**4.1 System model**

324 Wireless Communications and Networks – Recent Advances

**4. MIMO radar systems applied to wireless communications based on GASP**  Multiple-input multiple-output (MIMO) wireless communication systems have received a great attention owing to the following viewpoints: a) MIMO wireless communication systems have been deemed as efficient spatial multiplexers and b) MIMO wireless communication systems have been deemed as a suitable strategy to ensure high-rate communications on wireless channels (Foschini, 1996). Space-time coding has been largely investigated as a viable means to achieve spatial diversity, and thus to contrast the effect of fading (Tarokh, et al., 1998 and Hochwald, et al., 2000). We apply GASP to the design and implementation of MIMO wireless communication systems used space-time coding technique. Theoretical principles of MIMO wireless communication systems were discussed and the potential advantages of MIMO wireless communication systems are thoroughly considered in

MIMO architecture is able to provide independent diversity paths, thus yielding remarkable performance improvements over conventional wireless communication systems in the medium-high range of detection probability. As was shown in (Fishler, et al., 2006), the MIMO mode can be conceived as a means of bootstrapping to obtain greater coherent gain. Some practical issues concerning implementation (equipment specifications, dynamic range, phase noise, system stability, isolation and spurs) of MIMO wireless communication systems

MIMO wireless communication systems can be represented by *m* transmit antennas, spaced several wavelengths apart, and *n* receive antennas, not necessarily collocated, and possibly forwarding, through a wired link, the received echoes to a fusion center, whose task is to make the final decision about the signal in the input waveform. If the spacing between the transmit antennas is large enough and so is the spacing between the receive antennas, a rich scattering environment is generated, and each receive antenna processes *l* statistically independent copies of incoming signal. The concept of rich scattering environment is borrowed from communication theory, and models a situation where the MIMO architecture yields interchannel interference, eventually resulting into a number of independent random channels. Unlike a conventional wireless communication array system, which attempts to maximize the coherent processing, MIMO wireless communication system resorts to the fading diversity in order to improve the detection performance. Indeed, it is well known that, in conventional wireless communication array system, multiple access interference (MAI) of the order of 10 dB may arise. This effect leads to severe degradations of the detection performance, due to the high signal correlation at the array elements. This drawback might be partially circumvented under the use of MIMO wireless communication system, which exploits the channel diversity and fading. Otherwise, uncorrelated signals at the array elements are available. Based on mentioned above statements, it was shown in (Fishler, et al., 2006) that in the case of additive white Gaussian noise (AWGN), transmitting orthogonal waveforms result

Our approach is based on implementation of GASP and employment of some key results from communication theory, and in particular, the well-known concept that, upon suitably space-time encoding the transmitted waveforms, a maximum diversity order given by *m n* can be achieved. Importing these results in a wireless communication system scenario poses

(Fishler, et al., 2006).

are discussed in (Skolnik, 2008).

into increasingly constrained fluctuation of interference.

We consider MIMO radar system composed of *m* fixed transmitters and *n* fixed receivers and assume that the antennas as the two ends of the wireless communication system are sufficiently spaced such that a possible incoming message and/or interference provides uncorrelated reflection coefficients between each transmit/receive pair of sensors. Denote by ( ) *is t* the baseband equivalent of the coherent pulse train transmitted by the *i*-th antenna, for example,

$$s\_i(t) = \sum\_{j=1}^{N} a\_{i,j} p[t - (j-1)T\_p] \,, \qquad i = 1, \ldots, m \tag{40}$$

where *p*( )*t* is the signature of each transmitted pulse, which we assume, without loss of generality, with unit energy and duration *<sup>p</sup> τ* ;*Tp* is the pulse repetition time;

$$\mathbf{a}\_{i} = \begin{bmatrix} a\_{i,1'}, \dots, a\_{i,N} \end{bmatrix}^{T} \tag{41}$$

is an *N*-dimensional column vector whose entries are complex numbers which modulate both in amplitude and in phase the *N* pulses of the train, where ( )*<sup>T</sup>* denotes transpose. In the sequel, we refer to *<sup>i</sup>* **a** as the code word of the *i*-th antenna. The baseband equivalent of the signal received by the *i*-th sensor, from a target with two-way time delay *τ* , can be presented in the following form

$$\mathbf{x}\_{i}(t) = \sum\_{l=1}^{m} a\_{i,l} \sum\_{j=1}^{N} a\_{l,j} p[t - \tau - (j-1)T\_{p}] + n\_{i}(t) \quad , \qquad i = 1, \ldots, m \tag{42}$$

where , , 1, , *i l α i n* and *l m* 1, , , are complex numbers accounting for both the target backscattering and the channel propagation effects between the *l*-th transmitter and the *i*-th receiver; ( ) , 1, , *nt i n <sup>i</sup>* , are zero-mean, spatially uncorrelated, complex Gaussian random processes accounting for both the external and the internal disturbance. For simplicity, we assume a zero-Doppler target, but all the derivations can be easily extended to account for a possible known Doppler shift. We explicitly point out that the validity of the above model requires the narrowband assumption

