ABSTRACT ULTRA-WIDEBAND OFDM RADAR AND COMMUNICATION SYSTEM by Jonathan Paul Schuerger This paper examines the possibility of a dual use radar and communication system based on ultra-wideband orthogonal frequency division multiplexed (OFDM) waveforms. Theory for utilizing OFDM as radar signals is developed and analyzed. Signal performance is examined for a synthetic aperture radar imaging scenario and limitations on system resolution are considered. OFDM radar waveform electronic counter counter-measure (ECCM) capabilities are assessed for hostile environments and compared with several benchmark radar signals. Finally, the dual use system is implemented and experimental results for both radar and communication modes are presented in detail.
Transcript
ABSTRACT
ULTRA-WIDEBAND OFDM RADAR AND COMMUNICATION SYSTEM
by Jonathan Paul Schuerger
This paper examines the possibility of a dual use radar and communication systembased on ultra-wideband orthogonal frequency division multiplexed (OFDM)waveforms. Theory for utilizing OFDM as radar signals is developed and analyzed.Signal performance is examined for a synthetic aperture radar imaging scenario andlimitations on system resolution are considered. OFDM radar waveform electroniccounter counter-measure (ECCM) capabilities are assessed for hostile environments andcompared with several benchmark radar signals. Finally, the dual use system isimplemented and experimental results for both radar and communication modes arepresented in detail.
2.4 Model for SAR cross-range imaging. . . . . . . . . . . . . . . . . . . . . . 92.5 Cross range imaging in the synthetic aperture domain: (a) reference phase
history s0(ω, u); (b) received phase history sr(ω, u) for a target at yn =−10; (c) cross-range target function f(y). . . . . . . . . . . . . . . . . . . 11
2.6 Cross range imaging in the Doppler domain: (a) reference phase historyS0(ω, ku); (b) received phase history Sr(ω, ku) for a target at yn = −10;(c) cross-range target function f(y). . . . . . . . . . . . . . . . . . . . . . 13
3.1 OFDM signal spectrum S(f) for N sub-carriers. . . . . . . . . . . . . . . 15
4.1 OFDM signal and corresponding spectrum: (a) signal w/ all sub-bandson (b) spectrum w/ all sub-bands on ; (c) signal w/ one sub-band on; (d)spectrum w/ one sub-band on; (e) signal w/ random ratio of sub-bandson; (f) spectrum w/ random ratio of sub-bands on. . . . . . . . . . . . . 19
5.6 Simulated reconstructed images with jammer false targets: (a) OFDMsignal with IF jammer; (b) OFDM signal with DRFM jammer; (c) LFMchirp with IF jammer; (d) LFM chirp with DRFM jammer; (e) FH signalwith IF jammer; (f) FH signal with DRFM jammer. . . . . . . . . . . . . 42
5.8 Probability density functions for rACmax when using: (a) ternary vectorpopulation; (b) real number vector population. . . . . . . . . . . . . . . . 47
5.9 AC and XC probability density functions with a varying number of sub-bands using: (a) ternary vector population; (b) random real number vectorpopulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.10 Magnified plot of AC and XC probability density functions with a varyingnumber of sub-bands using: (a) ternary vector population; (b) random realnumber vector population. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.11 Using ternary vector population: (a) probability of detection; (b) proba-bility of false alarm; (c) magnified probability of detection; (d) magnifiedprobability of false alarm. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.12 Using real number vector population: (a) probability of detection; (b)probability of false alarm; (c) magnified probability of detection; (d) mag-nified probability of false alarm. . . . . . . . . . . . . . . . . . . . . . . . 51
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5.13 Threshold effect under jammer assumption with NO jammer: (a) proba-bility of detection; (b) probability of false alarm. . . . . . . . . . . . . . . 52
5.14 Threshold effect under NO jammer assumption with jammer: (a) proba-bility of detection; (b) probability of false alarm. . . . . . . . . . . . . . . 52
6.1 OFDM SDRC system block diagram. . . . . . . . . . . . . . . . . . . . . 556.2 Analog front end: (a) picture; (b) component list. . . . . . . . . . . . . . 566.3 Analog front end transmit and receive antennas: (a) picture; (b) typical
6.7 OFDM symbol recovery: (a) received raw data; (b) received filtered data;(c) transmitted symbol spectrum; (d) received symbol spectrum (red) over-layed with transmitted signal spectrum (black). . . . . . . . . . . . . . . 65
6.8 Sub-band configuration performance: (a) bit error ratio plot (b) imagequality, 57Mb/s (left), 79Mb/s (middle), and 97Mb/s (right). . . . . . . . 66
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Chapter 1
Introduction
Radar technology has been used in military applications since World War II to aid inthe detection of targets of interest [1, 2, 3]. In fact, it led to the eventual Allied vic-tory more so than the atomic bomb [4]. Early radar systems, however, were limitedto providing only distance (range) information of a target. Radar imaging theory onlybegan to surface in the mid 1960’s and grew into an important area of research in theearly 1970’s. Improved solid state electronics and processing methods have led to theradar imaging systems utilized today. Other types of imaging sensors such as optical andladar currently exist, however, in some scenarios they must have the ability to work indynamic, unfriendly environments. In reconnaissance and surveillance operations enemytargets may be purposefully or naturally masked in foliage. Other uncontrollable envi-ronmental conditions including rain, fog, clouds, and darkness can also hinder imagingcapability. Unlike optical and ladar imagers, radars use radio frequency (RF) signalswhich are immune to these types of environmental limitations making them a reliable,attractive option for use in combat scenarios.
Recently, combat situations have begun shifting from open battlefields to urban ter-rains causing reconnaissance and surveillance to become a much more challenging task.Very detailed enemy target position must be known to reduce civilian casualties and limitdamage to infrastructure (roads, buildings, houses, etc.) thus requiring the radar imagingsensors to produce high-resolution (HR) images. Synthetic aperture radar (SAR) imagingis commonly used when detailed information about a target area is required. It involvestransmitting signals at spaced intervals called pulse repetition intervals (PRI). The re-sponses at each PRI are collected and used to reconstruct a radar image of the terrain. Ingeneral, HR SAR images are generated by using ultra-wideband (UWB) waveforms forradar signal transmission. Often, wide bandwidths of radar waveforms were formed byemploying ultra-short pulses, such as Gaussian pulses. However, having constant pulseshape, these radars may be susceptible to certain types of deception jamming in scenarioswhere multiple pulse transmissions are mandatory, as is the case in SAR imaging. Decep-
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tion jammers attempt to manipulate transmitted radar signals in order to introduce falsetargets into the reconstructed HR SAR image [19]. Constant pulse shape also precludesusing these radars in situations where they could interfere with each other. Therefore, itis necessary to develop radar signals that employ wide bandwidths and at the same timehave the ability to quickly adapt to any potential adverse situations.
Orthogonal frequency division multiplexing (ODFM), a signal modulation schemecommonly utilized in commercial communications, shows great potential for use as arobust radar waveform. An OFDM signal is comprised of several orthogonal sub-carriersthat are simultaneously emitted over a single transmission path. Each sub-carrier occupiesa small slice of the entire signal bandwidth. Essentially, the sub-carriers partition thebandwidth of the signal into several smaller orthogonal blocks. The system has controlover the sub-carriers and, therefore, control over the spectrum of the waveform. Inradar scenarios, the spectrum can be manipulated to avoid any interference that maybe introduced from hostile and/or friendly systems. Because the signal is implementeddigitally, the bandwidth of the signal will be determined by the sampling rate of thedigit-to-analog converter. Recent advances in technology have increased sampling speedsallowing for OFDM waveforms employing ultra-wide bandwidths (500MHz and above) tobe generated accurately and at relatively lost costs. The result is a diverse signal capableof HR imaging.
The fast progression of technology has also made complex systems integration, suchas real-time sensor networks, a very real possibility. These networks are comprised ofseveral individual sensor platforms that must be able to collect and process data whilesimultaneously communicating the information to the other platforms in the network.This requires use of a dual mode signal, making OFDM waveforms ideal for this type ofscenario. OFDM signals utilized in these systems would have the ability to operate inthree separate modes; communication only, radar only and dual radar/communication.The communication only mode involves transmitting the OFDM signal directly to otherplatforms in the network. Standard OFDM data modulation techniques used in commoncommunication systems are employed where data rate optimization is the main priority.Alternatively, in when the system operates in the radar only mode pulse diversity isof key interest. Signals are formed in an attempt to eliminate all forms jamming andinterference. Dual mode operation requires a symbiotic relationship between radar andcommunication. Necessary trade offs must be made to ensure the system can adequatelyperform both processes. Depending on the situation, this may require a reduction in thedata rate, image resolution, and interference mitigation capabilities.
The purpose of this paper is to show that the well known OFDM communication signalcan also be used as a high resolution radar signal in a dual use system. The paper isfocused on use of OFDM for synthetic aperture radar (SAR) imaging purposes since muchis already known about OFDM for communication. However, OFDM communication
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theory will be briefly discussed and system communication experimental results will beanalyzed. Chapter 2 presents a detailed discussion on stripmap SAR imaging. Methods ofimage development and reconstruction used in later chapters are established. Chapter 3gives a brief description of OFDM waveform time domain and spectral properties. Also,basic communication methods and issues are discussed. Chapter 4 examines the theorybehind using OFDM as a radar signal. Software signal implementation is defined andapplied to standard radar signal analyzation techniques. Simulated SAR images arethen reconstructed for several imaging scenarios using OFDM radar signals and methodsdeveloped in Chapter 2. OFDM waveform electronic counter counter-measure (ECCM)capabilities are examined in Chapter 5 and compared to three benchmark radar signals.Statistical analysis in a jamming scenario with jamming penalization is also provided.Chapter 6 discusses the system implementation down to individual component properties.Radar and communication mode experimental results are presented and system resolutionand maximum data rate are obtained. Finally, Conclusions a made and future works areput forth.
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Chapter 2
Stripmap SAR Imaging
The principle idea behind SAR stems from the desire for high resolution images. Whilerange resolution is determined by the bandwidth of the radar signal (discussed in sec-tion 2.1), lateral resolution depends on the physical size of the antenna aperture. Increas-ing aperture size illuminates more of the surrounding scene with signals resulting in ahigher number of reflected signals. Each reflected signal carries with it information aboutthe scene. Therefore, increasing the number of reflected signals increases the amount ofinformation about the scene which in turn allows for higher resolution. This is similarto the way a larger lens increases resolution in a vision system. Unfortunately, high res-olution requires an impractical physical aperture size. An ingenious idea was developedto circumvent this issue. Instead of illuminating a scene with a stationary large apertureantenna, a much smaller aperture antenna would be moved along an imaginary aperturethe effect of which produces the same results utilizing a much more feasible physicalaperture. This is referred to as a synthetic aperture.
An imaging technique known as Stripmap SAR is most often used when generalinformation of a broad target area is needed, such as in reconnaissance and surveillance.The stripmap SAR model is shown in figure 2.1. In this approach a radar platform movesalong the synthetic aperture axis (i.e. parallel to the cross-range axis) transmittinga pulse at the equally spaced intervals u−L/2, u−L/2+1, . . . , uL/2−1, uL/2, where L is thelength of the synthetic aperture and is determined by the size of the target area andthe radar pulse repetition interval (PRI). The antenna radiation pattern is perpendicularto the movement of the platform and is fixed during the entire data collection process.At each transmission interval a different cross-range area is illuminated while the rangeinterval or swath remains fixed. Subsequent stripmap SAR data processing and imagingis then performed utilizing the theory of wavefront reconstruction introduced in [5]. Therange imaging, cross-range imaging, and image reconstruction algorithms presented inthis chapter sharply follow those given in [6]. It should be noted that signal samplingproduces a discrete waveform and, therefore, the continuous target area becomes a set of
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Figure 2.1: System model for stripmap SAR imaging.
discrete range bins xb = [x1, x2, . . . , xN ] and cross-range bins yb = [y−N/2, y−N/2+1, . . . , yN/2]as illustrated in figure 2.1. Although the number of range bins and cross-range bins areequal in this case, it is not necessary. The resolution of xb and yb depend on severalfactors and will be discussed in section 2.1 and section 2.2 respectively.
