Radar emission spectrum engineering Shannon D. Blunt 1 , John Jakabosky 1
1.1 Introduction
The topic of radar waveform design has been investigated for decades [1] and includes myriad contributions including various forms of frequency modulation (FM), binary and polyphase coding, recent multiple-input–multiple-output schemes and more. However, the impact of the radar transmitter is generally not considered as part of the waveform design process. Given the impact the trans- mitter has on the waveform, particularly for high-power systems and in light of stricter radar emission requirements [2,3], it has become necessary to address the actual physical signal that is launched from the antenna along with all the factors that contribute to the generation of this signal.
A holistic perspective for radar emission design necessitates a mathematical representation of the intended waveform that permits physical generation. Speci- fically, this representation must be continuous and relatively well contained
1Radar Systems & Remote Sensing Lab (RSL), University of Kansas, USA
spectrally (noting that, in theory, an idealized pulse is not bandlimited). Further, it is useful if this representation provides the means with which to parameterize the waveform for optimization of the resulting physical signal according to attributes such as range sidelobes and Doppler tolerance.
It is well known that a waveform should have a constant envelope to mitigate some of the distortion induced by the transmitter power amplifier (PA) as well as to maximize ‘energy on target’ for subsequent detection sensitivity. However, a waveform must also be differentiable, and thus continuous, with sufficient spectral containment to minimize the spectral shaping that is imposed by the transmitter, which can compound distortion. As they inherently meet these criteria, FM wave- forms such as the well-known linear FM (LFM) chirp have been widely used.
Of course, LFM is also known to possess rather high range sidelobes, which has led to the development of various non-linear FM (NLFM) waveforms that necessitate the identification of a suitable continuous phase/frequency function of time (see Section 5.2 of [1] for an overview). Many such methods are based on the principle of stationary phase [4–6], which relates the power spectral density (PSD) and the chirp rate at each frequency as a means to shape the waveform spectral content. Another class of NLFM is hyperbolic FM (HFM), otherwise known as linear period modulation [7,8], that is used in sonar and by many echo-locating mammals due to its Doppler invariance property. Further, HFM provides a rela- tionship between peak sidelobe level (PSL) and the waveform time-bandwidth product (BT) that serves as a useful performance benchmark [8] for all constant amplitude waveforms. Other design approaches include higher order polynomials [9], use of the Zak transform [10], application of Be´zier curves [11] and hybrid methods that also employ amplitude tapering on receive [12,13] (though such tapering also yields a signal-to-noise ratio, or SNR loss).
With the intent to obtain greater design freedom than could previously be achieved for NLFM, the notion of phase coding was also developed in which large- scale, parallelizable computing power can be applied to search for optimal codes having high dimensionality. This extensive litany of contributions includes Barker codes [14], P-codes [15,16], minimum peak sidelobe codes [17] and many others (see [1, Chapter 6]), with the goal of discovering ever longer codes with lower autocorrelation sidelobes (e.g., [18,19]).
Phase codes can be separated into the classes of binary codes and the more general polyphase codes, both of which have a structure involving a discrete sequence of phase values modulated onto rectangular subpulses (or chips). The the- oretically instantaneous transitions between adjacent chips corresponds to infinite bandwidth, thus requiring some practical means with which to implement codes on a physical radar system. For binary codes, the most common techniques with which to convert the code into a constant envelope, continuous waveform are derivative phase shift keying (DPSK) [20] and the biphase-to-quadriphase transformation [21], which is a form of minimum shift keying (MSK). Of the two, the latter is superior from a spectral containment standpoint [20], though both are still somewhat limited in design freedom due to being constrained to a binary phase constellation.
An alternative approach proposed to limit the spectral spreading of coded waveforms is to replace the rectangular chips with windowed (and thereby trun- cated) sinc kernel functions [20,22]. While this approach achieves excellent spec- tral containment (in [20] out-of-band suppression of as much as 100 dB was achieved), the sinc kernel also produces amplitude modulation (AM) that requires linear amplification. To generate these waveforms with non-linear amplification, and thus high power efficiency, the Chiriex out-phasing configuration comprising dual non-linear amplifiers followed by a summer was used [20], which represents a form of Linear amplification with Nonlinear Components (LINC) [23,24]. It has been experimentally demonstrated that high transmit power may be achieved for this type of configuration as long as adequate cross-calibration can be maintained between the two PAs.
More recently, an implementation meeting the criteria for the generation of physical waveforms was developed that facilitates the realization of arbitrary polyphase coding as a new NLFM waveform class denoted as polyphase-coded FM (PCFM) [25,26]. This implementation is a modified form of the continuous phase modulation (CPM) scheme [27] that is commonly used for aeronautical telemetry [28,29], deep-space communications [30] and forms the basis of the BluetoothTM wireless standard [31]. What these applications have in common are the dual requirements of power efficiency and spectral efficiency. The former is obtained by ensuring constant amplitude so that the transmitter PA can be operated in satura- tion, while the latter is achieved by bounding the instantaneous rate of phase change, both of which are inherently provided by the CPM structure.
By establishing the connection between phase coding and a resulting physically realizable waveform, the PCFM framework also provides the means with which to optimize the continuous waveform by searching over the high- dimensional space represented by the coding parameterization [32,33]. By direct extension, the underlying coding can likewise be optimized to account for the distortion the waveform encounters in the transmitter (via a transmitter model or actual hardware-in-the-loop) so as ultimately to optimize the physically emission [33,34]. In Section 1.2, this framework for generation and optimization of the physical emission is discussed along with a summary of various ways in which even greater design freedom can be achieved for the optimization of physical waveforms to contend with the potentially conflicting requirements of the sensing mission and spectral containment.
When addressing the physical radar emission it is observed that, with the use of spectrally efficient waveforms, the pulse rise/fall-times become the limiting factor on spectral containment due to the rather abrupt on/off nature of pulsed radars, specifically the high efficiency PAs that effectively behave like switches. As such, it becomes necessary to address the waveform and transmitter design jointly. In Section 1.3, recent work [35] is presented on the incorporation of a LINC archi- tecture into optimization of the PCFM-based physical emission so as to design the waveform within the context of pulse shaping for spectral containment while still maintaining high power efficiency.
In Section 1.4, another recent emission implementation/optimization scheme is described that relies on the relationship between the PSD and autocorrelation of the waveform to realize pulsed [36] and continuous wave (CW) [37] modalities, respectively, that achieve both good spectral containment and low range sidelobes with little to no SNR loss. As with PCFM-based approaches, experimental measurements demonstrate the efficacy of these spectral shaping methods. This formulation also permits joint optimization of the waveform with a low-loss amplitude taper, and it is likewise shown how in-band spectral gaps (avoidance regions) can be generated without incurring the sensitivity degradation that other- wise generally arises.