$$\frac{d\_{\text{max}}^m + d\_{\text{max}}^n}{c} << \frac{1}{B} \tag{43}$$

where *B* is the bandwidth of the transmitted pulse, max *md* and max *<sup>n</sup> d* denote the maximum spacing between two sensors at the transmitter and the receiver end, respectively. The signal ( ) *<sup>i</sup> x t* , at each of the receive elements, is matched filtered to the pulse *p*( )*t* by preliminary filter of the GD and the filter output is sampled at the time instants ( 1) *<sup>p</sup> τ k T* , *k N* 1, , . Thus, denote by ( ) *<sup>i</sup> x k* the *k*-th sample, i.e.,

$$\mathbf{x}\_i(k) = \sum\_{l=1}^{m} a\_{i,l} a\_{l,k} + n\_i(k) \, , \tag{44}$$

where ( ) *n k <sup>i</sup>* is the filtered noise sample. Define the *N*-dimensional column vectors

$$\mathbf{x}\_{i} = \begin{bmatrix} \mathbf{x}\_{i}(\mathbf{1}), \dots, \mathbf{x}\_{i}(\mathbf{N}) \end{bmatrix}^{\mathrm{T}} \tag{45}$$

and rewrite them as

$$\mathbf{x}\_{i} = \mathbf{A}\mathbf{u}\_{i} + \boldsymbol{\xi}\_{\text{PF}\_{i}} \; \prime \quad \text{i} = \mathbf{1}, \ldots, n \tag{46}$$

where

$$\boldsymbol{\xi}\_{\mathrm{P}\overline{\boldsymbol{\varepsilon}}\_{i}} = \left[ \boldsymbol{\xi}\_{\mathrm{P}\overline{\boldsymbol{\varepsilon}}\_{i}}(\mathbf{1}), \dots, \boldsymbol{\xi}\_{\mathrm{P}\overline{\boldsymbol{\varepsilon}}\_{i}}(\mathbf{N}) \right]^{\mathrm{T}},\tag{47}$$

$$\mathbf{a}\_{i} = [a\_{i,1'}, \dots, a\_{i,m}]^T \; , \tag{48}$$

and the ( ) *N m* -dimensional matrix **A**, defined in the following form

is an *N*-dimensional column vector whose entries are complex numbers which modulate both in amplitude and in phase the *N* pulses of the train, where ( )*<sup>T</sup>* denotes transpose. In the sequel, we refer to *<sup>i</sup>* **a** as the code word of the *i*-th antenna. The baseband equivalent of the signal received by the *i*-th sensor, from a target with two-way time delay *τ* , can be presented

( ) [ ( 1) ] ( ) , 1, ,

where , , 1, , *i l α i n* and *l m* 1, , , are complex numbers accounting for both the target backscattering and the channel propagation effects between the *l*-th transmitter and the *i*-th receiver; ( ) , 1, , *nt i n <sup>i</sup>* , are zero-mean, spatially uncorrelated, complex Gaussian random processes accounting for both the external and the internal disturbance. For simplicity, we assume a zero-Doppler target, but all the derivations can be easily extended to account for a possible known Doppler shift. We explicitly point out that the validity of the above model

> max max 1 *m n d d c B*

cing between two sensors at the transmitter and the receiver end, respectively. The signal ( ) *<sup>i</sup> x t* , at each of the receive elements, is matched filtered to the pulse *p*( )*t* by preliminary filter of the GD and the filter output is sampled at the time instants ( 1) *<sup>p</sup> τ k T* , *k N* 1, , .

> , , 1 ( ) ( ) *m i il lk i l x k α a nk*

> > , 1

where ( ) *n k <sup>i</sup>* is the filtered noise sample. Define the *N*-dimensional column vectors

, (42)

*md* and max

(43)

, (44)

*ii i* **x** *x xN* (45)

*<sup>i</sup> i i PF* **x A α ξ** *i , ,n* (46)

*PF PF PF* **ξ** *ξ ξ N* , (47)

,1 , [ ] , , *<sup>T</sup>* **α***i i im α α* , (48)

*T*

*<sup>n</sup> d* denote the maximum spa-

*x t α a p t τ j T nt i n*

, , 1 1

*i il lj p i*

*m N*

*l j*

requires the narrowband assumption

Thus, denote by ( ) *<sup>i</sup> x k* the *k*-th sample, i.e.,

and rewrite them as

where

where *B* is the bandwidth of the transmitted pulse, max

(1), , ( ) [ ]*<sup>T</sup>*

(1), , ( ) [ ] *ii i*

and the ( ) *N m* -dimensional matrix **A**, defined in the following form

in the following form

$$\mathbf{A} = \begin{bmatrix} \mathbf{a}\_1, \dots, \mathbf{a}\_m \end{bmatrix} \tag{49}$$

has the code words as columns. This last matrix is referred to as the code matrix. We assume that **A** is full rank matrix. It is worth underlining that the model given by (46) applies also to the case that space-time coding is performed according to (De Maio & Lops, 2007), namely, by dividing a single pulse in *N* sub-pulses. The code matrix **A** thus defines *m* different code words of length *N*, which can be received by a single receive antenna, thus defining the multiple-input single-output (MISO) structure, as well as by a set of *n* receive antennas, as in the present study.