2.1 Range Imaging
The range in SAR imaging is defined as the distance of the target perpendicular to themotion of the radar platform. The two dimensional range imaging system model is shown
Figure 2.2: Model for SAR range imaging.
in figure 2.2. For simplicity, a fixed cross-range is assumed. Each target within the
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antenna radiation pattern (beamwidth) will have a range x and a reflectivity σ associatedwith it so that x1, x2, x3, . . . , xN and σ1, σ2, σ3, . . . , σN correspond to range and reflectivityfor N targets present in the beamwidth. It is then useful to construct a range domainfunction for these N targets that can be described as,
f0(x) =N∑n=1
σnδ(x− xn) (2.1)
This function represents the ideal response from the N targets and is known as the idealrange function. If a transmitted radar pulse p(t) is illuminated on a target area with Ntargets then the received (echoed) signal will be the sum of the individual pulse responsefrom each target and has the form,
sr(t) =N∑n=1
σnp(t− tn) (2.2)
where tn = 2xnc
is the round trip delay, c being the speed of light. Observing equation (2.2)we see that the received signal is simply the sum of transmitted pulses delayed tn cor-responding to the time delay introduced by the target at range xn. The echoed signalin reality would take on an integral form as there are an infinite number of reflectors inthe target area. However, imaging algorithms will be carried out in the digital domainrequiring the use of a discrete Fourier transform (DFT) and so the discrete model is used.The Fourier transform of the echoed signal is,
Sr(ω) = P (ω)N∑n=1
σne−ω 2xn
c (2.3)
where P (ω) is the spectrum of the transmitted signal p(t) and the substitution for tn hasbeen made. By re-arranging equation (2.3) and taking the inverse Fourier transform wecan recover the ideal range function,
F−1(ω)
[Sr(ω)P (ω)
]= f0(x) (2.4)
remembering that f0(x) carries range and reflectivity information of the target area.Equation (2.4) is only applicable if the radar signal has infinite bandwidth, in reality asignal will have a finite bandwidth therefore a practical range recovery technique knownas matched filtering is performed. The matched filter can be described as,
sMx(t) = F−1(ω) [Sr(ω)P ∗(ω)]
= F−1(ω)
[N∑n=1
σn |P (ω)|2 e−ω2xnc
](2.5)
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where,psf(t) = F−1
(ω)
[|P (ω)|2
](2.6)
is called the point spread function from which the time domain representation of sM(t)can be formed as,
sMx(t) =N∑n=1
σnpsf(t− tn) (2.7)
where, again, tn has been substituted. Equation (2.7) appears similar to equation (2.1)and in fact if we substitute the range domain point spread function psf(2
c(x − xn)) for
the time domain point spread function psf(t− tn) we get,
sMx(t) =N∑n=1
σnpsf[2c
(x− xn)]
= f(x) (2.8)
where f(x) is the actual range function. The finite bandwidth of the signal producespsf(x − xn) and the target can therefore no longer be represented by δ(x− xn). Weknow from properties of Fourier analysis that widening signal bandwidth corresponds toa compression in time. From equation (2.8) we observe that compression of the pointspread function will result in a more resolved target range (i.e. the wider the bandwidththe closer the psf comes to a delta function). We can then conclude that transmitting awide bandwidth radar signal will result in an improved range resolution. For confirmationlet us consider the simple case of the sinusoidal pulse p(t) = rect( t
Tp)cosωt, where Tp is
the pulse duration determined by the rectangle function and rect( tTp
) = 1 for 0 < t < Tp
and zero otherwise. Figure 2.3(a) shows transmitted pulses for Tp = 1µs (top) andTp = 4µs (bottom) where ω = 2πf , f = 200MHz for both. The spectrum of the pulsewill be a sinc function with a width dependent on Tp and shifted by f as shown infigure 2.3(b). We can also observe from figure 2.3(b) that time domain pulse compressiondoes, in fact, widen the bandwidth as expected. Making use of equation (2.6) we obtainthe corresponding PSF for both pulses as shown in figure 2.3(c). Figure 2.3(d) showsthe range function resulting from the summation of point spread functions for four idealtarget reflections. When Tp = 1µs (top) all four target ranges are resolvable, however,for Tp = 4µs (bottom) it is only possible to discern the range for one of four targets thussubstantiating our previously derived relationship between bandwidth and resolution.Determining range bin resolution is relatively straight forward. If we say Tinit = 0 is theinitial sampling time and ∆t is the time sampling interval then,
∆x = c∆ts2 (2.9)
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(a) (b)
(c) (d)
Figure 2.3: Range resolution for pulsed sinusoid for tp = 1µs (top) and tp = 4µs (bot-tom): (a) time domain signals; (b) corresponding spectra; (c) corresponding PSF; (d)corresponding range function.
and,xi = Tinit + (i− 1)∆ts, i = 1, 2, 3, . . . , N (2.10)
where xi is the range value of the ith bin.
2.2 Cross-range Imaging
The cross-range in SAR imaging is defined as the target distance parallel to the movementof the radar platform. In other words, the platform moves along the cross-range axisduring the data collection process. The system model for cross-range imaging is shownin figure 2.4. Similar to range imaging, each target in the antenna radiation pattern willhave a cross-range y and reflectivity σ associated with it, so that y1, y2, y3, . . . , yN andσ1, σ2, σ3, . . . , σN correspond to the cross-range and reflectivity for N targets present inthe beamwidth. Referring to figure 2.1, the cross-range y = 0 is defined as the center ofthe target area (i.e. center of the cross-range axis). Any targets residing above y = 0 are
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Figure 2.4: Model for SAR cross-range imaging.
said to have a negative cross-range and vice versa for targets below y = 0. The reasonfor defining cross-range in this manner will become clear when deriving the phase historys(ω, u). The distance of the target from the platform in the cross-range domain is notonly dependent upon target position (xn, yn) but also the synthetic aperture position u.We want to attain a cross-range target function f(y) just as we obtained the range targetfunction f(x), however, we will go about it in a slightly different manner. The receivedechoed signal is the sum of individual pulse responses from each target and has the form,
sr(t, u) =N∑n=1
σnp(t− tdn) (2.11)
where tdn = 2c
√x2n + (yn − u)2 is the distance of each target (xn, yn) from the platform
when the radar is at synthetic aperture position u. If we consider transmitted radar pulsep(t) = ejωt then equation (2.11) becomes,
sr(t, u) = ejωtN∑n=1
σne−j2ktdn (2.12)
where k = ω/c is the wavenumber. Multiplying equation (2.12) by e−jωt we convertsr(t, u) to the baseband representation and obtain the expression for the phase history
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as,
sr(ω, u) =N∑n=1
σne−j2ktdn
=N∑n=1
σne−j2ω
c
√x2n+(yn−u)2
=N∑n=1
σne−jφn(tdn) (2.13)
where the functional dependencies are absorbed into φn(tdn). The cross-range targetfunction can be obtained via synthetic aperture domain matched filtering and is expressedas,
sMy(ω, u) = sr(ω, u) ∗ s0(ω,−u)
= f(y) (2.14)
where ∗ represents convolution. The function s0(ω, u) is called the reference phase history.It is constructed using y = 0 and is given as,
s0(ω, u) =N∑n=1
σne−j2ω
c
√X2c+(0−u)2 (2.15)
where Xc is the range at the center of the swath. Figure 2.5(a) shows s0(ω, u) whilefigure 2.5(b) shows sr(ω, u) for a target at the cross-range yn = −10. Both figureswere generated using the sinusoidal pulse p(t) = rect( t
Tp)cosωt from the example in the
previous section on range imaging. Equation (2.14) is then applied to the reference andreceived phase history and the resultant cross-range target function f(y) is obtained asshown in figure 2.5(c). We can also determine f(y) by matched filtering of the referenceand received phase history in the Doppler domain ku.
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(a) (b)
(c)
Figure 2.5: Cross range imaging in the synthetic aperture domain: (a) reference phasehistory s0(ω, u); (b) received phase history sr(ω, u) for a target at yn = −10; (c) cross-range target function f(y).
The Doppler domain representation is obtained by taking the Fourier transform of thesynthetic aperture domain phase history and is expressed as,
Sr(ω, ku) = Fu [sr(ω, u)] (2.16)
for the received phase history and,
S0(ω, ku) = Fu [s0(ω, u)] (2.17)
for the reference phase history. The rationale behind calling the ku-domain the Dopplerdomain comes from the fact that the time based u-domain measurements are dependenton the velocity of the radar platform. Matched filtering in the ku-domain is carried outand f(y) is obtained as,
sMy(ω, u) = F−1u [Sr(ω, ku)S∗0(ω, ku)]
= f(y) (2.18)
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Figure 2.6(a) shows the Doppler domain reference phase history while figure 2.6(b) showsthe received phase history with a target at the cross-range yn = −10. Since both S0(ω, ku)and Sr(ω, ku) are complex, the real part is used when plotting the functions. Figure 2.6(c)shows the resultant f(y) after equation (2.18) is applied. It can clearly be seen that bothmatched filtering approaches will yield the same result for f(y). However, the Dopplerdomain approach reduces signal processing time and will be implemented from here on.To determine the cross-range resolution we first consider the synthetic aperture samplinginterval ∆u. It can be shown that s(ω, u) is bandlimited to a bandwidth of ±2k, where kis the wavenumber. When sampling in the u-domain we must again satisfy the Nyquistrate and knowing that our highest spatial frequency component is 2k we can express ∆uas,
∆u = π
2k= λ
4 (2.19)
and since the synthetic aperture domain is directly mapped to the cross-range domain(i.e. y = u) the cross-range resolution is,
∆y = ∆u
= λ
4 (2.20)
The bin resolution for the cross-range y ∈ [−Y Y ] is then given as,
yi = (i− 1)∆y − Y, i = 1, 2, 3, . . . , N (2.21)
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(a) (b)
(c)
Figure 2.6: Cross range imaging in the Doppler domain: (a) reference phase historyS0(ω, ku); (b) received phase history Sr(ω, ku) for a target at yn = −10; (c) cross-rangetarget function f(y).
where N = 2Y + 1 and yi is the cross-range value of the ith bin. The derived cross-rangeresolution is overoptimistic. System limitations and spectral properties of the receivedsignal will limit the obtainable resolution. A more accurate estimate is given in [6] andis expressed as,
∆y = xλ
4Lcos2θ(0) (2.22)
where x is the range of the target, L is the synthetic aperture length, and θ(0) is theangle the target makes with the radar antenna when the antenna resides in the middleof the aperture. For simplicity, we will assume the ideal cross-range resolution given inequation (2.20) is sufficient.
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Chapter 3
UWB OFDM Waveform Modeling
As stated in the introduction, orthogonal frequency division multiplexing (OFDM) isa digital modulation scheme in which several orthogonal sub-carriers are transmittedsimultaneously over a single transmission path. These sub-carriers partition availablebandwidth into several compact orthogonal sub-bands. A continuous time analyticalOFDM waveform is described as,
sa(t) =N∑k=1
x(k)ej(2πkt/Tp+φ0(t)), 0 < t < Tp (3.1)
where x(k) is the kth data symbol in the vector x = [x(1)x(2)...x(N)], N is the numberof sub-carriers, and Tp is the pulse duration. The manner in which x is populatedvaries depending on the intended use of the waveform and will be discussed in a latersection. The kth sub-carrier will have a corresponding sub-band that occupies a narrowslice of bandwidth at a unique center frequency k∆f where ∆f = 1/Tp is the frequencyseparation of each sub-band. The sub-bands will overlap but the orthogonal nature ofthe waveform results in the peak of one sub-band corresponding to zeros for all othersub-bands. If we take the real part of equation (3.1) and assume φ0(t) = 0 then sa(t) hasthe form,
s(t) =N∑k=1
x(k)cos(2πk∆ft), 0 < t < Tp (3.2)
where ∆f has been substituted. The signal in equation (3.2) will be the form implementedin the actual radar system and will serve as a point of reference during signal discussionfor the remainder of the paper. We observe that s(t) is simply the summation of sinusoidalpulses with duration TP . The signal spectrum S(f) is obtained by applying a Fourier
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transform to sa(t) and is expressed as,
S(f) = F(t) [sa(t)]
=N∑k=1
x(k)∫ Tp/2
−Tp/2ej(2πk∆ft)e−jωtdt
=N∑k=1
x(k)sin(π (f − k∆f)Tp)π (f − k∆f)Tp
(3.3)
where ∆f has again been substituted. The result of equation (3.3) is expected if werecall the example from section 2.1. In that instance a single sinusoid gave rise to a singlesinc function so it is not surprising that the summation of multiple sinusoids producesa spectrum that is the summation of multiple sinc functions. The kth sinc functioncorresponds to the kth sub-band centered at k∆f . The spectrum S(f) for a signal withN sub-carriers is depicted in figure 3.1. It can be seen from the figure that the width ofthe main lobe of each sinc function is 2∆f and recalling that ∆f = 1/Tp we recognizethat the main lobe width is only dependent on the duration of the pulse. In the digitalimplementation of an OFDM waveform the number of sub-bands is directly related to thepulse duration. As the number of sub-bands increases the duration of the pulse increases.
