Notation and signal model

Robust direct data domain processing for MTI

2.2 Notation and signal model

Let us define the complex column vectorxcontaining the digitized complex vol- tages measured at a specific time (i.e. for a given range gate) at theN elements of an array and in M successive pulses. The overall signal received from the nth antenna element at themth pulse can be written as follows:

x_m;n¼asexp j2p nd

l sinqsþmf_s fr

þX^K1

i¼1

aiexp j2p nd

l sinqiþmfi

fr

þr_m;n (2.1)

whereas is the unknown complex amplitude of the target, qs is the target DOA (supposed to be known in the ideal case of perfect knowledge of target para- meters),dis the inter-element distance between the elements of the array,f_sis the target Doppler frequency (also supposed to be known in the ideal case of perfect knowledge of target parameters),f_r is the pulse repetition frequency (PRF) and l is the carrier wavelength. K1 is the number of narrowband interferences impinging on the array, ai, qi and f_i are the corresponding unknown complex amplitudes, DOAs and Doppler frequencies (i¼1;. . .;K1). Finally,r_m;n is the complex contribution of thermal noise at thenth antenna element at themth time instant.

The signal model defined in (2.1) can be conveniently reformulated in vec- torial form as follows:

x¼asa_sþX^K1

i¼1

aig_iþr (2.2)

wherea_sis theNM1 target space–time steering vector,g_iis theNM1 space–

time steering vector of the ith interference andrcontains the thermal noise con- tributions. As is apparent, both the useful target and the interference contributions are modelled as narrowband signal sources impinging on the antenna array from specific localized positions in the angle/Doppler plane. The thermal noise con- tribution can be considered as a white normal distributed random process.

Let us now define the two scalar complex variablesz_sandz_t as follows:

zs¼exp j2pd lsinqs

z_t¼exp j2p f_s fr

(2.3)

that are uniquely determined if the target parameters qs and f_s are assumed to be known. In addition, let us define Km ðMþ1Þ=2 and Kn ðNþ1Þ=2, which are the available DOF in the spatial and temporal domains, respectively.

Then, a space–time steering vector in the nominal target directionzcan be written as (see (2.4)):

z¼1 zs z²_s . . . z^K_sⁿ¹ zt ztzs . . . ztz^K_sⁿ¹ . . . z^K_t^m1 . . . z^K_t^m1z^K_sⁿ¹ (2.4) To retrieve the expression of the D³-STAP filter, it is useful to re-arrange the column vector xin a MN matrix form X, where xm;n is themth row and nth column element ofX. Let us also define the matrices:

X^ðK_m;n^mKnÞ¼Xðm:mþKm1;n:nþKn1Þ (2.5) wherem¼1;. . .;Km and n¼1;. . .;Kn. That is X^ðK_m;n^mKnÞ is a KmKn truncated version ofX, obtained takingKmrows starting from themth row andKncolumns starting from thenth column. In addition, letx^ðK_m;n^mKnÞbe a 1K_mK_nrow vector that contains the elements ofX^ðK_m;^m_n^KnÞread columnwise.

We observe that three sets of row vectors can be derived from the conventional single canceller principle (i.e. the 2-pulse Doppler filter), which contain inter- ference contributions only (the useful signal contribution is completely removed by the weighted subtraction), namely:

sx_m;n ¼x^ðK_m;n^mKnÞz¹_s x^ðK_m;nþ1^mKnÞ

tx_m;n ¼x^ðK_m;n^mKnÞz¹_t x^ðK_mþ1^mK_;n^nÞ

s;tx_m;n ¼x^ðK_m;n^mKnÞz¹_t z¹_s x^ðK_mþ1;nþ1^mKnÞ

(2.6)

wherem¼1;. . .;Kmandn¼1;. . .;Kn. The vectors in (2.6) are in total 3KmKmand they have dimensions 1KmKn each. They can be arranged in a 3KmKmKmKn

interference only matrixF₂, see also [4, 11 Chapter 12]. Specifically,F₂can be built as follows:

F₂¼

sx_1;1 ...

sx_Km;Kn

tx_1;1 ...

tx_Km;Kn

s;tx_1;1

...

s;tx_Km;Kn 2

66 66 66 66 66 66 66 66 66 66 64

3 77 77 77 77 77 77 77 77 77 77 75

(2.7)

A linear system matrixF can be obtained simply concatenating the row vectorz with the matrixF2. That is:

F¼ z F₂

" #

(2.8) By exploitingFin (2.8), theKmKn1 D³-STAP filterwis obtained by solving in the least squares sense the following linear system:

F w₁ w₂ ...

wKmKn

2 66 66 64

3 77 77 75¼

1 0 ...

0 2 66 66 4

3 77 77

5,Fw¼y (2.9)

As is apparent, the linear system in (2.9) tends to null the interference only matrix F₂, while preserving a non-zero look direction constraint in the nominal target direction (first row ofF). In addition, one should note that the linear system in (2.9) has to be solved in a least square sense, sinceFis generally non-squared.

It has also to be noticed that the D³-STAP filter deriving from solving (2.9) corresponds to the so-called forward (FW) implementation [11 Chap- ter 12], where the termFWcomes from the observation that vectorxis spanned from the first to the last element in the weighted subtractions in (2.6). As mentioned in [11 Chapter 12], an analogous D³-STAP filter design can be obtained by considering a reversed complex conjugated version of the received data vectorx, and then building signal free weighted subtractions like in (2.6), where the vector is now spanned from the last to the first element. The resulting D³-STAP filter will lead to the so-called backward (BK) implementation.

Moreover, FW and BK weighted subtractions can be combined leading to the forward–backward (FB) implementation which gives additional equations to the linear system (2.9).

It should be clear that any uncertainty in the knowledge of the target para- metersqsandf_swould directly affect the scalar quantities in (2.3) and consequently the matrix F. As a consequence, the matrix F₂ will no more contain only inter- ference contributions but also a useful signal contribution, and also the look direction constraint (first row ofF) will be no more exact. The final results of such a mismatch (e.g. of this error) is the so-called target self-nulling effect, see [11 Chapter 12, 13]. In other words, if the target parameters differ from the nominal ones, the filter vector determined through (2.9) will treat the true target as interference, thus nulling it. This is a major drawback of D³-STAP, since an uncertainty in the target parameters determination is always to be expected in real scenarios. The first idea proposed to overcome this problem is to synthesize mul- tiple constraints covering the region of uncertainty of target parameters [12]. This can be easily obtained by replacing one or more rows in the matrixFin (2.8) with

constraints of the form:

h

1 z_s;l z²_s;l . . . z^K_s;ⁿ_l¹ . . . z_t;l z_t;lz_s;l . . . z_t;lz^K_s;ⁿ_l¹ . . . z^K_t;l^m1 z^K_t;l^m1z_s;l . . . z^K_t;l^m1z^K_s;lⁿ¹

i (2.10)

where:

z_s;l¼exp j2pd lsinqs;l

z_t;l¼exp j2pf_s;l fr

(2.11)

qs;l andf_s;l being the DOA and the Doppler frequency values of thelth constraint, and replacing with non-zero entries the corresponding elements of the vectory. As apparent, every additional main beam constraint reduces the number of available DOFs reserved for the interference suppression. In the following section, a robus- tification of D³-STAP (i.e. RD³-STAP) is presented to overcome this problem.

As a final general comment, the D³-STAP approach has been here presented for the uniform linear array (ULA) case. Under this hypothesis, the derivation of the interference only quantities in ^sx_m;nand in ^s;tx_m;nis simplified. Nevertheless, a generalization to planar arrays with irregular sub-array structures is possible.