### **4.2 GD design for MIMO radar systems applied to wireless communications**

The problem of detecting a target return signal with a MIMO radar system can be formulated in terms of the following binary hypothesis test

$$\begin{cases} \mathsf{H}\_{0} \implies \mathbf{x}\_{i} = \mathsf{\xi}\_{PF\_{i}} \land & i = 1, \ldots, n \\ \mathsf{H}\_{1} \implies \mathbf{x}\_{i} = \mathbf{A} \mathbf{a}\_{i} + \mathsf{\xi}\_{PF\_{i}} \land & i = 1, \ldots, n \end{cases} \tag{50}$$

where , 1, , *PFi* **ξ** *i n* , are statistically independent and identically distributed (i.i.d.) zeromean complex Gaussian vectors with covariance matrix

$$E\left[\boldsymbol{\xi}\_{PF\_i}\boldsymbol{\xi}\_{PF\_i}^\*\right] = E\left[\boldsymbol{\xi}\_{AF\_i}\boldsymbol{\xi}\_{AF\_i}^\*\right] = \mathbf{M}\,\,\,.\tag{51}$$

Here [ ] *E* denotes the statistical expectation and( ) denotes conjugate transpose. The covariance matrix (51) is assumed positive definite and known. According to the Neyman-Pearson criterion, the optimum solution to the hypotheses testing problem (50) must be the likelihood ratio test. However, for the case at hand, it cannot be implemented since total ignorance of the parameters **α***<sup>i</sup>* is assumed. One possible way to circumvent this drawback is to resort to the generalized likelihood ratio test (GLRT) (Van Trees, 2003), which is tantamount to replacing the unknown parameters with their maximum likelihood (ML) estimates under each hypothesis. Applying GASP to the GLRT, we obtain the following decision rule

$$\frac{\max\_{\mathbf{a}\_1,\ldots,\mathbf{a}\_n} f(\mathbf{x}\_1,\ldots,\mathbf{x}\_n | \boldsymbol{\mathcal{H}}\_1,\mathbf{M},\mathbf{a}\_1,\ldots,\mathbf{a}\_n)}{f(\boldsymbol{\xi}\_{A\boldsymbol{F}\_1},\ldots,\boldsymbol{\xi}\_{A\boldsymbol{F}\_n} | \boldsymbol{\mathcal{H}}\_0,\mathbf{M})} \underset{\boldsymbol{\xi}}{\gtrless}\_{\boldsymbol{\mathsf{K}}\_0} \boldsymbol{K}\_{\boldsymbol{\mathsf{g}}}\,\prime \tag{52}$$

where <sup>1</sup> 1 1 (,,|,, ,, ) *n n f* **x x M** *H* **α α** is the probability density function (pdf) of the data under the hypothesis *H*<sup>1</sup> and 1 <sup>0</sup> ( ,, | ,) *AF AFn f* **ξ ξ** *H* **M** is pdf of the data under the hypothesis *H*<sup>0</sup> , respectively, *Kg* is a suitable modification of the original threshold. Previous assumptions imply that the aforementioned pdfs can be written in the following form:

$$f\left(\boldsymbol{\xi}\_{\boldsymbol{A}\boldsymbol{F}\_{1}},...,\boldsymbol{\xi}\_{\boldsymbol{A}\boldsymbol{F}\_{n}}\mid\boldsymbol{\mathsf{H}}\_{0},\boldsymbol{\mathsf{M}}\right) = \frac{1}{\pi^{\mathrm{N}n}\det^{n}\left(\boldsymbol{\mathsf{M}}\right)}\exp\left[-\sum\_{i=1}^{n}\boldsymbol{\mathsf{f}}\_{\boldsymbol{A}\boldsymbol{F}\_{i}}^{\*}\boldsymbol{\mathsf{M}}^{-1}\boldsymbol{\xi}\_{\boldsymbol{A}\boldsymbol{F}\_{i}}\right] \tag{53}$$

at the hypothesis *H*<sup>0</sup> and

$$f(\mathbf{x}\_1, \dots, \mathbf{x}\_n \mid \boldsymbol{\mathsf{H}}\_1, \mathbf{M}, \mathbf{a}\_1, \dots, \mathbf{a}\_n) = \frac{1}{\pi^{\mathrm{Nu}} \mathrm{det}^n(\mathbf{M})} \exp\left[ -\sum\_{i=1}^n (\mathbf{x}\_i - \mathbf{A} \mathbf{a}\_i)^\ast \mathbf{M}^{-1} (\mathbf{x}\_i - \mathbf{A} \mathbf{a}\_i) \right] \tag{54}$$

under the hypothesis *H*<sup>1</sup> , where det( ) denotes the determinant of a square matrix. Substituting (16) and (17) in (15), we can recast the GLRT based on GASP, after some mathematical transformations, in the following form

$$\sum\_{i=1}^{n} \boldsymbol{\xi}\_{AF\_i}^{\star} \mathbf{M}^{-1} \boldsymbol{\xi}\_{AF\_i} - \sum\_{i=1}^{n} \min\_{\mathbf{a}\_i} (\mathbf{x}\_i - \mathbf{A}\mathbf{a}\_i)^{\star} \mathbf{M}^{-1} (\mathbf{x}\_i - \mathbf{A}\mathbf{a}\_i) \Big\}\_{\begin{subarray}{c} \mathcal{H}\_0 \\ \mathcal{H}\_0 \end{subarray}}^{\mathcal{H}\_1} \boldsymbol{K}\_{\mathcal{g}} \,. \tag{55}$$

In order to solve the *n* minimization problems in (55) we have to distinguish between two different cases.