Figure 3.1: OFDM signal spectrum S(f) for N sub-carriers.
This relation will be shown in the following chapter. For now it is sufficient to simplyknow the relation.
3.1 OFDM Waveforms for Communication
OFDM waveforms are widely used in current commercial communication systems [7, 8, 9].Ideally, the orthogonality properties of the signal would allow different packets of data tobe transmitted on each sub-carrier without fear of interference. However, channel noiseand frequency dispersion gives rise to inter-channel interference (ICI). ICI is responsible
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for bit errors that results in corruption of data in the receiver. The number of sub-bandsin the signal is directly related to the sensitivity of ICI and, as mentioned previously, thewidth of the sub-bands is directly related to the number of sub-bands in the signal. Asthe number of sub-bands increases the pulse duration increases subsequently decreasingthe width of each sub-band. As the width decreases small shifts in frequency will have amuch more profound affect on the sub-bands which will result in increased ICI. However,increasing the number of sub-bands also increases the data rate. Therefore, it is importantto find the maximum number of allowable sub-bands for an acceptable bit error ratio(BER). Reduction of ICI for OFDM communications has been well studied in [10, 11,12, 13]. Several different sub-band data modulation techniques such as amplitude shiftkeying (ASK), phase shift keying (PSK), and pulse position modulation (PPM) are givenin [14, 15, 16]. These approaches increase the data rate by expanding the number of bitseach sub-band can represent. We will use the most basic method of data coding whichsimply involves populating data vector x with bit 1’s and bit 0’s. This method is knownas on-off keying (OOK). If x(k) = 1 then the kth sub-carrier exists in the signal and isdeemed ’on’. On the other hand, if x(k) = 0 then the kth sub-carrier is deemed ’off’.That is, for a data vector x = [1 0 1 0] we expect sub-bands centered at frequencies ∆fand 3∆f to be ’on’. By observing the existence and position of sub-bands in the receivedspectrum, we can reconstruct the transmitted data vector. ICI will play a role in notonly the number of sub-bands we can use at one time but also in available sub-bandfrequencies. Channel estimation must be performed to predict the behavior of the mediafrom transmitter to receiver. If the channel is estimated accurately the received signalcan be adjusted to account for the interference from the media which will improve theICI. Channel estimation is difficult and depends on the location and intended use of thesystem. For now, we only wish to prove that communication is possible with our system,therefore, the channel will be assumed to be Gaussian.
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Chapter 4
UWB OFDM SAR Imaging
4.1 UWB OFDM Radar Waveform Generation
The general model of an OFDM waveform from Chapter 3 will now be applied to theSAR imaging scenario. Since both simulated and actual signals are formed in the digitaldomain, we need to convert the continuous time signal into a discrete time signal rep-resentation. if we sample s(t) from equation (3.2) with sampling interval ∆ts we obtainthe baseband discrete time signal expression,
s[n] =N∑k=1
x(k)cos(2πk∆fn∆ts)
=N∑k=1
x(k)cos(2πkn/Ns), n = 0, 1, 2, . . . , Ns (4.1)
where Ns is the number of samples and ∆f = 1/Tp = 1/Ns∆ts has been replaced.The above is a perfectly valid expression for an OFDM signal. However, to reducecomputational time waveform construction will be performed in the frequency domain. Avector Sω of length Ns is first populated with either random or pseudo-random numbers.These numbers can range in value and can be real or complex. Each element in Sωrepresents a sub-band of the mathematical OFDM signal spectrum. That is, Sω of lengthNs will result in a signal with Nsub sub-carriers where Nsub is the number of sub-bandsand Ns = 2Nsub + 1. The vector Sω can then be mathematically described as,
Sω = [Xnf 0Xpf ] (4.2)
where Xpf = [x(1)x(2) . . . x(Nsub)] represents the positive sub-bands, the zero is the DCcomponent, and Xnf = (−1)[x(Nsub)x(Nsub − 1) . . . x(1)] is the number of negative sub-bands. Note that Xpf = x and that Xnf will be a negated and flipped version of Xpf .
17
An inverse Fourier transform is then applied to Sω to get the discrete time domain signal,
s[n] = F−1ω [Sω] , n = 0, 1, 2, . . . , Ns (4.3)
By implementing s[n] in the frequency domain first, we not only save computational timebut we can also easily control the number of sub-bands and more specifically which sub-bands are turned ’on’ or ’off’. Essentially, we have the ability to control the bandwidthof the signal. As mentioned in Chapter 3, if Xpf (k) = 0 the sub-band is deemed ’off’ and’on’ otherwise. The ability to control sub-band composition plays a large role in jammingscenarios and will be addressed in a later section.
4.2 MatLab Simulation of OFDM Waveform
Simulation of an OFDM signal is performed as described in section 4.1 except Sω must beconverted to the MatLab format Sω = [0Xpf Xnf ]. Figure 4.1 shows baseband OFDMsignals for three scenarios of interest:
• Only one sub-band in the signal is ’on’.
• All sub-bands in the signal are ’on’.
• A random ratio of sub-bands are ’on’.
Both the ratio and number of sub-bands are important considerations when using OFDMwaveforms as radar signals and will be thoroughly analyzed in section 4.3. In this instance,Xpf is randomly populated with either 1, 0,−1. The signals in figure 4.1 are generatedusing the following parameters: Nsub = 128, Ns = 257, and ∆ts = 1ns. The basebandbandwidth is B0 = 1/2∆ts = 500MHz, dividing by a factor of two ensures that we aresampling at the Nyquist rate to avoid aliasing. The signal duration is Tp = Ns∆ts = 256nsfrom which we calculate the sub-band spacing as ∆f = 1/Tp = 3.9MHz. Observingfigure 4.1(a) we see that when all sub-bands are ’on’ the OFDM waveform becomes a shortspike with a width equal to 2∆ts. In this case the entire bandwidth is occupied resultingin the best possible range resolution. However, the pulse is not unique which will causethe system to suffer in hostile environments. Figure 4.1(c) is simply a sinusoidal pulse andis the least desirable waveform to transmit in the SAR imaging radar scenario. Not only isthe pulse not unique, it also occupies only a very small fraction of the usable bandwidth,reducing it from 500MHz to 2∆f = 7.8MHz giving rise to very poor range resolution.Figure 4.1(e) shows an OFDM radar waveform when Xpf is populated randomly. Wecan observe that, in this case, the signal is unique and noise-like which will bode wellfor the system in jamming scenarios. We can also see from figure 4.1(f) that the signaloccupies the full bandwidth which maximizes the range resolution.
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(a) (b)
(c) (d)
(e) (f)
Figure 4.1: OFDM signal and corresponding spectrum: (a) signal w/ all sub-bands on(b) spectrum w/ all sub-bands on ; (c) signal w/ one sub-band on; (d) spectrum w/ onesub-band on; (e) signal w/ random ratio of sub-bands on; (f) spectrum w/ random ratioof sub-bands on.
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4.3 OFDM Radar Signal Analysis
4.3.1 Ambiguity Function for OFDM Waveform
The radar ambiguity function (AF) is a very important tool used to understand theperformance limitations of the chosen radar waveform. The AF is a symmetrical twodimensional function that assesses the uncertainty between the expected received radarsignal and versions of itself shifted in Doppler frequency and time. Ideally, for an AFplot we want there to exist only a single peak resembling the two dimensional deltafunction δ(v, τ) at zero Doppler and zero delay (i.e. AF(0,0)) as shown in figure 4.2. Inthis case there is no ambiguity between the expected signal and any shifted version ofitself. However, in reality signals will have a finite length and bandwidth thus causinguncertainty to appear in the AF, some examples of which are examined in [27]. The AFfor any practical waveform will have a main lobe whose maximum lies at AF(0,0) witha spread (width) in both Doppler and delay. The spread is the distance from AF(0,0)to the first zero of the respective axis and is determined by the signal duration Tp andbaseband bandwidth B0. The delay spread is τ0 = 1/B0 while the Doppler spread isv0 = 1/Tp. In our case, Tp is dependent on the number of sub-bands used in the OFDMsignal, therefore, we expect a signal with a larger number of sub-bands to produce amain lobe with a reduced Doppler spread and subsequently have less Doppler ambiguity.While v0 can be decreased, the fact that B0 is only dependent on the sampling rate and
Figure 4.2: Ideal ambiguity function δ(v, t).
not the number of sub-bands causes τ0 to remain fixed for all signals (with one exceptionthat will be discussed in a later section). We can then surmise that the only way toreduce main lobe delay ambiguity is to increase the sampling rate of the D/A converter.The ambiguity function derived and used in [27] is only applicable to narrowband signalsand therefore cannot be used for our UWB OFDM radar signal. Instead, we adopt the
20
wideband ambiguity function described in [37] that is mathematically expressed as,
AFwb(α, τ) =√|α|
∫ Tp/2
−Tp/2s(t)s∗(α(t− τ))dt (4.4)
where,α = c− v
c+ v
and v is the radial velocity of the radar platform to the target. Equation (4.4) takesinto account the Doppler shift of each sub-band individually whereas the narrowband AFrepresentation would have applied to only one sub-band causing all other sub-bands tobe Doppler shifted incorrectly. Substituting the analytical form of equation (3.2) intoequation (4.4) we obtain the AF for a wideband OFDM signal as,
AFofdm(α, τ) =√|α|
∫ Tp/2
−Tp/2
Nsub∑k=1
x(k)ej2πk∆ftNsub∑l=1
x(l)e−j2πl∆fα(t−τ)
dt (4.5)
rearrange to get,
AFofdm(α, τ) =√|α|
Nsub∑l=1
Nsub∑k=1
x(k)x(l)ej2πl∆fατ∫ Tp/2
−Tp/2ej2πl∆f(k−lα)tdt
and solving the integral we get,
AFofdm(α, τ) = Tp√|α|
Nsub∑l=1
Nsub∑k=1
x(k)x(l)ej2πl∆fατsinc(π∆f(k − lα)Tp) (4.6)
where,sinc(π∆f(k − lα)Tp) = sin(π∆f(k − lα)Tp)
π∆f(k − lα)TpEquation (4.6) shows explicitly the dependence of the AF on not only Nsub but also the
21
(a) (b)
(c) (d)
Figure 4.3: AF for OFDM radar signal, 65% fill ratio: (a) 32 sub-bands; (b) 64 sub-bands;(c) 128 sub-bands; (d) 256 sub-bands.
orientation of the sub-bands in the vector x. Figure 4.3(a)-(d) shows the AF (with peaksnormalized to one) for randomly generated wideband OFDM radar signals with 32, 64,128, and 256 sub-bands respectively. As expected, the OFDM signal containing 256sub-bands has the smallest Doppler ambiguity of the four signals. To better observe thechanges in the AF caused by changing the number of sub-bands, we need to look at thezero-Doppler and zero-delay cuts. That is, we wish to ”cut” the ambiguity functionsso that the Doppler cut and delay cut are equal to AF(0, τ) and AF(v, 0) respectively.Figure 4.4(a) shows the zero-delay cut for the four AF’s in figure 4.3. From the figurewe see that the ambiguity in Doppler decreases as the number of sub-bands increases.This follows the expected v0 for corresponding sub-bands derived earlier. Figure 4.4(b)shows the zero-Doppler cuts for the AF’s in figure 4.3. The figure confirms that becausethe bandwidth remains constant for any number of sub-bands, increasing the number ofsub-bands in the signal has no affect on the mainlobe delay ambiguity. From the above
22
(a) (b)
Figure 4.4: Cuts in: (a) delay; (b) Doppler.
analysis we can conclude that for minimum ambiguity we should generate the OFDMwaveform with the maximum allowable number of sub-bands. We must keep in mind,however, that increasing the number of sub-bands will increase the pulse duration (whichshould be minimized for radar processing purposes).
