MIMO Radar Limitations in Clutter G. J. Frazer∗ , Y. I. Abramovich∗ , and B. A. Johnson‡ ∗ ISR Division, DSTO, Edinburgh, SA, AUSTRALIA
[email protected] ‡ LM Australia Pty. Ltd. Edinburgh, SA, AUSTRALIA
Abstract— Recently discovered bounds for the clear-area available in the ambiguity domain for a MIMO radar waveform set are shown to be a significant limitation in radar applications having extended scatterer distributions. These limitations are demonstrated using an example from an experimental MIMO over-the-horizon radar. Two candidate waveform classes; timestaggered LFMCW, and frequency-staggered LFMCW, are suggested as possible MIMO waveforms in the case of an extended clutter scatterer environment. Each attains the best case bound on reduced clear-area in ambiguity. Limitations discussed in this paper impose significant restrictions to the applicability of MIMO techniques in other clutter-limited applications, such as airborne radar. Index Terms— MIMO radar, waveform design, ambiguity function, HF radar, over-the-horizon radar, experimental results
I. I NTRODUCTION In a series of papers [1]–[6] we have reported our investigations into the use of multiple-input multiple-output (MIMO) radar techniques [7], [8] for over-the-horizon radar (OTHR) [9]–[11]. Topics covered include; perspectives on how MIMO may be applied in OTHR, the impact of MIMO on transmitter sub-systems, using MIMO to determine the directionof-departure of a radar signal from a transmitter array, and, a demonstration of adaptive spatial clutter rejection using adaptation-on-transmit implemented using MIMO techniques. Here we provide further results based on the experiment discussed in [4], [5]. In particular, we provide experimental evidence demonstrating the impact of the MIMO radar ambiguity function volume and height bounds derived in [12]. The paper layout is as follows. In the following section a recently reported result concerning limitations introduced when applying MIMO waveform sets is reviewed. In the body of the paper we discuss the underlying experiment and present the results which are the main contribution of the work. We finish with further discussion and conclusions. II. C LEAR - SPACE IN THE MIMO A MBIGUITY FUNCTION In [12] the result from the 1965 paper by Price and Hofstetter [13] concerning bounds on the volume and height of the monostatic radar ambiguity function was extended for the case of MIMO waveform sets. To explain the connection with the results reported in this paper, we repeat and summarise the essential argument detailed in [12]. A conventional single-waveform monostatic or quasi-monostatic radar observing a point target located at
the range (delay) τ = 0 and Doppler frequency f = 0, together with clutter distributed over the region A with density 2 σc φ(τ, f ) for τ, f ∈ A (1) 0 for τ, f ∈ / A, has a performance that is quantified by the signal-to-clutter ratio [14] σs2 |χuu (0, 0)|2
qc = σc2
ZZ τ,f ∈A
where σs2 is signal power, and |χuu (τ, f )|2 is the symmetric form of Woodward’s well-known ambiguity function Z ∞ τ τ u(t − ) u∗ (t + ) exp[−j2πf t] dt (3) χuu (τ, f ) = 2 2 −∞ and u(t) is the radar waveform. It follows from (2) that the maximum radar performance in the presence of clutter is directly associated with the problem addressed by Price and Hofstetter [13] regarding limitations on the volume under the ambiguity function ZZ |χuu (τ, f )|2 dτ df. (4) V (A) = A
They demonstrated that if the ambiguity function has an impulse of volume V0 located at the origin, the region A is convex (in the τ -f plane), and every point in the region satisfies the condition (τ, f ) ∈ A ⇒ (−τ, −f ) ∈ A, then V (A) ≥
978-1-4244-2871-7/09/$25.00 ©2009 IEEE
1 V0 C(A) 4
(5) (6)
where C(A) denotes the area of the region A. The so-called “volume-clearance” condition relating to the ambiguity function is the maximum area in the range-Doppler plane that can be “cleared” of sidelobes. The “volumeclearance” condition immediately follows from (6), that is, if C(A) > 4 then |χuu (τ, f )|2 must contain some volume in A in addition to that of the impulse at the origin (V0 ). Moreover, it was demonstrated in [13], [15] that this upper bound of four on the clear area under |χuu (τ, f )|2 cannot be improved upon, since the periodic pulse train uT (t) =
The experiment was conducted under DSTO/RLM collaboration 290905.