*Case* 1: *N m* . In this case, the quadratic forms in (55) achieve the minimum at

$$
\hat{\mathbf{a}}\_{i} = (\mathbf{A}^{\ast}\mathbf{M}^{-1}\mathbf{A})^{-1}\mathbf{A}^{\ast}\mathbf{M}^{-1}\mathbf{x}\_{i} \quad \text{ } i = 1, \ldots, n \tag{56}
$$

and, as a consequence, the GLRT based on GASP at the main condition of GD functioning, i.e., equality in whole range of parameters between the transmitted information signal and refe-rence signal (signal model) in the receiver part, becomes

$$2\sum\_{i=1}^{n} \mathbf{x}\_{i}^{\ast} \mathbf{M}^{-1} \mathbf{A} (\mathbf{A}^{\ast} \mathbf{M}^{-1} \mathbf{A})^{-1} \mathbf{A}^{\ast} \mathbf{M}^{-1} \mathbf{x}\_{i} - \sum\_{i=1}^{n} \mathbf{x}\_{i}^{\ast} \mathbf{M}^{-1} \mathbf{A} \mathbf{A}^{\ast} \mathbf{M}^{-1} \mathbf{x}\_{i} + \sum\_{i=1}^{n} \boldsymbol{\xi}\_{\text{AF}\_{i}}^{\ast} \mathbf{M}^{-1} \mathbf{M}^{-1} \boldsymbol{\xi}\_{\text{AF}\_{i}} \geq\_{\mathbf{H}\_{0}}^{\mathsf{H}\_{1}} \mathsf{K}\_{\mathbb{S}} \quad \text{(57)}$$

*Case* 2: *N m* . In this case, the minimum of the quadratic forms in (55) is zero, since each linear system

$$\mathbf{A}\hat{\mathbf{a}}\_{i} = \mathbf{x}\_{i} \; , \quad \text{i} = \mathbf{1} \; , \ldots \; n \tag{58}$$

is determined. As a consequence the GLRT based on GASP at the main condition of GD functioning, i.e., equality in whole range of parameters between the transmitted information signal and reference signal (signal model) in the receiver part, becomes

$$\sum\_{i=1}^{n} \boldsymbol{\xi}\_{AF\_i}^\* \mathbf{M}^{-1} \mathbf{M}^{-1} \boldsymbol{\xi}\_{AF\_i} - \sum\_{i=1}^{n} \mathbf{x}\_i^\* \mathbf{M}^{-1} \mathbf{A} \mathbf{A}^\* \mathbf{M}^{-1} \mathbf{x}\_i \geq\_{\boldsymbol{\mathsf{M}}}^{\boldsymbol{\mathsf{H}}\_1} \boldsymbol{K}\_{\boldsymbol{\mathsf{g}}} \,. \tag{59}$$

#### **4.3 Performance analysis**

In order to define possible design criteria for the space-time coding, it is useful to establish a direct relationship between the probability of detection *PD* and the transmitted waveform, which is thus the main goal of the present section. Under the hypothesis *H*<sup>0</sup> , the left hand side of the GLRT based on GASP can be written in the following form

$$\sum\_{i=1}^{n} \boldsymbol{\xi}\_{AF\_i}^\* \mathbf{M}^{-1} \boldsymbol{\xi}\_{AF\_i} - \sum\_{i=1}^{n} \boldsymbol{\xi}\_{PF\_i}^\* \mathbf{M}^{-1} \boldsymbol{\xi}\_{PF\_i} \tag{60}$$

and, represents the GD background noise. It follows from (Tuzlukov 2005) that the decision statistic is defined by the modified second-order Bessel function of an imaginary argument or, as it is also called, McDonald's function with *m n* degrees of freedom. Thus, the decision statistic is independent of dimensionality *N* of the column vector given by (41) whose entries are complex numbers, which modulate both in amplitude and in phase the *N* pulses of the train. Consequently, the probability of false alarm *PFA* can be evaluated in the following form

$$P\_{FA} = \exp(-K\_{\mathcal{g}}) \sum\_{k=0}^{n} \frac{(K\_{\mathcal{g}})^k}{k!} \,. \tag{61}$$

This last expression allows us to note the following observations: a) the decision statistic is ancillary, in the sense that it depends on the actual interference covariance matrix, but its pdf is functionally independent of such a matrix; and b) the threshold setting is feasible with no prior knowledge as to the interference power spectrum, namely, the GLRT based on GASP ensures the constant false alarm (CFAR) property.

