveforms with random subband distribution a viable choice for usage in SAR scenarios with deception jamming.
The authors are grateful to the anonymous reviewers, whose valuable and thorough suggestions have resulted in the much improved paper. They also wish to thank Dr. Jon Sjogren of AFOSR for project support and program reviews. This work was supported by the U.S. Air Force Office of Scientific Research under Grant FA9550-07-1-0297.
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