Adaptive OFDM waveform design for spatio-temporal-sparsity exploited STAP radar
2.2 Sparse-measurement model
approach yields near-optimum performances by utilizing a substantially small number of secondary measurements. For example, in the ideal scenario, only two to five secondary data are found to be enough to produce a near-optimum perfor- mance. In the presence of temporal decorrelation, although we get a wider main- beam clutter notch, the near-optimum SINR-loss performance is still attained by using only two to five secondary measurements. A significant amount of improvement in performance due to the use of adaptive OFDM waveform is demonstrated by computing the ROCs for two different target responses. For example, in the presence and absence of temporal decorrelation effects, we respectively observe approximately 3 and 6 dB of improvement in detection performance.
The rest of the chapter is organized as follows. In Section 2.2, we first develop a parametric sparse-measurement model for OFDM-STAP radar. Then, in Section 2.3, we describe a sparse recovery technique to estimate the interference covariance matrix and to design the STAP filter weights. An adaptive OFDM waveform design algorithm is proposed in Section 2.4. We discuss the numerical results in Section 2.5.
Conclusions and possible future work are given in Section 2.6.
2.1.1 Notations
We present here some notational conventions that are used throughout this chapter.
We use math italic (a) for scalars, lowercase bold (a) for vectors and uppercase bold (A) for matrices. For a matrix A2Ckm, AT, AH and trf gA denote the transpose, conjugate transpose and trace ofA, respectively.Ik represents an iden- tity matrix of dimensionk. blkdiagð Þ forms a block-diagonal matrix with non- zero submatrices only on the main diagonal. Ref g is the real part and j jis the magnitude of a complex quantity. Among different types of vector norms, we consider the‘1and‘2norms, expressed ask k1andk k2, respectively. In addition, h;i,andare the inner product, Kronecker product and point-wise Hadamard product operators, respectively. For a random variablea,EðaÞand varðaÞ, respec- tively, denote the mean and variance ofa.
kðy;qÞ ¼ cosycosq^xþsinycosq^yþsinq^z, where ^x, ^y and ^z are the unit vectors of the Cartesian coordinate system.
In the following, we first describe the OFDM signal model and then develop a sparse-measurement model for a target located at a specific range gate and direc- tion. We also discuss in detail the effects of the interfering signals (clutter and thermal noise) on the received signal.
2.2.1 OFDM signal model
We consider a wideband OFDM signalling system with L active subcarriers, a bandwidth ofBHz, and a pulse duration ofTpseconds. Leta¼ ½a0;a1;. . .;aL1T represent the complex weights transmitted over the L subcarriers and satisfy PL1
l¼0 jalj2¼1. Then, the complex envelope of the transmitted signal can be represented as follows:
sðtÞ ¼XL1
l¼0
alej2plDft; for 0 t Tp (2.1)
where the subcarrier spacing is Df ¼B=ðLþ1Þ ¼1=Tp. Denoting fc to be the carrier frequency, a coherent burst ofN transmitted OFDM pulses [in a particular coherent processing interval (CPI)] is given by
esðtÞ ¼2 Re XN1
n¼0
sðtnTÞej2pfct
( )
(2.2)
z
Platform velocity Radar antenna array
x
Iso-range contours
y k(ψ,θ)
ψ θ
Figure 2.1 A schematic representation of the problem scenario
whereT is the pulse repetition interval (PRI). We point out here that, during the adaptive waveform design, we choose the spectral parameters of the OFDM waveform,al, in order to improve the STAP performance.
2.2.2 Sparse measurement model
We develop the OFDM-STAP measurement model similar to that in [2]. The only difference is that we use OFDM signalling technique, which gives rise to slightly different STAP models across different subchannels depending on the corre- sponding subcarrier frequencies. We consider that the target is at a far-field dis- tance r0 and along a direction ðyT;qTÞ and is moving with velocity vT. The distance r0 corresponds to a specific range gate, denoted by a round-trip delay t¼2r0=c, wherecis the speed of propagation. We further assume that the target has multiple scattering centres that resonate variably at different transmitted frequencies.
