Reconstruction and Compressed Sensing
5.2 CS THEORY
5.2.1 The Linear Model
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150 C H A P T E R 5 Radar Applications of Sparse Reconstruction SP subspace pursuit
SR sparse reconstruction
STAP space time adaptive processing SVD singular value decomposition TV total variation
ULA uniform linear array.
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FIGURE 5-2 A notional depiction of a collection geometry for RF tomography. Each of the antennas transmits RF signals whose echoes are received and processed by all of the nodes.
Some form of coding or time/frequency multiplexing is used to make the signals distinguishable at the receivers. This separability is depicted here with different coloring on the transmitted waveforms.
relationship between the signal of interest and the measured data can be captured in this framework. One key assumption is that multiple scattering by the target objects can be neglected, a common assumption referred to as the Born approximation2[4].
Intuitively, one can consider the matrix A as a dictionary whose k-th column, or atom, represents the data that would be collected if only a single element of xtruewere nonzero.
Our goal of inferring xtruefrom y can thus be viewed as the task of finding the dictionary entries that best describe the collected data. To make this idea concrete, we will briefly sketch two examples of expressing radar data in this framework. The reader is referred to the literature or earlier chapters in this series for more detailed radar data models.
5.2.2.1 Radio Frequency Tomography Example
We will now consider a frequency-domain example using several spatially distributed narrowband sensors. A radio frequency (RF) tomographic system illuminates a scene with several spatially distributed radar transmitters. The echoes from these sensors are received by a constellation of radar receivers for subsequent processing. Figure 5-2 depicts a notional scene with a network of these distributed sensors. Even traditional monostatic SAR can be easily viewed in a tomographic framework (see e.g., [5]).
Data from a set of M arbitrary-geometry bistatic scattering experiments will be pro- cessed jointly to image a scene. Each scattering experiment will use a receiver located
2Another method of model linearization is the Rytov approximation [3], which linearizes the nonlinear equation satisfied by the phase of the scalar field. For most radar applications the Born approximation is appropriate, and it is by far the most commonly used model. Thus, we will limit our discussion of linear approximations to the single-scattering setting. As we will show in Section 5.4.4, this will not preclude the consideration of multiple-scattering scenarios.
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at position rm ∈R3and a transmitter at tm ∈R3operating at wavenumber km =ωm/c= 2πfm/c, where c is the speed of propagation. The RF transmission frequency is given in hertz as fm. Data in this format can of course be synthesized from a pulsed system by match filtering and transforming the processed signal to the frequency domain. One can think of these data as phase history data for a multistatic SAR collection after dechirp or deramp processing [5,6].
Consider an isotropic point scatterer located at an arbitrary position q ∈R3. Assuming scalar wave propagation governed by the Helmholtz equation, the Green’s function (section 7.3 of [7]) describing the spherical wave emanating from a point source tmand impinging on the point q is given by
G(tm,q,km)= ej kmtm−q2 4πtm−q2
(5.2) To obtain the field received by a point source with an arbitrary transmit waveform am(k)=−∞∞ ˆam(t)ej ktdt, the frequency-domain Green’s function G(tm,q,km)is modu- lated by the complex-valued frequency-domain waveform3am(km); thus the field received at point q is am(km)G(tm,q,km). Similarly, if the scattering potential at q is given by x(q), then the measured field for scattering experiment m from the single point target would be ym =x(q)am(km)G(tm,q,km)G(q,rm,km) (5.3) The antenna patterns for the transmit and receive antennas are omitted for simplicity. If we linearize the scalar data model using the Born approximation [9] and discretize the scene into N voxels with locations qn and reflectivities x =[x(q1)x(q2) . . .x(qN)]T, then we can obtain the total measured field for experiment m as
ym = N n=1
x(qn)am(km)G(tm,qn,km)G(qn,rm,km) (5.4) We can express the complete data collection in the form of (5.1) by defining A∈CM×N with Amn =am(km)G(tm,qn,km)G(qn,rm,km)and denoting the additive measurement noise as e.
The inner products of the columns of A can be related to values of the familiar point spread function (PSF) from SAR imaging [10]. Under the far-field assumption t2 q2, the PSF reduces to the standard form encountered in parallel-ray computed tomography [6,11]. An example of this development in the time domain with arbitrary waveforms can be found in [12].
