Bandwidth Extrapolation of LFM Signals forNarrowband Radar Systems

Van Khanh NguyenNational Security and ISR Division

Defence Science and Technology OrganisationP.O. Box 1500, Edinburgh 5111, AUSTRALIA

Van.Nguyen@dsto.defence.gov.au

Mike D. TurleyNational Security and ISR Division

Defence Science and Technology OrganisationP.O. Box 1500, Edinburgh 5111, AUSTRALIA

Mike.Turley@dsto.defence.gov.au

Abstract— In this paper, we address the problem of improvingrange resolution in narrowband radar systems. By using linearfrequency modulation (LFM) waveform, we propose a methodof achieving range resolution improvement through extrapolatingthe bandwidth of the LFM waveform after it has been radiatedby the transmitter and reflected back from targets and clutterto the receiver of a radar system. The proposed method isbased on using linear prediction to extend the target frequencyresponse into spectral regions outside of the measured band.There are two key differences between the proposed techniqueand an existing bandwidth extrapolation technique designedfor wideband radar systems. Firstly, it uses correlation ratherthan stretch processing to obtain the range profile. This hasthe advantage of allowing the range depth of interest to be aslarge as the radar unambiguous distance, which is commonlyrequired in narrowband radar systems. Secondly, the bandwidthextrapolation process is carried out after Doppler processing,which allows the proposed technique to work effectively evenin the presence of strong stationary or slow moving clutter. Weevaluate the performance of the proposed method using real datacollected from an over-the-horizon radar system. We have shownthat the proposed technique is capable of improving the rangeresolution in the presence of stationary clutter that is 20 − 50dB stronger than the targets. In addition to range resolutionimprovement, we have observed that the proposed bandwidthextrapolation technique also improves the signal to noise ratio ofthe targets.

I. INTRODUCTION

Range resolution of a radar system is a measure of itsability to distinguish targets that are close in range. It isdefined as the minimum separation between two targets atwhich they can still be resolved as separated targets andis found to be inversely proportional to the bandwidth ofthe radar waveform. As a result, a simple way to improvethe range resolution of a radar system is to increase thebandwidth of the transmitted waveform. This may not alwaysbe possible due to the lack of spectrum availability and dueto hardware limitations of existing systems to cope withthe increased bandwidth. These factors motivate the need tofind ways of achieving range resolution improvement withouthaving to increase the transmitted waveform bandwidth. In[1], Cuomo proposes a method of achieving this goal forwideband radar systems by modeling the spectral responseof the measured data with a linear predictive filter and thenusing it to predict the spectral response at frequencies that

lie outside of the measurement bands. This method is knownas bandwidth extrapolation. Through simulations, it has beendemonstrated in [1] that a factor of 2 to 4 improvement inrange resolution can be achieved. The performance of thistechnique using field data is reported in [2]. A subsequentwork which exploits this idea to improve the resolution ofInverse Synthetic-Aperture radar (ISAR) images is reported[3]. This bandwidth extrapolation idea is further exploredin [4] and [5] for combining multiple disjoint frequencybands in ultrawide-band radars and frequency-hopped radars,respectively. Related works which use linear prediction toimprove the resolution of other dimensions can be found in[6] and [7]. In [6], the Doppler resolution is improved byextrapolating the measured data into the past and future. In[7], linear prediction is used to extrapolate the aperture of alinear array to improve the radar angular resolution.

The bandwidth extrapolation technique developed in [1]uses a linear frequency modulation (LFM) signal as thetransmit waveform and employs stretch processing as a basisof the preprocessing steps prior to bandwidth extrapolation.It is suited mainly for wideband radar systems where stretchprocessing is employed and the range depth of interest (RDOI)is much less than the radar unambiguous distance. In thispaper, we are interested in improving the range resolutionof narrowband radar systems. We particularly focus on therange resolution improvement method that does not place arestriction on the size of the RDOI since some narrowbandradar systems may require the RDOI to be as large as theradar unambiguous distance. An example of such systems isthe over-the-horizon (OTH) radar which operates at the highfrequency (HF) band with a typical waveform bandwidth inthe range of 10−30 kHz. Due to its wide coverage capability,the RDOI can be large as the radar unambiguous distance. Thisis often the case when the radar operates in an air mode (i.e.low time-bandwidth product) where a high pulse repetitionfrequency is used to avoid high velocity targets from causingDoppler aliasing. Due to the above requirement, the bandwidthextrapolation technique developed in [1] cannot be applied.In this paper, we propose a new bandwidth extrapolationtechnique for narrowband radar systems that allows the RDOIto be as large as the radar unambiguous distance. Moreover,the proposed technique carries out the bandwidth extrapolation

