**2. Problem Formulation**

Address a typical multi-static passive radar localization scenario as presented in Figure 1, where *M* non-cooperative transmitters located at **s**<sup>o</sup> t,*<sup>m</sup>* = [*x*<sup>o</sup> t,*m*, *<sup>y</sup>*<sup>o</sup> t,*m*, *<sup>z</sup>*<sup>o</sup> t,*m*] <sup>T</sup> (*m* = 1, 2, ... , *M*) are employed to illuminate the surveillance area, and *N* receivers located at **s**<sup>o</sup> r,*<sup>n</sup>* = [*x*<sup>o</sup> r,*n*, *y*<sup>o</sup> r,*n*, *z*<sup>o</sup> r,*n*] <sup>T</sup> (*n* = 1, 2, ... , *N*) are deployed to determine a single target's position denoted by **u**<sup>o</sup> = [*x*o, *y*o, *z*o] T. In fact, the exact positions of the transmitters and receivers might not be known, and only the inaccurate measured versions, i.e., **s**t,*<sup>m</sup>* = [*x*t,*m*, *y*t,*m*, *z*t,*m*] <sup>T</sup> and **s**r,*<sup>n</sup>* = [*x*r,*n*, *y*r,*n*, *z*r,*n*] T, are available for processing. Formulaically, we arrive at

$$\mathbf{s}\_{\mathbf{t},\mathcal{W}} = \mathbf{s}\_{\mathbf{t},\mathcal{W}}^{\bullet} + \Delta \mathbf{s}\_{\mathbf{t},\mathcal{W}} \tag{1}$$

$$\mathbf{s}\_{\mathbf{r},\mathcal{U}} = \mathbf{s}\_{\mathbf{r},\mathcal{U}}^{\bullet} + \Delta \mathbf{s}\_{\mathbf{r},\mathcal{U}} \tag{2}$$

where **Δs**t,*<sup>m</sup>* and **Δs**r,*<sup>n</sup>* are the position error of the *m*th transmitter and the *n*th receiver respectively and also referred to as position uncertainty. Stacking (1) and (2) with respect to all the transmitters and receivers, yields a 3(*M* + *N*)-by-1 transmitter and receiver position vector as

$$\mathbf{s} = \mathbf{s}^{\bullet} + \Delta \mathbf{s} \tag{3}$$

where **s** = [**s**<sup>T</sup> <sup>t</sup> , **<sup>s</sup>**<sup>T</sup> r ] <sup>T</sup> with **<sup>s</sup>**<sup>t</sup> = [**s**<sup>T</sup> t,1, **<sup>s</sup>**<sup>T</sup> t,2, ... , **<sup>s</sup>**<sup>T</sup> t,*M*] <sup>T</sup> and **<sup>s</sup>**<sup>r</sup> = [**s**<sup>T</sup> r,1, **<sup>s</sup>**<sup>T</sup> r,2, ... , **<sup>s</sup>**<sup>T</sup> r,*N*] <sup>T</sup> is the noisy transmitter and receiver position vector, **s**<sup>o</sup> = [(**s**<sup>o</sup> t ) T,(**s**<sup>o</sup> r ) T] <sup>T</sup> with **<sup>s</sup>**<sup>o</sup> <sup>t</sup> = [(**s**<sup>o</sup> t,1) T,(**s**<sup>o</sup> t,2) T, ... ,(**s**<sup>o</sup> t,*M*) T] T and **s**<sup>o</sup> <sup>r</sup> = [(**s**<sup>o</sup> r,1) T,(**s**<sup>o</sup> r,2) T, ... ,(**s**<sup>o</sup> r,*N*) T] <sup>T</sup> is the true transmitter and receiver position vector, and **Δs** = [**Δs**<sup>T</sup> <sup>t</sup> , **<sup>Δ</sup>s**<sup>T</sup> r ] <sup>T</sup> with **<sup>Δ</sup>s**<sup>t</sup> = [**Δs**<sup>T</sup> t,1, **<sup>Δ</sup>s**<sup>T</sup> t,2, ... , **<sup>Δ</sup>s**<sup>T</sup> t,*M*] <sup>T</sup> and **<sup>Δ</sup>s**<sup>r</sup> = [**Δs**<sup>T</sup> r,1, **<sup>Δ</sup>s**<sup>T</sup> r,2, ... , **<sup>Δ</sup>s**<sup>T</sup> r,*N*] <sup>T</sup> is the transmitter and receiver position error vector that can be assumed zero-mean Gaussian with covariance **Qs** without loss of generality.