(2)
|χuu (τ, f )|2 φ(τ, f ) dτ df
∞ X n=−∞
δ(t − nT )
(7)
has a clear area of four. This fundamental property of the ambiguity function implies that the maximum range-Doppler area that can be observed without ambiguity (“range-folding”, “Doppler-frequency aliasing”) is unity. In [12] it is shown that for a MIMO radar waveform set with cardinality K (i.e. K waveforms in the waveform set) then the result in (6) is modified and the clear area of four reduces to 4/K. More specifically, in [12] we defined the MIMO radar delay-Doppler ambiguity function (in contrast with the multistatic MIMO ambiguity functions provided in [16] and [17]) as K X K X |χjk (τ, f )|2 (8) |χK (τ, f )|2 ≡ j=1 k=1
with Z
∞
uj (t −
χjk (τ, f ) = −∞
τ ∗ τ ) u (t + ) exp[−j2πf t] dt (9) 2 k 2
where there are K waveforms uk (t) in the waveform set. The MIMO waveform set “volume-clearance” condition is now; if C(A) > 4/K, then |χK (τ, f )|2 must contain some volume in A in addition to the volume of the impulse at the origin. In the case of radar applications where the radar scatterer distribution (both clutter and targets) is other than a single point scatterer located at a given Doppler and delay then the implications of the clear area result changing from four to 4/K are profound. The ambiguity domain extent of the clutter and target scatterer distribution must now also be limited to 1/K otherwise measurements from one member of a K member waveform set will interact (i.e. be ambiguous) with one or more remaining members of the waveform set. In the following this circumstance is demonstrated using experimental data collected using a K = 14 waveform set in an OTHR application. III. E XPERIMENTAL MIMO OVER -T HE -H ORIZON R ADAR The results reported in this paper were derived from an experiment that investigated the application of MIMO waveform techniques to OTHR. The High Frequency (HF) OTHR used in the experiment was the JORN Laverton radar located in West Australia. See [11] for a description of JORN. The motivation for our work and experimental configuration we have used has previously been described in [4] and [5]. A short review is provided in the following section. A. Physical Arrangements The OTH radar transmitter is located approximately 100km east of the receiver system in central West Australia. Two radar transponders were deployed approximately 1200km north of the radar near the towns of Broome and Curtin. In this essentially monostatic geometry, these produce “target-like” returns at a specified Doppler shift and with enhanced radar cross section. Ionospheric soundings along the OTHR transmitter to Broome transponder and Curtin transponder to OTHR receiver paths were provided using oblique incidence sounding equipment, allowing for determination of the number and
range offset of propagating modes present in the “target-like” returns. The OTHR transmitter antenna array used to radiate the experimental waveform set consists of fourteen log-periodic dipole array (LPDA) elements arranged as an equi-spaced linear array. The array inter-element spacing is 12.5 m and the total aperture size is 162.5 m. Each member of the waveform set was radiated from a separate LPDA in the linear array. The transmitted MIMO radar signal comprising the waveform set was measured using synchronised radar receivers at Broome (for the one-way path) and at the OTHR receiver location (backscatter two-way path). For the analysis reported here, only a single receiver channel at the OTHR receiver was used (in which case we are strictly discussing MISO although the results apply directly to MIMO). B. Signal Model and Notation The MIMO based transmitter scheme consists of K transmitter antenna array elements located in a single array (spaced slightly greater than λ2 to avoid the issues discussed in [2]) concurrently transmitting K different radar waveforms u1 (t), . . . , uk (t), . . . , uK (t) of equal energy. A single radar receiver is located to receive the scattered waveform set. K X ak (θd )uk (t − τ0 )ej2πν0 t (10) Z(t) = η0 k=1
where η0 is a random complex scattering coefficient that is assumed identical for all waveforms, τ0 and ν0 are the delay and Doppler shift of the scatterer, and, aK (θd ) ≡ [a1 (θd ), . . . , aK (θd )]T
(11)