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after Doppler processing as opposed to the pulse by pulseapproached used in [1]. It involves separating the frequencyspectrum of the correlator output into groups of frequencyresponse having the same Doppler shift before applying thebandwidth extrapolation to each group separately. With thisapproach, the frequency response of moving targets will beremoved from that of the stationary clutter. Hence, it is capableof improving the range resolution of moving targets in thepresence of the strong stationary clutter.

II. PRELIMINARIES

A. Signal Model

The complex envelope of a single LFM pulse at basebandcan be described as

s(t) =

{ejπ(αt

2−Bt) 0 ≤ t ≤ To

0 otherwise(1)

where α = B/To is the sweep rate measured in Hertz persecond, B is the waveform bandwidth and To is the pulsewidth. The instantaneous frequency of this baseband signalgoes from −B

2 at t = 0 to B2 at t = To. A transmitted LFM

signal with P pulses can be written as

st(t) =P−1∑p=0

s(t− pT ) (2)

where T ≥ To is the interpulse period. This interpulse periodleads to an unambiguous distance of cT

2 metres where c is thespeed of light. The transmitted signal st(t) will be referred asa pulsed LFM signal when T > T0 and as an LFMCW signalwhen T = To. An advantage of the LFMCW signal over thepulsed LFM counterpart is that it allows the transmitter poweramplifiers to output at a constant power. The returned signalfrom K targets after it has been demodulated to baseband canthen be modeled as

r(t) =K∑

k=1

akst(t− τk) ej2πfk(t−τk) + c(t) + v(t)

=K∑

k=1

P−1∑p=0

aks(t− pT − τk) ej2πfk(t−τk) + c(t) + v(t)

(3)

where ak, τk and fk respectively represent signal amplitude,time delay and Doppler shift of the kth target. Here, v(t)is a zero-mean complex white Gaussian noise process withvariance σ2 and

c(t) =

Nc∑i=1

P−1∑p=0

cis(t− pT − τi)ej2πfi(t−τi) (4)

is the clutter return from Nc scatterers with ci, τi and firepresent amplitude, time delay and Doppler shift of the ith

scatterer. The Doppler shift fi is generally quite small andoften equal to zero. Note that in OTH radar environment theclutter distribution may be a continuum.

B. Linear-predictive data extrapolation

Let’s consider a data sequence x[0], x[1], · · · , x[N−1]. Thelinear-predictive data extrapolation method involves modellingthe signal x[n] as an auto-regressive (AR) process of order Q

x[n] =

Q∑q=1

λqx[n− q] + e[n] n = 0 · · ·N − 1 (5)

where λq are stable AR coefficients and e(n) is a white noiseprocess. The coefficients of the AR model can be estimatedfrom the data sequence x[0], x[1], · · · , x[N − 1]. A popularmethod to calculate the coefficients λq is the Burg algorithm.The AR coefficients can be used to extrapolate the data in theforward and backward directions according to

xF [n] =

Q∑q=1

λqx[n− q] n ≥ N (6)

xB [n] =

Q∑q=1

λ∗qx[n+ q] n < 0 . (7)