**Figure 1.** Practical scenario geometry of multi-static passive radar in the presence of transmitter/receiver position error and calibration targets.

Using the above notations, the distance from the *m*th transmitter to the target of interest is equal to

$$R\_{\mathbf{t},m}^{\bullet} = \|\mathbf{u}^{\bullet} - \mathbf{s}\_{\mathbf{t},m}^{\bullet}\|\tag{4}$$

the distance from the target of interest to the *n*th receiver is equal to

$$R^{\bullet}\_{\mathbf{r},\mathbf{u}} = \|\mathbf{u}^{\bullet} - \mathbf{s}^{\bullet}\_{\mathbf{r},\mathbf{u}}\|\tag{5}$$

and the baseline distance with respect to the mth transmitter and nth receiver is

$$R\_{\mathbf{t},m,\mathbf{r},\mathbf{u}}^{\bullet} = \|\mathbf{s}\_{\mathbf{t},\mu}^{\bullet} - \mathbf{s}\_{\mathbf{t},\mathbf{u}}^{\bullet}\|\tag{6}$$

According to this, the BR measurement with respect to the *m*th transmitter and *n*th receiver, i.e., the sum of the distances from the *m*th transmitter to the target and the target to the *n*th receiver, can be formulized as

$$\begin{split} r\_{\rm m,n} &= r\_{\rm m,n}^{\rm O} + \Delta r\_{\rm m,n} \\ &= R\_{\rm t,n}^{\rm O} + R\_{\rm r,n}^{\rm O} - R\_{\rm t,m,r,n}^{\rm O} + \Delta r\_{\rm m,n} \\ &= ||\mathbf{u}^{\rm O} - \mathbf{s}\_{\rm t,n}^{\rm O}|| + ||\mathbf{u}^{\rm O} - \mathbf{s}\_{\rm r,n}^{\rm O}|| - ||\mathbf{s}\_{\rm t,m}^{\rm O} - \mathbf{s}\_{\rm r,n}^{\rm O}|| + \Delta r\_{\rm m,n} \end{split} \tag{7}$$

where *r*<sup>o</sup> *<sup>m</sup>*,*<sup>n</sup>* = *R*<sup>o</sup> t,*<sup>m</sup>* + *<sup>R</sup>*<sup>o</sup> r,*<sup>n</sup>* <sup>−</sup> *<sup>R</sup>*<sup>o</sup> t,*m*,r,*<sup>n</sup>* represents the true BR with respect to the *m*th transmitter and *n*th receiver, **Δ***rm*,*<sup>n</sup>* is the BR measurement noise. Herein, it is important to emphasize that the time delay measurement comes from the cross correlation operation between the target signal and the direct path reference signal [7], and its error characteristics are not affected by the transmitter/receiver position error since the transmitter/receiver position error is not involved in the estimation of time delay. Thus, the BR measurement noise is only related to the time delay measurement noise, and not related to the transmitter/receiver position error. Obviously, there will be *MN* BR measurements to be produced with respect to the *M* transmitters and *N* receivers, which can be recast into a *MN*-by-1 vector as

$$\mathbf{r} = \mathbf{r}^0 + \Delta \mathbf{r} \tag{8}$$

where **r** = [**r**<sup>T</sup> <sup>1</sup> ,**r**<sup>T</sup> <sup>2</sup> , ... ,**r**<sup>T</sup> *M*] <sup>T</sup> with **<sup>r</sup>***<sup>m</sup>* = [*rm*,1,*rm*,2, ... ,*rm*,*N*] <sup>T</sup> is the BR measurement vector, **r**<sup>o</sup> = [(**r**<sup>o</sup> 1 ) T,(**r**<sup>o</sup> 2 ) T, ... ,(**r**<sup>o</sup> *M*) T] <sup>T</sup> with **<sup>r</sup>**<sup>o</sup> *<sup>m</sup>* = [*r*<sup>o</sup> *m*,1,*r*<sup>o</sup> *<sup>m</sup>*,2, ... ,*r*<sup>o</sup> *<sup>m</sup>*,*N*] <sup>T</sup> is the true BR vector, and **Δr** = [**Δr**<sup>T</sup> <sup>1</sup> , **<sup>Δ</sup>r**<sup>T</sup> <sup>2</sup> , ... , **<sup>Δ</sup>r**<sup>T</sup> *M*] <sup>T</sup> with **<sup>Δ</sup>r***<sup>m</sup>* = [**Δ***rm*,1, **<sup>Δ</sup>***rm*,2, ... , **<sup>Δ</sup>***rm*,*N*] <sup>T</sup> is the BR measurement noise vector, which is usually assumed follow a Gaussian distribution with zero-mean and covariance **Qr**.