is the K-variate transmit antenna array manifold (steering) vector for the direction-of-departure θd that will illuminate the scatterer. The received radar return Z(t) is processed as for conventional radar receive processing (matched filter, etc) K times, once for each waveform in the waveform set as the reference waveform. To provide optimum transmit beamforming performance the cross-ambiguity between the K members of the waveform set should be selected in consideration of the expected target and clutter scatterer distribution. Let zi,j,k be the received energy in the (i, j)th Dopplerdelay cell for the k th reference waveform. Assume i ∈ 1, . . . , Nd Doppler cells and j ∈ 1, . . . , Nr delay cells. Because of the element-space [1] MIMO approach used in this experiment, there is a one-to-one mapping between the k th waveform and the k th element in the linear equi-spaced transmitter array. For brevity write the transmitter spatial signal as zi,j = [zi,j,1 , . . . , zi,j,k , . . . , zi,j,K ]T for the case of K waveforms and elements in the transmitter array. A transmit beam in the azimuthal direction θ can therefore be formed for all or a selected subset of (i, j) Doppler-delay cells, for a given beamformer w(θ), as yi,j,θ = w(θ)H zi,j
(12)
For a conventionally formed beam w(θ) = [a1 , a2 · ejθ , . . . , aK · ej(K−1)θ ]T
(13)
where a = [a1 , a2 , . . . , aK ]T is some aperture window sequence selected to achieve desired beam sidelobe performance.
Waveform set
fWRF Hz
K
τAMB ms
A B
4 30
14 14
250 33.3
τAMB K
ms
17.9 2.4
TABLE I E XPERIMENT WAVEFORM PARAMETER SETS .
A Doppler-delay map measured using waveform set A is shown in Fig. 1. The map domain shows one full unambiguous Doppler interval (from -2Hz to 2Hz) and slightly more than one-half of the 4/K reduced delay ambiguous interval (12ms of 17.9ms). In this case the delay extent of the scatterer distribution is limited by ionospheric propagation and there is little evidence of delay-folded ambiguously measured clutter. The scatterer features of interest are the band of severe Doppler-spread clutter at all Doppler and delay 1.5ms to delay 4.5ms, and the location of multi-mode returns from the Broome (principle return at Doppler -1.01Hz and delay 8.64ms) and Curtin (principle return at Doppler -1.46Hz and delay 8.88ms) transponders. The corresponding Doppler-delay map measured using waveform set B is shown in Fig. 2. The map spans Doppler
10
8
time delay (ms)
IV. R ESULTS AND D ISCUSSION The two waveform sets used in the experiment comprised K = 14 waveforms each. All waveforms in both sets were based on a conventional linear frequency modulated continuous-wave (LFMCW) radar waveform. The K waveforms in each set were constructed by delaying the initial start time of each waveform with respect to the remaining members of the waveform set. We have called this waveform set timestaggered LFMCW. Waveform initial start-times were spaced equally in delay throughout one sweep repetition interval. This means that according to the previous 4/K result the clear-areain-ambiguity measurement property of both waveform sets is reduced by a factor of K = 14. For the particular waveform type we have used this reduced clear area in ambiguity manifests as reduced ambiguity interval in delay, while Doppler ambiguity remains unchanged. In other waveform set designs the reduced area in ambiguity will generally manifest as a different contraction of the clear-area in Doppler-delay. For example, an analogously defined frequency-staggered LFMCW waveform set [18]–[20] will retain the single set member delay ambiguity, however, the Doppler ambiguity will contract by the factor K. Regardless of the waveform parameters or type, the clear-area surrounding the impulse at the origin of the ambiguity function will contract by at least the waveform set cardinality K. The two waveform parameters sets used in our experiments reported here differed only in waveform repetition frequency (WRF) with the resultant key parameters listed in Table I. The unambiguous delay interval is listed for the one-waveform case for both the waveform sets as well as the effective unambiguous delay interval for the MIMO waveform set case (following clear-area-in-ambiguity contraction by K).
