*3.1. Waveform*

Most HF radars nowadays employ a variant of the linear FMCW waveform, ranging from a continuous signal, through interrupted FMCW, to FM pulses with a low duty cycle. Interrupted versions include notched sweeps as well as frequency-hopping and spaced sweep formats. In addition to the FMCW class, phase-coded pulse waveforms can still be heard. For most of these options, MIMO (multiple input, multiple output) implementations are possible.

When one moves from monostatic or quasi-monostatic to the bistatic case, several considerations need to be kept in mind. First, the separation of transmit and receive facilities greatly reduces the problem of self-interference, thereby expanding the waveform parameter space. For the moment, we set aside the case where radars in a stereoscopic configuration are sharing a common transmission frequency band. Second, it is well known that range-folded echoes pose a serious hazard for monostatic radars, arising from the combination of long-range propagation of HF radiowaves and the abundance of ionospheric and terrestrial scatterers. As illustrated in Figure 2, bistatic configurations offer a greater freedom with choice of waveform repetition frequency because the range-ambiguous zones of illumination are displaced from those of the receiving system. We note here that the use of non-repetitive waveforms is another tool for reducing this threat, though few HF radars presently employ such signals.

**Figure 2.** The problem of range-ambiguous echoes associated with periodic waveforms is greatly reduced with bistatic configurations. These enable one to steer the receiver beams over the desired ambiguity zone whilst rejecting unwanted returns.

Yet, even bistatic systems are advised to take account of the far-range illumination pattern as the magnitude of unwanted environmental echoes may be sufficient to disrupt through receiving array sidelobes. Third, an associated problem is the prospect of round-the-world (RTW) propagation—in powerful HF skywave radars, signals have been observed after three transits around the Earth. Of course, for low-power radars, background noise will almost invariably swamp RTW returns. Fourth, when pulsed waveforms are used, the fact that bistatic geometry couples time delay to angle-of-arrival may require that one implements a pulse-chasing capability [19], with its attendant penalties. Fifth, the spatial properties of bistatic resolution cells are well-known [20], but less attention has been paid to what we might call the Doppler sensitivity, ∂ω <sup>∂</sup>*<sup>v</sup>* , where ω is the Doppler shift and *v* is the target speed. To quantify this, recall that the bistatic Doppler shift of a target with velocity <sup>→</sup> *v* at location *x* given by

$$\boldsymbol{\omega} = -\frac{2\pi}{\lambda} \cdot \frac{d}{dt} \left( \boldsymbol{r}\_T^{\mathbf{x}} + \boldsymbol{r}\_x^{\mathbf{R}} \right) = -k \left( \boldsymbol{r}\_T^{\mathbf{x}} \cdot \overrightarrow{\boldsymbol{v}} + \overrightarrow{\boldsymbol{r}}\_x^{\mathbf{R}} \cdot \overrightarrow{\boldsymbol{v}} \right) = -k \left( \boldsymbol{r}\_T^{\mathbf{x}} + \boldsymbol{r}\_x^{\mathbf{R}} \right) \cdot \overrightarrow{\boldsymbol{v}} = -2k \cos \left( \frac{\varphi}{2} \right) \mathbf{v} \cdot \cos \beta \tag{4}$$

with ϕ the bistatic angle and β the target heading relative to the bisector axis; hence,

$$\frac{\partial \omega}{\partial v} = -2k \cos(\frac{\varphi}{2}) x \cos \beta \tag{5}$$

The Doppler sensitivity loss factor cos<sup>ϕ</sup> 2 is one component of the price we pay in return for whatever advantages we can extract from employing a bistatic configuration.