As presented in Figure 1, to alleviate the transmitter/receiver position error and enhance localization accuracy, *K* calibration targets located at **c**<sup>o</sup> *<sup>k</sup>* = [*x*<sup>o</sup> c,*k* , *y*<sup>o</sup> c,*k* , *z*<sup>o</sup> c,*k* ] <sup>T</sup> (*k* = 1, 2, ... ,*K*) are employed, and the BRs among the calibration targets and the transmitter-receiver pairs are also measured. Similarly, the exact positions of the calibration targets are not known to us, and the nominal versions denoted by **c***<sup>k</sup>* = [*x*c,*k*, *y*c,*k*, *z*c,*k*] <sup>T</sup> (*k* = 1, 2, ... ,*K*) are given as

$$\mathbf{c}\_{k} = \mathbf{c}\_{k}^{\diamond} + \Delta \mathbf{c}\_{k} \tag{9}$$

where **Δc***<sup>k</sup>* is the position error of the *k*th calibration target. Collecting (9) for all the *K* calibration targets forms a 3*K*-by-1 calibration target position vector as

$$\mathbf{c} = \mathbf{c}^{0} + \Delta \mathbf{c} \tag{10}$$

where **c** = [**c**<sup>T</sup> <sup>1</sup> , **<sup>c</sup>**<sup>T</sup> <sup>2</sup> , ... , **<sup>c</sup>**<sup>T</sup> *K*] <sup>T</sup> is the nominal calibration target position vector, **<sup>c</sup>**<sup>o</sup> = [(**c**<sup>o</sup> 1 ) T,(**c**<sup>o</sup> 2 ) T, ... ,(**c**<sup>o</sup> *K*) T] <sup>T</sup> is the true calibration target position vector, and **<sup>Δ</sup><sup>c</sup>** = [**Δc**<sup>T</sup> <sup>1</sup> , **<sup>Δ</sup>c**<sup>T</sup> <sup>2</sup> , ... , **<sup>Δ</sup>c**<sup>T</sup> *K*] <sup>T</sup> is the calibration target position error vector that is usually supposed to obey Gaussian distribution with zero-mean and covariance **Qc**. Herein, it should be pointed out that, the positions of calibration targets are generally considered to be more precise compared with those of the transmitters and receivers, although they are also contaminated by errors.

Then, the distance from the *m*th transmitter to the *k*th calibration target is given by

$$R^{\bullet}\_{\mathbf{c},k,\mathbf{t},\mathbf{m}} = \|\mathbf{c}^{\bullet}\_{k} - \mathbf{s}^{\bullet}\_{\mathbf{t},m}\|\tag{11}$$

and the distance from the *k*th calibration target to the *n*th receiver is given by

$$R^{\bullet}\_{\mathbf{c},k,\mathbf{r},n} = \|\mathbf{c}^{\bullet}\_{\mathbf{k}} - \mathbf{s}^{\bullet}\_{\mathbf{r},n}\|\tag{12}$$

Based on this, the BR measurement corresponding to the *k*th calibration target, *m*th transmitter and *n*th receiver can be modeled as

$$\begin{split} r\_{\mathbf{c},k,m,n} &= r\_{\mathbf{c},k,m,n}^{\mathbf{o}} + \Delta r\_{\mathbf{c},k,m,n} \\ &= R\_{\mathbf{c},k,\mathbf{t},m}^{\mathbf{o}} + R\_{\mathbf{c},k,\mathbf{r},n}^{\mathbf{o}} - R\_{\mathbf{t},m,\mathbf{r},n}^{\mathbf{o}} + \Delta r\_{\mathbf{c},k,m,n} \\ &= ||\mathbf{c}\_{\mathbf{k}}^{\mathbf{o}} - \mathbf{s}\_{\mathbf{t},m}^{\mathbf{o}}|| + ||\mathbf{c}\_{\mathbf{k}}^{\mathbf{o}} - \mathbf{s}\_{\mathbf{r},n}^{\mathbf{o}}|| - ||\mathbf{s}\_{\mathbf{t},m}^{\mathbf{o}} - \mathbf{s}\_{\mathbf{r},n}^{\mathbf{o}}|| + \Delta r\_{\mathbf{c},k,m,n} \end{split} \tag{13}$$