Transmit−beam fast−time v. Doppler map
6
4
2
0 −1.5
−1
−0.5
0
0.5
1
1.5
dBJu
Doppler (Hz)
Fig. 1. Doppler-delay map spanning Doppler from -2Hz to 2Hz and delay from 0ms to 12ms measured using a low WRF (4Hz) fourteen member (M = 14) waveform set. At this WRF and for M = 14 the delay scatterer domain is measured unambiguously. The scatterer features of interest are the band of severe Doppler-spread clutter at all Doppler and delay 1.5ms to delay 4.5ms, and the location of multi-mode returns from the Broome (principle return at Doppler -1.01Hz and delay 8.64ms) and Curtin (principle return at Doppler -1.46Hz and delay 8.88ms) transponders.
from -15Hz to 15Hz and delay from 7ms to 9.5ms which corresponds to the delay gate surrounding the Broome and Curtin transponder returns apparent in Fig. 1. However, in this case the use of a waveform set with K = 14 and fWRF = 30 Hz has meant that the Doppler-spread clutter at delay 1.5ms to delay 4.5ms shown in Fig. 1 is now ambiguously superposed on the delay gate containing the transponder returns. This delay folded clutter now totally obscures both transponders. The delay ambiguous behaviour observed in Fig. 2 can be further understood by considering the processed Doppler-delay map for the waveform B radiated signal measured locally to the transmitter array. This is shown in Fig. 3 with the Dopplerdelay map spanning Doppler from -15Hz to 15Hz and delay from 0ms to 9.5ms. In this case no scatterers are present (neither clutter nor transponder returns) and direct signal pointtarget-like returns can be observed spaced at a delay interval of 1/(fWRF ·K). Each peak in this Doppler-delay map corresponds to a different member of the waveform set and in an elementspace MIMO architecture these also correspond to the signal radiated from different antenna elements within the transmitter array. If the radar scatterer distribution extends in delay beyond the interval 1/(fWRF · K) then clutter from one waveform set member will be erroneously measured by one or more other waveform set members. Clutter located surrounding the delay of one ambiguity will be delay folded to regions of other ambiguities. Fig. 4 and Fig. 5 show spectrograms of the radiated signal local to the transmitter array, and the single receiver channel radar backscattered signal received at the OTHR receiver location, respectively. In the former, the fourteen individual members of the waveform set are clearly separated in delay. In the latter case, however, individual waveform set members
Fast−time v. Doppler map Reference waveform 01
Transmit−beam fast−time v. Doppler map 9.5
9 8
9
time delay (ms)
time delay (ms)
7
8.5
8
6 5 4 3 2
7.5
1
−10
−5
0
5
10
0
dBJu
−10
Doppler (Hz) (Pk=166.4 dBJu @ 0.2 Hz and 8.475 ms)
Fig. 2. Doppler-delay map spanning Doppler from -15Hz to 15Hz and delay from 7ms to 9.5ms measured using a medium WRF (30Hz) waveform set with fourteen members (M = 14). At this WRF and with M = 14 the unambiguous delay is 2.47ms. The Doppler-delay map shows one complete delay ambiguity centred on the delay of the two transponder returns. Banded Doppler-spread clutter is delay-folded into the delay region of the transponder returns which are now obscured (Broome at Doppler -1.01Hz and delay 8.64ms, and, Curtin at Doppler -1.46Hz and delay 8.88ms).
−5
0
5
10
dBJu
Pk=201.7 dBJu ( 0.0 Hz 2.5 ms) Reference waveform 01
Fig. 3. Doppler-delay map spanning Doppler from -15Hz to 15Hz and delay from 0ms to 9.5ms measured using a medium WRF (30Hz) waveform set with fourteen members (M = 14). This measurement was recorded at the antenna located in front of the transmitting array and does not contain delay (or Doppler) distributed clutter such as measured in the received backscatter signal. There are four delay ambiguities present for this delay interval. 15
frequency (KHz)
10
are no longer distinguishable following backscatter from the complex scatterer distribution illuminated by the OTHR.