Under the hypothesis *H*<sup>1</sup> , given **α***<sup>i</sup>* ,the vectors , 1, , *<sup>i</sup>* **x** *i n* , are statistically independent complex Gaussian vectors with the mean value <sup>1</sup> *<sup>i</sup>* **M A <sup>α</sup>** and identity covariance matrix. It follows that, given **α***<sup>i</sup>* , the GLRT based on GASP is no the central distributed modified secondorder Bessel function of an imaginary argument, with the no centrality parameter 1 1 *n i i i* **<sup>α</sup> AM A<sup>α</sup>** and degrees of freedom *m n* . Consequently, the conditional probability of detection *PD* based on statements in (Van Trees, 2003) and discussion in (Tuzlukov, 2005) can be represented in the following form

$$P\_D = \mathbf{Q}\_{m \times n} \left( \sqrt{2q}, \sqrt{2K\_{\mathcal{g}}} \right), \tag{62}$$

where

328 Wireless Communications and Networks – Recent Advances

<sup>1</sup> (,,|,,,, ) exp ( ) ( ) det ( )

under the hypothesis *H*<sup>1</sup> , where det( ) denotes the determinant of a square matrix. Substituting (16) and (17) in (15), we can recast the GLRT based on GASP, after some mathematical

1 1

*Case* 1: *N m* . In this case, the quadratic forms in (55) achieve the minimum at

**<sup>α</sup> <sup>ξ</sup> <sup>M</sup> <sup>ξ</sup> x A<sup>α</sup> MxA<sup>α</sup>** *<sup>H</sup>*

min( ) ( ) *<sup>i</sup> <sup>i</sup> <sup>i</sup>*

In order to solve the *n* minimization problems in (55) we have to distinguish between two

and, as a consequence, the GLRT based on GASP at the main condition of GD functioning, i.e., equality in whole range of parameters between the transmitted information signal and

1 11 1 1 1 1 1

*Case* 2: *N m* . In this case, the minimum of the quadratic forms in (55) is zero, since each li-

is determined. As a consequence the GLRT based on GASP at the main condition of GD functioning, i.e., equality in whole range of parameters between the transmitted inform-

1 1 1 1

In order to define possible design criteria for the space-time coding, it is useful to establish a direct relationship between the probability of detection *PD* and the transmitted waveform, which is thus the main goal of the present section. Under the hypothesis *H*<sup>0</sup> , the left hand si-

> 1 1 1 1 *<sup>i</sup> i i <sup>i</sup>*

*AF PF*

*AF PF*

**<sup>ξ</sup> M M <sup>ξ</sup> x M AA M x** *<sup>H</sup>*

*AF <sup>K</sup>*

*AFi i g*

 **x M A A M A A M x x M AA M x ξ M M ξ** *<sup>H</sup>*

*i i i i AF g*

*AF <sup>K</sup>*

2 () *<sup>i</sup> <sup>i</sup>*

*AF <sup>K</sup>*

**xx M α α x A<sup>α</sup> MxA<sup>α</sup>**

*π*

*n n Nn n i i i i*

**<sup>M</sup>** *<sup>H</sup>* (54)

*AF ii ii g*

1

11 1 <sup>ˆ</sup> ( ) , 1, , *i i i n* **<sup>α</sup> AM A AM x** (56)

ˆ , 1, , *i i* **Aα x** *i n* (58)

1 0

*H*

**<sup>ξ</sup> <sup>M</sup> ξ ξ <sup>M</sup> <sup>ξ</sup>** (60)

*i*

 

*n*

1 1

1 1

refe-rence signal (signal model) in the receiver part, becomes

1 1 1

ation signal and reference signal (signal model) in the receiver part, becomes

1 1 *<sup>i</sup> <sup>i</sup> n n*

*i i*

de of the GLRT based on GASP can be written in the following form

*n n*

*i i*

*i i i*

*n n n*

*i i*

*n n*

transformations, in the following form

1

*f*

different cases.

near system

**4.3 Performance analysis** 

1

1 0 . (55)

1 0 . (57)

*H*

. (59)

*H*

$$q = \sum\_{i=1}^{n} \mathbf{a}\_i^\star \mathbf{A}^\star \mathbf{M}^{-1} \mathbf{A} \mathbf{a}\_i \tag{63}$$

and (,) *<sup>k</sup> Q* denotes the generalized Marcum*Q* function of order *k*. An alternative expression for the conditional probability of detection *PD* , in terms of an infinite series, can be also written in the following form:

$$P\_D = \sum\_{k=0}^{\mathcal{O}} \frac{\exp(-q)q^k}{k!} \left[1 - \Gamma\_{inc}(\mathcal{K}\_{\mathcal{g}'}k + m \times n)\right],\tag{64}$$

where

$$\Gamma\_{inc}(p,r) = \frac{1}{\Gamma(r)} \Big|\_{0}^{w} \exp(-z) z^{r-1} dz \tag{65}$$

is the incomplete Gamma function. Finally, the unconditional probability of detection *PD* can be obtained averaging the last expression over the pdf of **α***<sup>i</sup>* , 1, , *i n* .

