**4.5 Simulation**

332 Wireless Communications and Networks – Recent Advances

*<sup>μ</sup>mN γσ λ <sup>λ</sup>*

. (79)

*<sup>σ</sup>* (80)

**AA I** , (83)

. (78)

1 1

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

1 1

*j j*

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

2

*a j*

*m m*

*m*

*<sup>σ</sup>*

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 &

<sup>1</sup> lg[ ] 1 ( 1) lg 1 1

*γσ λ m γ λσ*

lg[ ][ ] 1 ( 1) lg 1 ( 1)

<sup>2</sup> , 1, , *<sup>k</sup>*

*<sup>μ</sup><sup>N</sup> <sup>λ</sup> k m*

1 1

[ ]

1 1

4 Case 2 .

<sup>4</sup> ( ) Case 1

**AM A**

The last equation subsumes, as a relevant case, the set of orthogonal space-time codes. Inde-

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

4 2 4 *<sup>n</sup> a σ μN σ*

**A A M A A AA**

2( ) Case 1

Case 2 .

*a*

*γσ λ m γ μN*

2 2

*a j j a*

*m m a j <sup>j</sup> λ λ j j <sup>a</sup>*

2

, (77)

(81)

(82)

*<sup>n</sup>* **M I** *σ* , the above equation reduces to

2

max lg 1 ( 1) and [ ]

which, taking the logarithm, is equivalent

Thomas, 1991) to obtain

, , 1

*m*

1

implying that an optimum code must comply with the condition

 

In particular, if the additive disturbance is white, i.e., <sup>2</sup>

 

**AA**

2

*μN σ μN σ*

*a*

2

**I**

*a*

4

*n a n a*

2 4 2

**I**

ed, assuming *Nnm* , the condition (82) yields, for the optimum code matrix,

*σ μN σ σ μN σ*

*j*

1 1

**M AA M**

The equality in (79) is achieved if

*m*

The present section is aimed at illustrating the validity of the proposed encoding and detection schemes under diverse scenarios. In particular, we first assume uncorrelated disturbance, whereby orthogonal space-time codes are optimal. In this scenario, simulations have been run, and the results have been compared to the Chernoff bounds of the conventional GLRT receiver discussed in (De Maio & Lops, 2007) and to the GD performance achievable through a single-input single-output (SISO) radar system. Next, the effect of the disturbance correlation is considered, and the impact of an optimal code choice is studied under different values of transmit/receiver antenna numbers. In all cases, the behavior of the mutual information between the observations and the target replicas can be also represented, showing that such a measure is itself a useful tool for system design and assessment, but this analysis is outside of a scope of the present chapter.

Figure 8 represents the white Gaussian disturbance and assesses the performance of the GLRT GD. To elicit the advantage of waveform optimization, we consider both the optimum coded wireless communication system and the uncoded one, corresponding to pulses with equal amplitudes and phases. The probability of detection *PD* is plotted versus SCR assuming <sup>4</sup> 10 *PFA* and *nmN* 2 . This simulation setup implies that the Alamouti code is optimum in the sense specified by (82). For comparison purposes, we also plot the performance of the uncoded SISO GD. We presented the performance of the conventional GLRT to underline a superiority of GD employment.

Fig. 8. *PD* versus SCR; white Gaussian disturbance and disturbance with exponentially shaped covariance matrix ( *ρ* 95.0 ); ;10 2 <sup>4</sup> *PFA nmN* .

The curves highlight that the optimum coded wireless communication system employing the GD and exploiting the Alamouti code, achieves a significant performance gain with respect to both the uncoded and the SISO radar systems. Precisely, for *PD* 9.0 , the performance gain that can be read as the horizontal displacement of the curves corresponding to the analyzed wireless communication systems, is about 1 dB with reference to the uncoded GLRT GD wireless communication system and 5 dB with respect to the SISO GD. Superiority of employment GD with respect to the conventional GLRT wireless communication systems achieves 6 dB for the optimum coded wireless communication system, 8 dB for the uncoded wireless communication systems, and 12 dB for SISO wireless communication systems. It is worth pointing out that the uncoded wireless communication system performs slightly better the coded one for low detection probabilities. This is a general trend in detection theory, which predicts that less and less constrained fluctuations are detrimental in the high SCR region, while being beneficial in the low SCR region. On the other hand, the code optimization results in a more constrained fluctuation, which, for low SCRs, leads to slight performance degradation as compared with uncoded systems. The effect of disturbance correlation is elicited in Fig.8 too, where the analysis is produced assuming an overall disturbance with exponentially shaped covariance matrix, whose one-lag correlation coefficient *ρ* is set to 0.95. In this case, the Alamouti code is no longer optimum. The plots show that the performance gain of the optimum coded GLRT GD wireless communication system over both the uncoded and the SISO GD detector is almost equal to that resulting when the disturbance is

coded wireless communication system and the uncoded one, corresponding to pulses with equal amplitudes and phases. The probability of detection *PD* is plotted versus SCR assuming <sup>4</sup> 10 *PFA* and *nmN* 2 . This simulation setup implies that the Alamouti code is optimum in the sense specified by (82). For comparison purposes, we also plot the performance of the uncoded SISO GD. We presented the performance of the conventional GLRT to un-