III. THE PROPOSED BANDWIDTH EXTRAPOLATIONTECHNIQUE

In this paper, we propose a bandwidth extrapolation tech-nique that allows the RDOI to be as large as the unambiguousdistance. This technique requires the received signal to besampled at a rate at least equal to the original signal bandwidthB. As a result, it is more suited for radar systems whichemploy narrowband LFM waveforms. An example of suchsystems is the OTH radar where a waveform bandwidth in therange of 10 to 30 kHz is typically used. The principle ideabehind the proposed technique is to generate a signal that canbe used for bandwidth extrapolation but does not suffer theproblem of having the response of individual targets startingand ending at vastly different positions as in the case of thedechirped signal in stretch processing. To achieve this, wepropose to use the method of correlating the received signalwith the transmitted signal to obtain the range profile ratherthan the stretch processing method. This is also a commonpulse compression technique for processing narrowband LFMsignal. To see how the correlating method can avoid suchproblem, let’s begin by examining the correlator output inmore details. The correlator output at time τ will be givenby

y(τ) �∫ ∞

−∞s∗(t+ TM − τ) r(t) dt . (8)

where TM is a delay constant. By defining

h(t) = s∗(TM − t) , (9)

we havey(τ) =

∫ ∞

−∞h(τ − t) r(t) dt . (10)

Here, h(t) is the matched-filter and TM ≥ To is the delayconstant required to make the filter physically realizable.Thus, the correlator output y(t) is known as the matched

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filter output. Let Y (ω), H(ω), S(ω), R(ω), C(ω) and V (ω)denote the Fourier transform of y(t), h(t), s(t), r(t), c(t)and v(t), respectively. Since the matched filter output is justthe convolution of h(t) and r(t), the Fourier transform of thematched filter output will simply be given by

Y (ω) = H(ω)R(ω) . (11)

By evaluating the Fourier transform of h(t) and r(t), we have

H(ω) = S∗(ω)e−jωTM (12)

and

R(ω) =K∑

k=1

P−1∑p=0

BkS(ω−2πfk) e−jω(pT+τk)+C(ω)+V (ω)

(13)where Bk = ake

−j2πfkτk . Thus, the Fourier transform ofthe matched filter output can be re-written as the sum of thefrequency response of individual targets plus the clutter andnoise responses

Y (ω) =

K∑k=1

T (k)(ω) + S∗(ω) (C(ω) + V (ω)) e−jωTM (14)

where

T (k)(ω) �P−1∑p=0

BkS∗(ω)S(ω − 2πfk) e

−jω(pT+TM+τk) .

(15)is the frequency response of target k.

In a normal range-Doppler processing sequence, a window-ing function is then applied to the overall frequency responseY (ω) before it is inverse Fourier transformed. This windowingfunction has the effect of controlling the range sidelobe level.The output of the inverse Fourier transform is the matched-filter output of P pulses. By dividing the matched filteroutput into pulses, Doppler processing can then be applied toproduce the range-Doppler processed data. The range-Dopplerprocessed data at a Doppler bin can be considered as the rangeprofile at that particular Doppler shift.

With the correlation-based approach, we adopt the view thatthe range profile is a time domain signal. This is because therange profile at a Doppler bin is essentially equal to the sumof the matched filter output over P sweeps at that particularDoppler shift. This is in contrast to the stretch processingtechnique where the range profile is viewed as the frequencyspectrum of the dechirped signal. With this viewpoint, wewould need to extrapolate the frequency response into thelower and higher frequency bands in order to improve therange resolution. As can be seen from (15), the target time-delay τk only affects the phase of the target frequency re-sponse. It does not cause the frequency response to shift infrequency as in the stretch processing case. When all targetshave a zero Doppler shift, the frequency response of all targetswill be perfectly aligned. When the Doppler shift is non-zero,the frequency response will be shifted by fk Hertz. This is,however, relatively small compared to the overall bandwidth

Fig. 1. Processing sequence of the normal range-Doppler technique and theproposed bandwidth extrapolation techniques.

of the signal. Since the time-delay of a target only affectsthe phase of the frequency response and does not shift it infrequency, the RDOI can be extended to the full unambiguousdistance.