#### **4.4 Code design by information-theoretic approach**

In principle, the basic criterion for code design should be the maximization of the probability of detection *PD* given by (62) over the set of admissible code matrices, i.e.,

$$\text{arg } \max\_{\mathbf{A}} E\left[\mathbf{Q}\_{m \times n} \left(\sqrt{2\eta}, \sqrt{2K\_{\mathcal{g}}}\right)\right] = \text{arg } \max\_{\mathbf{A}} E\left[\mathbf{Q}\_{m \times n} \left(\sqrt{2\sum\_{i=1}^{n} \mathbf{a}\_i^\* \mathbf{A}^\* \mathbf{M}^{-1} \mathbf{A} \mathbf{a}\_i}, \sqrt{2K\_{\mathcal{g}}}\right)\right],\tag{66}$$

where arg max ( ) **<sup>A</sup>** denotes the value of **A**, which maximizes the argument and the statistical average is over **α***<sup>i</sup>* , 1, , *i n* . Unfortunately, the above maximization problem does not appear to admit a closed-form solution, valid independent of the fading law, whereby we prefer here to resort to the information-theoretic criterion supposed in (De Maio & Lops, 2007). Another way is based on the optimization of the Chernoff bound over the code matrix **A**. As was shown in (De Maio & Lops, 2007), these ways lead to the same solution, which subsumes some well-known space-time coding, such as Alamouti code and, more generally, the class of space-time coding from orthogonal design (Alamouti, 1998) and (Tarokh et al., 1999), which have been shown to be optimum in the framework of communication theory. In subsequent derivations, we assume that **α***<sup>i</sup>* , 1, , *i n* , are independent and identically distributed (i.i.d.) zero-mean complex Gaussian vectors with scalar covariance matrix, i.e.,

$$E\left[\mathbf{o}\_i \mathbf{o}\_i^\*\right] = \sigma\_a^2 \mathbf{I} \tag{67}$$

where <sup>2</sup> *<sup>a</sup> σ* is a real factor accounting for the backscattered useful power, and **I** denotes the identity matrix.

Roughly speaking, the GLRT strategy overcomes the prior uncertainty as to the target fluctuations by ML estimating the complex target amplitude, and plugging the estimated value into the conditional likelihood in place of the true value. Also, it is well known that, under general consistency conditions, the GLRT converges towards the said conditional likelihood, thus achieving a performance closer and closer to the perfect measurement bound, i.e., the performance of an optimum test operating in the presence of known target parameters. Diversity, on the other hand, can be interpreted as a means to transform an amplitude fluctuation in an increasingly constrained one. It is well known, for example that, upon suitable receiver design, exponentially distributed square target amplitude may be transformed into a central chi-square fluctuation with *d* degrees of freedom through a diversity of order *d* in any domain. More generally, a central chi-square random variable with 2*m* degrees of freedom may be transformed into a central chi-square with 2*m d* degrees of freedom. In this framework, a reasonable design criterion for the space-time coding is the maximization of the mutual information between the signals received from the various diversity branches and the fading amplitudes experienced thereupon. Thus, denoting by *I*(, ) **α X** the mutual information (Cover & Thomas, 1991) between the random matrices

$$\mathfrak{a} = [\mathfrak{a}\_1, \dots, \mathfrak{a}\_n] \tag{68}$$

and

330 Wireless Communications and Networks – Recent Advances

is the incomplete Gamma function. Finally, the unconditional probability of detection *PD* can

In principle, the basic criterion for code design should be the maximization of the probabili-

<sup>1</sup>

where arg max ( ) **<sup>A</sup>** denotes the value of **A**, which maximizes the argument and the statistical average is over **α***<sup>i</sup>* , 1, , *i n* . Unfortunately, the above maximization problem does not appear to admit a closed-form solution, valid independent of the fading law, whereby we prefer here to resort to the information-theoretic criterion supposed in (De Maio & Lops, 2007). Another way is based on the optimization of the Chernoff bound over the code matrix **A**. As was shown in (De Maio & Lops, 2007), these ways lead to the same solution, which subsumes some well-known space-time coding, such as Alamouti code and, more generally, the class of space-time coding from orthogonal design (Alamouti, 1998) and (Tarokh et al., 1999), which have been shown to be optimum in the framework of communication theory. In subsequent derivations, we assume that **α***<sup>i</sup>* , 1, , *i n* , are independent and identically distributed (i.i.d.) zero-mean complex Gaussian vectors with

*E qK E K*

**A A**

*<sup>a</sup> σ* is a real factor accounting for the backscattered useful power, and **I** denotes the id-