Then, the complex envelope of the received signal at the output of the lth subchannel is expressed as follows:
yðlÞ ¼zT;lalfðal;nlÞ þeðlÞ (2.3) wherezT;l is the target-scattering coefficient at thelth subchannel;
fðal;nlÞ ¼fDð Þ nl fSð Þal (2.4) is anMN1 space–time steering vector at thelth subchannel with
fDð Þ ¼nl 1;ej2pnl;. . .;ej2pðN1Þnl
h iT
(2.5) fSð Þ ¼al ej2pflt0h1;ej2pal;. . .;ej2pðM1ÞaliT
(2.6) representing the Doppler and spatial steering vectors, respectively; andeðlÞrepre- sents the interference along thelth subchannel. Here, the normalized spatial and Doppler frequencies are, respectively, defined as follows:
al¼flð1þbÞDt and nl¼flbT (2.7)
where fl¼fcþlDf is the lth subcarrier frequency; ð1þbÞ accounts for the stretching or compressing in time of the reflected signal due to the relative motion between the radar and target;b¼2hðvRvTÞ;ki=cis the relative Doppler shift;
andt¼ hd^y;ki=cis the interelement time delay. However, note that, in most of the practical scenarios,jbj 1, and therefore 1þb1.
Now, suppose instead of a specific pair ofðal;nlÞ, we consider all the possible combinations ofal;i;nl;j
fori¼1;2;. . .;Gaandj¼1;2;. . .;Gn. In other words, we discretize the spatio-temporal domain into G¼GaGn grid points (see Figure 2.2), whereGa andGn are the number of grids along the spatial and tem- poral axes, respectively. Then, a non-zero content from any such grid point would
suggest the presence of a scatterer at that particular spatial and temporal fre- quencies. Hence, we can rewrite the measurement model as follows:
yðlÞ ¼alFlxT;lþeðlÞ (2.8)
where
● Fl¼½flð1;1Þ flð1;2Þ flðGa;GnÞ is an MNG matrix containing all the possible combinations of spatial and Doppler steering vectors (for nota- tional simplicity, we writefðal;i;nl;jÞasflði;jÞ); and
● xT;lis aG1 sparse vector having only one non-zero entry corresponding to the target response at the true spatial and Doppler frequencies, i.e.,
xT;lði;jÞ ¼ zT;l; if al;i¼al and nl;j¼nl
0; otherwise (
Then, stacking allyðlÞs, we get the sparse-measurement model as
y¼YxTþe (2.9)
where
● y¼ ½yð0ÞT;. . .;yðL1ÞTT;
● Y¼blkdiagða0F0;. . .;aL1FL1Þ is an LMNLG sparse spatio-temporal measurement matrix;
Temporal response
Spatial response
Jamming
Clutter
Target
Normal ized Doppler frequency
Normalized spatial frequency
Figure 2.2 Spatio-temporal sparsity of the target and interference spectra
● xT ¼ ½xTT;0;. . .;xTT;L1T is an LG1 sparse vector having only L non-zero entries that are equal to the actual target-scattering coefficients
½zT;0;. . .;zT;L1T; and
● e¼ ½eð0ÞT;. . .;eðL1ÞTT is the interference vector.
The interference vectorecontains not only the thermal noise at the sensors but also the clutter returns. For an airborne radar, the main contribution to the clutter ori- ginates due to the ground reflections from all the azimuth directions. Though the ground has zero velocity, the ground-clutter is spread in both the angle and Doppler frequency due to the platform velocityvR. We represent the clutter returns (from a particular range gate) as a coherent summation of a large number (Nc) of clutter patches evenly distributed in azimuth angles yk;k¼1;2;. . .;Nc. Then, ignoring the effects of any ambiguous range gates and noticing that the target and clutter returns are affected in a similar way by the radar transmission, we construct a sparse representation of the interference model as follows:
e¼XNc
k¼1 YxC;kþn¼YxCþn (2.10)
where
● xC;k is an LG1 sparse vector havingL non-zero entries that correspond to the clutter returns from a specific azimuth angleyk;
● xC is anotherLG1 sparse vector with sparsity levelLNc, and it represents the overall clutter response; and
● nis the additive thermal noise component.