4.3.2 OFDM Peak Sidelobe Performance
Characterizing peak sidelobe (PSL) performance of a radar waveform is critical in deter-mining whether the signal will be acceptable for high resolution imaging. In our case,peak sidelobe refers the the maximum value residing outside of the mainlobe in the auto-correlation signal. We are concerned with the difference of the mainlobe peak and thehighest sidelobe peak. If the difference is too small, it will be difficult to discern actualtarget reflections from those that appear due to large sidelobe peaks. To examine theeffect the number of sub-bands and fill ratio (occupied bandwidth) has on the PSL, wewill examine the AF zero-Doppler cut, which is the auto-correlation of the OFDM signal.The cuts will be generated in the same manner to that of figure 4.4(b) for all possiblefill ratios. Because the signals are generated randomly, the occupied bandwidth will varysubsequently changing the PSL. Therefore, numerous simulation runs are required foreach fill ratio. Figure 4.5(a) shows the normalized PSL performance for a varying num-ber of signal sub-bands and fill ratio. The mean PSL and confidence intervals are given.The results are compared to the commonly used linear frequency modulated (LFM) radarsignal which will be derived in the following chapter. For now, knowledge of the LFM
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(a) (b)
Figure 4.5: OFDM radar signal: (a) peak sidelobe performance for varying number ofsub-bands and fill ratio; (a) peak correlation performance for varying signal lengths andfill ratio.
sidelobe performance is sufficient. A 500MHz bandwidth was used for both signals. Weobserve from the figure that for fill ratios greater than 50% the PSL performance of anOFDM waveform, for any number of sub-bands, surpasses that of the LFM signal. Also,as expected from the AF, the PSL decreases for an increasing number of sub-bands andif a PSL less than −20dB is required at least 128 sub-bands must be used in the OFDMsignal. In SAR imaging the receiver may not capture the entire reflected signal from atarget. In other words, the signal may be truncated depending on the sampling window ofthe receiver. The partial signal will expectedly have less power than a complete signal. Wenow consider how the auto-correlation peak is affected by the reduction in signal powerdue to truncation. Figure 4.5(b) shows the peak auto-correlation values for signals ofvarying lengths and fill ratio. As one could surmise, the peak correlation decreases as thesignal length decreases. From an imaging aspect, we know targets that reside at greaterdistances from the radar will return signals of smaller lengths and so we can concludethat targets farther away will be more difficult to depict in the SAR image. One simpleway to counter act the reduction in correlation is to choose a maximum target rangeand adjust the receiver sampling window so that a reflection from this maximum rangereturns a complete signal. It is also important to note from figure 4.5(b) that the PSLis independent of the signal length. Therefore, to maximize the difference between themainlobe peak and sidelobe peaks the length of the recovered signal must be maximized.
4.4 OFDM SAR Image Reconstruction
4.4.1 Range Imaging
The concept of range imaging described in section 2.1 will now be applied to the newlyformed UWB OFDM radar signal. The transmitted radar signal stx(t) is given in equa-
24
tion (3.2). The actual radar signal will be modulated with a carrier signal at somefrequency fc but all processing is done in the digital domain at the baseband bandwidth,therefore, equation (3.2) is sufficient. Also, for right now we will consider a only sin-gle target reflection located at some range R. The expected received radar signal is atruncated time delayed version of stx(t) and has the form,
srx(t) = stx(t− td) (4.7)
where td = 2R/c is the time delay introduced by the target. The truncation of srx(t)comes from the fact that the beginning of the swath represents t = 0, the time whensignal sampling begins, at which point analog-to-digital (A/D) converter will acquire Ns
samples. Only when the A/D converter has sampled for at least td will srx(t) begin beingsampled. If the A/D converter sampling rate is ∆ts then the number of samples lostdue to delay td is nl = dtd/∆tse. We then apply equation (2.5) to the transmitted andreceived signal to get,
sMx(t) = F−1 [Srx(ω)S∗tx(ω)] (4.8)
We will now consider the example of a single target at a range of 15 meters with nonoise. We form the OFDM signal using a randomly generated ratio of 128 sub-bandsresulting in a signal of length Ns = 257. The signal reflection at a range of 15 metersresults in nl = d2R/c∆tse = 100. Figure 4.6(a)-(b) shows the transmitted and receivedOFDM signals respectively. The matched filtered signal sMx(t) is given in figure 4.6(c).The range profile is generated by changing sMx(t) to sMx(x) using t = 2x/c. Fromequation (2.1) we recall that the range bin size ∆x is dependent on the sampling rate∆ts. Substituting ∆ts = 1ns for this case we arrive at ∆x = .15 meters. Although hard
25
(a) (b)
(c)
Figure 4.6: OFDM baseband signals: (a) transmitted; (b) received (range = 15 meters);(c) range profile for point target at 15 meters.
to depict from figure 4.6(c), the mainlobe has a width of 2∆x or .30 meters. The rangeresolution is determined solely by the bandwidth of a signal and is given as,
∆R = c
2B0(4.9)
where B0 is the baseband bandwidth of the signal. In our system B0 = 500MHz and whenplugged into equation (4.11) gives us ∆R = .30 meters which matches the mainlobe widthobserved in the range profile.
4.4.2 Cross-range Imaging
Similar to range imaging, we will now use concepts developed in section 2.2 for cross-rangeimaging using OFDM radar waveforms. The transmitted signal stx(t, u) now depends onthe radar platform position u of the radar platform. As in the previous section we willassume only a single point target. The received radar signal srx(t, u) is again a timedelayed version of stx(t, u) where the delay td is now dependent on the range, cross-range,and radar position as in equation (2.11). We then use equation (2.18) to obtain the
26
cross-range matched filtered response,
sMy(ω, u) = F−1 [Srx(ω, ku)S∗0tx(ω, ku)] (4.10)
Figure 4.7(a) shows the reference phase history while figure 4.7(b) shows the receivedphase history for a point target located at a cross-range of 10 meters. Because of sampling,the phase history is quantized. However, the quantized phase history is still sufficientbecause it is still a non-linear function of synthetic aperture position. Figure 4.7(c) is thecross-range profile obtained after matched filtering. We recall from equation (2.20) that
(a) (b)
(c)
Figure 4.7: Cross-range Imaging: (a) reference phase history; (b) received phase history(cross-range = 10 meters); (c) cross-range profile for a point target at 10 meters.
the cross-range resolution depends on the wavelength (i.e. the frequency), therefore,when forming the phase history we should use the highest frequency component of theOFDM signal. When the ideal cross-range resolution ∆y derived in equation (2.20) isapplied to this case we get ∆y = λ
4 = .64 = .15 meters. However, the mainlobe width
from figure 4.7(c) is .2 meters, meaning the ideal derivation is, in fact, an overoptimisticperformance or the cross-range resolution.
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4.4.3 Image Reconstruction
Now that range and cross-range profiles have been obtained, we now want to form areconstructed image of the target area. At each synthetic aperture position u an OFDMsignal is transmitted, received and matched filtered. The subsequent range profile isstored so that after matched filtering of the final transmitted and received signal thereexists a matrix containing range profiles for each synthetic aperture position where therows are the range profiles and the columns are the ranges. There is then an elementby element multiplication of the cross-range profile with each range column. Figure 4.8shows the reconstructed image for a point target at a range of 35 meters and a cross-range of 10 meters. Image reconstruction will now be applied to a multiple point targetscenario. We will examine point targets at ranges 5, 10, 20, and 30 meters. For simplicityall targets will have the same cross-range of -10 meters. Equation (4.7) is now expressedas,
srx(t) =N∑i=1
stx(t− tdi) (4.11)
where i = 1, 2, 3 . . . N are the number of target reflections at any given synthetic apertureposition. We expect four reflections from the targets at the four different ranges. Wethen carry out the matched filtering from equation (4.8) using the new received signalsrx(t) from equation (4.11). The cross-range matched filtering is performed exactly as wasdone in the one target example. the resulting reconstructed image is shown in figure 4.9.Notice that the maximum peak of the point target decreases as the distance increases.Again, as the distance increases, the truncation of the reflected signal increases which in
Figure 4.8: Reconstructed image of point target at a range of 35 meters and cross-rangeof 10 meters.
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Figure 4.9: Reconstructed image of point targets at ranges of 5, 10, 20, and 30 metersand cross-range of -10 meters.
turn will cause the matched filter output peak to decrease. A more complex but moreuseful example involves image reconstruction of an extended target. In this case, insteadof being a single point the target extends along the range or cross-range direction. Ex-tended targets more accurately portray actual targets one might expect to see in reallife scenarios. The methods of range and cross-range profiling are performed in the samemanner as the multiple point target scenario. Figure 4.10(a) shows a reconstructed imagefor an extended target with range of 15 meters and cross-range 5 to 15 meters while fig-ure 4.10(b) shows an image for an extended target with range 15 meters and cross-range-20 to -5 meters. It should be noted that actual target scattering depends on the angleof incidence, target material, target shape and the radar antenna radiation pattern.
(a) (b)
Figure 4.10: Reconstructed image for extended target with: (a) range of 15 meters andcross-range from 5 to 15 meters; (b) range of 15 meters and cross-range from -20 to -5meters.
These topics involve years of research on there own, therefore, these variables are notconsidered when simulating image reconstruction. We are more concerned in showing
29
that OFDM waveforms can, in fact, be utilized as a radar signal for high resolutionimaging. For the remaining chapters we will use extended target scenarios with constanttarget reflectivity.
30
Chapter 5
ECCM Capabilities in DeceptionJamming Scenarios
In some scenarios, enemy combatants may attempt to use electronic countermeasures(ECM) to confuse the radar by altering or degrading received signal characteristics re-sulting in a poor reconstructed image [17]. Several different methods of ECM exist andare classified as either suppressive or deceptive [18]. Spot jammers and swept jammersare two basic types of active suppression jamming that attempt to overwhelm the radarby transmitting high power waveforms in the same bandwidth making it more difficult forthe radar to recognize the signal. Spot jammers, used against specialized radars, trans-mit narrowband waveforms with focused power while swept jammers transmit widebandwaveforms that have the ability to simultaneously suppress multiple radars with varyingfrequencies. Deception jammers sense incoming radar signals and generate replicas thatsimulate target echoes in order to confuse radars, hindering the ability to identify truetargets from false targets. In range gate pull off (RGPO) the radar signal is replicatedwith controlled delay and amplified. Due to the amplification, the radar sees only thedelayed jamming signal and subsequently begins tracking the target at a false range [19].False target generation (FTG) is another commonly used form of deception jamming.Similar to RGPO, the radar signal is replicated and delayed to create a range offset. Thedelayed waveform is transmitted at the next expected arrival of the radar signal and isseen as an actual target during the correlation process. This type of FTG can be accom-plished using a digital radio frequency memory (DRFM) repeat jammer [20]. AnotherFTG option is to generate the replica waveforms by determining the instantaneous (IF) ofthe incoming radar signal. In attempt to suppress the effectiveness ECM radar engineershave developed electronic counter-countermeasures (ECCM). Several types of ECCM aregiven in [18, 21, 22]. Pulse repetition frequency (PRF) agility allows signals to be trans-mitted at non constant intervals denying jammers the ability to accurately predict thepulse repetition interval (PRI) causing the subsequent jammer signal to be transmitted at
31
times which will not affect the radar. Pulse diversity of radar signals is an effective ECCMtechnique against deception jammers. Two methods, Multi-tone phase modulation andslowly varying chirp rate of linear frequency modulated (LFM) chirps are explored in [20].Another method, involves coding signals in such a fashion that a transmitted waveformat an arbitrary PRI is orthogonal to the waveform at the previous PRI thus severelylimiting the effectiveness of deception jamming during the correlation process. OFDMwaveforms are an example of this type of coding scheme. As mentioned in Chapter 3,spectral components in OFDM signals are orthogonal to each other, therefore, by con-trolling the sub-bands we can ensure that any two waveforms emitted simultaneouslywill have minimal interference. Advances in sampling technology have made relativelylow cost ultra-wideband wave shaping using OFDM signals a possibility [23]. The widebandwidth along with excellent pulse diversity shows clear potential for UWB OFDMsignal use in deception jamming scenarios. Ultra short Gaussian monopulses [24, 25]allow for sub-meter resolution and exhibit good multipath performance. However, theconstant waveshape makes these pulses susceptible to certain forms of deception jamming(i.e. DRFM repeat jamming). LFM chirps given in [26, 27], while easily implemented,experience the same susceptibility to jamming as the Gaussian monopulse but becauseof the linear nature of the IF, the chirp is sensitive to a wider range of jammers (i.e.IF jamming). Frequency-hopping (FH) signals [28], similar to OFDM, change spectralcomposition at each PRI limiting the effectiveness of DRFM jammers. However, becausethe IF is constant, if the hopping interval is known this type of signal may be affected byIF jamming. This chapter investigates the ECCM capabilities of an UWB OFDM signaland the benchmark LFM chirp, and FH signal against an IF jammer and a DRFM repeatjammer. Jammer models will be constructed and quantitative and qualitative analysisfor this scenario will be performed.