5.2.2.2 The Ambiguity Function
We next present two moving target indication (MTI) examples. The first example models the data for a single pulse using a single radar channel to discriminate delay and Doppler.
The two-dimensional function describing the coupling between delay and Doppler for a radar waveform is known as the ambiguity function.
Consider the case of a radar with narrow fractional bandwidth and monostatic oper- ation, that is, co-located transmitter and receiver antennas. Working in the time domain,
3This is because Green’s function is a fundamental solution [8] to the scalar wave equation(∇2+k2)G=
−δ, whereδis the Dirac delta function.
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consider a complex baseband pulse, u(t), modulated in quadrature by a carrier with fre- quency, ω0 = 2πf0, to yield the transmit waveform, p(t) = Re{u(t)ejω0t}, where Re{}
denotes the real operator. The echo of the transmitted pulse waveform encodes backscatter energy from the illuminated scene. Assume the illuminated scene consists of scatterers at range r with radial velocity,v. We can parametrize the complex scene reflectivity in terms of delay and Doppler as x(τ, ω), whereτ(r)= 2rc is the round-trip propagation time, and ω(v) = 2ωc0v is the Doppler shift. The total received signal backscattered from the scene after quadrature demodulation is the baseband signal
yB(t)= x(τ, ω)u(t−τ)e−jωtdτdω+nB(t) (5.5) where the constant phase terms have been absorbed into the reflectivity, and nB(t)repre- sents the circular white complex Gaussian baseband noise.
Our goal is to determine the reflectivity function x(τ, ω)from the received data yB(t). This model can be discretized to obtain a set of equations in the form (5.1). If we discretize the scene reflectivity function x over delay and Doppler on a grid of points,{τm, ωm}, to produce the vector x and sample the received baseband signal yBand noise nB(t)at times {tm}to obtain y and e, we obtain the linear system of equations
y= Ax+e (5.6)
Each column ai of A thus represents the received waveform for a scatterer with given Doppler and range (i.e., delay), and all columns share the same2 norm, under the far- field assumption.
One simple approach to locating targets in the delay-Doppler plane is to derive a test statistic for a single hypothesized target at a particular delay-Doppler combination.
We will pursue this approach briefly to highlight a connection between CS theory and traditional radar practice. Recall that the likelihood ratio test statistic for the existence of a single target with delayτand Doppler frequencyωis the matched filter output [13],
χ(τ, ω)=
yB(t)u∗(t−τ)ejωtdt
= x(τ, ω)A(τ−τ, ω−ω)dτdω+
nB(t)u∗(t−τ)ejωtdt (5.7) where the radar ambiguity functionA(τ, ω)is given byA(τ, ω)= u(t)u∗(t−τ)ejωtdt . Thus, the output of the matched filter is the convolution of the reflectivity f(τ, ω)with the radar ambiguity functionA(τ, ω), plus a filtered copy of the baseband noise. The shape of the ambiguity function can be adjusted by varying the pulse waveform u(t). However, shaping of the ambiguity function is subject to a total volume constraint:
|A(τ, ω)|2dτdω= u(t)22 (5.8) Thus, while the waveform can be designed to reshape the ambiguity function to satisfy design criteria, improving the ambiguity response in one region will necessarily push energy into other parts of the delay-Doppler plane. For example, making the peak of the response narrower will typically raise the sidelobe levels. The sampled matched filter
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outputs can be written
χ = AHy (5.9)
The inner products of the columns of A, denoted by ai, are samples of the ambiguity func- tion:|aiHaj| = |A(τi−τj, ωi−ωj)|. This relationship is a crucial point in understanding the connection between CS and radar signal processing, as we will see in Section 5.2.5.
5.2.2.3 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 = cλ 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 yk ∈CJ, while the complete space-time snapshot will be denoted y∈CJ K, where the data have been pulse-wise concatenated, that is,
y=yT1 yT2 . . . yTKT
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 angleφ 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(cosθsinφ). 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 as ∈CJ, given by as(fs)=1 exp(j 2πfs) . . .exp(j 2π(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
fd = 2vT λ
wherevis 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
at(fd)=1 exp(j 2πfd) . . .exp(j 2π(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 fs and fd. 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 Ns ≥ J spatial frequency bins and Nd ≥ 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∈ CM×N. Neglecting range ambiguities, we can define the scene reflectivity function at a given range as x ∈ CN, 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, withαsparse. 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 operatorthat 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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