Roughly speaking, the GLRT strategy overcomes the prior uncertainty as to the target fluctuations by ML estimating the complex target amplitude, and plugging the estimated value into the conditional likelihood in place of the true value. Also, it is well known that, under general consistency conditions, the GLRT converges towards the said conditional likelihood, thus achieving a performance closer and closer to the perfect measurement bound, i.e., the performance of an optimum test operating in the presence of known target parameters. Diversity, on the other hand, can be interpreted as a means to transform an amplitude fluctuation in an increasingly constrained one. It is well known, for example that, upon suitable receiver design, exponentially distributed square target amplitude may be transformed into a central chi-square fluctuation with *d* degrees of freedom through a diversity of order *d* in any domain. More generally, a central chi-square random variable with 2*m* degrees of freedom may be transformed into a central chi-square with 2*m d* degrees of freedom. In this framework, a reasonable design criterion for the space-time coding is the maximization of the mutual information between the signals received from the various diversity branches and the fading amplitudes experienced thereupon. Thus, denoting by *I*(, ) **α X** the mutual

information (Cover & Thomas, 1991) between the random matrices

*m n g mn i i g*

*Q Q* **<sup>α</sup> AM A<sup>α</sup>** , (66)

arg max 2 , 2 arg max 2 , 2

1

<sup>2</sup> [ ] *<sup>E</sup> ii a <sup>σ</sup>* **α α <sup>I</sup>** , (67)

*i*

*n*

be obtained averaging the last expression over the pdf of **α***<sup>i</sup>* , 1, , *i n* .

ty of detection *PD* given by (62) over the set of admissible code matrices, i.e.,

**4.4 Code design by information-theoretic approach** 

scalar covariance matrix, i.e.,

where <sup>2</sup>

entity matrix.

$$\mathbf{X} = [\mathbf{x}\_1, \dots, \mathbf{x}\_k] = \mathbf{A}\mathbf{a} + \boldsymbol{\Xi} \tag{69}$$

the quantity to be maximized is

$$I(\mathfrak{a}, \mathfrak{X}) = H(\mathfrak{X}) - H(\mathfrak{X} \mid \mathfrak{a}) \,. \tag{70}$$

where

$$\Xi = [\xi\_1, \dots, \xi\_n]\_{\prime} \tag{71}$$

*H*( ) **X** denotes the entropy of the random matrix **Ξ** , and *H*(|) **X α** is the conditional entropy of **X** given **α** (Cover & Thomas, 1991). Exploiting the statistical independence between **α** and **X**, we can write (70) in the following form

$$H(\mathfrak{a}, \mathbf{X}) = H(\mathbf{X}) - H(\mathbf{X} \mid \mathfrak{a}) = H(\mathbf{X}) - H(\Xi) \tag{72}$$

where *H*( ) **Ξ** is the entropy of the random matrix **Ξ** . Assuming that the columns of **α** are i.i.d. zero-mean complex Gaussian vectors with covariance matrix <sup>2</sup> *<sup>a</sup> σ* **I** , we can write *H*( ) **X** and *H*( ) **Ξ** , respectively, in the following form:

$$H(\mathbf{X}) = \text{x} \lg[\left(\pi e\right)^{\mathsf{N}} \det(\mathbf{M} + \sigma\_a^2 \mathbf{A} \mathbf{A}^\*)] \tag{73}$$

and

$$H(\Xi) = \text{x} \lg \left[ \left( \pi e \right)^{\aleph} \det(\mathbf{M}) \right]. \tag{74}$$

As design criterion we adopt the maximization of the minimum probability of detection *PD* , which can be determined as the lower Chernoff bound, under an equality constraint for the average signal-to-clutter power ratio (SCR) given by

$$\text{SCR} = \frac{1}{Nmn} \text{E} \left[ \sum\_{i=1}^{n} \mathbf{a}\_i^\* \mathbf{A}^\* \mathbf{M}^{-1} \mathbf{A} \mathbf{a}\_i \right] = \frac{\sigma\_a^2}{Nm} \text{tr} \left( \mathbf{A}^\* \mathbf{M}^{-1} \mathbf{A} \right) = \frac{\sigma\_a^2}{Nm} \sum\_{j=1}^{m} \lambda\_j \,\,\,\tag{75}$$

where tr( ) denotes the trace of a square matrix and *<sup>j</sup> λ* are the elements or corresponding or-

dered (in decreasing order) eigenvalues of the diagonal matrix **Λ** defined by the eigenvalue decomposition **V ΛV** of the matrix 1 1 **M AA M** , where **V** is an *N N* unitary matrix. The considered design criterion relies on the maximization of the mutual information (70) under equality constraint (75) for SCR. This is tantamount to solving the following constrained minimization problem since *H*( ) **Ξ** does not exhibit any functional dependence on **A**.

$$\min\_{\lambda\_1,\ldots,\lambda\_m} \prod\_{j=1}^m \left[ \frac{1}{1 + \chi(\lambda\_j \sigma\_a^2 + 1)} \right]^n \quad \text{and} \quad \frac{\sigma\_a^2}{Nm} \sum\_{j=1}^m \lambda\_j = \mu \tag{76}$$

which, taking the logarithm, is equivalent

$$\max\_{\lambda\_1, \ldots, \lambda\_m} \sum\_{j=1}^m \lg\left[1 + \chi(\sigma\_a^2 \lambda\_j + 1)\right] \qquad \text{and} \qquad \sum\_{j=1}^m \lambda\_j = \frac{\mu mN}{\sigma\_a^2} \tag{77}$$

where *γ* is the variable defining the upper Chernoff bound (Benedetto & Biglieri, 1999).