Fig. 8. *PD* versus SCR; white Gaussian disturbance and disturbance with exponentially sha-

The curves highlight that the optimum coded wireless communication system employing the GD and exploiting the Alamouti code, achieves a significant performance gain with respect to both the uncoded and the SISO radar systems. Precisely, for *PD* 9.0 , the performance gain that can be read as the horizontal displacement of the curves corresponding to the analyzed wireless communication systems, is about 1 dB with reference to the uncoded GLRT GD wireless communication system and 5 dB with respect to the SISO GD. Superiority of employment GD with respect to the conventional GLRT wireless communication systems achieves 6 dB for the optimum coded wireless communication system, 8 dB for the uncoded wireless communication systems, and 12 dB for SISO wireless communication systems. It is worth pointing out that the uncoded wireless communication system performs slightly better the coded one for low detection probabilities. This is a general trend in detection theory, which predicts that less and less constrained fluctuations are detrimental in the high SCR region, while being beneficial in the low SCR region. On the other hand, the code optimization results in a more constrained fluctuation, which, for low SCRs, leads to slight performance degradation as compared with uncoded systems. The effect of disturbance correlation is elicited in Fig.8 too, where the analysis is produced assuming an overall disturbance with exponentially shaped covariance matrix, whose one-lag correlation coefficient *ρ* is set to 0.95. In this case, the Alamouti code is no longer optimum. The plots show that the performance gain of the optimum coded GLRT GD wireless communication system over both the uncoded and the SISO GD detector is almost equal to that resulting when the disturbance is

ped covariance matrix ( *ρ* 95.0 ); ;10 2 <sup>4</sup> *PFA nmN* .

derline a superiority of GD employment.

white. On the other hand, setting *nmN* 2 in (81), shows that, under correlated disturbance, the optimum code matrix is proportional to **M**: namely, an optimal code tends to restore the "white disturbance condition." This also explains why the conventional Alamouti code follows rather closely the performance of the uncoded GLRT GD wireless communication system.

The effect of number *n* of receive antennas on the performance is analyzed in Fig.9, where *PD* is plotted versus SCR for *mN* 8 , exponentially shaped clutter covariance matrix with *ρ* 95.0 , and several values of *n*. The curves highlight that the higher *n*, namely the higher the diversity order, the better the performance. Specifically, the performance gap between the case *n* 8 and the case of a MISO GLRT GD radar system (i.e., *n* 1 ) is about 2.5 dB, while, in the case of the conventional GLRT radar systems, is about 7 dB for *PD* 9.0 . A great superiority between the radar systems employing GLRT GD and conventional GLRT is evident and estimated at the level of 6 dB at *n* 8 and 10 dB in the case of a MISO (i.e., *n* 1 ) for *PD* 9.0 . Notice that this performance trend is also in accordance with the expression of the mutual information that exhibits a linear, monotonically increasing, dependence on *n*. The same qualitative, but not quantitative, performance can be presented under study of the number *m* of available transmit antennas on the GLRT GD wireless communication system performance.

Fig. 9. *PD* of optimum coded system versus SCR; disturbance with exponentially shaped covariance matrix ( *ρ* 95.0 ); and several values of *m*; 8;10 <sup>4</sup> *PFA mN* .

#### **4.6 Discussion**

We have addressed the synthesis and the analysis of MIMO radar systems employing the GD and exploiting space-time coding. To this end, after a short description of the MIMO radar signal model applied to wireless communications, we have devised the GLRT GD under the assumption of the additive white Gaussian disturbance. Remarkably, the decision statistic is ancillary and, consequently, CFAR property is ensured, namely, the detection threshold can be set independent of the disturbance spectral properties. We have also assessed the performance of the GLRT GD providing closed-form expressions for both *PD* and *PFA* . Lacking a manageable expression for *PD* under arbitrary target fluctuation models, we restricted our attention to the case of Rayleigh distributed amplitude fluctuation. The performance assessment that has been undertaken under several instances of number of receive and transmit antennas, and of clutter covariance, has confirmed that MIMO GD radar systems with a suitable space-time coding achieve significant performance gains over SIMO, MISO, SISO, or conventional SISO radar systems employing the conventional GLRT detector. Also, these MIMO GD radar systems outperform the listed above systems employing the conventional GD. Future research might concern the extension of the proposed framework to the case of an unknown clutter covariance matrix, in order to come up with a fully adaptive detection system. Moreover, another degree of freedom, represented by the shapes of the transmitted pulses could be exploited to further optimize the performance. More generally, the impact of space-time coding in MIMO CD radar systems to estimate the target parameters is undoubtedly a topic of primary concern. Finally, the design of GD and space-time coding strategies might be of interest under the very common situation of non-Gaussian radar clutter.
