Principles of Modern Radar - Volume 2 1891121537

5.2 CS Theory 155outputs can be writtenχ = A H y (5.9)The inner products of the columns of A, denoted by a i , are samples of the ambiguity function:|aiH a j |=|A(τ i − τ j ,ω i − ω j )|. This relationship is a crucial point in understandingthe connection between CS and radar signal processing, as we will see in Section 5.2.5.5.2.2.3 Multichannel ExampleNow consider processing multiple pulses with a multichannel phased array radar. In contrastto the previous example, assume that matched filtering has already been performedon the individual pulses in fast time and focus on modeling the target response in the slowtime4 and channel dimensions. This phased array will transmit a series of pulses steeredto a region of interest on the ground. 5 The echoes from these pulses will be received onmultiple channels connected to subarrays of the antenna. By coherently processing thesereturns, range, velocity, and angular bearing information can be extracted about movingtargets. An MTI system treats nonmoving objects as undesirable clutter and attempts tosuppress these returns using techniques like space-time adaptive processing (STAP) [15].Figure 5-3 depicts a notional MTI scenario.To be specific, consider a monostatic uniform linear array (ULA) consisting of Jchannels spaced equally at d meters apart. A coherent processing interval (CPI) for thissystem consists of data collected over K slow-time pulses with a sampling period of Tseconds and L fast time range bins. We shall assume that the system is narrowband (i.e., thebandwidth B ≪ f 0 ), where f 0 = c λ is the center frequency,6 and that pulse compressionhas already been performed. In addition, motion during a given pulse will be neglected. 7We will consider the data collected for a single range gate. The spatial-channel samplesfor pulse k will be denoted as a vector y k ∈ C J , while the complete space-time snapshotwill be denoted y ∈ C JK , where the data have been pulse-wise concatenated, that is,y = [ y T 1y T 2 ... yT K] TFor a CPI, we thus collect M = JK measurements of the unknown scene.4 Slow time refers to the relatively long intervals between successive pulses from a coherent radar. Fasttime refers to the time scale at which the electromagnetic pulse travels across the scene of interest, whichis the same as range up to a scale factor when propagation is in a homogenous medium, such as freespace.5 Our development here assumes that all the elements in the phased array transmit the same waveformduring a given pulse, with the exception of phase shifts to steer the beam in the desired illuminationdirection. A more general approach involves using distinct waveforms on each of the transmit channels.This multiple-input multiple-output (MIMO) approach has recently received significant attention; see forexample [14].6 This assumption allows time delays to be well approximated as phase shifts, which creates a correspondencebetween target velocity and the output bins of a fast Fourier transform (FFT) with respect toslow-time.7 This so-called stop-and-hop approximation is very reasonable for the short pulses associated with MTIplatforms [16].