5.1 Benchmark Signal Generation
An LFM chirp can be expressed as,
slfm(t) = Alfmejπ(2fc+kt)t, 0 < t < Tp (5.1)
where Alfm is the constant envelope used to equalize the energy, fc is the center fre-quency, k is the modulation (chirp) rate, and Tp is the pulse duration. Basic analog LFMtransmitter implementation initially requires that a pulsed sinusoid waveform at fc beamplified and passed through an up/down-chirp filter. Using a passive surface acous-tic wave (SAW) chirp filter as in [29] greatly reduces hardware complexity and decreasespower needed in the transmitter design. The chirped pulse is passed through a power am-plifier (PA) and transmitted through the antenna. The constant envelope (CE)-waveform
32
of the LFM chirp gives it high tolerances against nonlinearities requiring less stringentconstraints in the PA [30]. We now consider a first derivative (monocycle) Gaussian pulsewhich has the form,
sGuass(t) =(−AGuassσ3√
2πt
)e−
t22σ2 , 0 < t < Tp (5.2)
where AGauss is the constant used to equalize the energy and σ determines the width ofthe pulse. A simple, yet efficient, ultra-short ultra-wideband monocycle pulse generatoris given in [31] and consists of three basic blocks; the Gaussian pulse generator, thepulse shaping network, and the RC network. The final benchmark signal to analyze is afrequency-hopping (FH) radar signal which is given as,
sfh(t) = Afhej2π(fi+fc)t, i = 1, 2, 3, ... 0 < t < Tp (5.3)
where Afh is the constant used to equalize the energy, fc is the center frequency, and fi
are the frequencies determined by pseudo-random selections of i within a pre-determinedrange of frequencies. The transmitter consists of a clocked pseudo-random number (PN)generator that sends a number i to a frequency synthesizer which generates a sinusoid atfi. The sinusoid is mixed with the carrier sinusoid at frequency fc to generate a sinusoidalwaveform at (fi + fc). The new waveform is band-pass filtered to eliminate the (fi − fc)component acquired from mixing the two sinusoids. The waveform is then amplified andtransmitted. Similar to the LFM chirp the FH signal is a CE-waveform and exhibits thesame high tolerances to nonlinearities in the amplifier. Figure 5.1 shows the time domainsignals and corresponding spectra for the benchmark waveforms. We quickly observefrom the figure that all three signal spectra occupy the same 500MHz bandwidth whichequals the bandwidth of the OFDM waveform.
33
(a) (b)
(c) (d)
(e) (f)
Figure 5.1: Benchmark waveforms: (a) LFM time domain; (b) LFM spectrum; (c) Gaus-sian monopulse time domain; (d) Gaussian spectrum; (e) FH time domain; (f) FH spec-trum.
34
5.2 Deception Jammer Modeling
5.2.1 Instantaneous Frequency Estimator
The notion of instantaneous frequency (IF) developed primarily by Gabor and Villeinvolves generating the analytical signal representation of a waveform by applying theHilbert transform to a real signal s(t) to obtain z(t) = s(t) + jH [s(t)] from which the IFcan be derived. The analytic signal z(t) can also be found by taking the inverse Fouriertransform of S(f) to get an expression which provides a more simplistic way of extractingthe phase of the signal and is given as,
z(t) = A(t)ejφ(t) (5.4)
where A(t) is the time dependent amplitude and φ(t) is the phase function of the realsignal [32]. The continuous time IF of equation (5.4) has commonly been defined as,
ω̂(t) = dφ(t)dt
= φ′(t) (5.5)
However, examples given in [32, 33] show that equation (5.5) is not a suitable definitionin all cases, particularly the case of multi-component signals. It was stated in [33] thatequation (5.5) will give physically meaningful results only if the spectrum of the signal issymmetric about a center frequency. The UWB OFDM signal, LFM chirp, and FH signalall exhibit this characteristic. Therefore, equation (5.5) is sufficient for determining the IFof the waveforms in this scenario. A block diagram of an IF deception jammer is shown infigure 5.2; it is assumed the center frequency of the intercepted radar signal is known to thejammer. An arbitrary radar signal, sr(t) = Are
j(ωct+φ(t)) is first mixed with an exponentialat the known center frequency and filtered to remove undesirable components to obtainsmixed(t) = ej(ωct+φ(t)−ωct) = ejφ(t) which is simply a complex exponential containing thephase information of the intercepted waveform. The signal is sent through an I/Q detectorto recover the in-phase and quadrature components, I = cosφ(t) and Q = sinφ(t). TheI/Q channels enter a phase digitizer that determines the discrete
35
Figure 5.2: Block diagram of an instantaneous frequency deception jammer.
instantaneous phase φ̂n from the sampled In/Qn inputs and the resultant instantaneousphases are then stored into memory. A signal estimator block is then used to generatethe discrete waveform,
jr[n] = Ajej(ωc+ω̂dn)n
= ejω̂n (5.6)
where ω̂n = (ωc + ω̂dn) is the discrete IF of the waveform and ω̂dn is the discrete IFdeviation expressed as,
ω̂dn = φ̂n − φ̂n−1
Ts(5.7)
which is the discrete derivative calculated by using the current and previous instantaneousphases along with the time sampling interval Ts. The discrete waveform is then delayedto give the signal a false range offset and stored in memory. At the next predicted pulserepetition interval (PRI) the discrete waveform is sent through a D/A converter andtransmitted in the final form,
jr(t) = Ajej(ωc+ω̂d(t))t (5.8)
We now need to determine the IF for the test signals from so their respective jammersignal can be generated. The IF of the LFM chirp can be determined by first observingequation (5.1) and (5.4) from which we conclude that
φ̂lfm(t) = 2π(fct+ kt2) (5.9)
Applying equation (5.5) to the phase function the expression for the IF is given as,
ω̂lfm(t) = 2πkt+ ωc (5.10)
where 2πkt = ω̂dlfm(t) is the instantaneous frequency deviation of the LFM chirp. Com-
36
paring equation (5.8) to equation (5.10) we see that the IF generated by the jammerprecisely matches the theoretically defined IF for the LFM chirp. Deriving the IF for anOFDM signal is much more complex, so complex in fact, that we will only attempt toderive the IF for the case when all sub-bands in the signal are on. The OFDM waveformcan then be alternatively expressed as,
s(t) = 1− ej2π(N+1)∆ft
1− ej2π∆ft − 1
= γ1ej2πN∆ft − 1
= [γ1cos(2πN∆ft)− 1] + j [γ1sin(2πN∆ft)] (5.11)
where,γ1 = sin(2π(N + 1)∆ft)
sin(2π∆ft) (5.12)
The resulting expression for the instantaneous phase is then,
φ̂ofdm(t) = tan−1(
γ1sin(2πN∆ft)γ1cos(2πN∆ft)− 1
)(5.13)
the IF would then be calculated using equation (5.5) and (5.13). Finally, we examine theinstantaneous frequency of the FH signal. Similar to the LFM chirp, the IF of the FHsignal can be found by first examining equation (5.3) and (5.4) to get φ̂fh(t) = 2π(fi+fc)t.It is important to note that the frequency hopping interval τfh determines fi at any giventime t and is crucial in determining the IF of the FH signal. Assuming that τfh is knownthe instantaneous frequency can be expressed as,
ω̂fh(t) = 2πfi + ωc (5.14)
where ω̂fh only depends on fi. Figure 5.3 shows predicted jammer signals for the LFMchirp, UWB OFDM signal and FH signal. We observe that the jammer could easilypredict the signals for the LFM chirp as well as the FH signal (assuming the frequencyhopping interval is known). However, sporadic frequency variations in the UWB OFDMsignal prevent an accurate prediction.
5.2.2 DRFM-Based Architecture
A block diagram of a DRFM repeat jammer is shown in figure 5.4; it is assumed, as inthe previous section, that the center frequency intercepted waveform is known to the
Figure 5.4: Block diagram of a digital radio frequency memory repeat jammer.
jammer. A local oscillator generates an exponential at the known center frequency thatis then mixed with the intercepted radar signal sr(t) = Are
j(ωct+φ(t)). The basebandwaveform enters an A/D converter where it is sampled at the sampling interval Ts to pro-duce the discrete signal smixed[n] = ejφnn. An unavoidable product of sampling is signalquantization error, which if large enough can yield unacceptable results when generatingreplica waveforms. However, high bit-resolution reduces sampling speed subsequentlyreducing the instantaneous bandwidth of the DRFM jammer. A delay is introduced tothe discrete signal creating a false range offset by means of a controller and is then stored
38
in memory until the next predicted PRI. The memory should be high-speed and dualported to allow for simultaneous recording and replaying of signals [34]. It should also belarge enough to allow for storage of multiple delayed signals. The discrete delayed signaljr[n] = ejφn(n−nd) passes through a D/A converter and is mixed with an exponential atthe known center frequency resulting in the transmitted jammer signal expressed as,
jr(t) = Ajsr(t− td) (5.15)
where td is the time delay introduced to the jammer signal.
5.2.3 SAR Jammer Signal Modeling
We can use concepts developed in Chapter 2 to model the expected jammer signal receivedby the radar platform. The objective of the jammers is to introduce a time delay td to areplicated radar signal. This time delay can be represented as,
td =2√x2j + (yj − u)2
c(5.16)
where xj and yj are the jammer introduced range and cross-range positions of the falsetarget. Under the assumption that the jammers could form exact replicas of the transmit-ted radar signal then the jammer signals for the OFDM, LFM, FH and Gaussian radarsignals are given respectively as,
jofdm(t, u) =N∑k=1
σjej2πk(fc+∆f)
(t−
√x2j
+(yj−u)2
c
)(5.17)
jlfm(t, u) = σjejπ
[2fc+k
(t−
√x2j
+(yj−u)2
c
)](t−
√x2j
+(yj−u)2
c
)(5.18)
jfh(t, u) = σjej2π(fi+fc)
(t−
√x2j
+(yj−u)2
c
)(5.19)
jGuass(t, u) = − σj
σ3√
2π
t−√x2j + (yj − u)2
c
e− 12σ
(t−
√x2j
+(yj−u)2
c
)2
(5.20)
The reflectivity σj along with xj and yj are all determined by the jammer. For optimal
39
deception, the reflectivity of the false target should mimic that of a real target located atthe same position. Using the equations above we want to generate range profiles for boththe IF jammer and DRFM jammer case. For simplicity only point targets are considered.Figure 5.5 shows the range profiles for this scenario. Figure 5.5(a) shows that when anOFDM signal was transmitted neither jammer was able to introduce a false target intothe range profile. On the other hand, the LFM chirp, which does not change on a pulse
(a) (b)
(c) (d)
Figure 5.5: Range profiles of IF jammer (top) and DRFM jammer (bottom) generatedfalse target at 135 meters: (a) OFDM signal; (b) LFM chirp; (c) monocycle Gaussianpulse; (d) FH signal.
to pulse basis, produced range profiles that did contain a false target for both types ofjammer. As one might expect, the FH signal performed well against the DRFM jammerbut not the IF jammer. While the instantaneous frequency of the signal can be easilyreplicated, the signal varies every PRI giving it high resistance against the DRFM jammer.
