Principles of Modern Radar - Volume 2 1891121537

4.3 The MIMO Virtual Array 123separated by great distances). In a sense, the information observed by an array antenna is thedifference in path lengths between transmit/receive pairs as a function of angle of arrival.It can be shown, under reasonable assumptions, that a pseudo-bistatic transmit/receive pairis equivalent to a monostatic system that is located directly between the bistatic transmitterand receiver.This is valid as long as the target is in the far field; that is, the range to the targetis much greater than the size of the transmit and receive arrays. Let x T and x R denotethe position of a transmit element and receive element, respectively, given in Cartesiancoordinates relative to the center of the array. If u is a unit vector that points from the arraycenter to a target in the far field, and if the signal is narrowband relative to the size of thearray, then the receiver at x R observes essentially the same signal as would be observed atthe array center up to a phase shift. For this transmit/receive pair, the observed phase shiftis( )4π xT + x Rλ uT 2where λ is the wavelength corresponding to the radar center frequency. Observe that thisphase shift is the same as would be observed if the transmitter and receiver were bothlocated at (x T + x R ) /2, the point directly between them.The set of all virtual elements that correspond to a set of physical transmit/receiveelements is called the virtual array. This is similar to the coarray described in [4]. If the Mtransmit elements of a MIMO radar have positions described by x T,1 ,...,x T,M and the Nreceive elements by x R,1 ,...,x R,N , then the virtual array is the set of positions{ }xT,m + x R,n: m = 1,...,M; n = 1,...,N(4.1)2In general, if M transmitters (using M orthogonal waveforms) are used and N receivechannels are used, then the number of virtual phase centers will be MN. These virtualphase centers may not all be distinct depending on the array topology.In a phased array, there is essentially one transmit element since the transmitted signalsare perfectly correlated. After transmission, the radar cannot differentiate between signalstransmitted from different elements. So, the virtual array is generated only by a singlephysical transmit element, as in the upper left of Figure 4-2. Note that the length of thevirtual array is about half of the length of the physical phased array.Now, suppose that the radar has multiple transmit elements and that each is capableof transmitting a waveform that is orthogonal to all of the others. Two waveforms, φ 1 (t)and φ 2 (t), are said to be orthogonal if∫ ∞−∞φ 1 (t) φ2 ∗ (t) dt = 0 (4.2)Essentially, a filter matched to the waveform φ 1 will completely reject any contributiondue to φ 2 and vice versa. Since the transmitted signals are assumed to be orthogonal,the receiver is able to process the contribution of each transmitter independently; byapplying a filter matched to a particular transmitted waveform, the contributions from allof the other transmitted waveforms are removed due to their orthogonality. The impactof correlations between the waveforms when they are not perfectly orthogonal will bediscussed in subsequent sections.