Once the overall frequency response Y (ω) has been ob-tained, the proposed technique begins by applying an inverseFourier transform to the unweighted overall frequency re-sponse Y (ω) (i.e. without being weighted by the windowingfunction) to obtain the matched filter output for all P pulses.The matched filter output is then divided into pulses andDoppler processing is then applied to produce the range-Doppler processed data. As depicted in Fig. 1, the previoussteps are essentially the same as those involved in the normalrange-Doppler processing sequence, except that the frequencyresponse is not being weighted by a windowing function.Once the range-Doppler processed data is available, bandwidthextrapolation can then be applied to each Doppler bin ofthe range-Doppler processed data separately. This involvesapplying the Fourier transform to the range-Doppler processeddata at each Doppler bin to convert it into the correspond-ing frequency response. By doing so bin by bin, we haveeffectively separated the overall frequency response Y (ω) intogroups of frequency response with each group having thesame Doppler shift. The frequency response at each Dopplerbin will then be associated with an AR model so that it canbe extrapolated into the lower and higher frequency bands.The advantage of this post-Doppler approach over the pulseby pulse approach employed in [1] is that the frequencyresponse of moving targets will be removed from that of theclutter. Since each Doppler bin is associated by a differentAR model, the moving target frequency response can besuccessfully captured by the AR model. This approach canalso solve the problem of multiple targets where one target issignificantly stronger than others. In this case, extrapolationof the frequency response of weaker targets can still be doneas long as the strong target does not have the same Dopplershift as the weaker ones.

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IV. PERFORMANCE ASSESSMENT

We have conducted a series of experiments to evaluatethe performance of the proposed bandwidth extrapolationtechnique. In this paper, we present the results from two ofthese experiments. We used the JORN Laverton radar in all ofour experiments. This is an OTH radar system that utilizesthe ionosphere to reflect electromagnetic waves for targetdetection at very long ranges. For each of the experimentswe transmitted two LFMCW waveforms where one had twicethe bandwidth of the other. The performance of the waveformwith a larger bandwidth was used as the benchmark. Thetwo waveforms were transmitted by a linear antenna arrayof 28 elements as shown in Fig. 2. Due to the time-varyingnature of the ionosphere, the two waveforms were transmittedat the same frequency and immediately one after another toensure that both waveforms underwent similar propagationconditions. The returned signals were collected by a 3 kmlinear antenna array of 480 elements as shown in Fig. 3. Thereceived signals at the 480 elements were digitized and thenbeamformed using a conventional beamformer. To produce therange-Doppler map for the normal range-Doppler processingtechnique and the proposed technique, the output from abeam of interest was correlated with the reference waveformto generate the frequency response Y (ω). The processingsequences depicted in Fig. 1 were then applied to Y (ω) togenerate the range-Doppler processed data for the beam ofinterest. As this is an operational radar used by the Australianmilitary, the collected data are classified. In order for theresults of these experiments to be publicly released, we haveremoved the axis scales of all subsequent figures.

With the proposed technique, we set the order of the ARmodel to Q = N/4 and extrapolate the signal bandwidthby a factor of two. We compare the performance of theproposed technique with that of the normal range-Dopplerprocessing technique. It should be noted that it is not our aimto compare the performance of the proposed technique withthat of [1] as these two techniques have been designed for twodifferent operating conditions. The technique in [1] is moresuited for wideband radar systems with the RDOI being muchsmaller than the unambiguous distance. On the other hand,our proposed technique is more suited for narrowband radarsystems where the RDOI can be as large as the unambiguousdistance.

1) Experiment 1: In this experiment, the bandwidth of thetwo LFMCW waveforms were 12 kHz and 24 kHz. They bothhad 64 sweeps and a repetition frequency of 64 Hz. They weretransmitted at the carrier frequency of 16.854 MHz. Fig. 4shows range-Doppler maps for a particular radar dwell at theoutput of a conventional beam. The plot on the left is a range-Doppler map of the 12 kHz waveform produced by the normalrange-Doppler processing technique. In this plot, it shows thepresence of a moving target. Due to the multiple layer structurein the ionosphere, the electromagnetic waves emitted by thetransmit antenna array propagate to the target and back to thereceiver antenna array through various paths. As a result, the