5.3 Deception Jamming Reconstructed Images
Table 5.1 contains signal parameters for each signal used in simulation. Constant am-plitudes were given to each signal to ensure the waveform energies were approximatelyequal. The LFM chirp was given a constant amplitude of one its energy served as thebase energy. Since the OFDM and FH waveforms used in simulation were randomly gen-
40
erated at each PRI, average energies of both signals were used when determining theiramplitudes. The bandwidth of the Gaussian monopulse was determined by adjusting theduration of the pulse until the center frequency (250MHz) of the desired 500MHz band-width had the greatest magnitude. Unfortunately, this required that the pulse durationbe 4.2ns instead of the benchmark 256ns used for all other signals. The jammer sam-ples all signals at the same sampling rate and although Nyquist’s Theorem require onlysampling at only twice the bandwidth it is not uncommon in radar application to sampleup to four times the signal bandwidth. Figure 5.6 shows the simulated reconstructedimages for the benchmark signals when the radar is in the presence of an IF jammer anda DRFM repeat jammer. The signals created by the IF jammer and the DRFM jammerhad weak correlation with the transmitted OFDM signal subsequently causing the falsetarget ranges to be non-existent in the range profile. Therefore, the absence of a falsetarget in figure 5.6(a) and 5.6(b) is not unexpected. However, the LFM chirp had strongcorrelation with both jammer signals causing a false target presence in the generatedrange profiles. Figure 5.6(c) and 5.6(d) show that there are, in fact, false targets in bothreconstructed images. Since the FH signal varies at every PRI, the waveform recordedand retransmitted by the DRFM signal at the current PRI will always have weak cor-relation with the transmitted signal at all successive PRI’s resulting in the absence ofthe false target in figure 5.6(f). It was shown the IF jammer could replicate the FHsignal (if the hopping interval was known) which allows for strong correlation and as inthe case of the LFM chirp causes a false target to be present in the reconstructed imageas shown in figure 5.6(e). The results show that OFDM signals had the best overallperformance against IF jammer and the DRFM jammer. However, if the IF jammer isunable to predict the hopping interval for the FH signal we would expect no false targetin figure 5.6(f). Therefore, we can conclude that the OFDM signal performs at least aswell as the commonly used FH radar signal. The lack of pulse diversity makes the LFM
Table 5.1: Signal Parameters for Time-Frequency Simulations.
41
(a) (b)
(c) (d)
(e) (f)
Figure 5.6: Simulated reconstructed images with jammer false targets: (a) OFDM signalwith IF jammer; (b) OFDM signal with DRFM jammer; (c) LFM chirp with IF jammer;(d) LFM chirp with DRFM jammer; (e) FH signal with IF jammer; (f) FH signal withDRFM jammer.
42
chirp an undesirable choice against these types of jammers unless the chirp is furthermodulated. However, doing so requires more complex hardware leading to a heavier andmore expensive radar platform.
5.4 Statistical Performance of Randomly GeneratedOFDM Signals
In this scenario, SAR imaging is performed on a target area of interest containing aDRFM jammer. The imaging radar utilizes ultra-wideband (UWB) OFDM waveformswhere at each pulse repetition interval (PRI) a fresh waveform is randomly generated.The DRFM jammer residing in the target area will receive the transmitted pulse at eachinterval. Upon reception of the radar signal the jammer will attempt to replicate it in themanner mentioned in the previous section. It is assumed that the jammer can, at best,accurately replicate the transmitted radar signal from the previous PRI. The jammersignal is re-transmitted and received by the imaging radar at the next expected pulserepetition interval. That is, a radar transmitting signal sn(t) would expect to recover themimicked jammer signal sn−1(t) at interval n. The total received signal at n is the sumof the actual SAR signal and the DRFM signal and has the form,
sr(t, un) = sACTUAL(t, un) + sDRFM(t, un) (5.21)
As mentioned previously, image reconstruction relies heavily on MF of transmitted andreceived signals. From [20] we expect the matched filtered signal from equation (5.21) tohave the notation,
where sACTUAL(t, un) is the sum of actual target reflections of radar signal sn(t). Forsimplicity, if we assume only a single target reflection equation (5.22) can be expressedas,
where the first term in the expression is simply the auto-correlation (AC) of the currentradar signal. Recalling that the jammer signal is the previously transmitted radar signal,we observe that the second term in equation (5.23) is the cross-correlation (XC) betweentwo UWB OFDM signals. To determine the presence of a target, a maximum thresholdvalue λ is chosen for the MF output from equation (5.23) for which any value residingabove λ will be deemed a hit and any value below a miss. We are interested in the effectthe jammer signal will have during the MF process, therefore, we wish to determine the
43
probability of detection PD and the probability of false alarm PFA for two randomlygenerated UWB OFDM waveforms. However, this requires us to first find the probabilitydensity functions (PDF) of the maximum correlator output (MCO) for both the ACand XC cases using the random OFDM signals in noiseless and noisy scenarios. Fig. 1illustrates the concept via generalized plots of XC and AC PDFs for the MCO in absenceand in presence of AWGN.
(a) (b)
Figure 5.7: Generalized PDF’s for maximum correlator output: (a) correlator (matchedfilter) output without noise; (b) correlator (matched filter) output with AWGN.
5.4.1 Random Waveform Modeling
Revisiting Chapter 3 we recall the UWB OFDM waveform is given as,
s(t) =N∑k=1
x(k)ei(2πkt/Tp+φ0(t)), 0 < t < Tp (5.24)
where x(k) is the kth data symbol in the vector x = [x(1)x(2)...x(N)]. Each data symbolin the vector x corresponds to a sub-band present in the waveform. The manner inwhich x is populated (i.e. the values assigned to the sub-bands) depends on the modeof the system. For this scenario, we will analyze two methods of vector population foruse when the system is in the radar imaging mode. In the first method, each symbolin x is assigned a real number that follows a normal distribution. The second assignsonly a value of 1, 0,−1 (ternary) to each sub-band and follows a trinomial distribution ifrandom manner of sub-band coefficient assignment is assumed on a pulse-to-pulse basis.If we sample s(t) we can obtain the following baseband discrete time signal expression,
s[n] =N∑k=1
x(k)cos(
2πk2N + 1(n−N + 1)
), n = 1, 2, . . . , 2N + 1 (5.25)
44
We are interested in the maximum correlator output for both the auto-correlation andcross-correlation cases using random OFDM waveforms. If sT [n], sACTUAL[n], and sDRFM [n]correspond to the radar transmitted signal, radar actual received signal, and the jammersignal respectively, then we can derive the AC and XC as follows,
rAC [l] =N−l−1∑n=0
sT [n]sACTUAL[n− l], l = 1, 2, . . . , 2N − 1 (5.26)
and,
rXC [l] =N−l−1∑n=0
sT [n]sDRFM [n− l], l = 1, 2, . . . , 2N − 1 (5.27)
Observing equations (5.26) and (5.27) we see that they are the first and second termsrespectively of equation (5.22) in discrete form. We now wish to derive expressionsfor the probability density functions for the maximum outputs rXCmax and rACmax forequations (5.26) and (5.27) respectively, but first we must obtain an expression for theMCO XC and AC random variables. In the absence of noise rACmax is deterministic andequal to one after normalization. However, the maximum value rXCmax arises from thecross-correlation of two random signals and will therefore be a random variable itself. Thedistribution of rXCmax depends on the method of vector population. This scenario canbe visualized by referring to figure 5.7(a). White Gaussian noise (AWGN) is then addedto both rACmax and rXCmax causing the scenario shown in figure 5.7(b). The AWGN isconsidered to be the reduced noise remaining after signal correlation. The expressionsfor the MCO XC and AC random variables are given as,
RXC = rXCmax +Wr (5.28)
and,
RAC = rACmax +Wr
= 1 +Wr (5.29)
where again rACmax = 1 and Wr is the reduced WGN random variable found in thematched filter response. We now derive the probability density functions for equa-tions (5.28) and (5.29). The PDF for equation (5.29) is well known and given as,
PRAC (RAC) = N (1, σ2) (5.30)
which is a normal distribution with a mean of one and variance σ2. From [35], weknow that the PDF of equation (5.28) is the convolution between the probability densityfunctions of rXCmax and Wr. The PDF of Wr is simply PWr(Wr) = N (0, σ2). The XC
45
PDF is then,PRXC (RXC) = PrXCmax (rXCmax) ∗ N (0, σ2) (5.31)
where ∗ denotes convolution.
5.4.2 Statistical Simulation Results
We know wish to confirm the analysis of our scenario via simulation in MatLab. First,two random UWB OFDM waveforms are generated by populating x using one of themethods described in the previous section. Again, each element in x corresponds toa sub-band present in the OFDM signal. The mathematical spectrum of our signal iscomposed of both positive, negative, and DC frequency components. The vector x itselfonly represents the positive portion of the spectrum. To create the negative portion, wesimply copy and flip x. The final signal spectrum has the form S[ω] = [DC xp xn] whereS[ω] is in the necessary form for MatLab processing. The IFFT is then taken resultingin the time domain signal expression s[n]. The two signals, generated in this manner,are then matched filtered as to obtain the XC and AC. The maximum value rXCmax andrACmax are then chosen from the XC and AC respectively. The AC is normalized so thatrACmax = 1 and rXCmax is stored into an array. This process is carried out 105 timesso when completed we have the vector rXCmax = [rXCmax(1)rXCmax(2) . . . rXCmax(105)]which is an array of maximum cross-correlation values attained during simulation. A cu-mulative density function (CDF) was generated for rXCmax with a 103 bin size resultingin a ∆rXCmax = 10−3. The PDF PrXCmax (rXCmax) is then formed by taking the discretederivate of the CDF. The two separate methods of vector population will be used togenerate the UWB OFDM waveform to determine which performs better in the knownjamming scenario. The PDF PrXCmax (rXCmax) will be simulated for OFDM signals witha varying number of sub-bands. That is, the length x will vary with chosen lengths 64,128, 256, and 512. Figure 5.8(a) shows PrXCmax (rXCmax) for the ternary vector popula-tion method while figure 5.8(b) PrXCmax (rXCmax) when x is populated with random realnumbers. By observing figures 5.8(a) and 5.8(b) we see that for both methods for vectorpopulation as the number of sub-bands in the OFDM signal increases (i.e. length of xincreases) the mean and variance of PrXCmax (rXCmax) decreases subsequently reducing theoverall effectiveness of the jammer. We can conclude then that it is advantageous to
46
(a) (b)
Figure 5.8: Probability density functions for rACmax when using: (a) ternary vectorpopulation; (b) real number vector population.
maximize the number of sub-bands in the signal. However, the length of x is directlyrelated to the duration of the transmitted pulse. Thus, increasing the number of sub-bands will also increase image reconstruction processing time. As the number of sub-bands increases the vector sample size increases and the law of large numbers playsa larger role allowing for better distribution approximation of PrXCmax (rXCmax). Thiscan clearly be seen in figures 5.8(a) and 5.8(b) by the smoothing that takes place inthe distributions as the number of sub-bands increases. Figures 5.9(a) and 5.9(b) showPRAC (RAC) and PRXC (RXC) when PrXCmax (rXCmax) is generated using ternary and realnumber vector population respectively with a varying number of sub-bands in the ODFMsignal. Because the mean of PRAC (RAC) is always a deterministic one and does not dependon PrXCmax (rXCmax), it will remain the same for any number of sub-bands. All probabilitydensity functions shown in figure 5.9 are simulated using variance σ2 = .25 (SNR = 6dB)for the noise PDF. The small variance is chosen to show the effect PrXCmax (rXCmax) has onPRXC (RXC) as the signal-to-noise ratio becomes large. From equation (5.31) we know thatPRXC (RXC) is the result of convolution between PrXCmax (rXCmax) and PWr(Wr) (i.e. noisePDF). Since PrXCmax (rXCmax) is constant, the shape of PRXC (RXC) will depend largelyon the change in variance of PWr(Wr). As the variance increases, PWr(Wr) will becomewider and dominate the convolution causing PRXC (RXC) to appear normally distributedwith a variance equal to that of PWr(Wr). It is not until the width of PWr(Wr) approachesthe width of PrXCmax (rXCmax) that the shape PRXC (RXC) will begin to rely
47
(a) (b)
Figure 5.9: AC and XC probability density functions with a varying number of sub-bandsusing: (a) ternary vector population; (b) random real number vector population.