Since <sup>2</sup> lg[1 ( 1)] *<sup>a</sup> γ σ y* is a concave function of *y*, we can apply Jensen's inequality (Cover & Thomas, 1991) to obtain

$$\sum\_{j=1}^{m} \lg\left[1 + \chi(\sigma\_a^2 \lambda\_j + 1)\right] \le m \lg\left[1 + \chi\left(\frac{1}{m} \sum\_{j=1}^{m} \lambda\_j \sigma\_a^2 + 1\right)\right].\tag{78}$$

Moreover, forcing in the right hand side of (78), the constraint of (77), we obtain

$$\sum\_{j=1}^{m} \lg\left[1 + \chi(\sigma\_a^2 \lambda\_j + 1)\right] \le m \lg\left[1 + \chi(\mu N + 1)\right].\tag{79}$$

The equality in (79) is achieved if

$$
\lambda\_k = \frac{\mu N}{\sigma\_a^2} \quad , \quad \quad k = 1, \ldots, m \tag{80}
$$

implying that an optimum code must comply with the condition

$$\mathbf{M}^{-1}\mathbf{A}\mathbf{A}^\star\mathbf{M}^{-1} = \begin{cases} \frac{\mu\mathbf{N}}{\sigma\_a^2} \Big[ 2\mathbf{A}(\mathbf{A}^\star\mathbf{M}^{-1}\mathbf{A})^{-1}\mathbf{A}^\star - \mathbf{A}\mathbf{A}^\star \Big] & \text{Case 1} \\\\ \frac{\mu\mathbf{N}}{\sigma\_a^2}\mathbf{I} & \text{Case 2} \end{cases} \tag{81}$$

In particular, if the additive disturbance is white, i.e., <sup>2</sup> *<sup>n</sup>* **M I** *σ* , the above equation reduces to

$$\mathbf{AA}^\* = \begin{cases} \frac{4\sigma\_n^4 \mu N}{\sigma\_a^2} (\mathbf{A}^\* \mathbf{M}^{-1} \mathbf{A})^{-1} & \text{Case 1} \\\\ \frac{4\sigma\_n^4 \mu N}{\sigma\_a^2} \mathbf{I} & \text{Case 2} \end{cases} \tag{82}$$

The last equation subsumes, as a relevant case, the set of orthogonal space-time codes. Indeed, assuming *Nnm* , the condition (82) yields, for the optimum code matrix,

$$\mathbf{AA}^\* = \frac{4\sigma\_n^4 \mu N}{\sigma\_a^2} \mathbf{I} \,\,\,\tag{83}$$

i.e., the code matrix **A** should be proportional to any unitary *N N* matrix. Thus, any orthonormal basis of *<sup>N</sup>F* can be exploited to construct an optimum code under the Case 2 and white Gaussian noise. If, instead, we restrict our attention to code matrices built upon Galois Fields (GF), there might be limitations to the existing number of optimal codes. Deffering to (Tarokh, 1999) and to the Urwitz-Radon condition exploited therein, we just remind here that, under the constraint of binary codes, unitary matrices exist only for limited values of *N*: for 2 2 coding, we find the normalized Alamouti code (Alamouti, 1998), which is an orthonormal basis, with elements in GF (2), for <sup>2</sup> *F* .

Make some comments. First notice, that under the white Gaussian noise, both performance measures considered above are invariant under unitary transformations of the code matrix, while at the correlated clutter they are invariant with respect to right multiplication of **A** by a unitary matrix. Probably, these degrees of freedom might be exploited for further optimization in different radar functions. Moreover, (70) represents the optimum solution for the case that no constraint is forced upon the code alphabet; indeed, the code matrices turn out in general to be built upon the completely complex field. If, instead, the code alphabet is constrained to be finite, then the optimum solution (70) may be no longer achievable for arbitrary clutter covariance. In fact, while for the special case of white clutter and binary alphabet the results of (Tarokh, 1999) may be directly applied for given values of *m* and *n*, for arbitrary clutter covariance and (or) transmit/receive antennas number, a code matrix constructed on GF (*q*) and fulfilling the conditions (70) is no longer ensured to exist. In these situations, which however form the object of current investigations, a brute-force approach could consist of selecting the optimum code through an exhaustive search aimed at solving (66), which would obviously entail a computational burden ( ) *mN O q* floating point operations. Herein we use the usual Landau notation.*O n*( ) .; hence, an algorithm is *nO* )( if its implementation requires a number of floating point operations proportional to *n* (Golub & Van Loan, 1996). Fortunately, the exhaustive search has to be performed off line. The drawback is that the code matrix would inevitably depend on the target fluctuation law; moreover, if one would account for possible nonstationarities of the received clutter, a computationally acceptable code updating procedure should be envisaged so, as to optimally track the channel and clutter variations.