## *3.2. Transmitting Facility*

Central to the design of the transmitting facility is the orientation of the illumination pattern relative to that of the receiver. For any given location <sup>→</sup> *r* in the common zone, the radiated signal amplitude is proportional to *<sup>M</sup>*<sup>→</sup> *r <sup>T</sup> T*(θ,ϕ)*w*, or simply *<sup>M</sup>*<sup>→</sup> *r <sup>T</sup> T*(ϕ)*w* for HFSWR. The radar designer has the option to orient the maximum directive gain of the transmitting array towards that region in the receiving facility's field of view, which has been accorded the highest priority. More generally, for signal-to-noise dominated missions, we can formulate the HFSWR orientation problem as one of maximizing the figure of merit (FOM) of the priority-weighted pattern,

$$FOM = \max\_{\substack{\varphi\_0\\R}} \bigcup\_{\substack{\eta\\R}} P\left(\overrightarrow{\stackrel{\textstyle \cdot \cdot}{r}}\right) \widetilde{M}\_{\underset{\varGamma}{\rightarrow}}^{R} \widetilde{M}\_{\underset{\varGamma}{\rightarrow}}^{\overset{\textstyle \cdot \cdot \cdot}{T}} \widetilde{T}(q\nu, q\nu\_0)w \,d\stackrel{\textstyle \cdot \cdot}{r} \tag{6}$$

where ϕ<sup>0</sup> is the nominal boresight orientation of the transmit array and *P* → *r* represents the priority weighting over the receiver processing zone *R*.

A complication that arises with clutter-related missions of HFSWR is the phenomenon of multiple scattering [21,22]. This can corrupt the received echoes when the sea state is significant, so in addition to providing sufficient incident power density, a sophisticated transmit antenna design would attempt to minimize the associated contributions, relative to the echoes received via the primary propagation path. To do this requires a regional wave climatology but is otherwise straightforward.

#### *3.3. Propagation*

The involvement of distinct outbound and inbound propagation paths has major ramifications for HF skywave radar, with a lesser, though still observable, impact on HFSWR. For monostatic skywave radars, frequency management systems probe the ionosphere and determine (i) the frequency band providing adequate power density in the target zone, and (ii) some measure of the quality of the propagation channel [23]. With bistatic configurations, the frequency that works best for propagation from transmitter to target zone will often be poor for propagation from target zone to receiver; in this case, the optimum frequency will effect a compromise, and may, on occasion, take a highly non-intuitive value.

q

In order to quantify the impact on performance, we can exploit the geometrical congruence of a single bistatic signal path and two monostatic paths [24]. Figure 3 shows the concept underlying this technique.

**Figure 3.** The geometrical congruence of (**a**) a pair of monostatic radar observations, and (**b**) a single bistatic radar observation. To see the equivalence, simply imagine that the area shown as land is actually sea and the area shown as sea is actually land, whereby Figure 3b appears as a land-based bistatic radar.

The first emission travels from the monostatic radar at location X to the target zone at relative coordinates (*r*1, ϕ1), scatters, and returns to the radar. Using a scalar form of (1) for notational simplicity, the complex amplitude of the received signal is given by

$$s\_1 = R(\varphi\_1) M\_{\stackrel{X}{r\_1}}^X S \stackrel{\rightarrow}{(\stackrel{\rightarrow}{r\_1})} M\_X^{\stackrel{\rightarrow}{r\_1}} T(\varphi\_1) w \tag{7}$$

so the received power is |*s*1| 2. Now, write *M* → *r* 1 *<sup>X</sup>* <sup>=</sup> *aT* 1 *e i*ψ*<sup>T</sup>* <sup>1</sup> and *MX* → *r* 1 = *a<sup>R</sup>* 1 *e i*ψ*<sup>R</sup>* <sup>1</sup> . We can identify *aT* <sup>1</sup> as the one-way propagation amplitude loss factor for the outbound signal and *a<sup>R</sup>* <sup>1</sup> as the corresponding amplitude loss factor for the inbound signal. Power loss factors are then simply *aT* 1 <sup>2</sup> and *aR* 1 2 , and the propagation power loss for the two-way process is *aT* 1 2 . *aR* 1 2 .