The construction of this type of interference model makes it explicit here that we are dealing with the signal-dependent clutter model (i.e., transmit signal parameter aaffects the interference vectorethrough the matrixY), which is a more realistic representation than to assume uncorrelated clutter returns with the transmitted signal.
2.2.3 Statistical assumptions
To complete the description of our measurement model, we assume that the target is a small manmade object for whichxT is deterministic and unknown. On the other hand, the clutter returns from a particular patch atðyk;qÞcan be considered to be originated from a large collection of incoherent point scatterers, and hence applying the central limit theorem, we model the clutter returns as a circularly symmetric, zero-mean complex Gaussian process with unknown covariance matrix RC;k. Conventionally, the clutter returns from different patches are assumed to be uncorrelated to each other. Hence, overall, the clutter covariance matrix is expressed asRC¼PNc
k¼1RC;k. Next, we consider that the thermal noise compo- nents among different subchannels are uncorrelated; they are also both spatially and temporally uncorrelated. So, the noise component n is represented as another
circularly symmetric, zero-mean complex Gaussian process with covariance matrix s2nILMN. Hence, overall, the OFDM-STAP measurements are distributed as follows:
yCNðyT;RIÞ (2.11)
whereyT ¼YxT andRI ¼PNc
k¼1 RC;kþs2nILMN. 2.3 STAP filter design
In STAP, the received data are first processed with a linear filter having weightsw to yield a scalar output:
z¼wHy (2.12)
The primary objective of designing the STAP-filter weightswis to maximize the SINR at the filter output:
SINR¼jEðzÞj2
varðzÞ ¼jwHyTj2
wHRIw yHTR1I yT (2.13)
Next,zis fed to a detector that chooses one of the two possible hypotheses: the null hypothesis (target-free hypothesis) or the alternate hypothesis (target-present hypothesis), which are mathematically expressed as follows:
H0: yp¼ep
ys¼es;s¼1;2;. . .;Ns
(
H1: yp¼yT þep
ys¼es;s¼1;2;. . .;Ns (
8>
>>
>>
<
>>
>>
>:
(2.14)
where subscript ‘p’ suggests the primary range gate measurements, subscript ‘s’
denotes the secondary measurements, and Ns is the number of secondary range gates considered. Based on the statistical assumptions of the previous section, we know that the optimum detector comparesjzjwith a predefined threshold [1]. The threshold is determined to achieve a specified probability of false alarm (PFA) of the decision problem. Now, for a fixed value ofPFA, the probability of detection (PD) of such a detector is given by [1]
PD¼Q ffiffiffiffiffiffiffiffiffiffiffi pSINR
; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lnPFA
p
(2.15) whereQð;Þis the MarcumQ-function of order 1.
Following [1,10], we know that the maximum SINR and consequently max- imum value ofPDare attained when the optimal STAP-filter weight is
wopt/ RI 1yT (2.16)
In practical scenarios, since the knowledge ofRI is unknown, we have to estimate it using the measurements of the secondary range gates. To efficiently estimateRI, we first employ the sparse-recovery technique to obtain the estimates ofxC and then premultiply it with the known sparse-measurement matrixY. We explicitly denote the secondary measurements asys¼YxC;sþns, fors¼1;2;. . .Ns.
We employ a LASSO estimator [58] on these secondary measurements to obtain b
xC;s¼arg min
xC;s kysYxC;sk22þeCkxC;sk1 (2.17) witheC ¼s2n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cClnðLGÞ
p andcC being a tuning parameter. Then, we estimate the clutter covariance matrix as follows:
RbC¼ 1 Ns
XNs
s¼1
X
g2Cbs
^xC;s;g
2YgYHg
2 64
3
75 (2.18)
wherecCs is the non-zero support-set of the estimated clutter response from thesth secondary data; andYg denotes thegth column of Y. Hence, the estimate of the overall interference covariance matrix is given by RbI ¼RbCþhILMN, where h¼10s2nis chosen as 10 times the white noise level, similar to the approach of [63].