Principles of Modern Radar - Volume 2 1891121537

4.4 MIMO Radar Signal Processing 1314.4.4 MIMO GainThe array factor describes the pattern of an array antenna if each subarray was omnidirectional.The array factor of a MIMO radar in the direction θ when the beamformingweights are steered to the direction θ 0 is defined byf (θ; θ 0 ) = w (θ 0 ) H s (θ)√w (θ 0 ) H R e w (θ 0 )(4.27)When the optimal weights are used, this may be written in terms of the MIMO steeringmatrix A or, alternatively, in terms of the transmit/receive steering vectors a and b.f (θ; θ 0 ) =Vec {A (θ 0)} H ( R T φ ⊗ I )n Vec {A (θ)}√Vec {A (θ 0 )} H ( R T ) (4.28)φ ⊗ I n Vec {A (θ0 )}⎛⎞= ⎝a (θ 0) H R T φ√a (θ)( )⎠ b (θ0 ) H b (θ)√ (4.29)a (θ 0 ) H R T φa (θ 0 ) NIn general, the array factor will be complex valued. The gain of an antenna is themagnitude-squared of the array pattern in addition to the gain of the subarrays. Theantenna gain G (θ; θ 0 ) describes the increase in SNR that a target at angle θ will receiveif a beam is digitally steered in the direction θ 0 .If it is assumed that all of the transmit subarrays and receive subarrays of a MIMOarray antenna are identical, then the (two-way) gain of the MIMO radar isG (θ; θ 0 ) =⎛⎜⎝E TX (θ)∣∣∣a (θ 0 ) H R T φ a (θ) ∣∣2a (θ 0 ) H R T φa (θ 0 )} {{ }TransmitGain⎞( ∣⎟∣b (θ 0 ) H b (θ) ∣ 2 )⎠ E RX (θ)N} {{ }ReceiveGain(4.30)where E TX (θ) and E RX (θ) are the (one-way) subarray gains on transmit and receive,respectively. It is important to observe that the transmit gain is strongly dependent on thesignal correlation matrix, R φ . A similar development is presented in [13].4.4.5 Phased Array versus Orthogonal WaveformsWe have shown that the transmit gain of an array antenna can be controlled by design ofthe MIMO signal correlation matrix. The two extreme cases are considered: the phasedarray (rank-1 matrix) and orthogonal waveforms (full-rank matrix). These matrices weregiven in (4.21) and (4.22).The gain for these two cases can be computed using (4.30). The gains of the phasedarray, G PA , and of the radar using orthogonal waveforms, G ⊥ , are found to be⎛∣( ) ⎜∣a (˜θ ) ∣ ⎞H ∣∣2 ( ∣0 a (θ)⎟∣b (θ 0 )G PA θ; ˜θ H b (θ) ∣ 2 )0 ,θ 0 = ⎝E TX (θ)⎠ E RX (θ)(4.31)MN⎛∣⎜∣a ( ) ∣ ⎞H ∣∣2 (θ 0 a (θ)⎟G ⊥ (θ; θ 0 ) = ⎝E TX (θ)M 2 ⎠ E RX (θ)∣ b (θ 0 ) H b (θ) ∣ 2 )N(4.32)