Fig. 2. JORN radar transmitter array

Fig. 3. JORN radar receiver array

received signal contained multiple echoes from the target. Thismultipath effect led to the target having a number of peaks inthe range-Doppler map as can be seen in Fig. 4. In addition tothe moving target, the received signal contained echoes from ameteor trial, which are evident by their spread in Doppler, andstrong ground backscatter clutter returns. These clutter returnswere 20−50 dB stronger than the target and distributed acrossall ranges. They all had a Doppler shift near 0 Hz and formeda vertical stripe in the middle of the range-Doppler map. Themiddle plot is a range-Doppler map of the 12 kHz waveformproduced by the proposed technique with an extrapolationfactor of 2. A comparison of these two plots clearly showsthe range resolution improvement of the proposed techniquedespite the presence of the strong stationary clutter. This ispossible due to the fact that the bandwidth extrapolation step isapplied to each Doppler bin independently. With this approachthe frequency response of the target is removed from the clutterfrequency response, which allows it to be correctly modeledby an AR process. As a result, the target frequency responsecan be extrapolated to the lower and higher frequency bands.This improvement enables the multiple paths of the movingtarget to be clearly resolved. It can also be observed that theresolution of the meteor returns is also improved. The ploton the right in Fig. 4 is a range-Doppler map of the 24 kHz

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Fig. 4. Experiment 1 - Range Doppler maps produced by the normal range-Doppler processing technique and the proposed technique.

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Fig. 5. Experiment 2 - Range Doppler maps produced by the normal range-Doppler processing technique and the proposed technique.

waveform produced by the normal range-Doppler processing,which is used to benchmark the proposed technique. From therange-Doppler maps of the middle and right plots, it can beobserved that the range resolution of the proposed techniquewith an extrapolation factor of 2 is comparable to that ofthe normal range-Doppler processing technique with twicethe signal bandwidth. Due to the time-varying nature of theionosphere, the propagation paths had changed slightly duringthe transmission of the two waveforms. The energy of thepropagation paths during the transmission of the 24 kHz wave-form had been reduced slightly when compared to that duringthe transmission of the 12 kHz waveform. In addition, therewere also changes in the energy of some propagation pathsin relative to others. However, all six different propagation

paths can still be observed for both waveforms. Another visibledifference between the range-Doppler maps in the middle andright plots is that the meteor echoes were not present in thereceived signal of the 24 kHz waveform. This is due to theshort-lived nature of the meteor echoes.

Apart from the range resolution improvement, it is alsoobserved that the proposed technique has increased the signalto noise ratio (SNR) of the target for each propagation pathby 2−2.8 dB. The reason for this SNR gain can be explainedby looking at the windowing function applied to the frequencyresponse to control the range sidelobe level. The blue curve(envelop of the yellow area) in Fig. 6 is an example windowused for normal range processing of the 12 kHz signal andthe black curve is the corresponding window applied to the

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frequency response that has been extrapolated by a factor of 2.The rectangular window represents the case when the originalfrequency response is not being weighted by any windowingfunctions. The area under each window can be considered asthe amount of energy retained by that window. As a result,the area between the rectangular window and the blue curverepresents the amount of energy lost due to applying thewindowing function in the normal processing sequence. Byextrapolating the frequency response to double its originalbandwidth and then applying the windowing function, theproposed technique has recovered part of this lost energy. Theportion of energy of the original signal that has been recoveredis shown in red. Furthermore, the extrapolated parts of thespectrum also provide additional energy which is representedby the green areas. Hence, the energy gained by performingthe bandwidth extrapolation is the sum of these two portions(i.e. red plus green portions). Unlike the area of the redportion which is upper bounded by the area between theblue curve and the rectangular window, the area of the greenportion increases as the extrapolation factor is increased. Asa result, one would intuitively expect the SNR of a target tokeep increasing by increasing the extrapolation factor. This is,however, only true for strong targets where their frequencyresponses can be correctly modelled by the AR process. Inthe case of a weak target, the target frequency response maynot be captured by the AR process and hence will not beextrapolated. The energy gain for the weak target will thenmainly come from the red portion. In fact, the extrapolatedregion in that case would contain mainly noise, which canhave a negative effect on the overall SNR of the target. Forthis reason, it is advisable to keep the extrapolation factor low.We have found that the proposed technique works quite wellwith an extrapolation factor in the range of 2 to 3. One shouldremember that the SNR gain is an attractive by-product ofthe proposed bandwidth extrapolation technique. The primarybenefit is still the range resolution improvement.