A second observation is then made in a different direction, to a target zone at coordinates (*r*2, ϕ2),

$$\vec{r}\_2 s\_2 = R(\vec{\varphi}\_2) \vec{M}\_{\vec{r}\_2}^X S \begin{pmatrix} \vec{r}\_2 \\ \vec{r}\_2 \end{pmatrix} \vec{M}\_X^{\vec{r}\_2} T(\vec{\varphi}\_2) w \tag{8}$$

with two-way propagation loss *aT* 2 2 . *aR* 2 2 , as shown in Figure 3a. Now imagine that there is a transmitter at location (*r*1, ϕ1) and a receiver at (*r*2, ϕ2) as shown in Figure 3b; that is, a bistatic radar configuration interrogating the region previously occupied by the monostatic radar. The complex amplitude for this case is given by

$$\mathbf{s}\_3 = \mathcal{R}(q\mathbf{\hat{z}})M\_X^{\vec{r}\_2}\mathcal{S}(X)M\_{\vec{r}\_1}^X T(q\mathbf{\hat{z}})w \tag{9}$$

where we have taken the orientation of the imagined arrays to be parallel to those of the monostatic system. Now, the propagation paths satisfy reciprocity, *M<sup>X</sup>* → *r* 1 = *M* → *r* 1 *<sup>X</sup>* and *M* → *r* 2 *<sup>X</sup>* <sup>=</sup> *<sup>M</sup><sup>X</sup>* → *r* 2 . Further, the gain patterns of the transmit and receive arrays are strongly determined by the array apertures but vary only weakly with steer angle over moderate departures from boresight. Thus, we can write *T*(ϕ2) = α *T*(ϕ1) and *R*(ϕ2) = β*R*(ϕ1) where 0.87 < α, β < 1 for a radar whose arrays each steer over a 60◦ arc. Substituting in (9), and invoking (7) and (8),

*s*<sup>3</sup> = *R*(ϕ2)*M* → *r* 2 *<sup>X</sup> <sup>S</sup>*(*X*)*MX* → *r* 1 *T*(ϕ1)*w* = *s*2 3 = *R*(ϕ2)*M* → *r* 2 *<sup>X</sup> <sup>S</sup>*(*X*)*M<sup>X</sup>* → *r* 1 *T*(ϕ1)*w*.*R*(ϕ2)*M* → *r* 2 *<sup>X</sup> <sup>S</sup>*(*X*)*MX* → *r* 1 *T*(ϕ1)*w* = *S*(*X*) *R*(ϕ2)*M<sup>X</sup>* → *r* 2 *M<sup>X</sup>* → *r* 1 *T*(ϕ2) <sup>α</sup> *w*.*R*(ϕ1)*M* → *r* 2 *<sup>X</sup> M* → *r* 1 *<sup>X</sup> T*(ϕ1)*w* = *S*(*X*) *R*(ϕ2)*M<sup>X</sup>* → *r* 2 *M* → *r* 2 *X T*(ϕ2) <sup>α</sup> *<sup>w</sup>*.β*R*(ϕ1)*M<sup>X</sup>* → *r* 1 *M* → *r* 1 *<sup>X</sup> T*(ϕ1)*w* = *<sup>S</sup>*(*X*) *S* → *r* 1 *S* → *r* 2 . <sup>β</sup> α*R*(ϕ1)*MX* → *r* 1 *S* → *r* 1 *M* → *r* 1 *<sup>X</sup> <sup>T</sup>*(ϕ1)*w*.*R*(ϕ2)*M<sup>X</sup>* → *r* 2 *S* → *r* 2 *M* → *r* 2 *<sup>X</sup> T*(ϕ2)*w* = *<sup>S</sup>*(*X*) *S* → *r* 1 *S* → *r* 2 <sup>β</sup> α . <sup>√</sup>*s*1*s*<sup>2</sup> <sup>≈</sup> *<sup>S</sup>*(*X*) *S* → *r* 1 *S* → *r* 2  . <sup>√</sup>*s*1*s*<sup>2</sup> (10)

The magnitudes of *S* → *r* 1 and *S* → *r* 2 can be estimated by inversion of the respective Doppler spectra, or even approximated at zero cost by assuming fully developed seas—typically valid for HF frequencies above 15 MHz. The steer directivity loss factor <sup>β</sup> <sup>α</sup> ≈ 1 so its effect is insignificant compared with the variability of the other terms. Thus, from measurements of the returned clutter power from monostatic observations *s*<sup>1</sup> and *s*2, we can predict the echo power for the bistatic configuration observation *s*<sup>3</sup> for an arbitrary specified scattering coefficient *S*(*X*). One point to note here is that we have simplified the discussion by ignoring the polarization domain; this is not a significant issue for HFSWR and can be avoided in the skywave radar case by a combination of spatial and temporal averaging.