where **Δ***r*c,*k*,*m*,*<sup>n</sup>* represents measurement noise in *r*c,*k*,*m*,*n*, *r*<sup>o</sup> c,*k*,*m*,*<sup>n</sup>* <sup>=</sup> *<sup>R</sup>*<sup>o</sup> c,*k*,t,*<sup>m</sup>* <sup>+</sup> *<sup>R</sup>*<sup>o</sup> c,*k*,r,*<sup>n</sup>* <sup>−</sup> *<sup>R</sup>*<sup>o</sup> t,*m*,r,*<sup>n</sup>* represents the true BR with respect to the *k*th calibration target, *m*th transmitter and *n*th receiver. Collecting (13) for the set of *K* calibration targets, *M* transmitters and *N* receivers, results in a *KMN*-by-1 vector as

$$\mathbf{r}\_{\mathbb{C}} = \mathbf{r}\_{\mathbb{C}}^{\mathbb{O}} + \Delta \mathbf{r}\_{\mathbb{C}} \tag{14}$$

where **r**<sup>c</sup> = [**r**<sup>T</sup> c,1,**r**<sup>T</sup> c,2, ... ,**r**<sup>T</sup> c,*K*] <sup>T</sup> with **<sup>r</sup>**c,*<sup>k</sup>* = [**r**<sup>T</sup> c,*k*,1,**r**<sup>T</sup> c,*k*,2, ... ,**r**<sup>T</sup> c,*k*,*M*] <sup>T</sup> and **<sup>r</sup>**c,*k*,*<sup>m</sup>* <sup>=</sup> [*r*c,*k*,*m*,1,*r*c,*k*,*m*,2, ... ,*r*c,*k*,*m*,*N*] <sup>T</sup> denotes the BR measurement vector from the calibration targets, **r**o <sup>c</sup> = [(**r**<sup>o</sup> c,1) T,(**r**<sup>o</sup> c,2) T, ... ,(**r**<sup>o</sup> c,*K*) T] <sup>T</sup> with **<sup>r</sup>**<sup>o</sup> c,*<sup>k</sup>* = [(**r**<sup>o</sup> c,*k*,1) T,(**r**<sup>o</sup> c,*k*,2) T, ... ,(**r**<sup>o</sup> c,*k*,*M*) T] <sup>T</sup> and **<sup>r</sup>**<sup>o</sup> c,*k*,*<sup>m</sup>* = [*r*o c,*k*,*m*,1,*r*<sup>o</sup> c,*k*,*m*,2, ... ,*r*<sup>o</sup> c,*k*,*m*,*N*] <sup>T</sup> denotes the corresponding true value vector, **Δr**<sup>c</sup> = [**Δr**<sup>T</sup> c,1, **<sup>Δ</sup>r**<sup>T</sup> c,2, ... , **<sup>Δ</sup>r**<sup>T</sup> c,*K*] T with **Δr**c,*<sup>k</sup>* = [**Δr**<sup>T</sup> c,*k*,1, **<sup>Δ</sup>r**<sup>T</sup> c,*k*,2, ... , **<sup>Δ</sup>r**<sup>T</sup> c,*k*,*M*] <sup>T</sup> and **<sup>Δ</sup>r**c,*k*,*<sup>m</sup>* = [**Δ***r*c,*k*,*m*,1, **<sup>Δ</sup>***r*c,*k*,*m*,2, ... , **<sup>Δ</sup>***r*c,*k*,*m*,*N*] <sup>T</sup> denotes the

corresponding error vector, which is presumed to be a Gaussian random vector with zero mean and covariance **Qr**c.

Now, the purpose of this work is to determine the target position from the noisy BR measurements and the inaccurate transmitter/receiver positions. In particular, the calibration targets with known position and the corresponding BR measurements are also available to reduce the transmitter/receiver position error and improve localization accuracy.
