Principles of Modern Radar - Volume 2 1891121537

128 CHAPTER 4 MIMO Radarauto-correlations of the M transmitted waveforms, {φ m (t)}. The element in row m andcolumn m ′ describes the response of waveform m when the matched filter correspondingto waveform m ′ is applied. This element of the matrix may be written(R φ ) m,m′∫ ∞= φ m (t) φm ∗ ′ (t) dt (4.18)−∞Since a phased array is a special case of MIMO radar, it also possesses a MIMOsignal correlation matrix. Let φ 0 (t) be the radar waveform used by a phased array system.Each subarray will transmit this signal but with a phase shift applied to steer the beam ina particular direction. To steer a beam in the direction ˜θ 0 , the transmitted signals may bewritten PA (t) = a ∗ (˜θ 0)φ0 (t) (4.19)where a (θ) is the transmit steering vector corresponding to the direction θ. To satisfy theenergy constraint of (4.3), the signal φ 0 (t) must be normalized so that∫ ∞−∞|φ 0 (t)| 2 dt = 1 M(4.20)The signal correlation matrix is found to beR φ/PA = 1 M a∗ (˜θ 0) a∗ (˜θ 0) H(4.21)This steering angle, ˜θ 0 , must be chosen by the phased array before transmitting since thisis a form of analog beamforming. As the phased array scans its transmit beam from dwellto dwell, the signal correlation matrix will change, but it will remain rank-1.Note that the correlation matrix is scaled so that the trace (the sum of the diagonalelements) of the correlation matrix is 1. This is required so that the signal-to-noise ratioremains γ regardless of the structure of the MIMO correlation matrix. In effect, thisenforces a constant transmit power between designs to allow reasonable comparisons.Consider now the case where each subarray transmits one of a suite of orthogonalwaveforms; the correlation matrix is full-rank. If orthogonal waveforms are used, eachwith equal power, then the correlation matrix is a scaled identity matrix,R φ/⊥ = 1 M I M (4.22)Another example is when the array is spoiled on transmit. To cover a larger area,the transmit beamwidth may be increased by applying a phase taper across the array orby transmitting out of a single subarray. In the latter case, the transmitted waveforms areconsidered to be identically zero for all but one subarray. This approach is referred toas spoiling on transmit since it effectively spoils the transmit beampattern of the phasedarray by trading peak gain for a wider coverage area. The signal correlation matrix of thespoiled phased array, R φ/Spoil , is rank-1, just as in the unspoiled phased array case. In thecontext of synthetic aperture radar (SAR), as well as in synthetic aperture sonar, an arraythat uses a single element on transmit and multiple elements on receive is referred to as aVernier array.