Fig. 6. Windowing functions applied to the frequency response to controlrange sidelobe.

2) Experiment 2: In this experiment, the bandwidth of thetwo LFMCW waveforms were 10 kHz and 20 kHz. They bothhad 64 sweeps and a repetition frequency of 50 Hz. They weretransmitted at the carrier frequency of 17.642 MHz. Similar toExperiment 1, we plot the range-Doppler maps of the 10 kHzwaveform using both the normal processing technique and theproposed technique. We also plot the range-Doppler map of

the 20 kHz waveform using the normal processing techniqueand use it as the benchmark. The three range-Doppler maps areshown in Fig. 5. In the collected data, there were five targetsclearly present with each having a different Doppler shift. Theywere all very close in range. As can be seen from Fig. 5,the proposed bandwidth extrapolation technique improves therange resolution of all five targets. Similar to Experiment 1,the time-varying nature of the ionosphere caused the energy ofthe propagation paths during the transmission of the 20 kHzwaveform to be lower than that during the transmission of the10 kHz waveform.

V. CONCLUSIONS

In this paper, we have proposed a new bandwidth ex-trapolation technique that is capable of improving the rangeresolution for narrowband radar systems. Unlike an existingtechnique designed for wideband radar systems that has arestriction on the size of the range depth of interest (RDOI),the proposed technique allows the RDOI to be as large as thewaveform unambiguous distance. This capability is achievedby using the correlation approach to generate the range profileof the targets as opposed to using the stretch processingtechnique. In addition, the proposed technique carries out thebandwidth extrapolation after Doppler processing. As a result,the proposed technique is capable of improving the rangeresolution in the presence of strong stationary or slow movingclutter. We have verified the performance of the proposedbandwidth extrapolation technique using real data collectedby an over-the-horizon (OTH) radar of the Jindalee OperationRadar Network (JORN). We have shown that the proposedtechnique is capable of improving the range resolution evenin the presence of clutter that is 20 − 50 dB stronger thanthe targets. In addition, we have found that the proposedbandwidth extrapolation technique improves the SNR of thetargets by an amount of 2− 2.8 dB.

ACKNOWLEDGMENT

The authors wish to thank Mr Jim Ayliffe for his involve-ment with the data collection.

REFERENCES

[1] K. M. Cuomo, “A bandwidth extrapolation technique for improved rangeresolution of coherent radar data,” MIT Lincoln Laboratory Project ReportCPJ-60, Rev. 1, DTIC #ADA-258462, Dec. 1992.

[2] S. L. Borison, S. B. Bowling and K. M. Cuomo, “Super-resolutionmethods for wideband radar,” The Lincoln Laboratory Journal, vol. 5,pp. 441-461, 1992.

[3] T. G. Moore, B. W. Zuerndorfer and E. C. Burt, “Enhanced imagery usingspectral-estimation-based techniques,” The Lincoln Laboratory Journal,vol. 10, pp. 171-186, 1997.

[4] K. M. Cuomo, J. E. Piou and J. T. Mayhan, “Ultrawide-band coherentprocessing,” IEEE Trans. Antennas adn Propagation, vol. 47, pp. 1094-1107, June 1999.

[5] J. Yu and J. Krolik, “Multiband chirp synthesis for frequency-hoppedFMCW radar,” Asilomar 2009, pp. 1315-1319.

[6] M. D. E. Turley, “Signal processing techniques for marintime surveillancewith skywave radar,” 2008 International Conference on Radar, Adelaide,Australia, pp. 241-246, Sept 2-5, 2008.

[7] D. N. Swingler and R. S. Walker, “Line-array beamforming using linearprediction for aperture interpolation and extrapolation,” IEEE Trans.Acoust., Speech, Signal Processing, vol. 37, pp. 16-30, Jan. 1989.

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