5
0
−5
V. C ONCLUSIONS −10
The experimental results presented in the paper provide an effective demonstration of an important limitation of MIMO radar. The interaction between the MIMO waveform set cardinality K and the intended radar scatterer distribution comprising targets and clutter is more demanding than for conventional single waveform radars. This is not a property of a particular waveform set, but is a fundamental property of the MIMO waveform ambiguity function. Of course, particular waveform set designs may be more or less suitable for particular scatterer distributions, since the detail of exactly how the reduced 4/K clear-area in ambiguity is matched to the intended scatterer distribution remains a design choice available to the radar designer. Appropriate waveform set selections based on time-staggered LFMCW or frequency-staggered LFMCW, with suitable selection flexibility in fWRF and K, appear to cover most cases found in OTHR. Any conventional (i.e. nonMIMO) radar application that is presently forced to manage ambiguity in Doppler, or delay, or both, using high, medium and low pulse repetition frequency approaches (such as airborne microwave radar) will be particularly susceptible to the limitations introduced by the 4/K property of the MIMO ambiguity function. R EFERENCES [1] G. J. Frazer, Y. I. Abramovich, and B. A. Johnson, “Spatially waveform diverse radar: Perspectives for high frequency OTHR,” in Proceedings of the IEEE Radar Conference. Boston, MA, USA: IEEE, Apr 17–20 2007, pp. 385 – 390.
−15 0.02
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0.08
0.1
0.12
0.14
0.16
0.18
0.2
time (s): fs=62.50 KHz fc=11.208 MHz
Fig. 4. Spectrogram showing a segment (0.2s) of the radiated waveform set as measured locally to the transmitter facility. The fourteen individual members of the waveform set are clearly separated in delay. Amplitude variation between waveform set members arises due to the measurement antenna being located in front and beyond the end of the transmitting array at 74deg with respect to boresight.
[2] B. A. Johnson, Y. I. Abramovich, and G. J. Frazer, “HF skywave spatially diverse waveform radar: Transmitter subsystem implementation issues,” in Proceedings of the Antennas, Radar and Wave Propagation (ARP) Symposium. Montreal, Canada: IASTED, May 30 - June 1 2007, pp. 208–213. [3] G. J. Frazer, B. A. Johnson, and Y. I. Abramovich, “Orthogonal waveform support in MIMO HF OTH radars,” in Proceedings of the Waveform Design and Diversity Conference. Pisa, Italy: IEEE, June 2007, pp. 1–5. [4] G. J. Frazer, Y. I. Abramovich, and B. A. Johnson, “Recent results in MIMO Over-the-Horizon Radar,” in Proceedings of the IEEE Radar Conference. Rome, Italy: IEEE, May 2008. [5] ——, “Use of adaptive non-causal transmit beamforming in OTHR: Experimental results,” in Proceedings of the IEEE International Radar Conference, Adelaide, Australia, September 2008. [6] ——, “HF skywave MIMO radar: The HiLOW experimental program,” in Proceedings of the 42th Asilomar Conference on Signals, Systems and Computers. Pacific Grove, CA, USA: IEEE, Oct 2008. [7] J. Dorey and G. Garnier, “The RIAS pulsed synthetic-antenna radar,” L’Onde Electrique, vol. 69, pp. 36–44, Dec. 1989. [8] F. C. Robey, S. Coutts, D. Weikle, J. C. McHarg, and K. Cuomo, “MIMO radar theory and experimental results,” in 38th Asilomar Conference on Signals, Systems and Computers, vol. 1, 2004, pp. 300–304. [9] M. I. Skolnik, Radar Handbook, 3rd ed. McGraw-Hill, 2008. [10] S. B. Colegrove, “Project Jindalee: From bare bones to operational
15
frequency (KHz)
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−15 0.02
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time (s): fs=31.25 KHz fc=11.208 MHz
Fig. 5. Spectrogram showing a segment (0.2s) of the received backscattered radar waveform set. The time delay between members of the waveform set is not sufficient to separate waveform set members in the presence of earth return clutter. The delay depth of the clutter is greater than the waveform set delay separation.
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