on PrXCmax (rXCmax). By observing figures 5.9(a) and 5.9(b), we see that the peak densityvalue of PRXC (RXC) for any number of sub-bands is slightly less than the peak densityvalue of PRAC (RAC) in this case. This is expected and results from widening that occursduring convolution. As the noise variance decreases the peak density value of PWr(Wr)will increase, but until the peak of PWr(Wr) is greater than the peak of PrXCmax (rXCmax)the peak density value of PRXC (RXC) will depend mostly on the peak of PWr(Wr). Fig-ures 5.10(a) and 5.10(b) are magnified plots of figures 5.9(a) and 5.9(b) respectively. Theplots are presented to show the threshold λ, which was previously derived as the crossingof PRAC (RAC) and PRXC (RXC). Figures 5.10(a) and 5.10(b) also more clearly show theeffect PrXCmax (rXCmax) has on the mean of PRXC (RXC). Essentially, the plots
(a) (b)
Figure 5.10: Magnified plot of AC and XC probability density functions with a varyingnumber of sub-bands using: (a) ternary vector population; (b) random real number vectorpopulation.
confirm that the mean of PRXC (RXC) is the equal to the mean of PrXCmax (rXCmax) forcorresponding sub-bands. They also show that as the number of sub-bands in the OFDMwaveform increases the threshold will decrease subsequently increasing the PD and de-
48
creasing the PFA. At large variances the noise PDF is much wider than PrXCmax (rXCmax)and, because the two are convolved, the noise PDF will dominate until the width of thenoise PDF is reduced due to the decreasing variance. Figures 5.11(a) and 5.11(b) showthe probability of detection (PD) and probability of false alarm (PFA) respectively for theternary vector population scenario with a varying number of sub-bands. The thresholdshifts as the noise variance decreases and therefore we must recalculate the threshold forevery SNR value. Since we simulate using discrete vectors, we must choose a thresholdvalue that may not correspond to the exact crossing of PRAC (RAC) and PRXC (RXC). Thiswill cause both the PD and PFA curves in figure 5.11 to appear
(a) (b)
(c) (d)
Figure 5.11: Using ternary vector population: (a) probability of detection; (b) probabilityof false alarm; (c) magnified probability of detection; (d) magnified probability of falsealarm.
slightly rougher than if we could choose the exact λ for the PDFs. We want to considerthe SNR value at which the PD and PFA reach .95 and .05 respectively. We will considerthese values of the PD and PFA the minimum values for acceptable system performance.Observing figures 5.11(a) and 5.11(c) we see that when using 512 sub-bands we reach aPD of .95 when the SNR = 5.4dB. However, when using only 64 sub-bands the PD doesnot reach .95 until the SNR = 6dB, a .6dB increase. Figures 5.11(b) and 5.11(d) showthat the PFA behaves similar to the PD in this scenario. Using 512 sub-bands, the PFA
49
hits .05 when the SNR = 5.55dB. On the other hand, when using 64 sub-bands the PFAhits .05 at the SNR = 6.25dB, an increase of .75dB. Figures 5.12(a) and 5.12(b) showthe probability of detection and probability of false alarm respectively for the randomreal number vector population scenario with a varying number of sub-bands. Figure 5.12is similar to figure 5.11 in the sense that varying the sub-bands has a slightly greateraffect on the PFA. Also, as in figure 5.11, the threshold must be chosen from a discretevector causing slightly rough curves. In figures 5.12(a) and 5.12(c) we observe that thePD reaches .95 at an SNR = 5.4dB and SNR = 6dB when using 512 and 64 sub-bandsrespectively, an increase of .6dB. These results match those obtained when using ternaryvector population. The PFA crosses .05 at an SNR = 5.55dB and SNR = 6.25dB whenusing 512 and 64 sub-bands respectively, an increase of .75dB. We can conclude fromthese results that both ternary and real number vector population perform identically(when comparing the PD and PFA at the acceptable values). This grants us the freedomto populate the vector in either manner depending on the transmit mode which addsto the overall robustness of the system. Also, increasing the number of sub-bands usedin the signal will have only a slight impact on the PD and PFA. The SNR differs by amaximum of only .75dB for the range of sub-bands used in the scenario. This, again, isfor observation of the PD and PFA at the acceptable values.
50
(a) (b)
(c) (d)
Figure 5.12: Using real number vector population: (a) probability of detection; (b) prob-ability of false alarm; (c) magnified probability of detection; (d) magnified probability offalse alarm.
5.4.3 Threshold Effect on Jammer Assumption
In the previous section we adjusted the detection threshold λ the optimize the PD andPFA in the presence of a jammer. We now must consider the scenario when the existenceof a jammer is assumed but our assumption is incorrect. In this case PRXC (RXC) is nolonger effected by PrXCmax (rXCmax) and equation (5.31) becomes,
PRXC (RXC) = N (0, σ2)
= PWr(Wr) (5.32)
The detection threshold is again the crossing of PRAC (RAC) and PRXC (RXC), however,the mean of PRXC (RXC) becomes zero which shifts the PDF left reducing λ. To avoidconfusion, we will call the jammer adjusted threshold λ′ and the threshold with no jam-mer λ. We now want to determine the PD and PFA using both thresholds to determinewhether it is more beneficial to assume a jammer presence and use λ′ or whether theassumption creates too large of a negative affect on the PD. That is, we will analyze thePD and PFA using λ and λ′ when a jammer does NOT exist. The worst case scenario willbe examined and requires using an OFDM signal with 64 sub-bands which was shown to
51
be most effected by the jammer in the previous section. Figures 5.13(a) and 5.13(b) showthe PD and PFA respectively for this scenario. We now observe the PD and PFA shown inthe figures at the acceptable values given in the previous section. Using λ′ over λ whenno jammer is present we find that the SNR at which the PD reaches the acceptable valueincreases by .9dB. On the other hand, the SNR at which the acceptable value is reachedfor the PFA decreases by .7dB. It is clear that the PD is effected more than the PFA
when choosing λ′. Since we are concerned with false targets being introduced, we mayconsider the .2dB additional increase in SNR for the PD allowable and therefore chooseto use λ′. However, before making any decisions, we must first explore the alternativescenario when a jammer presence is NOT assumed but the assumption is incorrect. Thatis, we will analyze the PD and PFA using λ and λ′ when a jammer exists. From equa-tion (5.30) we know that PRAC (RAC) is jammer independent and so we expect the PD toremain unchanged from figure 5.13(a). Figures 5.14(a) and 5.14(b) show the PD and PFArespectively for this alternative scenario.
(a) (b)
Figure 5.13: Threshold effect under jammer assumption with NO jammer: (a) probabilityof detection; (b) probability of false alarm.
(a) (b)
Figure 5.14: Threshold effect under NO jammer assumption with jammer: (a) probabilityof detection; (b) probability of false alarm.
52
We observe from figure 5.14(a) that the PD is, in fact, that given in figure 5.13(a) and thatusing λ′ again results in the PD reaching the acceptable value .9dB later. However, theSNR at which the acceptable value is reached for the PFA decreases by 1.3dB. Choosingλ′ in this alternative scenario actually effects the PFA more than the PD and results in a.4dB additional decrease in the SNR for the PFA.
We conclude from the outcome of both assumption scenarios that using λ′ as the stan-dard detection threshold is more beneficial. Although the PD is reduced by an additional.2dB (compared to the PFA) when NO jammer exists, choosing λ′ causes an additionalimprovement in the PFA of .4dB when a jammer does exist. The PFA improvement istwo times the decline in the PD. Considering we are concerned with the introduction offalse targets, we are willing to take the possible hit in the probability of detection for thetwo fold improvement in the probability of false alarm.
53
Chapter 6
System Implementation andExperimental Results
6.1 Hardware Implementation
The radar system itself is implemented as a software defined radar and communication(SDRC) system. Instead of using analog components to shape signal parameters wedesign the waveform using software on a PC. This method gives us a dynamic rangefrom which signal parameters can be chosen. Although we utilize OFDM waveforms, therobustness of the system allows any type of signal to be used for transmission. However,the parameters of the chosen signal must reside within the limitations of the digital-to-analog (D/A) converter and analog front end (AFE) components. Signal processingfor our SDRC system is also performed utilizing software. This grants us the abilityof performing multiple processing tasks simultaneously providing the possibility of dualradar and communication mode. Essentially, the system can operate in both modes atthe same time. A block diagram of the OFDM system is given in figure 6.1. The OFDMtransmit signal with chosen parameters is generated on PC1 in MatLab as described insection 4.2. The system allows for a variable number of sub-bands to be used at anygiven time. The discrete time domain signal is sent to a D/A converter that samples at arate of 1 gigasample per second which, adhering to the Nyquist sampling rate, gives us
54
Figure 6.1: OFDM SDRC system block diagram.
a system bandwidth of 500MHz. The bandwidth can be increased simply by replacingthe D/A converter with a faster sampling version. The newly formed analog signal isheterodyned with a free running oscillator at a frequency of 7.5GHz, the result of whichis the continuous time OFDM radar/communication signal. Ideally the signal would thenpass through a power amplifier, however, the 33dB power amplifier is currently beingbypassed to ensure the system operates below the allowable FCC threshold. The signalis transmitted through a horn antenna with a 15dB power gain. Any reflected signal isreceived by another horn antenna with the same properties and amplified by two 24dBultra-low noise amplifiers. The received signal is again heterodyned with the 7.5GHzfree running oscillator and reduced to baseband. The baseband signal is sampled by ananalog-to-digital (A/D) converter that samples at a rate equal to the D/A converter (1gigasample per second). The discrete signal is then sent to the PC2 for post processing.
6.1.1 System Analog Front End
The components of the AFE are the limiting factors in signal parameterization (other thansignal bandwidth), therefore, a detailed description of each component will be presentedso a defined parameter range can be obtained. Figure 6.2(a) is the AFE of the radarsystem and figure 6.2(b) is a component list. Each component is given a letter thatcorresponds to the letters in figure 6.2(a).
55
(a) (b)
Figure 6.2: Analog front end: (a) picture; (b) component list.
Two power supplies (B) give the necessary 15v DC operating voltage to the 24dB (A)and 33dbB (F) power amplifiers while the third power supply (B) gives the necessary12v DC operating voltage to the free running oscillator. The input to the power suppliesmust be an AC voltage ranging from 85-264 volts. The maximum output voltage andcurrent of the power supplies are 15v DC and 2A respectively. The output voltage isvariable from 10-15v DC. The two 24dB (A) ultra-low noise amplifiers on the receive sidecan operate with frequencies ranging from 2-8GHz which is acceptable for our OFDMradar/communication signal that has a maximum frequency of 8GHz. The free runningoscillator (C) is set to 7.5GHz but has a mechanical tuning variation of +/−100Mhz. Toensure the system transmit signal is within the allowable 8GHz ceiling of the amplifiers(B), the free running oscillator (B) should be set to a frequency of 7.4GHz. This shouldprevent any small variations in oscillator frequency from affecting the amplification ofthe received signal. The two mixers (E) are specified to take intermediate signals withfrequencies ranging from 0-4GHz and local oscillator frequencies ranging from 4-12GHz.The output of the mixers (E) must be within the frequencies 4-12GHz. The transmissionand receive horn antennas are shown in figure 6.3(a). The antennas are connected to theAFE through low noise antenna cables (G). Figure 6.3(b) shows the gain pattern for
56
(a) (b)
Figure 6.3: Analog front end transmit and receive antennas: (a) picture; (b) typical gainpattern.
the antennas used in our system. From the figure, we observe that the we can achievea typical gain of 15dB with a 20◦ antenna radiation angle. Both antennas reside onadjustable tripods and can be spaced up to 6 meters apart if necessary for communicationmode tests (each low noise antenna cable is 3 meters).
6.2 Radar Mode Experimental Results
6.2.1 Range Profile Results
The test objectives are to construct a range profile for a multiple number of targets anddetermine the resolution of the current system. The targets used in experimentation weretrihedral corner reflectors with 1-foot square sides, resulting in an RCS of approximately2 m2. Estimated signal power at the target is about 30 dBm, which is compliant withthe FCC spectral mask for UWB-OFDM transmissions. Indoor short-range experimentswere conducted and it was determined that a target placement of 1.1 meters from theantennas would allow for far field testing. Range profiles were obtained via matchedfiltering of MatLab simulated signals and received sampled signals. Three separate testswere conducted:
• Transmission and reception of OFDM waveforms with only one target present.Check matched filtered response to ensure target detection and generate targetrange profile.
• Transmission and reception of OFDM waveforms with two targets present. Ob-tain range profiles for varying target distances and ensure that the actual targetseparation matches the distance in the range profile. Determine system resolution.
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• Transmission and reception of OFDM waveforms with three targets present. Ob-tain range profiles for varying target distances and ensure that the actual targetseparation matches the distance in the range profile.
