Multichannel Example

Now consider processing multiple pulses with a multichannel phased array radar. In con-trast to the previous example, assume that matched filtering has already been performed on the individual pulses in fast time and focus on modeling the target response in the slow-time⁴ and channel dimensions. This phased array will transmit a series of pulses steered to a region of interest on the ground.⁵ The echoes from these pulses will be received on multiple channels connected to subarrays of the antenna. By coherently processing these returns, range, velocity, and angular bearing information can be extracted about moving targets. An MTI system treats nonmoving objects as undesirable clutter and attempts to suppress 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 J channels spaced equally at d meters apart. A coherent processing interval (CPI) for this system consists of data collected over K slow-time pulses with a sampling period of T seconds and L fast time range bins. We shall assume that the system is narrowband (i.e., the bandwidth B f0), where f₀ = ^c_λ is the center frequency,⁶and that pulse compression has already been performed. In addition, motion during a given pulse will be neglected.⁷ We will consider the data collected for a single range gate. The spatial-channel samples for pulse k will be denoted as a vector y_k ∈ C^J, while the complete space-time snapshot will be denoted y∈ C^{J K}, where the data have been pulse-wise concatenated, that is,

y=^y^T₁ y^T₂ . . . y^T_K^T

For a CPI, we thus collect M= J K measurements of the unknown scene.

4Slow time refers to the relatively long intervals between successive pulses from a coherent radar. Fast time refers to the time scale at which the electromagnetic pulse travels across the scene of interest, which is the same as range up to a scale factor when propagation is in a homogenous medium, such as free space.

5Our development here assumes that all the elements in the phased array transmit the same waveform during a given pulse, with the exception of phase shifts to steer the beam in the desired illumination direction. 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 for example [14].

6This assumption allows time delays to be well approximated as phase shifts, which creates a corre-spondence between target velocity and the output bins of a fast Fourier transform (FFT) with respect to slow-time.

7This so-called stop-and-hop approximation is very reasonable for the short pulses associated with MTI platforms [16].

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Pulse

Channel

Range

(a) (b)

FIGURE 5-3 (a) A simple depiction of an MTI data collection scenario. The targets of interest are the moving vehicles on the road network. (b) A notional rendering of the data cube collected during a CPI. Each element in the cube is a single complex-valued sample.

The three dimensions are channels, pulses, and range. Similar diagrams are used to describe MTI data sets, along with much more detail, in [15,16].

At a given range, the response of the array over the CPI to a point target can be characterized with a space-time steering vector. First, consider a target response arriving at elevation angle θ and azimuth angle φ as measured by J spatial channels. Since the array is linear in the azimuth plane, there is a conical ambiguity in the arrival direction of a given signal characterized by the cone angleθc = cos⁻¹(cos θ sin φ). The spatial frequency observed by the array for a given cone angle is then

fs = d λcosθc

The spatial steering vector for a given cone angle is then a_s ∈ C^J, given by as( fs) =^1 exp( j2π fs) . . . exp( j2π(J − 1) fs)^T

where we have selected the first element as the zero-phase reference for the array. Similarly, we can define the normalized Doppler frequency as

f_d = 2vT λ

wherev is the velocity of the target. The temporal steering vector describing the response of a single element across K time samples to a target at normalized Doppler fd is then given as the length K vector

a_t( fd) =^1 exp( j2π fd) . . . exp( j2π(K − 1) fd)^T

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The combined space-time steering vector for a target is then given as the Kronecker product of the temporal and spatial steering vectors, that is,

a( fs, fd) = at( fd) ⊗ as( fs)

For a given range bin, the vector a( fs, fd) represents the data that the radar would collect over a CPI if only a single target having unit scattering amplitude were present at the angle-Doppler location encoded by f_s and f_d. Specifically, the steering vector a( fs, fd) corresponds to a single column of the A matrix for this MTI problem.

Let us discretize the frequency variables into N_s ≥ J spatial frequency bins and N_d ≥ K Doppler frequency bins spaced uniformly across the allowed ranges for each variable to obtain N = NsNd unique steering vectors. We can organize the resulting steering vectors into a matrix A∈ C^M×N. Neglecting range ambiguities, we can define the scene reflectivity function at a given range as x ∈ C^N, where the rows of x are indexed by angle-Doppler pairs⁸( fs, fd). We then obtain the linear relationship between the collected data and the scene of interest as

y= Ax + e

where e in this case will include the thermal noise, clutter, and other interference signals.

A more realistic formulation would include measured steering vectors in the matrix A.

Thus, we see that the data for a multichannel pulsed radar problem can be placed easily into the framework (5.1).

5.2.2.4 Comments

The overall message is that most radar signal processing tasks can be expressed in terms of the linear model (5.1). Additional examples and detailed references can be found in [17].

We should also mention that both of these examples have used the “standard” basis for the signal of interest: voxels for SAR imaging and delay-Doppler cells for MTI. In many cases, the signal of interest might not be sparse in this domain. For example, a SAR image might be sparse in a wavelet transform or a basis of canonical scatterers.⁹As another example, in [18], the authors explore the use of curvelets for compressing formed SAR images. Suppose that the signal is actually sparse in a basis such that x = α, withα sparse. In this case, we can simply redefine the linear problem as

y= Aα + e (5.10)

to obtain a model in the same form with the forward operator A. Indeed, many of the early CS papers were written with this framework in mind with A = , where  is the measurement operator, and is the sparse basis. In this framework, one attempts to define a measurement operator that is “incoherent” from the basis . We will not dwell on this interpretation of the problem and refer the interested reader to, for example, [19].

8Note that x in this context is the same image reflectivity function used in common STAP applications [15].

9Selection of the appropriate sparse basis is problem dependent, and we refer the reader to the literature for more detailed explorations.

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