Principles of Modern Radar - Volume 2 1891121537

4.4 MIMO Radar Signal Processing 125The peak transmit power of the radar and other elements of the radar range equation willbe accounted for elsewhere. Each of these M transmitted waveforms will reflect off targetsin the scene, and the echoes will be observed by each of the N receivers.If the narrowband assumption is valid (i.e., the array is small relative to the bandwidthof the signal), then each subarray observes the same signal up to a phase shift. It is theinformation encoded in this phase shift from subarray to subarray that allows beamformingto be implemented. For a particular wavelength and for a particular angle of interest θ,we can form the corresponding steering vector that describes these phase shifts. In theMIMO case, we will need two steering vectors: the length-M transmit steering vector,a (θ); and the length-N receive steering vector, b (θ). Assume that each element of thesteering vectors is unit-magnitude. Consequently, we have‖a (θ)‖ 2 = M, ‖b (θ)‖ 2 = N (4.4)Note that, for this analysis, we assume that the signals are narrowband, but this modelcould be extended to include wideband effects.We now seek a model for the data observed from a single point target. Consider atarget at an angle θ that has a signal-to-noise ratio (SNR) of γ and a bulk phase shift ofψ. The signal observed by receiver n due to this target may be writteny n (t; θ) = √ γ e iψ b n (θ)M∑a m (θ) φ m (t) + η n (t) (4.5)m=1where a m (θ) and b n (θ) are, respectively, elements of the transmit steering vector a (θ)and the receive steering vector b (θ) corresponding to the angle θ, and η n (t) is receivernoise. Of course, a radar operates in three-dimensional space, which requires that twoangles be specified. In this treatment, only one angle, θ, is considered for clarity.This data model includes only a single target and assumes that the processor hasknowledge of the range of the target. In general, this is not a practical assumption, sinceone purpose of a radar system is to measure the range to a target. This is accomplishedby hypothesizing a number of target ranges and processing each range bin independently.So, if a target is present in a particular range bin, then the assumption of known range ismeaningful.The signal presented in (4.5) describes the observation of one of the N receive subarrays.The data observed by all of the receivers may be written more compactly by groupingthe data so that each element of a (column) vector corresponds to a receive subarray. Let (t) be a (column) vector where each of the M elements is the waveform transmitted bya particular subarray.⎛ ⎞ (t) =⎜⎝φ 1 (t).φ M (t)⎟⎠ (4.6)The signal observed by a MIMO radar corresponding to a target at an angle θ is writteny (t; θ) = √ γ e iψ b (θ) a (θ) T (t) + (t) (4.7)where b and a are the receive and transmit steering vectors, respectively. We can definethe MIMO channel matrix, A (θ), in terms of the transmit steering vector, a (θ), and the