HF skywave radars routinely collect backscatter ionograms (BSI) over the arc of coverage, typically out to a range of 5000–6000 km, so there is a wealth of propagation data available from which to derive statistical predictions that can be used for bistatic system design. A representative BSI is shown in Figure 4, with the instantaneous range depth marked for a nominal radar frequency of 15 MHz. Assuming a slow variation with azimuth, both *r*<sup>1</sup> and *r*<sup>2</sup> need to lie between 1400 km and 2300 km. Figure 5 shows an instance of an inferred sub-clutter visibility (SCV) map computed for a representative radar network (the monostatic input data are real but not obtained from these radars).

**Figure 4.** A backscatter ionogram—a map of echo strength as a function of (group) range and radar frequency. The dashed lines show, for a representative frequency, how the outbound and inbound group ranges must both lie in the band indicated for the system to operate successfully.

**Figure 5.** A map showing a single instance of the predicted bistatic sub-clutter visibility (clutter-to-noise ratio) in the overlap region of two skywave radars, as inferred using the geometrical congruence technique from a single azimuthal scan recorded with a separate monostatic radar.

While it is necessary to exceed some target-specific threshold of power density in order to achieve detection, for slow-moving targets, such as ships, that may not be sufficient. The presence of multimode propagation and phase path fluctuations associated with field line resonances and other ionospheric disturbances can blur the Doppler spectrum of the radar returns and thereby obscure the desired echoes. This raises the question: Can we extend the analysis discussed in the preceding paragraphs so as to obtain statistical information on the phase path modulation spectrum over bistatic paths?

The answer is a qualified 'yes'. Techniques to estimate and then correct for phase path variations have been developed and installed in operational systems since the 1980s [12,13] so the individual phase path modulation time series are available for each leg of the synthesized bistatic path. A rudimentary synthesis approach would simply concatenate the phase modulation histories, then halve them, but that could introduce Doppler spreading due to phase discontinuity at the junction point. A superior method involves first phase-shifting the second half to ensure phase continuity and then applying a conjugate taper weighting around the junction to affect a smooth first derivative.

This approach works for the most important class of fluctuations, where the spatial scale is of the order of 102 km, and latitude-dependent, being linked to the geomagnetic field line resonances (FLR) that are observed as micro-pulsations at ground level. At times, other dynamical processes cause fluctuations over much smaller spatial scales. Figure 6 illustrates these two types of modulation: Each frame shows the measured phase fluctuation time series over a two-way skywave channel. In Figure 6a, the modulation estimated from the echoes originating in four individual range cells spaced over a range depth of about 150 km shows a high degree of spatial correlation, suggesting that the outbound and inbound legs of a bistatic skywave radar observation would experience related modulation sequences. In contrast, when other types of modulation prevail, the paths can experience uncorrelated and often more erratic modulations. In Figure 6b, the cells shown are spaced over a total of only 20 km, yet the modulation patterns are quite distinct. In both cases, we can construct a simulated bistatic path resultant modulation sequence, using the ideas of the previous paragraph, but only for the former type can we hope to associate the observed modulation with the known properties of geophysical wave processes in the ionosphere.

**Figure 6.** Phase modulation sequences measured over skywave propagation paths; (**a**) an example of a field line resonance modulation, with slow spatial variation over a distance of 150 km, and (**b**) an example where the modulation arises from other geophysical mechanisms, with spatial decorrelation occurring within 20 km.

It is perhaps apposite to note here that an operationally significant relative of the problem of joint path optimization is the converse—the selection of frequencies that guarantee strong propagation over one path and little over the other for the same radar frequency. Such a bistatic configuration has direct relevance to the detection of nonlinear target echoes and the ability to suppress sea clutter by many tens of dB. A description of this scheme can be found in [25].