Figures 6.4(a) and 6.4(b) show the transmitted and received (after noise reduction) signalsrespectively when only one target is present in the target area. The target resides 1.57meters from the radar antennas. Figure 6.4(c) shows the obtained range profile for thisscenario. The difference from peak to the highest sidelobe is approximately 17 dB whichis 3dB better than direct transmission to the receiver. This improvement may seem
(a) (b)
(c)
Figure 6.4: Range profile with one target present: (a) transmitted radar signal; (b)received radar signal; (c) range profile.
surprising; however, the random nature of the signal will cause the difference to vary forevery transmission. Again, this can be controlled through sub-band manipulation. Inthe second scenario we introduced a second target to the target area. Both targets wereidentical corner reflectors. The first target was, again, located 1.57 meters from the radarantennas. The distance of the second target was varied with respect to the first targetand several range profiles were obtained. The receiver A/D converter records a sampleevery nanosecond. Therefore, using the equation ∆t = 2∆x
c, we calculate the theoretical
range bin resolution to be ∆x = .15 meters. In other words, every sample corresponds
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Second Target from: Antennas/First TargetFigure Actual Distance (meters) Expected Samples Binned Distance (meters)
Table 6.1: Two target range profiling scenario with varying target distance.
to a distance of .15 meters. Any target reflection that lies in between ∆x will not beseen until the following sample. Also, using the equation ∆R = c
2B we calculate thetheoretical system resolution to be .30 meters, where the signal bandwidth B = 500MHz.Essentially, this states that two targets must be a minimum of .30 meters apart to bediscerned. Table 6.1 shows the experimental parameters for several different locations ofthe second target. The second column corresponds to the expected number of samplesrecorded before the reflected signal from the second target is seen. We want to comparethe number of expected samples between the first and second target with the actualnumber of samples present in the range profile. This will allow us to determine if thereceiver is sampling at the correct rate. Fig. 5(a)-(f) show the range profiles for the twotarget scenario corresponding to Table 6.1. All of the range profiles are magnified at thepeaks so the number of samples between peaks can more clearly be seen. Notice in Fig.5(a) that the two peaks are distinguishable and the number of samples between peaks istwo which matches the expected number from the table. After range binning, the targetsare a distance of 30cm apart. We can also observe from Fig. 5(a) that the value between
59
(a) (b)
(c) (d)
(e)
Figure 6.5: Range profiles with the second target at distance: (a) 28cm; (b) 43cm; (c)59cm; (d) 104cm (e) 127cm.
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the two peaks does not fall to zero. This tells us that our actual system resolution is notquite as good as the theoretical resolution. However, in Fig. 5(b) the peaks are threesamples apart (also matching the expected number of samples) and the value betweenpeaks does fall close to zero. We can conclude that our actual system resolution is atworst 45cm and, with some signal manipulation to reduce sidelobes, we can expect toreach our theoretical system resolution of 30cm. From observation of Fig. 5(a)-(f), wesee that the expected number of samples between peaks does match the actual numberof samples. The expected number of samples from the antenna to the second targetwould ideally be used to determine the target distance from the antenna. However, dueto synchronization issues between the transmitter and receiver the time that receiversampling begins varies. One crude way to determine the actual target range is to sendthe transmit signal directly to the receiver while simultaneously sending it through theradar. We then perform matched filtering of the transmitted and received signals usingthe first peak as the reference point. The number of samples between the reference peakand the target peak is measured. Using the expected number of samples from the antennato the target (as given in the table) we can determine the signal travel time through theradar analog front end by simple subtraction of the samples. Adopting this technique,we have determined that the front end delay is 26 samples or 26 nanoseconds. Themaximum distance from peak to highest sidelobe in any of the figures is approximately15dB which does not meet the 20dB or better requirement for high resolution imaging.Again, improvements can be made through signal manipulation. In the final scenariowe introduced a third target into the target area. All three targets were identical cornerreflectors. The first target was, again, located 1.57 meters from the radar antennas. Thedistances of the second and third target were varied with respect to the first target andseveral range profiles were obtained. Table 6.2 shows the experimental parameters forseveral different locations of the second and third target. Fig. 6(a)-(c) show the rangeprofiles for the three target scenario corresponding to Table 6.2. All of the range profilesare magnified at the peaks so the number of samples between peaks can more clearly be
Table 6.2: Three target range profiling scenario with varying target distance.
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(a) (b)
(c)
Figure 6.6: Range profiles for the three target scenario with second and third targets atdistance: (a) 41cm and 171cm; (b) 120cm and 154cm; (c) 80cm and 160cm.
seen. As with the two target scenario we wish to compare the expected number of samplesfrom Table 6.2 with the actual number present in the range profiles. From observing Fig.6(a)-(c), we conclude that the expected number of samples for both the second and thirdtarget match the actual sampling spacing in all of the figures. As with the two targetscenario the difference from the smallest peak to the highest sidelobe does not satisfy the20dB or better requirement. The x-axis is chosen to show the sample spacing betweenthe peaks of the range profiles. To calculate the distance between the peaks we wouldsimply use the equation x = ct
2 , where t is the is binned into intervals of one nanosecond(for one sample).
6.3 Communication Mode Experimental Results
The test objectives are to show that communication can be performed using OFDMwaveforms with our current system implementation. As mentioned in section 3.1 we willuse on-off keying (OOK) as the data modulation technique where each sub-band willcorrespond to either a bit 1 or bit 0. For the purposes of communication, the OFDM
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Configuration Enabled Sub-bands (even only) Total Sub-bands Data Rate (Mb/s)Nen1 16-60 19 57Nen2 10-60 26 79Nen3 0-64 32 97
Table 6.3: Sub-band configurations
waveform is known as a symbol and has a length denoted as,
Lsym = N
B(6.1)
where N is the number of sub-bands and B is the signal bandwidth. A 64 sub-bandOFDM signal will be used for all communication experiments, therefore, Lsym = 128ns.If all 64 sub-bands are used as data carriers the maximum bit per symbol rate is 64.That is, we could transmit 64 bits of data for every OFDM symbol sent. Unfortunately,it was discovered that systematic in-band noise produced by the AFE caused sub-bandsat certain frequencies to become unusable. It was necessary to reduce the number ofsub-bands used as data carriers from N to Nen (the actual number of sub-bands used asdata carriers). To avoid data corruption due to ICI, we also introduced a guard bandbetween sub-bands equal to 2∆f , the length of one sub-band. In other words, we turnedoff every other sub-band in the symbol which further reduced Nen. It was determinedthat sub-bands in the frequency ranges 0-100MHz and 450-500MHz greatly interferedwith neighboring sub-bands even with guard bands present. We suspect that the A/Dand D/A converters, which again samples at 1 gigasamples per second, distort the signalat these frequencies. To ensure optimal data recovery, we turned off sub-bands residingin these frequency ranges. In communications, it is also necessary to introduce a guardinterval between successive OFDM symbol transmissions. The guard interval for ourOFDM symbols was chosen to be 4 times the length of the typical delay spread of UWBsignals in our environment [36], resulting in a guard interval length of Lguard = 200ns.The data rate of our system can then expressed as,
Drate = Nen
Lguard + Lsym(6.2)
where Lguard and Lsym are constants in this scenario. The only means of increasing ordecreasing Drate is to change Nen. Three different sub-band configurations were testedand their properties are given in Table 6.3. We now want to determine the bit errorratio (BER) for each sub-band configuration from Table 6.3 at different transmissiondistances. The usable sub-bands in the OFDM symbol were randomly populated with azero or a one corresponding to bit 0 and bit 1. The symbol was then transmitted directlyto the receiver. By receiving a single OFDM symbol, our spectral resolution is one sample
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per sub-band. Since the guard band width is exactly that of a sub-band, we sample eachguard band. To recover the data in these configurations, the energy in enabled sub-bandscan be compared to the energy in adjacent guard bands where a detection threshold wasthen used to determine a bit 1 or bit 0. The transmitted data sequence was compared tothe received data sequence to determine the BER which is given as,
BER = Nincorrect
Nen
(6.3)
where Nincorrect is the number of incorrect data bits. Several thousand pulse sequencesfor each configuration were transmitted, received, and analyzed for BER performance atseveral different transmission distances. Figure 6.7 shows the data recovery process forone of these transmissions. Figure 6.7(a) is the received raw data sampled by the A/Dconverter. The OFDM symbol of interest is circled in red. It can be seen that the raw datacontains large valued random oscillations outside of the symbol interval. Through timedomain filtering, we removed these oscillations to retain only the OFDM symbol whichis shown in figure 6.7(b). The spectrum of this signal will be used for data recovery.Figure 6.7(c) displays the mathematical spectrum of the actual transmitted symbol. Byobserving the real (positive) frequencies, we see that in this particular transmission only7 sub-bands are ’on’. Figure 6.7(d) shows the received symbol spectrum (red) with thetransmitted spectrum (black) overlayed on top. All 7
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(a) (b)
(c) (d)
Figure 6.7: OFDM symbol recovery: (a) received raw data; (b) received filtered data;(c) transmitted symbol spectrum; (d) received symbol spectrum (red) overlayed withtransmitted signal spectrum (black).
sub-bands enabled in the transmit symbol are present in the received spectrum. Inthis instance, we obtained perfect data recovery (the BER = 0). The final BER wascalculated by averaging the BER from each transmission at a particular distance. Fig-ure 6.8(a) shows the BER for each sub-band configuration and varying distances. Noticethat only Nen1 has a low BER for all distances. This occurs because Nen1 is the onlyconfiguration that deals with the symbol issues mentioned above. We then used thesame the configurations from Table 6.3 to send an image of a printer from transmitterto receiver. Figure 6.8(b) shows the resultant images for Nen1 (left), Nen2 (middle), Nen3
(right). The printer can be clearly depicted when use Nen1 which is expected from thelow BER in figure 6.8(a) for this configuration. Using Nen2, the printer can still be dis-cerned, however, some noise has been introduced into the image. We observe that usingconfiguration Nen3 will result in a very poor reconstructed image at the receiver. In thiscase, no information can be gathered from the image. We can then conclude that
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(a)
(b)
Figure 6.8: Sub-band configuration performance: (a) bit error ratio plot (b) image quality,57Mb/s (left), 79Mb/s (middle), and 97Mb/s (right).
if precise details of an image are needed than we should use Drate = 57Mb/s. On theother hand, if only general image information is required then using Drate = 79Mb/s isacceptable. Configuration Nen3 should never be used. Of course, after system improve-ments are made the results for these data rates will improve and using Nen3 may be apossibility.
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Chapter 7
Conclusions and Future Work
This paper developed theory for using ultra-wideband OFDM waveforms in high resolu-tion SAR imaging. Range and cross-range resolution was explored with varying signalparameters. It was determined that for our system implementation the maximum rangeand cross-range resolution was .30 meters and .20 meters respectively. The resolutionwas first obtained through simulation using MatLab software and then experimentallyconfirmed. Sidelobe performance of OFDM waveforms was analyzed and determined tobe dependent on the number of sub-bands used in the signal and independent of signallength. Sidelobe peaks were also found to be dependent on sub-band distribution whichcauses only certain distributions acceptable by our standards. The ECCM capabilities ofOFDM radar waveforms were addressed and compared with three benchmark radar sig-nals. In a deception jamming scenario, it was demonstrated that OFDM waveforms willperform better than constant envelope LFM chirps and Gaussian monopulses and couldperform at least as well as the FH signal. It was then shown that OFDM waveforms be-haved remarkably well against DRFM jammers and that a jammer adjusted threshold isthe optimal detection threshold for SAR imaging. The system implementation was givenin detail down to the component level and suggestions for performance improvementswere presented. We experimentally obtained range profiles for a both the two and threetarget scenario. System range resolution was then inferred from range profile plots andcompared to theoretical resolution derivation. The communication aspect of the systemwas examined and data transmission using on-off keying was tested. It was determinedthat systematic noise along with D/A and A/D converter sampling error reduced thedata rate of the system. We reached our initial goals by showing that our system couldbe used for both radar and communication at a proof of concept level. Several futuresystem and theoretical objectives include:
1. Experimentally obtain cross-range profile using current system implementation anddetermine actual system cross-range resolution.
2. Reduce systematic noise and improve communication data rate.
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3. Analytically assess sub-band distribution problem to minimize peak sidelobes forpurposes of increasing image resolution.
4. Operate the current system in the dual use mode.
5. Field test SDRC system.
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